#### Half Yearly Model Question Paper 2019

10th Standard EM

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Maths

Time : 02:30:00 Hrs
Total Marks : 100

Part I

Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

14 x 1 = 14
1. If there are 1024 relations from a set A = {1, 2, 3, 4, 5} to a set B, then the number of elements in B is

(a)

3

(b)

2

(c)

4

(d)

8

2. If the sequence t1,t2,t3...are in A.P. then the sequence t6,t12,t18,....is

(a)

a Geometric Progression

(b)

an Arithmetic Progression

(c)

neither an Arithmetic Progression nor a Geometric Progression

(d)

a constant sequence

3. Which of the following should be added to make x4 + 64 a perfect square

(a)

4x2

(b)

16x2

(c)

8x2

(d)

-8x2

4. In figure CP and CQ are tangents to a circle with centre at O. ARB is another tangent touching the circle at R. If CP=11 cm andBC =7 cm, then the length of BR is

(a)

6 cm

(b)

5 cm

(c)

8 cm

(d)

4 cm

5. A straight line has equation 8y = 4x + 21. Which of the following is true

(a)

The slope is 0.5 and the y intercept is 2.6

(b)

The slope is 5 and the y intercept is 1.6

(c)

The slope is 0.5 and the y intercept is 1.6

(d)

The slope is 5 and the y intercept is 2.6

6. The value of is $si{ n }^{ 2 }\theta +\frac { 1 }{ 1+ta{ n }^{ 2 }\theta }$ equal to

(a)

$ta{ n }^{ 2 }\theta$

(b)

1

(c)

$cot^{ 2 }\theta$

(d)

0

7. The angle of elevation of a cloud from a point h metres above a lake is $\beta$. The angle of depression of its reflection in the lake is 45°. The height of location of the cloud from the lake is

(a)

$\frac { h\left( 1+tan\beta \right) }{ 1-tan\beta }$

(b)

$\frac { h\left( 1-tan\beta \right) }{ 1+tan\beta }$

(c)

h tan(45°-$\beta$)

(d)

none of these

8. Given that sinθ=$\frac{a}{b}$, then cosθ is equal to

(a)

$\frac { b }{ \sqrt { { b }^{ 2 }-{ a }^{ 2 } } }$

(b)

$\frac { b }{ a }$

(c)

$\frac { \sqrt { { b }^{ 2 }-{ a }^{ 2 } } }{ b }$

(d)

$\frac { b }{ \sqrt { { b }^{ 2 }-{ a }^{ 2 } } }$

9. A shuttle cock used for playing badminton has the shape of the combination of

(a)

a cylinder and a sphere

(b)

a hemisphere and a cone

(c)

a sphere and a cone

(d)

frustum of a cone and a hemisphere

10. If S1 denotes the total surface area at a sphere of radius ૪ and S2 denotes the total surface area of a cylinder of base radius ૪and height 2r, then

(a)

S1=S2

(b)

S1>S2

(c)

S1<S2

(d)

S1=2S2

11. A cylinder 10 cone and have there are of a equal base and have the same height. what is the ratio of there volumes?

(a)

3:1:2

(b)

3:2:1

(c)

1:2:3

(d)

1:3:2

12. If the mean and coefficient of variation of a data are 4 and 87.5% then the standard deviation is

(a)

3.5

(b)

3

(c)

4.5

(d)

2.5

13. Which of the following is incorrect?

(a)

P(A) > 1

(b)

0 ≤ P(A) ≤ 1

(c)

P(ф)=0

(d)

P(A)+P($\bar { A }$)=1

14. A number x is chosen at random from -4, -3, -2, -1, 0, 1, 2, 3, 4 find the probability that |x|≤4

(a)

0

(b)

1

(c)

$\frac{1}{2}$

(d)

$\frac{1}{9}$

15. Part II

Answer any 10 questions. Question no. 28 is compulsory.

10 x 2 = 20
16. If f(x)=3x-2, g(x)=2x+k and if f o g = f o f, then find the value of k..

