MATHEMATICS-XII mathsNmethods - 1 - IMPORTANT QUESTIONS FOR CBSE 2019-20 SUB :- MATHEMATICS (041) Note: Please go through concepts of each and every topics and then try to solve with open mind to observe the structure of problems given for two sections. For Section C(4 marks) 1. Let R be the relation on N X N defined by (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation. Sol.For the relation R defined by (a, b) R (c, d) if ad=bc on N X N. Reflexivity: Let (a, b) ∈ N X N. (a, b) R (a, b) as ab = ba (Multiplication is commutative on N) ⇒ (a, b) R (a, b) ⩝ (a, b)∈ N X N. ∴ R is reflexive relation on N X N. Symmetry: Let (a, b) R (c, d) ⇒ ad = bc ⩝ a, b, c, d ∈ N ⇒ cb = da ⩝ a, b, c, d ∈ N (Multiplication is commutative on N) ⇒ (c, d) R (a, b) ⩝ a, b, , ∈ N. ∴ R is symmetric relation on N X N. Transitivity: Let (a, b) R (c, d) ⇒ ad = bc ⩝ a, b, c, d ∈ N … (i) & (c, d) R (e, f) ⇒ cf = de ⩝ c, d, e, f ∈ N … (ii) Multiplying (i) and (ii) we get adcf = bcde Divide by cd both sides ⇒ af = be (Multiplication is commutative on N) ⇒ (a, b) R (e, f) ⩝ a, b, c, d, e, f ∈ N ∴ R is transitive relation on N X N. ∵ R is reflexive, symmetric and transitive so R is equivalence relation. OR Show that the relation R in the set R of real numbers, defined as R = {(a, b): a, b Є R and a≤" # } is neither reflexive, nor symmetric nor transitive. (NCERT) (Delhi 2010) (AI 2010) Sol. For the relation R = {(a, b): a, b Є R and a ≤" # } in the set R of real numbers. Reflexivity: ’ ( ∈ R, ) ’ ( * # = ’ + but ’ ( ≰ ’ + ⇒( ’ ( , ’ ( )∉ R ⇒ R is not re.lexive relation. Symmetry: 1≤ (2) # ⇒ (1, 2) ∈ R but (2, 1) )∉ R as 2 ≰ (1) # ⇒ R is not symmetric relation. Transitivity: As 63 ≤ (4) # ⇒ (63, 4) ∈ R and (4, 2) ∈ R as 4 ≤ (2) # but (63, 2) ∉ R as 63 ≰ (2) # ∴ R is not transitive relation on R. OR Let A = {1,2,3,…,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 ∈A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)]. (NCERT Exemplar) (Delhi 2014) (SP 15) Sol. For the relation R defined by (a, b) R (c, d) if a + d = b + c for a, b, c, d ∈ A on the set A = {1, 2, 3,…, 9} Reflexivity: Let (a, b) ∈ A X A. (a, b) R (a, b) as a + b = b + a (Addition is commutative on N) ⇒ (a, b) R (a, b) ⩝ (a, b)∈ A X A. ∴ R is reflexive relation on A. Symmetry: Let (a, b) R (c, d) ⇒ a + d = b + c ⩝ a, b, c, d ∈ A ⇒ c + b = d + a ⩝ a, b, c, d ∈A (Addition is commutative on N) ⇒ (c, d) R (a, b) ⩝ a, b, , ∈ A. ∴ R is symmetric relation on A. Transitivity: Let (a, b) R (c, d) ⇒ a + d = b + c ⩝ a, b, c, d ∈A
15
Embed
MATHEMATICS-XII maths Nmethods - CBSEGuess...MATHEMATICS-XII mathsNmethods - 1 - IMPORTANT QUESTIONS FOR CBSE 2019-20 SUB :- MATHEMATICS (041) Note: Please go through concepts of each
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
MATHEMATICS-XII mathsNmethods
- 1 -
IMPORTANT QUESTIONS FOR CBSE 2019-20
SUB :- MATHEMATICS (041)
Note: Please go through concepts of each and every topics and then try to solve with open mind to observe the
structure of problems given for two sections.
