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General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
Question 1
• Q1

The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

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• Q2

If a tower 30 m high, casts a shadow $10\sqrt{3}$ m long on the ground, then what is the angle of elevation of the sun?

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• Q3

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP.

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• Q4

What is the common difference of an A.P. in which a21 – a7 = 84?

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• Q5

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.

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• Q6

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

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• Q7

A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.

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• Q8

If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y.

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• Q9

Find the value of p, for which one root of the quadratic equation px2 – 14x + 8 = 0 is 6 times the other.

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• Q10

For what value of n, are the nth terms of two A.Ps 63, 65, 67,... and 3, 10, 17,... equal ?

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• Q11

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

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• Q12

A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

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• Q13

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region.

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• Q14

In what ratio does the point divide the line segment joining the points P(2, –2) and Q(3, 7)? Also find the value of y.

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• Q15

Water in a canal, 5·4 m wide and 1·8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation?

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• Q16

In the given figure, two concentric circles with centre O have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region.

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• Q17

The dimensions of a solid iron cuboid are 4·4 m × 2·6 m × 1·0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe.

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• Q18

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius on its circular face. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

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• Q19

How many terms of an A.P. 9, 17, 25, ... must be taken to give a sum of 636?

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• Q20

If the roots of the equation (a2 + b2) x2 – 2(ac + bd) x + (c2 + d2) = 0 are equal, prove that $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\mathrm{c}}{\mathrm{d}}.$

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• Q21

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k.

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• Q22

Construct a triangle ABC with side BC = 7 cm, ∠B = 45°, ∠A = 105°. Then construct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of the ∆ABC.

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• Q23

Two different dice are thrown together. Find the probability that the numbers obtained have

(i) even sum, and
(ii) even product.

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• Q24

In the given figure, XY and X'Y' are two parallel tangents to  circle with centre O and another tangent AB with point of contact C, is intersecting XY at A and X'Y' at B. Prove that ∠AOB = 90°.

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• Q25

In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3·5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

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• Q26

Prove that the lengths of two tangents drawn from an external point to a circle are equal.

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• Q27

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms.

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• Q28

Solve for x :

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• Q29

A takes 6 days less than B to do a work. If both A and B working together can do it in 4 days, how many days will B take to finish it?

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• Q30

From the top of a tower, 100, high, a man observes two cars on the opposite sides of the tower and in same straight line with its base, with angles of depression 30° and 45°. Find the distance between the cars. [Take $\sqrt{3}$ = 1.732]

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• Q31

In the given figure, O is the centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90°. Find the area of the shaded region.

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