1 D.A.V PUBLIC SCHOOLS ODISHA ZONE - II SAMPLE QUESTION PAPERS 2016-2017; CLASS- XII SUB:-MATHEMATICS F.M:-100 TIME: - 3Hrs General instructions: - i) All questions are compulsory ii) This question paper consists of 29 questions divided into four sections. Section A Comprises of 4 questions of one mark each, section B comprises of 8 questions of two marks each ,section C comprises of 11 questions of four marks each and section D contains 7 questions of six marks each. SECTION – A 1. Give examples of two functions and g such that g o f is injective but g is not injective. 2. If and A is not invertible then what is the value of λ? 3. If and then write the vector of magnitude 6 in the direction of 4. Let * is a binary operation on R + , defined as . Find the value of x for which 2*(x*5)=10 SECTION – B 5. Solve for x: 6. Prove that the determinant of an odd order skew symmetric matrix is 0. 7. If then prove that 8. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1% 9. Evaluate 10. Form the differential equation of the family of circles having centre on Y-axis and radius 3 units. 11 If the vectors are coplanar, then find λ. 12. If A and B are two events such that and then find P(A) SECTION – C 13. Let and Find AB and hence solve the system of equations 14. Examine the differentiability of the function OR Find the value of a,b,c so that the function f(x) defined below is continues at x=0 F(x) = 15. If , prove that 16. Find the equation of tangents to the curve that are parallel to the line OR Find the intervals in which the function is i) Increasing ii) Decreasing 17. A Jet of an enemy is flying along the curve . A soldier is placed at the point (3, 2). What is the nearest distance between the soldier and the Jet? Write one important role of a citizen to protect his country. 18. Integrate 19. Solve differential equation OR Solve differential equation 20. For any three vectors prove that 21. Prove that the lines are perpendicular if 22. Four balls are drawn without replacement from a box containing 8 red and 4 white balls. Find the probability distribution of the number of red balls drawn. 23. A man is known to speak the truth 3 out of 5 times. He throws a die and reports that is a 6. Find the probability that it is actually a 6..
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D.A.V PUBLIC SCHOOLS ODISHA ZONE - IISAMPLE QUESTION PAPERS 2016-2017; CLASS- XII
SUB:-MATHEMATICS F.M:-100 TIME: - 3Hrs General instructions: -i) All questions are compulsoryii) This question paper consists of 29 questions divided into four sections. Section A Comprises of 4 questions of one mark each, section B comprises of 8 questions of two marks each ,section C comprises of 11 questions of four marks each and section D contains 7 questions of six marks each. SECTION – A1. Give examples of two functions and g such that g o f is injective but g is not injective.2. If and A is not invertible then what is the value of λ?3. If and then write the vector of magnitude 6 in the direction of 4. Let * is a binary operation on R+, defined as . Find the value of x for which 2*(x*5)=10
SECTION – B5. Solve for x: 6. Prove that the determinant of an odd order skew symmetric matrix is 0.7. If then prove that 8. Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 1%9. Evaluate 10. Form the differential equation of the family of circles having centre on Y-axis and radius 3 units.11 If the vectors are coplanar, then find λ.12. If A and B are two events such that and then find P(A)
SECTION – C13. Let and Find AB and hence solve the system of equations 14. Examine the differentiability of the function OR Find the value of a,b,c so that the function f(x) defined below is continues at x=0 F(x) = 15. If , prove that 16. Find the equation of tangents to the curve that are parallel to the line
OR Find the intervals in which the function is i) Increasing ii) Decreasing17. A Jet of an enemy is flying along the curve . A soldier is placed at the point (3, 2). What is the nearest distance between the soldier and the Jet? Write one important role of a citizen to protect his country.18. Integrate 19. Solve differential equation OR Solve differential equation 20. For any three vectors prove that 21. Prove that the lines are perpendicular if 22. Four balls are drawn without replacement from a box containing 8 red and 4 white balls. Find the probability distribution of the number of red balls drawn.23. A man is known to speak the truth 3 out of 5 times. He throws a die and reports that is a 6. Find the probability that it is actually a 6..
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SECTION – D24. Let be defined as Show that f is invertible. Also find.
OR Let and a binary operation * on A is defined as . Show that * is both commutative and associative. Find the identity if any.25. If and , then prove that a = b = c
OR If then prove that 26. Sketch the graph of y= │x+3│ and evaluate the area under the curve y= │x+3│ above x axis and between x = -6 to x=0. 27. Evaluate
OR Evaluate as a limit of a sum.28. Find the vector equation of the plane passing through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane Also show that the plane thus obtained contains the line 29. A company produces soft drinks that have a contract which requires that a minimum of 80 units of chemical A and 60 units of chemical B go into each bottle of the drink. The chemicals are available in prepare mixed packets from two different suppliers. Supplier S has a packet of mix of 4 unit of A and 2 unit of B that costs ₹10. The supplier T has a packet of mix of 1 unit of A and 1 unit of B that costs ₹4. How many mixed packets mixed from S and T should the company purchase to honour the contract requirement and yet minimize cost? Make a LPP and solve graphically. Do you think drinking soft drinks are good for health? ************************************************
CHAPTER-WISE QUESTIONSRELATIONS AND FUNCTIONS
ONE /TWO MARK QUESTIONS1. Let be two functions defined as and g then find
Ans:2. Let N be the set of natural numbers and a relation R is defined over N as .Check for Symmetry and Transitivity of R. [Ans :Neither Symmetric nor transitive]3. Do you think , defined as is bijective? Justify your answer.
[Ans: No, Not one-one]4. Determine whether the operation * on R defined by is a Binary operation.5. Find for defined as [Ans:6. Let * be the binary operation on Q, defined as . Find the identity element if any.7.Let be a binary operation, defined by , then find the value of .
8.Let be a binary operation on given by . Write the value of .9. Let be a binary operation on given by . Write the value of .10. If and , find .11. If and given by and , find .12. Let and is a relation in given by . Is symmetric? Give reasons.13. Is the function defined by one-one? Give reasons14. Is the function defined by onto? Give reasons.15. Let , where set A={1,2,3} and set B={a,c} defined as Find
16.If is an invertible function, find the inverse of . 17.Let be a binary operation, defined by , then find the value of .
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18. Let be a binary operation on given by . Write the value of .19.Let be a binary operation on given by . Write the value of .20.If and , find .
21.If and given by and , find .22.Let and is a relation in given by . Is symmetric? Give reasons.
23.Is the function defined by one-one? Give reasons24. Is the function defined by onto? Give reasons.25. Let , where set A={1,2,3} and set B={a,c} defined as Find .26. .Give an example to show that the relation R in the set of natural number,defined by
R= is not transitive . 27..Write the number of all one-one functions from the set A= to itself . 28..Give an example of a relation which is reflexive , symmetric , but not transitive . 29.Write the number of equivalence relations in the set containing(1,2) and (2,1) . 30..Let * is a binary operation on Q defined as a*b = ab/5 . Write the identity element for * ,if any .
31.Let be two functions defined as and g then find 32.Let N be the set of natural numbers and a relation R is defined over N as . Check for Symmetry and Transitivity of R.34.Do you think , defined as is bijective? Justify your answer. 35.Determine whether the operation * on R defined by is a Binary operation.
36.Find for defined as [Ans:37.Let * be the binary operation on Q, defined as . Find the identity element if any.
38. Let f : Q Q: f(x) =2x +3. Then, f -1(y) =?
39. a *b = a+b – 2. Then find identity element and inverse.40. f(x) = 3x + 5 and g(x) = 2x -3, Find fog, gof.41. f(x) = sin x + cos x, Find its domain and range.42. So that all parallel lines are equivalence relation.43. f(x)=3x +5 / 2x – 1 is 1,1 function on Q.44. a *b = a+b – ab where a and b are in z is a binary operation. Prove it.45. f(x) = 3x +4 / x + 1. Find its inverse.46. if (fx) = (x2 -1) and g(x) = 2x +3 then find (g o f)(x)=?
FOUR/ SIX MARKS QUESTIONS1. Prove that the inverse of an equivalence relation is also an equivalence relation.2. Show that the relation R in set A = { x Z : 0 x 12} given by R = { (a, b) : |a - b| is a multiple of 4} is an
equivalence relation. Find the set of all elements related to 1. 3. 3. Let N be the set of all natural numbers and R be the relation on N X N defined by (a,b) R (c,d) iff ad (b + c)
= bc (a + d). Examine whether R is an equivalence relation on N X N ? (HOTS)4. Check whether the relation R in real numbers set defined by R = { (a,b) : a b3} is reflexive, symmetric or transitive.5. Let A = {1,2, …………. ,9} and R be the relation in A X A defined by (a,b) R (c,d) iff a + d = b + c for (a,b), (c,d) in A X A.
Prove that R is an equivalence relation and also obtain the equivalence class [ (2,5) ].6. Let a relation R on the set A of real numbers be defined as (a,b) R 1 + ab > 0 a,b A. Check if R is an equivalence
relation or not.7. Show that the function f : R { x ∈ R : -1 < x < 1} defined by f(x) = , x R is one – one and onto function.
(HOTS)8. Let A = N X N and let * be a binary operation on A defined by (a,b) * (c,d) = (ad + bc, bd) (a,b) , (c,d) N X N. Check
that i) * is commutative. ii) * is associative .Find the identity element of * in A , if exists. 9. Let A = R X R and * be a binary operation A defined by (a,b) * (c,d) = (a + c, b + d). Prove that * is commutative and
associative. Find the identity element for * on A also find the inverse element of in A.10. Consider: R+ [-9, ) given by ) = 5x2 + 6x – 9 where R+ is the set of all non negative real numbers. Prove that f is
invertible and find.
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11. If f(x) = and g(x) = 1-x; -2x1, then find fog.12. If f:R→R , f(x) = x2-5x+4 and g:R→R, g(x) = logx, then find the value of (gof)(2).
13. Let be defined by , Show that is invertible. Find
14. If , show that for all . What is the inverse of f?15. Show that given by , is bijective. 16. Let and . Consider the function defined by . Show that is one-one and onto and hence find .17. Consider the binary operations and defined as and for all . Show that is commutative but not associative, is
associative but not commutative.18. Let and be a binary operation on A defined by . Show that is commutative and associative. Also, find the identity
element for on A, if any.19. Let N be the set of all natural number and R be the relation in NXN defined by (a,b) R(c,d) if ad = bc ,show that R is
the equivalence relation .20. Show that the function f: R− R defined by f(x) = 2x3 – 7 for x R is a bijective function .21. Let * be a binary operation on Q defined by a*b = a+ b- ab ; a,b Q show that (i) associative and (ii) commutative . 22. If R1 and R2 be two equivalence relations on set A, prove that R1R2 is also an equivalence relation on set A .What shall
you tell about the equivalence of R1R2 .23. Let f: N R be a function defined as f(x) = 4x2 + 12x + 15 show that f : N− S where S is the range of f is invertible , find
the inverse of f .24. Let f : W−W be defined as f(n) = n-1 ,if n is odd . show that f is invertible .Also find f-1.n+1 , if n is even 25. Consider the binary operations * and 0 on R defined as a* b = and a0b = aa , b€R .Show that * is commutative but
not associative , where as 0 is associative but not commutative . Further show that a ,b€ R a* ( b0 c) = (a*b) 0 (a* c) .26. Let A = R- and B = R- .Consider the function f: AB defined by f(x) = . Show that f is one-one and onto and hence find
f-1 .27. A binary operation * on the set is defined as a*b = a + b , if a+b 6 a+ b – 6 if a+b6 28. Show that zero is the identity for this operation and each element a of the set is invertible with 6-a, being the inverse
of a .
29. Let T be the set of all triangles in a plane with R a relation in T given by R = {( T1, T2 ) : T1 T2 }. Show that R is a equivalence relation.
