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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

If the distance between the points (4, k) and (1, 0) is 5, then what can be the possible values of k?

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• Q2

The ratio of the height of a tower and the length of its shadow on the ground is $\sqrt{3}:1$. What is the angle of elevation of the sun?

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• Q3

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

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• Q4

A number is chosen at random from the number –3, –2, –1, 0, 1, 2, 3. What will be the probability that square of this number is less then or equal to 1?

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• Q5

In the given figure, PA and PB are tangents to the circle from an external point P. CD is another tangent touching the circle at Q. If PA = 12 cm, QC = QD = 3 cm, then find PC + PD.

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• Q6

Draw a line segment of length 8 cm and divide it internally in the ratio 4:5.

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• Q7

Find the roots of the quadratic equation $\sqrt{2}{x}^{2}+7x+5\sqrt{2}=0$.

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• Q8

Find how many integers between 200 and 500 are divisible by 8.

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• Q9

Find the value of k for which the equation x2 + k(2x + k − 1) + 2 = 0 has real and equal roots.

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• Q10

In the figure, AB and CD are common tangents to two circles of unequal radi. Prove that AB = CD.

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• Q11

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

(i) have a sum less than 7

(ii) have a product less than 16

(iii) is a doublet of odd numbers.

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• Q12

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2 ∠OPQ.

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• Q13

The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is find the value of y.

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• Q14

Find the sum of n terms of the series $\left(4-\frac{1}{n}\right)+\left(4-\frac{2}{n}\right)+\left(4-\frac{3}{n}\right)+..........$

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• Q15

Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.

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• Q16

In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the shaded region.

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• Q17

If the equation (1 + m2) x2 + 2mcx + c2a2 = 0 has equal roots then show that c2 = a2 (1 + m2).

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• Q18

If the pth term of an A. P. is q and qth term is p, prove that its nth term is (p + qn).

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• Q19

A solid metallic sphere of diamter 16 cm is melted and recasted into smaller solid cones, each of radius 4 cm and height 8 cm. Find the number of cones so formed.

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• Q20

The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If height of the tower is 50 m, find the height of the hill.

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• Q21

In the given figure, the side of square is 28 cm and radius of each circle is half of the length of the side of the square where O and O' are centres of the circles. Find the area of shaded region.

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• Q22

In a hospital used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If tank is filled completely then what will be the height of standing water used for irrigating the park. Write your views on recycling of water.

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• Q23

A chord PQ of a circle of radius 10 cm substends an angle of 60° at the centre of circle. Find the area of major and minor segments of the circle.

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• Q24

Peter throws two different dice together and finds the product of the two numbers obtained. Rina throws a die and squares the number obtained. Who has the better chance to get the number 25.

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• Q25

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

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• Q26

Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream.

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• Q27

If $a\ne b\ne 0$, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.

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• Q28

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

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• Q29

If the pth term of an A. P. is $\frac{1}{q}$ and qth term is $\frac{1}{p}$, prove that the sum of first pq terms of the A. P. is $\left(\frac{pq+1}{2}\right).$

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• Q30

An observer finds the angle of elevation of the top of the tower from a certain point on the ground as 30°. If the observe moves 20 m towards the base of the tower, the angle of elevation of the top increases by 15°, find the height of the tower.

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• Q31

A cone of radius 10 cm is divided into two parts by a plane parallel to its base through the mid-point of its height. Compare the volumes of the two parts.

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