For Section C(4 marks)
1. Let R be the relation on N X N defined by (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation.
Sol.For the relation R defined by (a, b) R (c, d) if ad=bc on N X N.
Reflexivity: Let (a, b) ∈ N X N. (a, b) R (a, b) as ab = ba (Multiplication is commutative on N) ⇒ (a, b) R (a, b) ⩝ (a, b)∈ N X N. ∴ R is reflexive relation on N X N. Symmetry: Let (a, b) R (c, d) ⇒ ad = bc ⩝ a, b, c, d ∈ N ⇒ cb = da ⩝ a, b, c, d ∈ N (Multiplication is commutative on N) ⇒ (c, d) R (a, b) ⩝ a, b, �, � ∈ N. ∴ R is symmetric relation on N X N. Transitivity: Let (a, b) R (c, d) ⇒ ad = bc ⩝ a, b, c, d ∈ N … (i)
& (c, d) R (e, f) ⇒ cf = de ⩝ c, d, e, f ∈ N … (ii)
Multiplying (i) and (ii) we get adcf = bcde Divide by cd both sides
⇒ af = be (Multiplication is commutative on N)
⇒ (a, b) R (e, f) ⩝ a, b, c, d, e, f ∈ N
∴ R is transitive relation on N X N. ∵ R is reflexive, symmetric and transitive so R is equivalence relation.
OR
Show that the relation R in the set R of real numbers, defined as R = {(a, b): a, b Є R and a≤ "#} is neither
reflexive, nor symmetric nor transitive. (NCERT) (Delhi 2010) (AI 2010)
Sol. For the relation R = {(a, b): a, b Є R and a ≤ "#} in the set R of real numbers.
Reflexivity: '( ∈ R, )'(*# = '+ but
'( ≰ '+ ⇒ ('( , '() ∉ R ⇒ R is not re.lexive relation. Symmetry: 1≤ (2)# ⇒ (1, 2) ∈ R but (2, 1) ) ∉ R as 2 ≰ (1)# ⇒ R is not symmetric relation. Transitivity: As 63 ≤ (4)#⇒ (63, 4) ∈ R and (4, 2) ∈ R as 4 ≤ (2)# but (63, 2) ∉ R as 63 ≰ (2)#
∴ R is not transitive relation on R.
OR
Let A = {1,2,3,…,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 ∈A. Prove that
R is an equivalence relation. Also obtain the equivalence class [(2, 5)]. (NCERT Exemplar) (Delhi 2014) (SP 15)
Sol. For the relation R defined by (a, b) R (c, d) if a + d = b + c for a, b, c, d ∈ A on the set
A = {1, 2, 3,…, 9}
Reflexivity: Let (a, b) ∈ A X A. (a, b) R (a, b) as a + b = b + a (Addition is commutative on N) ⇒ (a, b) R (a, b) ⩝ (a, b)∈ A X A. ∴ R is reflexive relation on A. Symmetry: Let (a, b) R (c, d) ⇒ a + d = b + c ⩝ a, b, c, d ∈ A ⇒ c + b = d + a ⩝ a, b, c, d ∈A (Addition is commutative on N) ⇒ (c, d) R (a, b) ⩝ a, b, �, � ∈ A. ∴ R is symmetric relation on A. Transitivity: Let (a, b) R (c, d) ⇒ a + d = b + c ⩝ a, b, c, d ∈A
MATHEMATICS-XII mathsNmethods
- 2 -
& (c, d) R (e, f) ⇒ c + f = d + e ⩝ c, d, e, f ∈A
⇒ a + d + c + f = b + c + d + e ⩝ a, b, c, d, e, f ∈A
⇒ a + f = b + e ⩝ a, b, c, d, e, f ∈ A
⇒ (a, b) R (e, f) ⩝ a, b, c, d, e, f ∈ A
∴ R is transitive relation on A. ∵ R is reflexive, symmetric and transitive so R is equivalence relation.