30. Show that the relation R defined by ( a, b ) R ( c, d ) => a + d = b + c on the set N x N is an equivalence relation.31. Prove that the relation R in the set A = { 1, 2, 3, 4, 5 } given by R = { ( a, b ) : | a – b | is even }, is an equivalence
relation.32. Let Z be the set of all integers and R be the relation on Z defined as R = {( a, b ) : a, b ε Z, and ( a – b ) is divisible by 5
}. Prove that R is an equivalence relation.33. Let * be a binary operation on Q, defined by a * b = , show that * is commutative as well as associative. Also, find its
identity, if it exists.34. Show that the relation S in the set R of real numbers, defined as S = {( a, b ) : a, b ε R and a b3 } is neither reflexive,
nor symmetric, nor transitive.35. Show that the relation S in the set A = { x ε Z : 0 x 12 } given by S = {( a, b ) : a, b ε Z, | a – b | is divisible by 4 } is an
equivalence relation. Find the set of all elements related to 1.36. Consider f : R+ [ -5, ) given by f(x) = 9x2 + 6x – 5. Show that f is invertible with f-1 (y) = ( .37. A binary operation * on the set { 1, 2, 3, 4, 5 } is defined as : a * b = 38. Show that zero is the identity for this operation and each element a. ( 0 ) of the set is invertible with 6 – a, being
the inverse of a.39. Consider f : R+ [ 4, ) given by f (x) = x2 + 4. Show that f is invertible with the inverse f-1 of f given by f-1 (y) , where R+
is the set of all non-negative real numbers.40. Consider the binary operations * : R x R R and o : R x R R defined as a * b = |a – b| and a o b = a for all a, b R.
Show that * is commutative but not associative, o is associative but not commutative.41. Show that the function f in A = R – {} defined as f(x) = is one-one and onto. Hence find f-1.
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42. Prove that the relation R in the set A = {5, 6, 7, 8, 9} given by R = {(a, b) : |a – b| is divisible by 2}, is an equivalence relation. Find all elements related to the element 6.
43. Let A = R X R and * be a binary operation A defined by (a,b) * (c,d) = (a + c, b + d). Prove that * is commutative and associative. Find the identity element for * on A also find the inverse element of in A.
44. Let A = {1,2, …………. ,9} and R be the relation in A X A defined by (a,b) R (c,d) if a + d = b + c for (a,b), (c,d) in A X A. Prove that R is an equivalence relation and also obtain the equivalence class [ (2,5) ].
45. Let A=N N and * be the binary operation on A defined by (a,b)*(c,d)=(a+c, b+d) Show that * is commutative and associative. Find the identity element for * on A, if any.
46. Show that the function f in A=R- defined as f(x) = is one-one and onto. Hence, find (x)47. Show that the relation R in the set A={x:X€z,0≤x≤12} given by={(a,b):|a-b| is divisible by 4} is an equivalence relation
find the set of all the lements elated to1.48. let * be a binary operations on Q defined by a*b=a+b - ab; a,b Q show that
i) associative and ii) commutative49. If R1 and R2 be two equivalence relations on a set A, prove that R1 R2 I s also50. equivalence relation on A. What shall you tell about the equivalence ofR1 R2 (justify).
Let be a function defined as Show that51. is Invertible, where S is the range of . Hence find inverse of.
5. Let A = {1,2,3,…9} and R be a relation in A×A defined by (a,b) R (c,d)52. if a+d = b+c for all (a,b) , (c,d) in A×A . Prove that R is an equivalence53. relation .
54. Show that the function defined as is one – one55. and onto function. Do you think it is invertible? If so then find
56. Define a binary operation * on A={0,1,2,3,4,5,} as a*b= , show that 0 is the identity for this operation and each elements ‘a’ of this set is invertible with 6-a being the inverse of ‘a’.
57. Show that f : N N given by f(x)= , is bijective(both one-one and onto).
58. If , show that for all . What is the inverse of f?59. Show that given by , is bijective. 60. Let and . Consider the function defined by . Show that is one-one and onto and hence find .61. Consider the binary operations and defined as and for all . Show that is commutative but not associative, is
associative but not commutative.62. Let and be a binary operation on A defined by . Show that is commutative and associative. Also, find the identity
element for on A, if any.INVERSE TRIGONOMETRIC FUNCTION
( one /two marks)2. Write the principal valu of cos-1(cos 6800)3. Find the value of sin-1{ cos( }4. What is the other branch of sec-1 x ? 5. Write the value of tan( 2 tan-1 (1/5)).6. If Sin (sin-1 1/5 + cos-1x ) =1 ,then find the value of x.7. Write the principal value of: .8. Write the principal value of: 9. Write the principal value of: 10. Write the principal value of: .11. Show that .
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12. Solve for : 13. Evaluate: .14. Evaluate :.15. Prove that: .16. Prove that:.17. Write the principal value of a) b) ) .18. Find the domain of .19. Find the value of .20. Write the value of - 2 ) .21. If + ) = 1 , then find the value of x . 22. Prove that tan ( + cos-1 ) + tan ( - cos-1 ) = 23. Solve for x, tan-1 (x+1) + tan-1 (x-1) =8/1124. Prove that sin-1 (12/13) + cos-1 (4/5) +tan-1 (63/16) = π25. Solve for x, tan-1 () + tan-1 () = π /426. Prove that sin-1 (.) + sin- (.) + sin-1(.) = π/227. Prove that cos-1 = x/228. Prove the following: tan-1 ( ) + tan-1 ( ) = cos-1 ( )
29. Prove the following: cos-1 ( ) + sin-1 ( ) = sin-1 ( )30 Prove that tan-1[ 31. Prove the following: 2tan-1 ( tan-1 (1/5) + tan-1(1/8) =32. Prove that tan-1 (33. Prove the following: sin-1 (1/3 ) = sin-1 ( )34. Prove that sin-1 ( ) + sin-1 ( ) = cos-1 ( ) = tan-1 ( )35. If y = cot-1 ( -tan-1 ( , then prove that sin y = tan2 ( )36. If cos-1 (x/2) + cos-1 (x/3) =α ,then prove that 9x2 -12xy cosα +4y2 = 36 sin2α .37. Solve for x : sin-16x + sin-16x = 38. Prove that : 4tan-1 - tan-1 = 39. Prove that : cot( - 2cot-1 3) = 740. Prove that : sin(2tan-1 ) + cos(tan-1 2) = 14/1541. Solve for x : If cos(tan-1 x) = sin(cot-1 )
Solve FOR X:42. sin-1 (1-x) - 2 sin-1 x = . (Hots ,Ans-0)43. Sin-1x+sin-1(1-x) = cos-1x 44.45.46. + = (Ans- )47. .= 0 .48. + = . 49. tan (cos -1 x) = sin (cot -1 1/2) 50. sin-1 x + sin-1 2x = /3
SECTION – B (Four mark questions)
51. Find the values of x which satisfy the equation - 2 .
52. Solve for x , 2 = , 0 x .
53. Solve the following equation ) + ) = .
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54. Prove that ) + ) = 2 ) .55. Show that 56. Solve the equation = 0 .
57. Prove that: = .
58. Prove that : = +
59. Solve for x: = .60. If sin (sin–11/2 +cos–1x)=1 ,then find the value of x.61. Write the range of one branch of sin–1x other than the principal branch.
62. What is the principal value of + ?
63. Prove the following: sin-1 (1/3 ) = sin-1 ( ).
64. Solve for x : sin-16x + sin-16 x = - 65. If cos-1 (x/2) + cos-1 (x/3) = α,then prove that 9x2 -12xy cosα + 4y2 = 36 sin2α .17. If + + = π ,
prove that x2 + y2 +z2 + 2xyz = 1 .
66. Solve the following equation: 67. ( Four marks)
68. If y = cot-1 - tan-1 , prove that sin y = tan2 x/2 (HOTS)69. Prove that ½ tan-1 x = cos-1 () 70. Show that cos (2 tan-1 1/7) = sin (4 tan-1 1/3)71. Prove that 2 tan-1 1/5 + sec-1 + 2 tan-1 1/8 = 72. If cos-1 x/2 + cos-1 y/3 = , prove that 9x2 – 12xy cos + 4y2 =36 sin2 (HOTS)73. If cos-1x + cos-1y + cos-1z = , then show that x2 + y2 + z2 + 2xyz = 1(HOTS)74. Prove that sin-1 (8/17) + sin-1 (3/5) = cos-1 (34/85)75. Prove that : - sin-1 (1/3) = sin-1 76. Prove that :tan ( ½ sin-1 ¾) = 77. Show that : 78. Simplify :79. Evaluate : tan-1 1+tan-12 +tan-13 (Ans- )80. Solve :
1. sin-1 (1-x) - 2 sin-1 x = . (Hots ,Ans-0)2. Sin-1x+sin-1(1-x) = cos-1x
81.82.83. + = (Ans- )84. = 0 .85. + = . (Ans-1/4)86. tan (cos -1 x) = sin (cot -1 1/2) (Ans- )87. sin-1 x + sin-1 2x = /3 (Ans- )
88. Prove that: .89. Prove that: 90. Prove that: .91. Prove that: 92. Write in the simplest form: .
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93. Write in the simplest form: .94. Write in the simplest form: .95. If , then prove that
96. Find the values of x which satisfy the equation - 2 .
97. Solve for x , 2 = , 0 x .
98. Solve the following equation ) + ) = .
99. Prove that ) + ) = 2 ) .100. Show that 101. Solve the equation = 0 .
102. Prove that: = .
103. Prove that : = +
104. Solve for x: = .105. If sin (sin–11/2 +cos–1x)=1 ,then find the value of x.106. Write the range of one branch of sin–1x other then the principal branch.
107. What is the principal value of + ?
108. Prove the following: sin-1 (1/3 ) = sin-1 ( ).
109. Solve for x : sin-16x + sin-16 x = - 110. If cos-1 (x/2) + cos-1 (x/3) = α,then prove that 9x2 -12xy cosα + 4y2 = 36 sin111. If + + = π , prove that x2 + y2 +z2 + 2xyz = 1 .
112. Solve the following equation: MATRICES AND DETERMINANTS
( one /two marks)1. If then find the value of x .( Ans-2)2. If is a singular matrix then find value of x.(Ans-4/3)3. IfA is a square matrix of order 3x3 then find the value of .(Ans. 8|A|)4. If A IS an invertible square matrix of order n then| adj A| is equal to …………….. (Ans. |A|n-1
5. Let A is a square matrix of order 3 and |A| =10 , then find |adj A| (Ans. 100)6. A matrix has 18 elements. Write the possible orders of a matrix.7. If and .8. Find and , if +.9. If , find the values of and such that .10. Construct a matrix, whose elements are given by .11. If and , prove that .12. By using elementary transformations, find the inverse of the matrix .13. Express the matrix as a sum of symmetric and skew-symmetric matrix.
a. DETERMINANTS14. Evaluate, .15. For what value of the matrix has no inverse.16. If then find the value of if .17. Using the properties of the determinants, prove that:
a.
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18. Using the properties of the determinants, prove that:
19. Using matrices solve the following system of linear equation:a. and
20. Using matrices solve the following system of linear equation:a. and
21. If , prove that Hence find 22. Show that the points qnd do not lie on a straight line for any value of .
23 If , find the value of x and y.24. If , find the value of y.25. If , find x.26. If A is a square matrix such that A2 = A, then write the value of ( I + A )2 -3A.27. If x, write the value of x.28. If A is a square matrix of order 3 such that |adj A | = 64, find |A|.
4 MARKS 29. If A = , find x and y such that A2 – x A + y I = 0, where I = unit matrix.30. If A = .32. If A = , find K if A2 = 8A + KI, where I = unit matrix.33. If f(x) = x2 – 4x + 1, find f(A), where A = .34. 5. Two schools A and B decided to award prizes to their students for three values Honesty (x), punctuality (y) and
obedience (z). School A decided to award a total of Rs 15,000 for the three values to 4, 3 and 2 students respectively, while school B decided to aw3ard Rs 19,000 for the three values to 5, 4 and 3 students respectively. If all the three prizes together amount to Rs 5,000, then
a. (i) Represent the above situation by a matrix equation and form linear equation using matrix multiplication.b. (ii) Which value you prefer to be rewarded most and why?
35. 6. A store in a mail has three dozen shirts with SAVE ENVIROMENT printed two dozen shirts SAVE TIGER printed and five dozen shirts with GROW PLANTS printed. The cost of each shirt is Rs 595, Rs 610 and Rs 795 respectively. All these items were sold in a day. Find total collection of the store using matrix method. Which shirt you would like to buy and why.
36. If A = , find A -1, using elementary row operation.CHAPTER – 4
DETERMINANTSMarks
37. Find x, if ,38. Find the value of p, such that the matrix is singular .39. Find the value of x, such that the points (0,2), (1,x) and (3,1) are collinear .40. If for matrix A ,|A| =3, find|5A|, where matrix A is of order 2x2.41. A is a non –singular matrix of order 3x3 and |A| =-4. Find |adjA|.42. Given a square matrix A of order 3x3 , such that |A| =12 ,find the value of |A adjA|.
