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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 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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• Q2

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

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• Q3

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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• Q4

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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• Q5

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

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• Q6

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

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• Q7

Prove that tangents drawn at the ends of a diameter of a circle are parallel to each other.

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• Q8

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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• Q9

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

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• Q10

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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• Q11

If the mth term of an A. P. is $\frac{1}{n}$ and nth term is $\frac{1}{m}$ then show that its (mn)th term is 1.

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• Q12

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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• Q13

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

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• Q14

The $\frac{3}{4}$th part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water is emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.

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• Q15

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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• Q16

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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• Q17

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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• Q18

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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• Q19

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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• Q20

A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/h.

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• Q21

Construct an isosceles triangle with base 8 cm and altitude 4 cm. Construct another triangle whose sides are $\frac{2}{3}$ times the corresponding sides of the isosceles triangle.

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• Q22

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

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• Q23

The ratio of the sums of first m and first n terms of an A. P. is m2 : n2. Show that the ratio of its mth and nth terms is (2m−1):(2n−1).

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• Q24

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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• Q25

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

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• Q26

The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volumes of the two parts.

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• Q27

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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• Q28

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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• Q29

The angle of elevation of a cloud from a point 60 m above the surface of the water of a lake is 30° and the angle of depression of its shadow in water of lake is 60°. Find the height of the cloud from the surface of water.

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• Q30

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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• Q31

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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