Equivalence class [(2,5)] is {(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}.
OR
Determine whether the relation R defined on the set R of all real numbers as R={(a,b): a,b∈ 8 and 9 − " + √3 ∈ <, where S is the set of all irrational numbers}, is reflexive , symmetric and transitive.
OR
Let f: N�Z be defined as
f(n) = =>?'( , @ℎBC C DE F��?>( , @ℎBC C DE BGBCH for all n∈N. State whether the function is bijective. Justify your answer.
Sol. Injectivity : Let I' and I( ∈ N such that f(I') = f(I().
Case I. Let both numbers I' and I( are odd then f(I') = f(I() ⇒ KL?'( = KM?'( ⇒ I' = I( ⩝ I',I( ∈ N ∴ f is 1-1 function.
Case II. Let both numbers I' and I( are even then f(I') = f(I() ⇒ ?KL( = ?KM( ⇒ I' = I( ⩝ I',I( ∈ N ∴ f is 1-1 function.
So in both the cases f is 1-1.
Surjectivity : Case I. Let n is an odd natural number then y = f(n) ⇒ y = >?'( ⇒ 2y= n-1 ⇒ n = 2y+1 ∴ Each positive integer is an image of odd natural number. Case II. Let n is an odd natural number then y = f(n) ⇒ y = ?>( ⇒ 2y= -n⇒ n = -2y ∴ Each negative integer is an image of even natural number.
Combining both the above cases, RR = Co-domain of f.
∴ f is onto.
∵ f is 1-1 and onto function. ∴ f is bijective function.
OR
MATHEMATICS-XII mathsNmethods
- 3 -
Let f: N →R be a funcEon defined as f(x) = 4x2 +12x+15, show that f : N →S, where S is range of f, is inverEble find f
-
1. (NCERT) (AIC 2013) (SP 15)
Sol. Given f: N →R, f(x)= 4 x2 +12x+15. Let y=f(x) ⇒ y= 4 x
Let y= f(x) ⇒d = K?(K?# ⇒ y(x − 3) = x − 2 ⇒ xy − 3y = x − 2 ⇒ x(y − 1) = 3y − 2 ⇒ x = #Z?(Z?' is defined ⩝ y∈B ⇒ f is into function. ∵ f is 1-1 and onto function. ∴ f is bijective function ⇒ f is invertible f-1(x)=#K?(K?' from the above equation.
2. Find matrix X so that X )1 2 34 5 6* = )−7 −8 −92 4 6 *. (MT 15) (F 17) o)1 −22 0 *q
OR
If A= r1 −12 −1s and = r9 1" −1s and (t + u)( = t( + u(, then find the values of a and b. (F 15) (" = 4, 9 = 1)
OR
On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8
children less, every one would have got ₹ 10 more. However, if there were 16 children more, every one would
have got ₹ 10 less. Using matrix method, find the number of children and the amount distributed by Seema.
What values are reflected by Seema’s decision? (ER 16) (32, ₹ 30; to help needy people)
OR
If A = r 3 1−1 2s . Show that A2 -5A +7 I =0 and hence find (i) A
-1 (ii) t# x(D) y(z − 'z'z #z {|
OR
Express r 3 1−1 2s as sum of symmetric and skew symmetric matrices.
OR
MATHEMATICS-XII mathsNmethods
- 4 -
Prove that }"� − 9( �9 − "( 9" − �(�9 − "( 9" − �( "� − 9(9" − �( bc − a( �9 − "(} is divisible by a+b+c and find the quotient.