4 MARKS43. If A=, show that A’ A-1 = 44. Prove that =(ab+bc+ca)2 .45. Prove that = ( b2 –ac) (ax2 +2bxy+cy2 ).46. Prove that =3abc-a2 –b2 –c2 .47. Prove that =4a2 b2 c2 .48. Prove that =2(a +b)(b +c)(c +a) .49. Prove that =(x-y)(y-z)(z-x)(x +y +z).50. 8. Given two matrices A = and B = ,Verify that BA =6I, Use the result to solve the system x-y =3 , 2x+3y+4z =17 ,y
+2z =7 .
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51. Prove that =(x +y+ z)(x-z)2 .52. Prove that =2(a +b+ c)3 .53. A school wants to award its students for the value of honesty, regularity, and hard work with a total cash award of Rs.
6,000 .Three times the award money for hard work added to that given for honesty amounts to Rs 11,000. The award money given for honesty and hard work together is double the one f=given for regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values, namely, honesty, regularity and hard work, suggest one more value which the school must include for awards.
54. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more values which the management of the colony must include for awards.
55. Given A = , B= , Find BA and use this to solve thea. system of equations , , .
56. Using properties Prove that = 3abc-a2 –b2 –c2 .57. Prove that = 4a2 b2 c2 .58. Using properties of determinates,
a. Prove that = ( 1+
59. Prove that = 2(a +b+ c)3 .
MATRICES AND DETERMINANTSSECTION – B & C (Four & Six mark questions)
60. Prove that = a3
61. If A = and B = , then verify that = B-1A-1.
62. If A= , show that A2 - 5A +7I=0, Hence find A-1.
63. If A= and I is the identity matrix of order 2 show that
64. I+A=(I–A)
65. If F(x)= ,show that F(x)F(y)=F(x+y)66. Using elementary transformations, find the inverse of the matrix
67. For the matrix A = ,show that
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68. Using properties determinants. Prove that
69. Using properties of determinates, prove that:
= (a+b+c) (a – b)(c – b)
70. Prove without expanding that =abc+ ab+ bc+ ca
71. =abc or is a factor of determinant.a. Using properties of determinants ,prove that
72. =9y2(x+y).a. Using properties of determinants prove that:
73. =
74. 13. If A = ,a≠1 then prove by induction that An= 75. 14. For wellbeing of orphanage, three trusts A, B and C was donated 10%, 15% and 20% of their total fund Rs. 200000,
Rs.300000 and Rs. 500000 respectively. Using matrix multiplication finds the total amount of money received by orphanage by three trusts. By such donations, which values are generated?
76. A school wants to award its students for the value of honesty, regularity, and hard work with a total cash award of Rs. 6,000 .Three times the award money for hard work added to that given for honesty amounts to Rs 11,000. The award money given for honesty and hard work together is double the one f=given for regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values ,namely, honesty, regularity and hard work, suggest one more value which the school must include for awards.
77. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more values which the management of the colony must include for awards.
78. 17. A trust caring for handicapped children gets Rs. 30000 every month from its donors. The trust spends half of the funds received for medical and educational care of the children and for that it charges 2% of the spent amount from them and deposits the balance amount in a private bank to get the money multiplied so that in future the trust goes on functioning regularly. What percent of interest should the trust gets from the bank to get a total of Rs. 1800 every month? Using matrix method, find the rate of interest. Do you think people should donate to such trusts?
79. 18. There are 2 families A and B .There are 4 men ,6 women and 2 children in family A, and 2 men, ,2 women and 4 children in family B, The recommended daily amount of calories is 2400 for 1900 for women ,1800 for children and 45
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gms of proteins for men ,55 grams for women and 33 grams for children. Represent the above information using matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of 2 families .What awareness can you create among people about the balanced diet from this question.
80. Two schools Aand B decided to award prizes to their students for three values honesty(x) , punctuality(y) and obedience (z) . School A decided to award a total of Rs. 11,000 for the three values to 5,4, and 3 students respectively , while school B decided to award Rs. 10, 700 for the three values to 4,3,and 5 students respectively .If all the three prizes together amount to Rs. 2700 , then using matrix method find the values of x,y, and z . Which value you prefer to be rewarded most and why .
81. Given A = , B= , Find BA and use this to solve the82. system of equations , , .
83. 21.Prove that =3abc-a2 –b2 –c2 .
84. 22.Prove that =4a2 b2 c2 .85. 23. Using properties of determinates,
86. Prove that =( 1+
87. 24. Prove that =2(a +b+ c)3 .
88. 25. Prove that = 3abc – a2 – b2 – c2 .MATRICES
89. A matrix has 18 elements. Write the possible orders of a matrix.90. If and .91. Find and , if +.92. If , find the values of and such that .93. Construct a matrix, whose elements are given by .94. If and , prove that .95. By using elementary transformations, find the inverse of the matrix .96. Express the matrix as a sum of symmetric and skew-symmetric matrix.
DETERMINANTS97. Evaluate, .98. For what value of the matrix has no inverse.99. If then find the value of if .100. Using the properties of the determinants, prove that:
a.
101. Using the properties of the determinants, prove that:i.
102. Using matrices solve the following system of linear equation:a. and
103. Using matrices solve the following system of linear equation:
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a. andIf , prove that Hence find Show that the points qnd do not lie on a straight line for any value of .
(Four & Six mark questions)Prove that = a3
104. If A = and B = , then verify that = B-1A-1.105. If A=, show that A2 - 5A +7I=0, Hence find A-1.
1. If A= and I is the identity matrix of order 2 show that106. I+A=(I–A)107. If F(x)= ,show that F(x)F(y)=F(x+y)108. Using elementary transformations, find the inverse of the matrix 109. For the matrix A =,show that HOTS
1. Using properties determinants. Prove that110.
111. Using properties of determinates, prove that: = (a+b+c) (a – b)(c – b)
112. Using properties prove that =abc+ ab+ bc+ ca=abc or is a factor of determinant.
1. Using properties of determinants ,prove that113. =9y2(x+y).
1. Using properties of determinants prove that:114. =
115. If A = ,a≠1 then prove by induction that An= 116. For wellbeing of orphanage, three trusts A, B and C was donated 10%, 15% and 20% of their total fund Rs.
200000, Rs.300000 and Rs. 500000 respectively. Using matrix multiplication finds the total amount of money received by orphanage by three trusts. By such donations, which values are generated? HOTS
117. A school wants to award its students for the value of honesty, regularity, and hard work with a total cash award of Rs. 6,000 .Three times the award money for hard work added to that given for honesty amounts to Rs 11,000. The award money given for honesty and hard work together is double the one f=given for regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values ,namely, honesty, regularity and hard work, suggest one more value which the school must include for awards.
118. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more values which the management of the colony must include for awards. A trust caring for handicapped children gets Rs. 30000 every month from its donors. The trust spends half of the funds received for medical and educational care of the children and for that it charges 2% of the spent amount from them and deposits the balance amount in a private bank to get the money multiplied so that in future the trust goes on functioning regularly. What percent of interest should the trust gets from the bank to get a total of Rs. 1800 every month? Using matrix method, find the rate of interest. Do you think people should donate to such trusts? HOTS
121. There are 2 families A and B .There are 4 men ,6 women and 2 children in family A, and 2 men, ,2 women and 4 children in family B, The recommended daily amount of calories is 2400 for 1900 for women ,1800 for children and 45 gms of proteins for men ,55 grams for women and 33 grams for children. Represent the above information using matrices. Using
14
matrix multiplication, calculate the total requirement of calories and proteins for each of 2 families .What awareness can you create among people about the balanced diet from this question. HOTS
122. Two schools Aand B decided to award prizes to their students for three values honesty(x) , punctuality(y) and obedience (z) . School A decided to award a total of Rs. 11,000 for the three values to 5,4, and 3 students respectively , while school B decided to award Rs. 10, 700 for the three values to 4,3,and 5 students respectively .If all the three prizes together amount to Rs. 2700 , then using matrix method find the values of x,y, and z . Which value you prefer to be rewarded most and why .
123. Given A = , B= , Find BA and use this to solve the124. system of equations , , .125. Using properties Prove that =3abc-a2 –b2 –c2 .126. Prove that =4a2 b2 c2 .127. Using properties of determinates,
Prove that =( 1+128. Prove that =2(a +b+ c)3 .129. Show that the matrix satisfies the equations and hence find .130. Solve the following system of equation by matrix method .131. Use product to solve the system of equations .132. If find using solve the system of linear equations .133. Find the inverse of by using elementary transformations.134. The sum of three numbers is 6. If we multiply the third number by3 and add second to it, we get 11. By adding
1st and 3rd numbers we get double of the 2nd number. Represent the information algebraically and find the numbers using matrix method.
135. Solve the following system of equations by matrix method where and .and .136. An amount of Rs 5000 in put into three investments at the rate of interest of 6%,7%,8% per annum
respectively. The total annual income is Rs 358. If the combined income from the first two investments in Rs 70 more than the income from the third. Find the amount of each investment by matrix method.
137. Compute if and .138. Suppose and then find BA and use this to solve the system of
equations and 139. 36. Using the properties of determinants prove that :-
a. i. = 2abc (a + b + c)3
b. ii. = (1 + a2 + b2)3
c. iii. = (1 + pxyz) (x - y) (y - z) (z - x)
d. iv. = abc + ab + bc + ca
e. v. = 4a2 b2 c2
f. vi. = (a + b + c) (a2 + b2 + c2)
g. vii. = a3 + b3 + c3 – 3abch. 37. If a, b, c are all + ve and distinct then prove that the value of is always negative.i. 38. If a b c and = 0, then using properties of determinants, prove that a + b + c = 0j. 39. If = 0, then find the value of + + where (p )
k. 40. Find the maximum value of :- i. = (where )
CONTINUITY & DIFFERENTIABILITY.
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ONE/TWO MARKS QUESTIONS1. If f(x) = , then find f”(2) . [ ans - ]2. The function f(x) = , if x≠
a. 3, if x= , Is continuous at x= , then find k .i. [ans. K = 6]
3. If y = ) ,then find [ans..] i. 1 if , x [ a = 3, b = -8]
4. If f(x) = ax + b , if 3 x 5 ,i. 7 if 5 x . Determine the values of a & b , so that f(x) is continuous.
5. Find , if + = 1 , [ ans. = -1]a. 4/ 6 marks questions
6. If y = show that (1 + x2) - 2 = 0 .7. Verify Rolle’s Theorem for the function: f(x) = sinx + cosx, x [0, .8. If = a(x – y . prove that, = .9. If x = a cos3 , y = asin 3 , find the value of
i. [ ans . |sec
10. If f(x), defined by the following, is continuous at x = 0, find the values of a, b and c.f(x) = [ ans. a = -3/2 , b R – {0} , c = ½ ]
11. Verify Lagrange’s Mean value theorem for the following : f(x) = x2 + 2x + 3, x [4,6]12. Find the value of a for which the function f defined as:
a. f(x)= a sin (x+1), x0i. , x> 0, is continuous at x=0. [ ans. a = ½]
13. If y = sin(m ), prove that (1 - ) - x + m2y = 0 .14. If y = , prove that ( 1 –x2 ) + y = 0.15. If Y = ( Find .
i. [ ans.( sec2 x log sinx + 1) +b. (sec x. tan x. log cos x – sec x . tan x)
16. If the function f defined by 1. f(x) = ax2+bx+c, x 1 is derivable2. bx+2, x> 1
b. then find the value of a and b.17. . f(x)= , x < 5 /2
i. a , x= 5 /2 is continuousii. , x> 5/2
b. at x=5/2, find the value of a and b.18. 13. . If y = log (19. Prove that (x2+1) + xy+1 =020. 14. 8. If y= xcotx + , find 21. If x= a(-sin), y=a(1+cos), find .22. Differentiate w.r.t. x, sin-1[]23. If y=xx, then prove that -2-= 0.24. Verify mean value theorem for the function f(x)= sin x- sin 2x in [0,π].25. Find the points on the curve y=x3-3x, where the tangent to the curve is parallel to the chord joining (1, -2) and (2,2).26. At what point on the curve y=(cot x-1) in [0, 2 π] is the tangent parallel to x-axis.27. If y=(tan-1x)2, prove that (1+x2)2y2+2x(1+x2)y1=2.28. 22.If the function f defined by f(x)= |x-3| +|x-4| then show that f is not differentiable at x=3 and x=4.
