www.sakshieducation.com www.sakshieducation.com Half-Yearly Examinations - Maths Paper-II Model Paper-I Part-A Time: 2 Hours Maximum Marks: 35 Section-I Group-A (Geometry, Analytical Geometry, Statistics) 1. Prove that the tangents at the ends of a diameter of a circle are parallel. 2. Find the point on x–axis which is equidistant from (2,3) and (4,–2) 3. Find the equation of straight line passing through the points (4,–7) and (1,5) 4. Write the merits of Arithmetic Mean? Group-B (Matrices, Computing) 5. Show that AB ≠ 0, BA=0, If . 6. A matrix D has an inverse. D –1 = Find D. 7. Write the characteristics of a computer. 8. Define the i) Algorithem ii) Flow chart Section-II 9. Find the distance between the centres of two cicles whose radii are 5 cm and 7 cm having three common tangents? 3 4 1 2 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ 1 0 B= 0 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ a 0 0 A= 1 0 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦
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Half-Yearly Examinations - Maths Paper-II
Model Paper-I
Part-A
Time: 2 Hours Maximum Marks: 35
Section-I
Group-A
(Geometry, Analytical Geometry, Statistics)
1. Prove that the tangents at the ends of a diameter of a circle are parallel.
2. Find the point on x–axis which is equidistant from (2,3) and (4,–2)
3. Find the equation of straight line passing through the points (4,–7) and (1,5)
4. Write the merits of Arithmetic Mean?
Group-B
(Matrices, Computing)
5. Show that AB ≠ 0, BA=0, If..
6. A matrix D has an inverse. D–1= Find D.
7. Write the characteristics of a computer.
8. Define the i) Algorithem ii) Flow chart
Section-II
9. Find the distance between the centres of two cicles whose radii are 5 cm and 7 cm having
three common tangents?
3 41 2
⎡ ⎤⎢ ⎥⎣ ⎦
1 0B =
0⎡ ⎤⎢ ⎥⎣ ⎦a
0 0A =
1 0⎡ ⎤⎢ ⎥⎣ ⎦
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11. Find the median of the following observations 1.8, 4.0, 2.7, 1.2 , 4.5, 2.3, 3.1 and 3.7 .
12. Maximise the objective function at (0,120) and (80, 40).
13. One end of the diameter of a circle is (2,3) and the centgre is (–2,5). Find the co-ordinate
of the other end of the diameter.
14. Define programming language.
Section-III
Group-A
15. State and prove Pythagoras Theorem.
16. The point G(0,6) is the centroid of the triangle, two of whose vertices are A(–4,4), B(6,12)
Find the co-ordinates of the third vertex. Show that area of Δ ABC = 3(area of Δ AGB).
17. Find the area of the triangle enclosed between the coordinate axes and the line passing
through (8,–3) and (–4, 12).
18. Marks scored by 100 students in a 25 marks unit test of mathematics is given below. Find
the median.
Marks 0-5 5-10 10-15 15-20 20-25
Students 10 18 42 23 7
Group-B
(Matrices, Computing)
19. Given that and (A+B)2 = A2+B2. Find a,b.
20. Solve the following equations using matrix inversion method
1 1 a 1 A = ,B =
2 1 b 1−⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦
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21. What are the different boxes used in a flow chart? Describe their functions in details?
22. Write an algorithm and draw a flow chart to pick largest number of the three given number?
Section-IV
(Polynomials, Linear Programming)