OR
Prove the following, using properties of determinants:
}a + b + 2c 9 "� b + c + 2a "� 9 � + 9 + 2"} = 2(a+b+c)3 (Foreign10), (DelhiC 2012), (Delhi 2014)
OR
Using properties of determinants, prove the following :
}b + c 9 − " 9� + 9 b − c "9 + " � − 9 �} = 3abc- a3- b
3- c
3 (AIC 2012) (SP2 17)
OR
Prove the following, using properties of determinants:
If I, d, ~ are in GP, then using properties of determinants, show that }�I + d I d�d + ~ d ~0 �I + d �d + ~} =0, @ℎB�B I ≠ d ≠ ~ and p is any real number. (SP 15)
OR
If cos 2� = 0, then find the value of } 0 cos � sin �cos � sin � 0sin � 0 cos �}(. (Exemplar) )'(*
OR
If } 9 " − I � − ~9 − I " � − ~9 − I " − d � } = 0, then using properties of determinants, find the value
of WK + SZ + X� where I, d, ~ ≠ 0 (DelhiC 17) (2)
OR
MATHEMATICS-XII mathsNmethods
- 5 -
If x ,y and z are unequal and }I I( 1 + I#d d( 1 + d#~ ~( 1 + ~#} = 0, prove that 1+xyz = 0. (B 17)
OR
Let g(�) = } cos � � 12 sin � � 2�sin � � � }, then find lim�→� e(�)�M . (Exemplar) (0)
OR
If � is real number find the maximum value of } 1 1 11 1 + sin � 11 + cos � 1 1}. (Exemplar) )'(*
OR
If ∆= } 1 9 9(9 9( 19( 1 9 } = −4 then find the value of }9# − 1 0 9 − 9V0 9 − 9V 9# − 19 − 9V 9# − 1 0 }. (SP 18) (16)
(Hint use |A| = |9��t|()
OR
If 9 + " + � ≠ 0 and }9 " �" � 9� 9 "} = 0, then using properties of determinants, prove that 9 = " = �. (Bh
15) (MT 15) (AIC 17)
OR
Using properties of determinants, prove that ��(W]S)MX � �9 (S]X)MW 9" " (X]W)MS
�� = 2(9 + " + �)# (SP1 17)
OR
If p≠ 0, q ≠ 0 and } � � �� + �� � �� + r�� + � �� + r 0 } = 0 then, using properties of determinants, prove that at
least one of the following statements is true: (a) p, q, r are in G. P., (b) α is a root of the equation �I( + 2�I + � = 0. (SP1 17)
3. f(x)= =� ��� K�?(K , Dg I ≠ �( 3 , Dg I = �(H, find k for which f(x) is continuous.
OR
Discuss the differentiability of the function g(I) = =2I − 1, I < '(3 − 6I, I ≥ '(H at I = '(
OR
Find ‘a’ and ‘b’, if the function given by g(I) = �9I( + ", I < 12I + 1, I ≥ 1 H is differentiable at I = 1. (SP 18) (9 = 1, " = 2 )
OR
Show that the function g(I) = 2I − |I|, x ϵ R , is continuous but not differentiable at I = 0. OR
Determine the value of ‘a’ and ‘b’ such that the following function is continuous at = 0:
MATHEMATICS-XII mathsNmethods
- 6 -
f(x) = ��� f]��� f���(�]')f , if − π < I < 02, x = 02 ���� ¡ ?'¢f , if x > 0 H
¤(9 = 0, " may be any real number other than zero )¦
OR
For what value of k is the following function continuous at = − �̂ ?
�¬�¬� '?X®VKKM , @ℎBC I < 09, @ℎBC I = 0√K¯('^]√K )?V , @ℎBC I > 0H 9C� g DE �FC�DC°F°E 9� I = 0, gDC� �ℎB G9±°B Fg 9.
(AIC 2012), (DelhiC 2013) (8)
OR
If f(x)=
�¬�¬�®²> (W]')K]®²> KK , @ℎBC I < 0�, @ℎBC I = 0UK]SKM?√KSUK³ , @ℎBC I > 0 H 9C� g DE �FC�DC°F°E 9� I = 0, gDC� 9, " 9C� �.