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HOTS QUESTIONS29. , If Y = ( x+ )n . Then prove that : 30. If x + y = 0 ,-1< x<1 and x ≠y, then prove that : 31. If , then prove that : 32. Then prove that =
33. Then prove that = 34. f(x) = -1 ≤x<0
i. , 0 ≤ x ≤ 1 , is continuous in [ -1 , 1 ] , then find the value of p. [ ans . p = -1/2 ]
35. Find , so that the functioni. is continuous at .
36. Find the relationship between and such that the function is a continuous function.i.
37. For what value of , is the function continuous at :a.
38. Find all the points of discontinuity of the function defined bya.
39. Discuss the continuity of the function at.40. Find of the function 41. Find of the function .42. Find of the function 43. If find at 44. Differentiate, with respect to 45. If show that .
CHAPTER - 5CONTINUITY DIFFERERTIABILITY
4 MARKS46. Find k, so that the function is continuous at x = 547. For what value of k, so that the function is continuous at x=248. Determine the constants a and b, such that the function is continuous at x=249. Find the values of a and b such that the function defined by is continuous.50. Find the relationship between a and b is continuous at x = 3.51. Find the value of k so that the function is continuous at x = 52. The function is continuous at x=1, find the values of a and b.
DIFFERENTIATION4 MARKS
53. Find for the following:54. 1. y = 2. y = 3. y = 55. 4. y= xx + (sin x)x 5. (cos x)y = (sin y)x 6. Y = cos-1[2x+1/1+4x]56. 7. y= 8. Y= 57. 9. If y = A emx + B enx , prove that d2y/dx2 - (m+n) dy/dx + mny = 0.58. 10. If log y = tan-1, show that (1+x2)y2 + (2x + 1)y1 = 0.59. 11. If x+ y = 0, for -1< x < 1, show that dy/dx = -1/(1+x)2
60. 12. If y = 3cos(log x) + 4sin(log x), show that x2 (d2y/dx2) + x(dy/dx) + y = 0.61. 13. If x= a(- sin), y = a(1+cos), find d2y/dx2.62. 14. If sin y = x sin(a+y), prove that dy/dx = sin2(a+y)/sin a.63. 15. If y = (tan-1 x)2, prove that (x2 + 1)2 (d2y/dx2) + 2x(x2 + 1) dy/dx = 2.
17
64. 16. If y = sin-1x, show that (1-x2) d2y/dx2 - x dy/dx = 0.65. 17. If x = 66. 18. If yx = ey-x, prove that dy/dx = ( 1+log y )2/log y.67. If xpyq = (x + y)p+q, prove that (i) dy/dx = y/x and (ii) d2 y/dx2.68. If xy = ex-y, prove that dy/dx = log x/(1 + log x)2.
ROLLE’S THEOREM AND LAGRANE’S MEAN VALUE THEOREM4 MARKS
69. f(x) = sin 2x in [0, π]70. f(x) = cos x + sin x on [0, 2π].71. f(x) = sin x2 – 5x + 6 in [2, 3].72. It is given that for the function f(x) = x3 – 6x2 + px + q on [1, 3], Rolle’s theorem holds with c = 2 + . Find the values of p
and q.73. f(x) = x + 1/x in 1 3.74. f(x) = (x – 1)(x – 2)(x – 3) in [0, 4].75. f(x) = x3 + x2 – 6x in [-1, 4].76. f(x) = 1/4x – 1 in [1, 4].77. f(x) = sin x – sin 2x in [0, π].
CHAPTER – 5CONTINUTY AND DIFFERENTIABILITYSECTION – B( 2 marks each)
78. If y = tan -1 then prove that = + 79. If x changes from 4 to 4.01 then find the approximate change of log x80. Find derivative y = tan -1
(Four & Six mark questions)81. If, then show that .
82. If y = log , prove that ( 1+ x2) d2y/dx2 + x = 0 .83. a) If y=(tan –1 x)2 , show that (x2+1)2y” +2x(x2+1)y’=2
a. b). If F(x)= ,x≠ .find the values of so that f(x) becomes continuous at x=84. Find a,b and c for which the function
85. is continuous at x=0
86. Find the value of k so that the function is continuous at x =
87. The function is continuous at x=1,88. find the values of a and b.89. Examine the following function (x) for continuity at x = 1 and differentiability
90. at x = 2,(x) = 91. Verify mean value theorem for the function f(x) = (x-3)(x-6)(x-9) in [3,5].
18
Le For what value of a,f is continuous at x 90. Find the value of `a` for which the function (x) defined as :
91. (x) = , is continuous at x = 0.
92. If cos y=xcos(a+b) with a≠ ±1 prove that 93. If, then show that .94. If and , Prove that
95. If 96. 16. If y = Show that: (x2+1)2y2 + 2x (x2 +1) y1 = 2.
97. If + = a3(x3 – y3), prove that : .98. If y = A emx + B enx , prove that d2y/dx2 - (m+n) dy/dx + mny = 0.99. If log y = tan-1, show that (1+x2)y2 + (2x + 1)y1 = 0.
100. If x + y = 0, for -1< x < 1, show that dy/dx = -1/(1+x)2
101. If y = 3cos(log x) + 4sin(log x), show that x2 (d2y/dx2) + x(dy/dx) + y = 0.102. If x= a( - sin ), y = a(1+cos ), find d2y/dx2.103. If sin y = x sin(a+y), prove that dy/dx = sin2(a+y)/sin a.104. If y = (tan-1 x)2, prove that (x2 + 1)2 (d2y/dx2) + 2x(x2 + 1) dy/dx = 2.105. If y = sin-1x, show that (1-x2) d2y/dx2 - x dy/dx = 0.
106. If y = log , prove that ( 1+ x2) d2y/dx2 + x = 0 .
107. If y = , then show that ( 1- x2 ) d2y/dx2 - 3x - y = 0 .108. If yx = e y - x, prove that dy/dx = ( 1+log y )2/log y.109. If xpyq = (x + y)p+q, prove that (i) dy/dx = y/x and (ii) d2 y/dx2.110. If then show that .
a. CONTINUITY AND DIFFERENTIABILITY111. Find , so that the function
i. is continuous at .112. Find the relationship between and such that the function is a continuous function.
i.113. For what value of , is the function continuous at :
a.114. Find all the points of discontinuity of the function defined by
a.115. Discuss the continuity of the function at.116. Find of the function 117. Find of the function .118. Find of the function 119. If find at 120. Differentiate, with respect to
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121. If show that .
CHAPTER – 6APPLICATION OF DERIVATIVES
4 MARKS RATE OF CHANGE
1. A stone is dropped into a quiet lake and waves move in circles at a speed of 5 cm per second. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
2. The length x of a rectangle is decreasing at the rate of 5 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.
3. Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand-cone increasing, when the height is 4 cm?
4. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
5. A spherical balloon is being inflated by pumping in 16 cm3/sec of gas. At the instant when balloon contains 36π cm3 of gas, how fast is its radius increasing?
INCREASING AND DECREASING FUNCTIONS4 MARKS
6. Find the intervals(s) in which function f(x) is an increasing or decreasing:7. f(x) = x3 -12 x2 +36x +17 2. F(x) =2 x3 -9x2 +12x +15 3. F(x) = x3 + 1/x3 , x 08. Find the intervals in which the function f given by f(x) = sin x +cos x , 0 , is strictly increasing or decreasing 9. Prove that y = – is an increasing function of in [ 0, π/2 ].10. Show that y = log ( 1 + x ) – , x . – 1 is an increasing function of x, throughout its domain.
TANGENTS AND NORMALS4 MARKS
11. Show that the curves x = y2 and xy = k cut at right angles, if 8k2 = 1.12. Find the equation of tangents to the curve x = sin 3t, y = cos 2t, at t = .
(i) Find the equation of the normal at the point ( am2 , am3 ) for the curve ay2 = x3.(ii) For the curve y = 4x3 – 2x5, find all points at which the tangent passes through the origin.(iii) Prove that the curves y2 = 4ax and xy = c2 cut at right angle, if c4 = 32a4.(iv) Find the equations of tangents to the curve y = x3 + 2x + 6 which are perpendicular to the
linea. x + 14y + 4 = 0.
13. Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8.14. Find the equation of the tangents to the curve x = a sin3 , y = b cos3 = .
APPROXIMATIONS 4 MARKS
(i) Using differential, find the approximate value of .(ii) Using differential, find the approximate value of f(2.01), where f(x) = 4x3 + 5x2 + 2.
5 3. 6. ( 82 )1/4
6 MAXIMA AND MINIMAa. 4 MARKS
1. Show that the right circular cone of least curved surface and given volume has an altitude equal to times the radius of the base.
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2. A wire of length 28 m is to be cut into two half pieces. One of the pieces is to be made into a circle and the other into a square. What should be the lengths of two pieces so that the combined area of the square and the circle is minimum?
3. Prove that the volume of the greatest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.
4. Show that the rectangle of maximum area that can inscribed in a circle of radius r is a square of side .
5. Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height h is .
6. Show that the semi-verticle angle of a right circular cone of given total surface area and maximum volume is sin-1
7. Show that a right circular cylinder which is open at the top, and has a given surface area will have the greatest volume if its height is equal to the radius of its base.
8. If the length of three sides of a trapezium other than the base are equal to 10 cm each, then find the maximum area of the trapezium.
9. Show that of all the rectangles of given area the square has the smallest perimeter.10. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter
of the window is 12 m, find the dimensions of the rectangle that will produce the largest area of the window.
11. A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimensions of the rectangle so as to admit maximum light through the whole opening.
12. Show that the height of a closed right circular cylinder of given surface and maximum volume is equal to the diameter of the base.
13. A right circular cylinder is inscribed in a given cone. Show that the curved surface area of cylinder is maximum when diameter of cylinder is equal to radius of base of cone.
14. An open box, with a square base, is to be made out of a given quantity of metal sheet of area c2. Show that the maximum volume of the box is c3/6.
15. Find the area of greatest rectangle that can be inscribed in an ellipse x2/a2 + y2/b2 = 1.b. CHAPTER – 6
c. APPLICATIONS OF DERIVATIVESi. SECTION – B (Four mark questions)
1. Find the equation of the tangent to the curve at the point where it cuts the x-axis.
2. Using differential, find the approximate value of 7 3.A particle moves along the curve 6y=x3+2 find the points on the curve at which y co-ordinates is changing 8 times
as fast as the x co-ordinates.
8 4.Prove that y= - is an increasing of function in(0, ).
9 5.Find the equations of the tangents to the curves y= which is parallel to the line 4x- 2y+5=010 6.Find the angle between the curves y2=x and x2=y.
63. Prove that the curves x y = 4 and x2 + y2 = 8 touch each other.64. The length x of a rectangle is decreasing at the rate of 5 cm/minute.
11 When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter,12 (b) the area of the rectangle.
65. Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms13 a cone on the ground in such a way that the height of the cone is always14 one-sixth of the radius of the base. How fast is the height of the sand-cone15 increasing, when the height is 4 cm?
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66. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
67. Find the interval in which the function given by (x) = sin x + cos x,16 0 is strictly increasing or strictly decreasing.17 12. Find the intervals in which the following functions are increasing or Decreasing:18 i.f(x)=8 + 36x + 3x2 – 2x3.19 ii.f(x)= – 2x3 – 9x2 – 12x + 1.20 iii.f(x)=x4 - 4x3 + 4x2 + 15.
68. Find the intervals in which the function f given by f(x) = sin x +cos x ,21 0 , is strictly increasing or decreasing .
69. Prove that y = – is an increasing function of in [ 0, π/2 ].70. Find a point on the curve y = , at which the tangent is parallel to the
22 chord joining the points (2,0) and (4,4).
71. Find the equation of the tangent to the curve x = , y = at t = 72. Prove that the curve x = y2 and xy = k cut at right angles if 8k2 = 1 .
73. Using differential, find the approximate value of .23 19.Using differential, find the approximate value of f(2.01), where24 f(x) = 4x3 + 5x2 + 2.
74. Find the intervals in which the function f(x) = x3 + is1. increasing (ii) decreasing.2. SECTION – C (Four mark questions)
25 Show that the right circular cone of least curved surface and given volume has an altitude equal to times the radius of the base.
26 A wire of length 28 m is to be cut into two half pieces. One of the pieces is to be made into a circle and the other into a square. What should be the lengths of two pieces so that the combined area of the square and the circle is minimum?
