# Sample Paper – Math 10th CBSE SA 2

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**SAMPLE PAPER/MODEL TEST PAPER**

**SUBJECT – MATH 10th CBSE SA 2**

**Section – A**

1. If the centroid of the triangle formed by (7, x), (y, -6) and (9, 10) is at (6, 3), then

(x, y) is:

(a) 4 cm (b) 3 cm (c) √3 cm (d) 2 cm

2. If the sum of n terms of an A.P. be 3n2 + n and its common difference is 6, then its first term is:

(a) 1 (b) 4 (c) 3 (d) 2

3. PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∟POR = 120^{0} then ∟OPQ is:

(a) 30^{0 } (b) 45^{0} (c) 90^{0} (d) 60^{0}

4. In a single throw of a pair of dice the probability of getting the sum a perfect square is:

(a) 1/6 (b) 7/36 (c) 1/18 (d) 2/9

5. If the centroid of the triangle formed by (7, x), (y, -6) and (9, 10) is at (6, 3), then (x, y) is:

(a) (5, 4) (b) (5, 2) (c) (-5, -2) (d) (4, 5)

6. The circumference of a circle is 100 cm. The side of a square inscribed in the circle is:

(a) 100√2/π cm (b) 100/π cm (c) 50 √2/π cm (d) 50 √2 cm

7. If ax^{2} + bx + c = 0 has equal roots, then C =

(a) –b^{2}/2a (b) b/2a (c) –b/2a (d) – b^{2}/4a

8. The volume of the greatest sphere that can be cut be cut off from a cylinder big of wood base radius 1 m and height 5 cm is:

(a) 10/3 π (b) 5 π (c) 20/3 π (d) 4/3 π

9. Which of the following can not be probability of an event?

(a) 9/4 cm (b) 4/9 cm (c) 9/2 cm (d) 2/9 cm

10. The distance between two parallel tangents of a circle of radius 4 cm:

(a) 4 cm (b) 3 cm (c) 2 cm (d) 1 cm

**Section – B**

11. A cylinder, cone and a hemisphere of equal base have the same height. What is their ratio in their volumes?

12. For what value of P, are the points (2, 1) (P,-1) and (-1, 3) collinear.

13. A letter is chosen at random from the word ‘MATHEMATICS’. Find the probability of choosing letter A.

14. Find the sum of all 3 – digit natural numbers which are divisible by 13.

15. Solve for x:

X^{2} + x – (α + 2) (α + 1) = 0

16. For what value of n is the nth term of the following two APs the same:

(a) 69, 68, 67, …..

(b) 1, 7, 13, 19, …..

17. Triangle ABC is an isosceles triangle in which AB = AC circumscribed about a circle Show that BC is bisected at the point of contact.

18. What is the ratio of the areas of a circle and equilateral triangle whose diameter and a side are respectively equal.

**Section – C**

19. Draw a triangle ABC with sides BC= 6 cm, AB =5 cm and ∟ABC = 60^{0 }. Then construct a triangle whose sides are ¾ of the corresponding sides of triangle ABC.

20. The perimeter of an isosceles triangle is 32 cm. If each equal side is 5/6 times the base. Find the area of the triangle.

21. Two circle touch internally at a point P and from a point T on the common tangent at P. Tangent segment TQ,TR are drawn to the circles. Prove that TQ = TR

22. The sum of 5th and 9th term of an A.P. is 72 and the sum of 7th and 12th terms is 97. Find the A.P.

23. The vertices of a triangle are (-1, 3) ; and (5, 1). Find the length of medians through vertices (-1, 3) and (5, 1).

24. A motor boat whose speed is 15 km/h in still water goes 30 km downstream and comes back in a total time of 4 hours 30 minutes. Find the speed of the stream.

25. A box contains 20 balls bearing numbers 1, 2, 3, 4, ……., 20. A ball is drawn at random from the box. What is the probability that the number on the ball is?

(a) not divisible by 10.

(b) an odd number

(c) divisible by 2 or 3

(d) prime number

26. The point R divides the line segment AB, where A (-4, 0) and B (0, 6) are such that AR = ¾ AB. Find the co – ordinate R.

27. A toy is in the form of a cone mounted on a hemisphere with same radius. The diameter of the base of the conical portion is 7 cm and the total height of the toy is 14.5 cm. Find the volume of the toy. [Use π = 22/7]

28. A toy is in the form of a cone mounted on a hemisphere of common base radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy. [Use π = 22/7]

**Section – D**

29. Construct a triangle ABC with BC = 6 cm, ∟A = 60^{0 }and median AD through A is 5 cm long. Construct a triangle A’BC’ similar to triangle ABC, with BC’ = 8 cm. Write steps of construction.

30. The interior angle of a polygon are A.P. The smallest angle is 120^{0} and the common difference is 5^{0}. Find the number of sides of polygon.

31. The sum of the radius of the base and the height of a solid cylinder is 37 cm. If the total surface area of the solid cylinder is 1628 cm2, find the volume of the cylinder.

(Take π = 22/7)

32. There are two poles, one each on either bank of river just opposite to each other one pole is 60 m high from the top of this pole, the angle of depression of the top and the foot of the other pole are 30^{0} and 60^{0} respectively. Find the width of the river and the height of the other pole.

33. Solve for x:

2 (x^{2} +1/x^{2}) – 3 (x – 1/x) – 4 = 0

34. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

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