´9 = − 32 , " ∈ 8 − µ0¶, � = 12·
4. If I = 9 sin ��, d = " cos ��, then show that (9( − I()d ¸MZ¸KM + "( = 0. (SP 17)
OR Differentiate log )�]¢ ��� f�?¢ ��� f* with respect to x. OR Differentiate log (I®²>K + cot( I) with respect to x. (SP14) )I®²>K )�FE I ±F» I + ®²> KK * − 2�F� I �FEB�(I*
OR
If d = log )√I + '√K*( , then show that I(I + 1)(d( + (I + 1)(d' = 2. (SP 18)
OR
If d = log¼I + √I( + 1½ , ��FGB �ℎ9� (I( + 1) ¸MZ¸KM + I ¸Z¸K = 0
Find the position vector of the foot of the perpendicular and the perpendicular distance from the point P with
position vector 2ı̂ + 3ȷ̂ + 4kÝ to the plane �Ø. (2ı̂ + ȷ̂ + 3kÝ)−26 = 0. Also, find image of P in the plane. (CR 16)
)ã = '( , Öð = 3ı̂ + z
( ȷ̂ + ''( kÝ; �DE� = √'V
( ; Äå9»B = �4, 4, 7*
10. Find the mean number of heads in three tosses of a fair coin.
OR
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability
distribution of the number of diamond cards drawn. Also, find the mean and the variance of the distribution.
)ñB9C = '( , Var = #
+*
OR
Four bad oranges are accidentally mixed with 16 good ones. Find the probability distribution of the number
of bad oranges when two oranges are drawn at random from this lot. Find the mean and variance of the
distribution. )M = (Y , V = 'VV
VzY*
OR
A bag contains (2n+1) coins. It is known that n of these coins have a head on both sides where as the rest of
the coin are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss
results in a head is #'V(, find the value of n.
OR
For 6 trials of an experiment, let X be a binomial variate which satisfies the relation 9 P�X = 4 =P�X = 2. Find the probability of success. )8�( + 2� − 1 = 0, � = '
V* (RU 15)
OR
A person wants to construct a hospital in a village for welfare. The probabilities are 0.4 that some bad element
oppose this work, 0.8 that the hospital will be completed, if there is no any oppose of any bad element and 0.3 that the hospital will be completed, if bad element oppose. Determine the probability that the
construction of hospital will be completed. )#Y*
11. A shopkeeper sells 3 types of flower seeds A1,A2 and A3. They are sold as a mixture where the proportion are
4:4:2 respectively. The germination rates of 3 types of seeds are 45%, 60% and 35%. Calculate the probability.
(a) Of a randomly chosen seed to geminate.
(b) That it is of the type A2, given that a randomly chosen seed does not germinate. (SP14)
)0.49, '^Y'* (NCERT Exemplar)
A factory has two machines A and B. Past record shows that Machine A produced 60% of the items of output
and machine B produced 40% of the items. Further 2% of the items produced by machine A and only 1%
produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at
random from this and is found to be defective. What is the probability that it was produced by machine B?
OR
MATHEMATICS-XII mathsNmethods
- 12 -
Bag I contains 1 white, 2 black and 3 red balls; Bag II contains 2 white, 1 black and 1 red balls; Bag III contains
4 white, 3 black and 2 red balls. A bag is chosen at random and two balls are drawn from it with replacement.
They happen to be one white and one red. What is the probability that they come from Bag III. ) ^V'\\*
OR
Bag I contains 3 red and 4 black balls and bag II contains 4 red and 5 black balls. One ball is transferred from
bag I to Bag II and then 2 balls are drawn at random (without replacement) from bag II. The balls so drawn are
found to be both red in colour. Find the probability that the transferred ball is red. (SP 14, G 15) )Y\*
OR
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and
are found to be hearts. Find the probability of the lost card being of hearts. (DelhiC 2012) )''Y�*
OR
If a machine is correctly set up, it produces 90% acceptable items. If it is incorrectly set up, it produces only 40%
acceptable items. Past experience shows that 80% of the set ups are correctly done. If after a certain set up, the
machine produces 2 acceptable items, find the probability that the machine is correctly setup. (0.95)
OR
There are 3 coins. One is 2 headed coin another is a biased coin that comes up heads 75% of the times
and third is also a biased coin that comes up tail 40% of the times. One of the 3 coins is chosen at random
and tossed, and it shows heads, what is the probability that it was the 2 headed coin?)(�Vz * (AI 2014)
OR
A letter is known to have come from either TATANAGAR or CALCUTTA . On the envelop just 2 consecutive
letters TA are visible. What is the probability that letter has come from
TATANAGAR ? ) z''* (NCERT Exemplar)
OR
An urn contains 3 red and 5 black balls. A ball is drawn at random, its colour is noted and returned to the
urn. Moreover, 2 additional balls of the colour noted down, are put in the urn and then two balls are
drawn at random (without replacement) from the urn. Find the probability that both the balls drawn are
of red colour. (B 15) )'+*
For Section D(6marks)
12. Ten students were selected from a school on the basis of values for giving awards and were divided into three
groups. First group comprises hard workers, second group comprises honest and law abiding students and
third group contains vigilant and obedient students. Double the number of students of the first group added
to the number in second group gives 13, while the combined strength of the first and second group is 4 times
that of the third group using matrix method, find the number of students in each group.