27 Prove that the volume of the greatest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.
28 Show that the rectangle of maximum area that can inscribed in a circle of radius r is a square of side .29 Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right circular
cone of height h is .
30 Show that the semi-vertical angle of a right circular cone of given total surface area and maximum volume is sin-1 31 Show that a right circular cylinder which is open at the top, and has a given surface area will have the greatest
volume if its height is equal to the radius of its base.32 If the length of three sides of a trapezium other than the base are equal to 10 cm each, then find the maximum
area of the trapezium.
33 9. Show that the height of cylinder of maximum volume that can be inscribed in a sphere of radius is . Also find the maximum volume.
34 10. If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, then show that
the area of the triangle is maximum, when the angle between the given side and the hypotenuse is .15. . A given quantity of metal is cast into a half cylinder with a rectangular base
35 and semi circular ends. Show that the total surface area is minimum, if the
22
36 ratio of the length of the cylinder to the diameter of its semicircular ends is37 .
16. An open box, with a square base, is to be made out of a given quantity of
38 metal sheet of area c2. Show that the maximum volume of the box is c3/6 .17. Find the area of greatest rectangle that can be inscribed in an ellipse
39 x2/a2 + y2/b2 = 1.18. A point on the hypotenuse of a right angled triangle is at distances
40 a and b from the sides. Show that the length of the hypotenuse is
41 at least .19. An Apache helicopter of enemy is flying along the curve given by y= x2 +7 .A soldier placed at (3,7) ,wants to shoot
down the helicopter when it is nearest to him. Find the nearest distance.
20. Show that the height of a cylinder of maximum volume that can be inscribed in a sphere of radius R is . Also find the maximum volume.
21. A window is in the form of a rectangle above which there is a semicircle. If the perimeter of the window is p cm , show
that the window will allow the maximum possible light only when the radius of the semicircle is cm42 APPLICATION OF DERIVATIVES
43 VSA (1 mark )44 The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of perimeter of the square. 45 The radius of the circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?46 If the radius of a soap bubble is increasing at the rate of 1 cm sec. At what rate its volume is increasing when the
radius is 1 cm.47 A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the
radius of the circular wave is 10 cm, how fast is the enclosed area increasing?48 Find the value of for which the function is strictly increasing.49 Write the interval for which the function is decreasing.50 Find the rate of change of the total surface area of a cylinder of radius r and height h with respect to radius when
height is equal to the radius of the base of cylinder.51 Find the rate of change of the area of a circle with respect to its radius. How fast is the area changing w.r.t. its
radius when its radius is 3 cm?52 If a manufacturer’s total cost function is , where is the out put, find the marginal cost for producing 20 units.53 Find for which is strictly increasing on 54 SA (1 marks )55 A balloon which always remain spherical is being inflated by pumping in 900 cubic cm of a gas per second. Find the
rate at which the radius of the balloon increases when the radius is 15 cm.56 A man 2 meters high walks at a uniform speed of 5 km/hr away from a lamp post 6 metres high. Find the rate at
which the length of his shadow increases.57 Water is running out of a conical funnel at the rate of 5 cm3 /sec. If the radius of the base of the funnel is 10 cm
and altitude is 20 cm, find the rate at which the water level is dropping when it is 5 cm from the top.58 The length x of a rectangle is decreasing at the rate of 5 cm/sec and the width y is increasing as the rate of 4
cm/sec when x = 8 cm and y = 6 cm. Find the rate of change of (a) Perimeter (b) Area of the rectangle.59 Sand is pouring from a pipe at the rate of 12c.c/sec. The falling sand forms a cone on the ground in such a way that
the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when height is 4 cm?
60 The area of an expanding rectangle is increasing at the rate of 48 cm2 / sec. The length of the rectangle is always equal to the square of the breadth. At what rate is the length increasing at the instant when the breadth is 4.5 cm?
61 A ladder 5 metres long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 2 cm/sec. How fast is its height on the wall decreasing when the foot of the ladder is 4 metres away from the wall?
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62 LA (4marks)63 Prove that the surface area of solid cuboid of a square base and given volume is minimum, when it is a cube.64 Show that the volume of the greatest cylinder which can be inscribed in a right circular cone of height and semi-
vertical angle is Show that the right triangle of maximum area that can be inscribed in a circle is an isosceles triangle.
65 A given quantity of metal is to be cast half cylinder with a rectangular box and semicircular ends. Show that the total surface area is minimum when the ratio of the length of cylinder to the diameter of its semicircular ends is
66 A jet of an enemy is flying along the curve . A soldier is placed at the point What is the nearest distance between the soldier and the jet?
67 Find a point on the parabola which is nearest to the point 68 A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each cover and
folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum.
69 A window in the form of a rectangle is surmounted by a semi circular opening. The total perimeter of the window is 30 metres. Find the dimensions of the rectangular part of the window to admit maximum light through the whole opening.
70 An open box with square base is to be made out of a given iron sheet of area 27 sq. meter, show that the maximum value of the box is 13.5 cubic metres.
71 A wire of length 28 cm is to be cut into two pieces. One of the two pieces is to be made into a square and other in to a circle. What should be the length of two pieces so that the combined area of the square and the circle is minimum?
72 Show that the height of the cylinder of maximum volume which can be inscribed in a sphere of radius is Also find the maximum volume.
73 Show that the altitude of the right circular cone of maximum volume that can be inscribed is a sphere of radius is 4 .
74 APPLICATIONS OF DERIVATIVES75 FOUR MARKS QUESTIONS
76 Using differentials find the value of [Ans0.1925]77 The two equal sides of an isosceles triangle with fixed base b cm are decreasing at the rate of 3cm/sec. how fast is
the area decreasing when each of the equal sides is equal to the base.[Ans:]
78 If the length of a simple pendulum is decreased by 2%, find the percentage decrease in its periods T, where T=2 π.[1%]
79 Water is leaking from the conical funnel at the of 5 cm3/sec. if the radius of the base of the funnel is 10cm and the height is 20 am, find the rate at which the water level is dropping, when it is 5 cm from the top.
[Ans:]80 Show that the curves and cut orthogonally if 81 Find the value of p for which the curves and ) cut each other at right angles.
[Ans: 82 Find the intervals on which the function f(x)= , (x0) is (i) increasing, (ii) decreasing.
[Ans: Increasing in ( decreasing in 83 Prove that all points of the curve y2=4a[x + a sin] at which the tangent is parallel to the axis of x, lie on a parabola.
(HOTS)84 Find the equation of the tangent to the curve which is parallel to the line
Ans:85 Find the intervals in which the function f defined by f(x)= sin x+ cos x , 0x2 π. Is (i) Increasing (ii) Decreasing
[Ans: (i) (ii) ]86 Find the values of x, for which is an increasing function. Also find the points on the curve where the tangent is
parallel to x-axis. [Ans: (0,0),(2,0) and (1,1)]
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87 Fin all points of local maxima ans minima and the corresponding maximum and minimum values of the function f where f(x)= sin4x + cos4x. [Ans: inc , Dec ]
88 If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given. Show that the area of the triangle is maximum when the angle between them is (π/3)
89 Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.90 If the length of three sides of a trapezium other than base are equal to 10cm each then find the area of the
trapezium when it is maximum. [Ans: 91 16. An open box with square base is to be made out of a given quantity of sheet of area C2. Show that the
maximum volume of the box is C3/6.92 Prove that the semi vertical angle of the right circular cone of given volume and least curved surface area is Cot-1.93 AB is a diameter of a circle and C is any point on the circle. Show that the area of the triangle ABC is maximum,
when it is isosceles.94 Find the point on the curve which is nearest to the point (-1,2)95 Find the area of the greatest isosceles triangle that can be inscribed in the ellipse + =1 with its vertex at
one end of the major axis. (HOTS) [Ans; 96 Find the area of the greatest rectangle that can be inscribed in the ellipse + =1
(HOTS) [Ans: 2ab]97 Show that the maximum volume of cylinder which can be inscribed in a cone of height ‘h’ is and semi vertical
angle α is 98 .A given quantity of metal is to be cast into a half circular cylinder (i.e with rectangular base and semicircular
ends)Show that in order ,that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of it’s circular ends is
99 Show that the semi vertical angle of a right circular cone of given surface area and maximum volume is Sin-1(1/3).100 Show that of all the rectangle with a given perimeter the square has the largest area.101 A wire of length 25m is to be cut into two pieces .One of the wires is to be made into a square and other into a
circle. What should be the length of two pieces so that the combined area of the square and the circle is minimum .
102 An isosceles triangle of vertical angle is inscribed in a circle of radius a . Show that the area of the triangle is maximum when
103 Find the condition for which the curves - =1 and intersect orthogonally. i. (HOTS) [Ans:
104 Find the equation of all tangents to the curve that are parallel to the line (HOTS) [Ans 105 Prove that the curves and touch each other.106 43. A wire of length 25m is to be cut into two pieces .One of the wires is to be made into a square and other into a
circle. What should be the length of two pieces so that the combined area of the square and the circle is minimum .
107 44. Show that of all the rectangle with a given perimeter the square has the largest area.45. Show that the semi vertical angle of a right circular cone of given surface area and maximum volume is Sin-1(1/3).
108 46. AB is a diameter of a circle and C is any point on the circle. Show that the area of the triangle ABC is maximum when it is isosceles.
109 47. Prove that the semi vertical angle of the right circular cone of given volume and least curved surface area is Cot-
1/.
CHAPTER-7 INTEGRALS (1/2 Marks questions)
1. Evaluate. A log 2
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2. If (P+sinx)dx =+k then find value of p. A.cosx3. Evaluate . A.24. Evaluate A.Sinx-xcosx+k5. Evaluate dx. A. +4.5 +K6. Evaluatecosx dx A . +k7. Find dx. A - +K8. Evaluate)2 dx. A.x + k9. A.log +k10. dx A.3
SECTION – B&C (Four & Six mark questions)11. Evaluate: dx.12. Evaluate:13. Evaluate dx.14. Evaluate dx.15. Evaluate.16. Evaluate. dx.17. Evaluate:. Sin7x dx 18. Evaluate: dx. HOTS19. Evaluate : HOTS20. Evaluate:.21. Evaluate:.22. Evaluate: HOTS23. Evaluate: dx.24. Evaluate: 25. Evaluate: 26. Evaluate: HOTS27. Evaluate: 28. Evaluate: dx , By using limit of sums.29. Evaluate: dx.30. Prove that 31. Evaluate by limit of sum: (i). (ii).32. 33. 34. Evaluate HOTS35. Evaluate: 36. 37. 38. 39. 40.41.42.43.44.45.46.47.48. Where 49. as a limit of sum50.CHAPTER – 7
26
INTEGRALSINDEFINITE INTEGRALS1 MARKS 51. dx. 52. dx. 53. dx.54. dx. 55. dx. 56. dx. 57. dx.58. dx.59. dx.60. 4 MARKS61. dx. 62. dx. 63. dx.64. dx. 65. dx. 66. 67. 68. dx. 69. dx.70. dx. 71. dx. 72. .73. dx. 74. . 75. dx.76. dx. 77. . 78. dx.79. dx. 80. 81. DEFINITE INTEGRALS1 MARKS 82. 83. 84. 85. x dx 86. 87. dx 88. If find the value of k.89. If write the value of a.
4 MARKS 90. 91. 92. 93. dx.94.
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95. dx 96. dx.97. . 98. dx 99. dx 100. dx101. Prove that 102. Prove that + ) dx =
LIMIT OF SUMS5 MARKS
103. 104. 105. 106. 107.
CHAPTER – 7INTEGRALSINDEFINITE INTEGRALS1 MARKS
108. dx. 109. dx. 110. dx.111. dx. 112. dx. 113. dx. 114. dx.115. dx.116. dx.117.
4 MARKS118. dx. 119. dx. 120. dx.121. dx. 122. dx. 123. 124. 125. dx. 126. dx.127. dx. 128. dx. 129. .130. dx. 131. . 132. dx.133. dx. 134. . 135. dx.136. dx. 137. 138.
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DEFINITE INTEGRALS1 MARKS
139. 140. 141. 142. x dx 143. 144. dx 145. If find the value of k.146. If write the value of a.
4 MARKS 147. 148. 149. 150. dx.151. 152. dx 153. dx.154. . 155. dx 156. dx 157. dx158. Prove that 159. Prove that + ) dx =
LIMIT OF SUMS6 MARKS
160. 161. 162. 163. 164.
a. INTEGRALSSLA (1/2 marks )Evaluate the following integrals.