17. LetA= {1,2, 3, 4} and B = {-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Let R = {(1, 3), (2, 6), (3, 10), (4, 9)} $\subseteq$ A x B bea relation. Show that R is a function and find its domain, co-domain and the range of R.

18. Find the geometric progression whose first term and common ratios are given by
a = 256 , r = 0.5

19. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

20. Solve 2x − 3y = 6, x + y = 1

21. Show that  $\triangle$PST~$\triangle$PQR

22. In figure OA· OB = OC·OD
Show that $\angle A=\angle C\quad and\quad \angle B=\angle D$

23. Find the equation of a line passing through the point (3, - 4) and having slope $\frac { -5 }{ 7 }$

24. prove that $\sqrt { \frac { 1+cos\theta }{ 1-cos\theta } }$ =cosec $\theta$+cot$\theta$

25. An observer 1.5 metres tall is 20.5 metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from eye of the observer.

26. Find the diameter of a sphere whose surface area is 154 m2.

27. A hemispherical section is cut out from one face of a cubical block (Fig.7.42) such that the diameter l of the hemisphere is equal to side length of the cube. Determine the surface area of the remaining solid.

28. Find the standard deviation of the data 2, 3, 5, 7, 8. Multiply each data by 4. Find the standard deviation of the new values.

29. Find the standard deviation of 30, 80, 60, 70, 20, 40, 50 using the direct method.

30. Part III

Answer any 10 questions. Question no. 42 is compulsory.

10 x 5 = 50
31. Forensic scientists can determine the height (in cms) of a person based on the length of their thigh bone. They usually do so using the function h(b)=2.47b+54.10 where b is the length of the thigh bone.
(i) Check if the function h is one – one
(ii) Also find the height of a person if the length of his thigh bone is 50 cms.
(iii) Find the length of the thigh bone if the height of a person is 14796 cms.

32. Let A = {1, 2, 3, 4, 5}, B = N and f: A $\rightarrow$B be defined by f(x) = x2. Find the range of f. Identify the type of function.

33. Find the first term of a G.P. in which S6 = 4095 and r = 4

34. Determine the AP whose 3rd term is 5 and the 7th term is 9.

35. Discuss the nature of solutions of the following quadratic equations.
x2 + 2x + 5 = 0

36. A two digit number is such that the product of its digits is 12. When 36 is added to the number the digits interchange their places. Find the number.

37. In Figure, O is the centre of a circle. PQ is a chord and the tangent PR at P makes an angle of 50o with PQ. Find $\angle$POQ,

38. In $AD\bot BC$ prove that AB2+CD2 = BD2+AC2

39. If the points A(6, 1), B(8, 2), C(9, 4) and D(P, 3) are the vertices of a parallelogram, taken in order. Find the value of P.

40. A pole 5 m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a point ‘A’ on the ground is 60° and the angle of depression to the point ‘A’ from the top of the tower is 45°. Find the height of the tower. ($\sqrt3$=1.732)

41. The shadow of a tower, when the angle of elevation of the sum is 45o is found to be 10 metres, longer than when it is 60o. find the height of the tower

42. The ratio of the volumes of two cones is 2:3. Find the ratio of their radii if the height of second cone is double the height of the first.

43. Find the number of coins, 1.5 em is diameter and 0.2 em thick, to be melted to form' a right circular cylinder of height 10 ern and diameter 4.5 cm.

44. If P(A).=037, P(B).=0.42, P(A∩B) =009 then find P(AUB).

45. Part IV

2 x 8 = 16
1. If R = {(a, -2), (-5, b), (8, c), (d, -1)} represents the identity function, find the values of a, b, c and

2. Find the value of ‘a’ for which the given points are collinear.
(a, 2 – 2a), (– a + 1, 2a) and (– 4 – a, 6 – 2a)

1. In the rectangle WXYZ, XY+YZ=17 cm, and XZ+YW=26 cm.
Calculate the length and breadth of the rectangle

2. If sin3$\theta$+ycos3$\theta$=sin$\theta$cos$\theta$ and x sin$\theta$ =ycos$\theta$, then prove that x2+y2=1.