''� ô−8 10 −216 −10 42 0 −2õ , I = 5, d = 3, ~ = 2
OR
Using elementary transformations, find the inverse of the matrix A= ô8 4 32 1 11 2 2õ and use it to solve the
Find the distance of the point (3,4,5) from the plane I + d + ~ = 2 measured parallel to the line 2I = d = ~. (DelhiC 2012) �λ = −4, Point of intersection is �1, 0, 1, distance = 6 unit
OR
Find the distance of the point −2ı̂ + 3ȷ̂ − 4kÝ from the line �Ø = ı̂ + 2ȷ̂ − kÝ+ λ (ı̂ + 3ȷ̂ − 9kÝ) measured parallel
to the plane: I − d + 2~ − 3 = 0. )√Y\( *
OR
Find the equation of the plane through the line of intersection of the planes I + d + ~ = 1 and 2I + 3d +4~ = 5 and twice of its y-intercept is equal to three times its z-intercept. (DelhiC 17) �λ = −1, x + 2y + 3z =4
MATHEMATICS-XII mathsNmethods
- 15 -
17. A firm has to transport atleast 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is ₹ 400 and each small
van is ₹ 200. Not more than ₹ 3,000 is to be spent daily on the job and the number of large vans cannot
exceed the number of small vans. Formulate this problem as a LPP and solve it graphically given that the
objective is to minimize cost. OR
A company produces two different products. One of them needs ¼ of an hour of assembly work per unit, 1/8
of an hour in quality control work and ₹ 1.2 in raw materials. The other product requires 1/3 of an hour of
assembly work per unit, 1/3 of an hour in quality control work and ₹ 0.9 in raw materials. Given the current
availability of staff in the company, each day there is at most a total of 90 hours available for assembly and 80
hours for quality control. The first product described has a market value (sale price) of ₹ 9 per unit and the
second product described has a market value (sale price) of ₹ 8 per unit. In addition, the maximum amount of
daily sales for the first product is estimated to be 200 units, without there being a maximum limit of daily sales
for the second product. Formulate and solve graphically the LPP and find the maximum profit.
OR
If a young man rides his motor-cycle at a speed of 25 km/hr, he has to spend ₹ 2 per km on petrol with very
little pollution in the air. If he drives his motor-cycle r at a speed of 40 km/hr, the petrol cost increases to ₹ 5
per km and rate of pollution also increases. He has a maximum of ₹ 100 to spend on petrol and travel a
maximum distance in one hour time. Express this problem as an LPP and solve it graphically.
What value is indicated in this question? (DelhiC 2014)
´Max Distance = 30 km at )Y�# , V�# *· ;Indication of Value: Vehicle should be driven at a moderate speed to
decrease the pollution.
OR
An aeroplane can carry a maximum of 200 passengers. A profit of ₹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 ₹ 400 is made on each
economy class 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 airline’s fund. The airline reserves at least 20 seats
for executive class. However, at least 4 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 maximize the profit for
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? ( Max profit = ₹64,000 at x=40,y=160.) (Foreign 2013)