165.166.167. dx168.169.170.171.172.173.174.
a. SA (4 marks)175.176. 177.178. 7.
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179.180. 181.182. 183.184.
LA (6 marks)185.186.187.188.189.190. Evaluate the following integrals as limit of sums:191.192.
CHAPTER – 7INTEGRALSSECTION – B&C (Four & Six mark questions)
193. Evaluate: dx.
194. Evaluate:
195. Evaluate dx.
196. Evaluate dx.
197. Evaluate .
198. Evaluate. dx.
199. Evaluate: .
200. Evaluate: dx.
201. Evaluate :
202. Evaluate: .
203. Evaluate: .
204. Evaluate:
205. Evaluate: dx.
206. Evaluate:
30
207. Evaluate: .
208. Evaluate:
209. Evaluate:
210. Evaluate:
211. Evaluate: dx , By using limit of sums.
212. Evaluate: dx.
213. Prove that
214. Evaluate: dx
215. Evaluate by limit of sum: (i). (ii).
(iii) (iv).
216. Evaluate
217. Evaluate:
APPLICATION OF INTEGRALSLA (marks)
1. Find the area enclosed by circle .2. Find the area of region bounded by 3. Find the area enclosed by the ellipse .4. Find the area of region in the first quadrant enclosed by –axis, the line and the circle .5. Find the area of region 6. Prove that the curve y = and, x = divide the square bounded by
i. into three equal parts. 7. Using integration, find the area of the region bounded by the triangle whose vertices are andand.
CHAPTER - 8APPLICATION OF INTEGRALSMARKS
8. Find the area of the shaded region enclosed between the two circles :9. x2 + y2 = 1 , (x – 1)2 + y2 = 1.10. 2. Using integration find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x.11. Using integration find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle
x2 + y2 = 32.12. Find the area of the region lying between the parabolas y2 = 4ax and x2 = 4ay, where a > 0.13. Using integration find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.14. Find the area of the region included between the parabola y = 3x2/4 and the line 3x – 2y + 12 = 0.15. 7. Using integration find the area of the region bounded by the triangl ABC, where vertices A, B and C are (-1, 1) (0,
5) and (3, 2) respectively.16. Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y.17. Sketch the graph of y = | x + 3 | and evaluate the area under the curve y = | x + 3 | above x-axis and between
x = -6 to x = 0.
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18. 10. Using the method of integration find the area of the region bounded by the lines 3x – 2y + 1 = 0, 2x + 3y – 21 =0 and x – 5y + 9 = 0.
19. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B(4, 5) and C(6, 3).
20. Find the area of the region bounded by the parabola y = x2 and y = |x|.
Applications of integrals ( 4 & 6 MARKS )21. Find the area of the region bounded by the lines . 22. Using integration find the area enclosed by parabola and the line .23. Find the area of the region using integration
i. ( Ans- )24. Using integration find the area of ,the equations of whose sides are given by respectively.25. Find the area of the region .
i. (HOTS ,Ans-)26. Using integration find the area of the following region .
i. (HOTS, ANS- )27. Find the area of the region in the 1st quadrant enclosed by x-axis, the line and the circle .28. Find the area of the region bounded by the linesand .29. Prove that the curves and divide the area of the square bounded by and into three equal parts.30. .Using integration find the area of the region bounded by the following curves 31. .32. 11.Find the area of the region enclosed between the two circles :33. x2 + y2 = 4 and (x-2)2 + y2 =4 ( Ans- )34. 12 . Find the area of the circle which is interior to the parabola x2 =4y35. 13. A sign board in the shape of a square I x I + I y I = 1 A message is written on the sign 36. board “ keep your environment clean and green” . Find the area bounded by the curve37. I x I + I y I = 1. Identify the value being conveyed by the message on the sign board.
i. (value based and HOTS ANS-16/3).38. Draw a rough sketch of the given curve y = 1 + | x + 1 |, x = -3 , 39. x = 3 , y= 0 and find the area of the region bounded by them40. using integration.41. Determine the area under the curve y = included42. between the lines x =0 and x =a43. A farmer has a piece of land. He wishes to divide equally in his two sons to maintain peace 44. and harmony in the family . If his land is denoted by area bounded by the curves y2 = 4x , 45. x=4 and to divide the area equally . He draw a line x = a then what is the value of a? What 46. is the importance of equality among the people? ( value based )
47. 17.Find the area lying above x-axis and included between the circle x2+y2=8x and 48. inside the parabola y2=4x. ( Hots ,Ans- )49. Using the method of integration find the area of the triangle ABC, co-ordinate of whose
vertices are A(2,0) ,B (4,5) and C(6,3).( Ans-7 sq unit)50. Using Integration, find the area of the region in the first quadrant enclosed by the x-axis , the
line and the circle 51. Find the area of the region enclosed between the two circles: x2 + y2 = 4 and (x – 2)2 + y2 = 4.52. Find the area of the circle x2+ y2 = 16 which is exterior to the parabola y2 = 6x by using Integration.53. Using integration find the area of the region (HOTS)54. Find the area of the region {(x, y) : 0 y x2+ 1, 0 y x + 1, 0 x 2} (HOTS) ( Ans-23/6)55. Find the area between x- axis, the curve x = y2 and its normal at the point (1, 1)
i. (HOTS , Ans- 11/12)
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56. Given that is directly proportional to the square of x and when x = 2, then find the equation of the curve at (4, 2). Find the area of the region bounded by the curve between the lines y = 1 and y = 3 using integration. (HOTS ,Ans-
CHAPTER - 8APPLICATION OF INTEGRALSMARKS
57. Find the area of the shaded region enclosed between the two circles :x2 + y2 = 1 , (x – 1)2 + y2 = 1.
58. Using integration find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x.59. Using integration find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle
x2 + y2 = 32.60. Find the area of the region lying between the parabolas y2 = 4ax and x2 = 4ay, where a > 0.61. Using integration find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.75. Find the area of the region included between the parabola y = 3x2/4 and the line 3x – 2y + 12 = 0.76. Using integration find the area of the region bounded by the triangl ABC, where vertices A, B and C are (-1, 1) (0, 5) and (3, 2) respectively.77. Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y.
78. Sketch the graph of y = | x + 3 | and evaluate the area under the curve y = | x + 3 | above x-axis and between x = -6 to x = 0.79. 10. Using the method of integration find the area of the region bounded by the lines 3x – 2y + 1 = 0, 2x + 3y – 21 =0
and x – 5y + 9 = 0.80. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B(4,
5) and C(6, 3).81. Find the area of the region bounded by the parabola y = x2 and y = |x|.
CHAPTER – 8APPLICATIONS OF INTEGRALS
SECTION – C (Six mark questions)
82. Find the area of the shaded region enclosed between the two circles :i. x2 + y2 = 1 and (x – 1)2 + y2 = 1.
83. Using integration find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x.84. Using integration find the area of the region in the first quadrant enclosed85. by the x-axis, the line y = x and the circle x2 + y2 = 32.86. Find the area of the region lying between the parabolas y2 = 4ax
i) and x2 = 4ay, where a > 0.87. Using integration find the area of the region bounded by the parabola
i) y2 = 4x and the circle 4x2 + 4y2 = 9.88. Find the area of the region included between the parabola y = 3x2/4 and the line 3x – 2y + 12 = 0.89. Sketch the graph of y = | x + 3 | and evaluate the area under the curve y = | x + 3 | above x-axis and between x
= -6 to x = 0.90. Using the method of integration find the area of the triangle ABC, coordinates of whose vertices are A(2, 0), B(4,
5) and C(6, 3).
91. Find the area of the region using integration {(x,y);x2+y2≤2ax, y2 ax,;x 0, y 0,}
92. 10.Find the area of smaller region bounded by the ellipse + =1 and
93. the straight line + =11. Find the area of the region bounded by the parabolas y2=4ax and x2=4ay
94. 12.Using integration find the area of the region
95.
33
96. 13.Using integration find the area of the region bounded by the parabola97. y2 =4x and the circle 4x2 +4y2=9
1. Find the area of the region enclosed between the two circles:98. x2 + y2 = 4 and (x – 2)2 + y2 = 4.
1. Find the area of the circle x2+ y2 = 16 which is exterior to the parabola99. y2 = 6x by using Integration.
1. Using integration find the area of the region included between the parabola100. and the line
1. Using the method of integration, find the area of the region bounded by101. lines 2x + y = 4 , 3x – 2 y = 6 and x – 3y + 5 = 0.102. 18. Find the area of the region .
103. Find the area of the smaller region bounded by the ellipse104. and the straight line x/3+y/2 = 1.105. Using method of integration find the area bounded by the curve |x|+|y|=1
i. APPLICATION OF INTEGRALSLA (marks)
106. Find the area enclosed by circle .107. Find the area of region bounded by 108. Find the area enclosed by the ellipse .109. Find the area of region in the first quadrant enclosed by –axis, the line and the circle .110. Find the area of region 111. Prove that the curve y = and, x = divide the square bounded by into three equal parts. 112. Using integration, find the area of the region bounded by the triangle whose vertices are andand.
i) DIFFEREANTIAL EQUATION113. Form the differential equation representing the family of ellipse
i) having foci on x axis and center at the origin.114. Solve the following differential equation
i) ex tan y dx + (1- ex)Sec² y dx =0115. Solve the differential equation x Cos(x/y)dy/dx= y Cos(x/y)+x, x≠0.116. Solve the differential equation (x²-1)dy/dx +2xy=2/(x²-1) .117. Solve the initial value problem- (x-Sin y) d y = tan y dx =0, y(0)=0.118. Find the particular solution of the differential equation
i) (1+x²)dy/dx = (e^(mtan^(-x) )-y) given that y=1when x=0.119. Solve the differential equation √(1+x²+y²+x²y²) ≠ x ydy/dx =0.120. Solve (xᶟ-3xy²) dx = (yᶟ-3x²y) d y.121. Solve (1+y²)(1+log x) dx +x d y=0 being given that y=1 when x=1.122. Find the differential equation of family of all circles in second quadrant and touching the co-ordinate axis.123. Solve (x log x) dy/dx+y=2/xlog x.124. Solve dy/dx+y tan x=2x+x² tan x.
CHAPTER – 10VECTOR ALGEBRA
1 MARKS 1. Find a unit vector in the direction of = 3î - 2ĵ + 62. Find what value of are the vectors 2 + + and - 2 + 3 perpendicular to each other?3. If P (1, 5, 4) and Q (4, 1, -2) , find the direction ratios of4. If || = 2, || = and =, find the angle between 5. If is a unit vector and = 80, then find
34
i. 4 MARKS6. The scalar product of the vector with the unit vector along the sum of vectors
i. and is equal to one. Find the value of λ.7.8. Find a unit vector perpendicular to each of the vectors where .9. If two vectors are such that then find the value of
i. .10. Using vectors, find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).11. If vector are such that is
i. perpendicular to then find the value of λ.12. If are three vectors such that | = 5, || = 12 and || = 13, and , find the value of 13. Let Find a vector which is perpendicular to
i. both 14. If then find the value of , so that are
i. perpendicular vectors.15. Dot product of a vector with vectors are respectively 4, 0 and 2.
i. Find the vector.
VECTORSVSA (1 mark)
16. What are the horizontal and vertical components of a vector aofmagnitude 5 making an angle of 150° with the direction of -axis.
17. When is ?18. What is the area of a parallelogram whose sides are given by and?19. What is the angle between and , if and.20. Write a unit vector which makes an angle of with -axis and with-axis and an acute angle with -axis.21. If A is the point (4, 5) and vector has components 2 and 6 along-axis and -axis respectively then write point B.22. What are the direction cosines of a vector equiangular with co-ordinateaxes?
SA (4/6marks)23. and . If lies in the plane of and , then find the value of .24. Prove that angle between any two diagonals of a cube is 25. Let and , and are unit vectors such that and theangle between and is
then prove that = .26. Prove that the normal vector to the plane containing three points withposition vectors , and , andlies in the
direction of vector and .27. For any three vectors , and prove that ,and are coplanar.
VECTOR ALGEBRAONE/TWO MARK QUESTIONS
28. Find the unit vector in the direction of [Ans:29. If , Find the angle between and [Ans: 30. Find , if the vectors , and are co planar
i. [ Ans: λ = 7]31. Find if for a unit vector , [ Ans 32. If and then what will be the angle between and .