23. Draw a circumcircle to a ABC with measures AB= 4 cm, BC= 4 cm, and AC= 6 cm.
24. Construct a triangle ABC in which BC= 5cm, A= √70° and median AD through A= 3.5cm.
PART-B
Marks : 30×½=5
1. ΔABC ∼ Δ DEF, m ∠A + m∠B = 130° then ∠f = –––––––––––. ( )
A) 130° B) 140° C) 50° D) 40°
2. The line y= mx+c intersect the x–axis at the point ––––––––. ( )
A) (0,C) B) (C,0) C) (–c/m, 0) D) (0, –c/m)
3. The slope of the line joining (4,6) and (2,–5) is ––––––––––– . ( )
A) 6/5 B) –2/4 C) 5/6 D) 11/2
4. The histogram consists of ––––––––––– . ( )
A) sectors B) triangles C) Squares D) rectangles
5. The median of the scores 13, 23, 12, 18, 26, 19 and ––––––––––– . ( )
A)14 B) 26 C) 13 D) 18
6. The arthmetic mean of a+2, a, a–2 is –––––––––––. ( )
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7. If and A= B then p and x are –––––––––––.( )
A) p= 6, x= 2 B) p=2, x=6 C) p=3, x=4 D) p=4, x=3
8. If then the order of AT is = –––––––––––. ( )
A) 3×2 B) 2×2 C) 2×3 D) 3×3
9. The father of computer –––––––––––. ( )
A) Pascal B) Bill gates C) Charles Babbage D) Newin
10. Vaccum tubes are used in –––––––– generation of computers. ( )
A) fourth B) First C) Second D) Third
Answers : 1. C 2. C 3. D 4. D 5. D
6. B 7. A 8. A 9. C 10. B
II. Fill in the blanks with suitable words. Marks : 10×½=5
11. Angle in a semi circle is –––––––.
12. Basic proportionality theorem is also known as ––––––––– theorem.
13. The slope of a line perpendicular to 2x+3y=4 is –––––––.
14. If A.M. of 3, 5, 9, x, 11 is 7 then x = –––––––.
15. Formula for calculation the median of frequency distribution is ———.
16. The angle between the lines x–2=0 and y+3=0 is ––––––––.
1 2 3A =
4 5 6⎡ ⎤⎢ ⎥⎣ ⎦
3 4 3 4 A = , B =
6 p 2⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦x
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18. , If A has not multiplicative inverse then x= ––––––––.
19., then AB = ––––––––.
20. Expand C.P.U. ––––––.
Answers
11. Right angle 12. Thales
13. 3/2 14. 7
15. x2+2x–15=0 16. 90
17. 45 18. √a + √b
19. 1/13 20. 4
III. Match the following. Marks : 10×½=5
Group-A Group-B
21. The height of the equilateral ( ) A) 2
triangle of side 2√3 is
22. If C= 90° in Δ ABC and
a =3, b=4, then C= ( ) B) y–y1 = m(x–x1)
23. Slope and point form of ( ) C) a
a line.
24. The equation of y-axis is ( ) D) 3a
25. A.M. of a–d, a, a+d is ( ) E) 3
[ ]5A , B x y
2⎡ ⎤
= =⎢ ⎥⎣ ⎦
4 xA
x 9⎡ ⎤
= ⎢ ⎥⎣ ⎦
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G)
H) –5
Answers : 21. E 22. F 23. B 24. A 25. C
Group-A Group-B
26. ( ) A) 1
27. ( ) B) 5
28. If = 0 then a = ( ) C) A–1.B–1
29. (AB)–1 ( ) D) ∝
30. Computer ( ) E) cos θ
F) B–1.A–1
G) sin θ
H) An electronic machine
Answers : 26. G 27. A 28. B 29. F 30. H
2a 56 3
2Tanπ
2sec 1sec
θ −θ
x y 1a b
+ =
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Key
Section-I
Group-A
1. Prove that the tangents at the ends of a diameter of a circle are parallel.
A. Given: Let 'O' be the centre of the circle and be a diameter. Let and be the two
tangents drawn at A and B to the circle with centre 'O'
R.T.P. //
Proof: ∠A = ∠B = 90° –1) (Tangent is perpendicular to the diameter at the point of contact)
Let and be two lines and be a transversal then A and ∠B = 90° + 90° = 180°
(since From 1)
If two lines are cut off a transversal and a pair of interior
angles So formed are supplementary then the two lines are
parallel.
//
2. Find the point on x-axis which is equidistant from (2,3) and (4,–2)?
A. Let the required point be (x,0)
Distance between (x,0) and (2,3) = Distance between (x,0) and (4,–2)
2x 8x 16 4= − + +2x 4x 4 9⇒ − + +
2(x 4) 4= − +2(x 2) 9⇒ − +
2(x 4) 4= − +2(x 2) 9⇒ − +
2 2(x 4) 0 ( 2)= − + − −2 2(x 2) (03)∴ − +
BDAC
ABBDAC
BDAC
BDACAB
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B D
CA
O
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x2–4x+13 = x2–8x+20
⇒ –4x+13+8x–20 = 0 ⇒ 4x–7 = 0
x = 7/4 ∴ The required point is (7/4, 0)
3. Find the equation of straight line passing through the points (4,–7) and (1,5)
A. Equation of a line passing through two points is