[Ans
FOUR MARKS QUESTIONS
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33. For any three vector .Prove that.34. Find a vector whose magnitude is 3 units and which is perpendicular to each of the vector and .35. If makes an angle with and has magnitude 3 then prove that the angle between and each of is.36. are the position vector of the vertices of ABC. Find an expression for the area of ABC. And hence deduce the
condition for the points A, B and C are collinear. (HOTS) i. [Ans
37. If and find a vector such that and .38. In any prove the cosine formula: . (HOTS)39. Show that the four points A, B, C and D with position vectors and 4( respectively are co-planar.40. Let be three vectors of magnitude 3, 4 and 5 respectively. If each one is perpendicular to the sum of the other two
vectors, prove that.41. If and are two unit vectors inclined at angle then prove that.42. For any vector prove that (HOTS)43. If are vectors such that and and , then prove that 44. If are unit vectors and . Angle between is 30o , then prove that (HOTS)45. Show that are coplanar iff are co-planar.46. A girl walks 4km towards west. Then she walks 3km in a direction 30o east of north and stop. Determine the girl’s
displacement from her initial point of departure. i. [ Ans : -
47. If are mutually perpendicular vectors of equal magnitudes, then show that is equally inclined to 48. For any three vector .Prove that .49. Find a vector whose magnitude is 3 units and which is perpendicular to each of the vector and .50. Decompose the vector into two vectors and one of which is perpendicular to the vector .51. If makes an angle with and has magnitude 3 then prove that the angle between and each of is .52. If and find a vector such that and .53. In any prove the cosine formula : .54. Show that the four points A,B,C and D with position vectors and 4( respectively are co-planar.55. If and then what will be the angle between and .56. Let be three vectors of magnitude 3 ,4 and 5 respectively. If each one is perpendicular to the sum of the other two
vectors, prove that .57. If and are two unit vectors inclined at angle then prove that .
CHAPTER – 11THREE DIMENSIONAL GEOMETERY1 MARKS
1.Find the direction cosines of the line passing through the following points: ( -2, 4, -5), (1, 2, 3).2.Write the vector equation of the following line: .3.Write the intercept cut off by the plane 2x + y – z = 5 on x-axis.4.What are the direction cosines of a line, which makes equal angles with the coordinate axes?5.Find the distance of the plane 3x – 4y + 12z = 3 from the origin.6.Find the direction cosines of the line: .
4 AND 6 MARKS5
7.Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line .8.Find the equation of the line passing through the point P(4, 6, 2) and the point of intersection of the line and the plane x + y –z = 8.9.Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
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10. Find the shortest distance between the following two lines: = ( 1 + λ)î + (2 – λ)ĵ + (λ + 1)k ; and = (2î – ĵ – k) + (2î + ĵ + 2k).11. Find the equation of the plane determined by the points A (3, -1, 2), B (5, 2, 4) and C (-1, -1, 6). Also find the distance of the point P (6, 5, 9) from the plane.
11. Show that the lines are coplanar. Also find the equation of the plane containing the lines.12. Find the points on the line at a distance of 5 units from the point P (1, 2, 3).13. Find the length and the foot of the perpendicular from the point P (7, 14, 5) to the plane
a. 2x + 4y – z = 2. Also find the image of point P in the plane.14. Find the vector equation of the plane passing through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the
plane x – 2y + 4z = 10. Also show that the plane thus obtained contains the line = -î + 3ĵ + 4k + λ (3î - 2ĵ – 5k).15. Show that the lines, = 3î+ 2ĵ – 4k + λ (î + 2ĵ + 2k) ; = 5î - 2ĵ + (3î + 2ĵ + 6k) are intersecting. Hence, find their point of
intersection.16. Find the vector equation of the plane through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x –
2y + 4z = 10.17. Show that the lines = (î + ĵ – k) + λ (3î – ĵ) and = (4î –k) + (2î + 3k) are coplanar. Also find the equation of the plane
containing them.18. What are the direction cosines of a line which makes equal angles with the coordinate axis.
i. [ ans . ]19. Find the Cartesian equation of the line which passes through the point ( - 2, 4, -5 ) and is parallel to the line .
[ ans. ] 20. Find the position vector of a point A , in space such that OA is inclined at 600 to OX and at 450 to OY and
i. | OA | = 10 units. [ ans. Position vector of A =( 5 , 5 , 5 )] .21. If a unit vector makes an angle of with x- axis , and with y – axis , and an acute angl e with z- axis, then find the
value of . [ ans . ]
22. If P (2, 3, 4) is the foot of perpendicular from origin to a plane, write the equation of the plane. i. [ ans. 2x+3y+4z = 29]
23. Find the angle between pair of lines and .24. Find the shortest distance between the lines and ).25. Find the equation of the plane passing through the intersection of the planes and through the point .26. Find the co-ordinates of the point where the line through and crosses .27. 10.Find the equation of the plane passing through the point and perpendicular to each the planes and .28. Find the distance of the pointfrom the point of intersection of the line and the plane .29. Show that the lines and are co-planer. Also find the equation of the plane containing these lines.30. Show that the line whose vector equation is is parallel to the plane whose vector equation is . Also find the
distance between them.31. Find the co-ordinates of foot of perpendicular drawn from the point or the lines .
32. Find the distance of the point from the plane measured parallel to the line .
33. Find the image of the point on the line . 34. Find the vector equation of the points and and perpendicular to the plane .
Also show that the plane thus obtained contains the line .35. Find the equation of the plane passing through the intersection of the planes and through the point .36. 19.Find the equation of the plane passing through the point and perpendicular to each the planes and .37. 20.Find the distance of the point from the plane measured parallel to the line .38. Find the image of the point on the line . 39. Find the vector equation of the points and and perpendicular to the plane .Also show that the plane thus obtained
contains the line .40. Show that the lines and are co-planer. Also find the equation of the plane containing these lines.
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41. A line makes angles α, β, ϒ, and with the diagonals of a cube , prove that a. Cos2 α + Cos2 β + Cos2 ϒ + Cos2 = 4/3.
42. Prove that ,if a plane has the intercepts a , b ,c , and is a distance p units from the origin then prove that : a -2 + b -2 + c -2 = p -2.
43. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x-3y+4z -6 =0 ans ( , , )
a. HOT QUESTION44. Show that the line joining the origin to the point ( 2,1,1 ) is perpendicular to
the line determined by the points (3,5,-1) and ( 4,3,-1) .45. Find the equation of the line passing through the point (3,0,1) and parallel to the 46. Planes x+2y = 0 and 3y-z=0 .47. Find the equation of the plane passing through the points (2,1,-1 ) and ( -1,3.4)
and perpendicular to the plane x-2y +4z = 10 .48. Find the distance of the point ( -2,3,-4 ) from the line = = , measured49. Parallel to the plane 4x + 12y – 3z + 1 =0 .50. Find the perpendicular distance of the point (1,0,0 ) from the line = = .
Also find the coordinates of foot of perpendicular and equation of perpendicular.
CHAPTER – 12LINEAR PROGRAMMING
7 MARKS 1. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has space for at the most 20 items . A fan costs him Rs 360 and a sewing machine Rs 240 . He expects to sell a fan at a profit of Rs 22 and a sewing machine for a profit of Rs 18.. Assuming that he can sell all the items that he buys, how should he invest his money to maximise his profit? Solve it graphically.2. If a young man rides his motorcycle at 25 km/hour, he had to spend Rs 2 per km on petrol. If he rides at a faster speed of 40 km/hour, the petrol cost increases to Rs 5 per km. He has Rs 100 to spend on petrol and wishes to find , what is the maximum distance he can travel in one hour. Express this as an LPP and solve it graphically.3. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 400 is made an each first class ticket and a profit of Rs 300 is made on each second class ticket. The airline reserves at least 20 seats for first class. However at least, four times as many passengers prefer to travel by second class than by first class. Determine how many tickets of each type must be sold to maximise profit for the airline. Form an LPP and solve it graphically.4. A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit and F2 costs Rs 6 unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of F2 contains is 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimal cost for diet that contains of mixture of these two foods and also meets the minimal nutritional requirements. 5. One kind of flour cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of other ingredients used in making the cakes. Formulate the above as a linear programming and solve graphically.6. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftmens time in its making while a cricket bat takes 3 hours of machine time and 1 hours of craftmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftmans time. If the profit on a racket and on a bat is Ra 20 and Rs 10 respectively, find the number of tennis racket and cricket bats that the factory must manufacture to earn the maximum profit. Make it as LPP and solve it graphically. 7. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs
25,000 and Rs 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250
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units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70lakhs and his profit on the desktop model is Rs 4,500 and on the portable model is Rs 5,000. Make an LPP and solve it graphically.
8. A company produces softdrinks that has a contract which requires that a minimum of 80 units of the chemical A and 60 units of the chemical B go into each bottle of drink. The chemicals are available in prepared mix packets from two different suppliers. Supplier S had a packet of mix of 4 units of A and 2 units of B that costs Rs 10. The supplier T has a packet of mix unit of A and 1 unit of B that costs Rs 4. How many packets of mixes from S and T should the company purchase to honour the contract requirements and yet minimise cost? Make an LPP and solve graphically.9. A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from crops A and B per hectare are estimated as Rs 10,500 and Rs 9,000 respectively. The control weeds, a liquid herbicide has to be used for crops A and B at the rates of 20 litres and 10 litres per hectare respectively. Further not more than 800 liters of herbicide should be used in order to protect fishes and wild life using a pond which collects drainage from this land. Keeping in mind that the protection of fish and wild life is more important than earning profit, how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wild life is utmost necessary to preserve the balance in environment?10. An aeroplane can carry a maximum of 200 passengers. A profit of Rs 500 is made on each executive class ticket out of which 20% will go to the welfare fund of the employees. Similarly a profit of Rs 400 is made on each economy ticket out of which 25% will go for the improvement of facilities provided to economy class passengers. In both cases, the remaining profit goes to the airlines fund. The airline reserves at least 20 seats for executive class. However at least four times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the net profit of the airline? Make the above as an LPP and solve graphically. Do you think, more passengers would prefer to travel by such an airline than by others?
ii. Linear Programming (chapter-12)11. Corner points of the feasible region determined by the system of linear constraints , are (0,3),(1,1) and (3,0) . Let Z
= px + qy , where p,q> 0 .Find the condition in p and q , so that the minimum of Z occurs at (3,0) and (1,1) .12. Minimize Z = x+2y , Subject to constraints are 2x+y ≥ 3 , x+ 2y ≥ 6 and x,y ≥ 0 . Show that the minimum of Z occurs
at more than two points .13. Maximize Z= -x + 2y , subject to constraints are x≥ 3 , x + y≥ 5 , x + 2y ≥ 6 and x , y ≥0 .14. A co-operative society of farmers has 50 hec of land to grow two crops A and B . The profits from crops A and B
per hectare are estimated as Rs 10,500 and Rs 9000 , respectively . To control weeds , a liquid herbicide has to be used for crops A and B at the rate of 20L per hec and 10L per hec respectively .Further not more than 800L , herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land .Keeping in mind that the protection of fish and other wildlife is more important than earning profit.How much land should be allocated to each crop so as to maximise the total profit . Formulate the above as an LPP and solve it graphically . Do you agree with the message that the protection of wildlife is atmost necessary to preserve the balance in environment .
15. A manufacturing company makes two types of teaching aids A and B of mathematics for class XII . Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing .For fabricating and finishing ,the maximum labour hours available per week are 180 and 30 ,respectively . The company makes a profit of Rs80 on each piece of type A and Rs120 on each piece of type B . How many pieces of type A and type B should be manufactured per week to get a maximum profit .Make it as an LPP and solve graphically . What is the maximum profit per week .
16. A factory owner wants to purchase two types of machines A and B for his factory .The machine A requires an area of 1000 m2 and 12 skilled men for running it and its daily output is 50 units whereas the machine B requires area of 1200 m2 and 8 skilled men for running it and its daily output is 40 units . If area of 7600 m2 and 72 skilled men are available to operate the machines , how many machines of each type should be bought to maximise the daily output . What should be qualities of good machine .
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17. One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat . Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat, assuming that there is no shortage of other ingredients used in making the cakes . Formulate the above as a linear programming problem and solve it graphically .
18. A company manufactures two types of screws A and B . All the screws have to pass through a threading machine and a slotting machine .A box of type A screw require 2 min on the threading machine and 3 min on the slotting machine .A box of type B screws requires 8 min on the threading machine and 2 min on the slotting machine . In a week , each machine is available for both.On selling these screws , the company gets a profit of Rs100 per box on type A screws and RS170 per box on type B screws . Formulate this problem as an LPP given that the objective is to maximise profit . Solve the LPP and determine the maximum profit to the manufacturer .
19. A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate .He has 30 and 17 units of workers( male and female) and capital respectively, which he uses to produce two types of goods A and B .To produce 1 unit of A , 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce 1 unit of B . If A and B are priced at Rs100 and Rs120 per unit respectively , how should he use his resources to maximise the total revenue . From the above as an LPP and solve it graphically .Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should he paid at the same rate .
20. A manufacturer produces nuts and bolts . It takes 1 h of work on machine A and 3 h on machine B to produce package of nuts .It takes 3 h on machine A and 1 h on machine B to produce a package of bolts .He earns a profit of RS17.50 per package on nuts and Rs7 per package on bolts .How many packages of each should be produced each day so as to maximise his profits,if he operates his machines for atmost 12 h a day . Formulate above as LPP and solve it graphically .LINEAR PROGRAMMING
21. A manufacturer makes two products, A and B. Product A sells at $200 each and takes 1/2 hours to make .Product B sells at $300 each and take 1 hour to make. There is a permanent order for 14 units of products A and 16 units of product B. A working week consists of 40 hours of production and the weekly turnover must be less than $10000 .If the profit on each of product A is $20 and on product B $30, then how many of each should be produced so that the profit is maximum? Also find the maximum profit.
22. A diet is to contain at least 80 units of vitamin and 100 units of minerals. Two foods F1 andF2 are available. Food F1 costs $5 per unit and F2 cost $6 per unit. One unit of foods F1 contains three units of vitamin A and four units of minerals. One unit of F2 contains 6units of vitamin A and three units of minerals. Formulate these a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirement.
23. There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% of phosphoric acid and F2 consist of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crops .If F1 costs $6 per kg and F2 cost $5per kg. Determine how much of each type of fertilizers should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
24. David wants to invest at most 12000 in bonds A and B. According to the rules he has to invest at least $2000 in bond A and at least $4000 in bond B. If the rates of interest on bonds A and B respectively are 8% and 10% per annum, Formulate the problem as L.P.P. and solve it graphically for maximum interest. Also determine the maximum interest receive in a year.
25. One kind of cakes requires of flour and 25 gram of fats and another kind of cakes requires 100 grams of flour and 50 grams of fats. Find he maximum no of cakes which can be made 5 kg of flour and 1 kg of fat assuming that there is no shortage of ingredient used in makings the cakes. Formulate the above L.P.P. and solve it graphically.
26. 6. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has space for at most 20 items. A fan cost him Rs. 360 and a sewing machine Rs. 240. His expectation is that he can sell a fan at a profit of Rs . 22 and a sewing machine at a profit Rs. 18. Assuming that he can sell all the items that he can buy , how should he invest his money in order to maximize his profit? Formulate this as a linear programming problem and solve it graphically. (Max. profit Z = Rs. 392 at x=8 , y=12)
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27. A small firm manufacturers gold rings and chains. The total number rings and chains manufactured per day is at most 24.It takes one hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain Rs. 190. Find the number of rings and chains that should be manufactured per day so as to earn the maximum profit. Make it as a LPP and solve it graphically.(gold rings =8 , chains=16)
28. A cooperative society of farmers has 50 hectares of land to grow two crops A and B. The profits from A and B per hectare are estimated as Rs. 10, 500 and Rs. 9,000 respectively. To control weeds , a liquid herbicide has to be used for crops A and B at the rate of 20 litres and 10 litres per hectare respectively. Further not more than 800 litres of herbicides should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit , how much land should be allocated to each crop so as to maximize the total profit? Form an LPP from the above and solve it graphically. Do you agree with the message that the protection of wildlife is ut most necessary to preserve the balance in environment? ( Value Based)( Max.profit- Rs. 495000 for crop A 30 hectares and crop B 20 HECTARES)
29. An Aeroplane can carry a maximum of 200 passengers . A profit of Rs. 500 is made on each executive class ticket out of which 20% will go to welfare fund of the employees . Similarly a profit of Rs. 400 is made on each economy ticket out of which 25% will go for the improvement of facilities provided to economic class passenger. In both cases the remaining profit goes airlines fund. The airline reserve at least 20 seats for executive class . However at least 4- times as many passengers prefer to travel by economy class then by the executive class. Determine how many tickets of each type must be sold in order to maximize the net profit of the airline. Make the above as an LPP and solve graphically. Do you think more passengers prefer to travel by such an airline than by others? ( Value Based)( 40 – Executive class , 160- economic class)
30. A Manufacturing Company makes two types of teaching aids A and B of Mathematics for class-XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing . For fabricating and finishing the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs,. 80 on each piece of type A and Rs. 120 on each piece in type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit ? Make it as an LPP and solve graphically. What is the maximum profit per week?( Max. profit for week = Rs. 1680)
31. A factory owner wants to purchase two types of machines A and B for his factory . The machine A requires an area of 1000 m2 and 12 skilled men for running it and its daily output is 50 units whereas the machine B requires area of 1200 m2 and 8 skilled men for running it and its daily output is 40 units . If area of 7600 m2 and 72 skilledmen are available to operate the machines . How many machines of each type should be bought to maximize the daily output? What should be the qualities of good machine? (Value Based) ( A =4 , B =3 MAX. = 320 )
32. A manufacturer considers that men and women workers are equally efficient so he pays them at the same rate . He has 30 and 17 units of workers ( male and female) and capital respectively , which he uses to produce 2 types of goods A and B . To produce one unit of A , 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce 1 unit of B . If A and B are priced at Rs. 100 and Rs. 120 per unit respectively . Then how should he use his resources to maximum the total revenue? Formulate the above as an LPP and solve it graphically. Do you agree with this view of the manufacture that men and women workers are equally efficient and so should be paid at the same rate? ( value based)
33. A firm has to transport 1200 package using large vans which can take 200 packages each and small vans which can take 80 packages each . The cost for engaging each large van is Rs. 400 and each small van is Rs. 200. Not more than Rs. 3000 is to be spent on the job and the number of the large vans cannot exceed the number of small vans. Formulate this as an LPP given that the objective is to minimize the cost . What will be the minimum cost?
34. A new cereal , formed of a mixture of bran and rice contains at least 88 g of protein and at least 36 mg of iron . Knowing that bran contains 80 g of protein and 40 mg of iron per kg and the rice contains 100 g of protein and 30 mg of iron per kg . Find the minimum cost of producing 1 kg of this new cereal if bran costs Rs. 28 per kg and rice cost Rs. 25 per kg . ( HOTS)
35. A manufacturer of electronic circuits as a stock of 200 resistors , 120 transistors and 150 capacitors and is required to produce 2 types of circuits A and B . Type A requires 20 resistors and 10 transistors and 10 capacitors. Type B
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requires 10 resistors , 20 transistors and 30 capacitors . If the on type A circuit is Rs. 50 and that on type on B circuits is Rs. 60 . formulate this problem as LPP so that the manufacturer can maximize his profit. ( Ans :- Max Z = 50x + 60y , subject to 2x + y ≤ 20 , x + 2y ≤ 12 , x + 3y ≤ 15 , x≥ 0 , y≥ 0)
36. A man rides his motorcycle at speed of 50 km/hr. He has to spend Rs. 2 per km on petrol . If he rides at a faster speed of 80 km/hr, the petrol cost increases to Rs. 3 per km. He has at most Rs. 120 to spend on petrol and one hours time . He wishes to find the maximum distance that he can travel? Express as LPP. ( Z = x+y , 2x +3y ≤12-0 , 8x+5y≤400 , , x≥ 0 , y≥ 0)
37. A manufacturer produces to models of bikes –Model X and Model Y . Model X takes 6 men hours to make per unit, while Model Y takes 10 men hours per unit . There is a total of 450 men hours available per week. Handling and marketing cost are Rs. 2000 and Rs. 1000 per unit of models X and Y respectively. The total funds available for these purposes are Rs. 8000 per week. Profits per units for Model X and Y are Rs. 1000 and Rs. 500 respectively . How many bikes for each Model should be manufacturer produce so as to yield a maximum profit? Find the maximum profit. (Ans. X=25 Y=30 and profit 40,000)
CHAPTER – 13 ( PROBABILITY) ONE MARK QUESTIONS
1. Evaluate P( AB) , If 2P(A) = P(B) = and P() = .2. Three events A,B and C have probabilities ,,and respectively . If P(AC) = and P(BC) = , then find the values of P(C/B)
and P(A/ C/ ) .3. Given two independent events A and B such that P(A) = 0.3 , P(B) = 0.6 , Find P( A and not B) . 4. If P(A) = ,P(B) = and P(AB) = ,then evaluate P(A/B) .5. Find the variance of the Binomial distribution B(5,) .
FOUR MARKS QUESTIONS 6. A,Band C shot to hit a target . If A hits 4 times in 5 trails ; B hits 3 times in 4 trails and C hits 2 times in 3 trails . What
is the probability that the target is hit . If hitting the target depends upon their concentration only then whose concentration is more . What is the importance of concentration in doing a work .
7. In a hurdle race , Rita has to cross 10 hurdles . The probability that he will clear each hurdle is .What is the probability that he knock down fewer than 2 hurdles . Do you think that sports play an important role in student life8
8. The probabilities of two students A and B coming to school in time are and respectively . Assuming that their coming to school is independent , find the probability of only one of them coming to school in time . Write at least one advantage of coming to school in time .
9. There are two bags , bag I and bag II . Bag I contains 2 white and 4 red balls and bag II contains 5 white and 3 red balls . One ball is drawn at random from one of the bags and is found to be red .Find the probability that it was drawn from bag II .
10. A speaks truth in 60% of the cases and B in 90 % of the cases. In what percentage of cases are they likely to state the same fact .Which value of A is lacking should improve upon .
(Four & Six mark questions)11.A and B throw a dice alternatively till one of them gets a number greater than four and wins the game. If A starts the
game what is the probability of B’s winning?12. A committee of 4 students is selected at random from a group of 8 boys and 4 girls on the committee. Calculate the probability that there are exactly 2 girls on the committee.13. A urn contains 3 red and 5 black balls. A ball is drawn at random its colour is noted and returned to the colour noted down are put into the urn and then two balls are drawn at random (without replacement) from the urn. Find the probability that both the balls are of red colour.14. How many times must a man toss a coin so that the probability of having at least one head is more than 80%?15. A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of five steps he is one step away from the starting point.16.In a hurdle race Ritam has to cross 10 hurdles. The probability that he will clear all the hurdles is. What is the probability that he will knock down fewer than 2 hurdles? Do you think that sports play an important role in
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student’s life?(Ans- )17. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and Standard deviation of the number of kings.(Ans-18. Out of 9 outstanding students of a school, there are 4 boys and 5 girls. A team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected.19. In a game, a man earns a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find expected value of the amount he wins/loses.20. In answering questions on a MCQ test with 4 choices per question, a student’s known the answer guesses or copies the answer. Let is the probability that he knows the answer, is the probability that he copies it...Assuming that the student who copies the answer will be correct has the probability, what is the probability that the student knows the answer given that he answered it correctly? HOTS
21. Assume that the chances of a patient having a heart attack is 40%, assuming that Meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces the risk by 25%.At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random, suffers heart attack. Find the probability that the patient followed a course of meditation and yoga. VBQ(Ans-14/29)
22. The probabilities of A, B and C solving a problem are , and respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them can solve it.
23. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of die.
24. There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up tails 25% of the times and the third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows head, what is the probability that it was from the two headed coin? HOTS
25. Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards.
26. A manufacturer has three machine operators A,B and C . The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7 % defective items respectively. A is in the job for 50% of the time, B is In the job for 30% of the time and C is in the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A? HOTS
27. Suppose that the reliability of a HIV test is specified as follows:28. Of people having HIV, 90% of the test detects the diseases but 10% go undetected. Of people free of HIV, 99% of
the tests are judge HIV –ive but 1% are diagnosed as showing HIV +ve. From a large population of which only 0.1% has HIV , one person is selected at random, given the HIV tests and the pathologist reports him/her as HIV +ve. What is the probability that the person actually has HIV +ve ? HOTS (Ans-)
29. In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian? What value is reflected in this question? VBQ
30. 10% of the bulbs produced in a factory are of red color and 2% are red and defective. If one bulb is picked up at random, determine the probability of its being defective if it is red.
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