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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

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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• Q2

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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• Q3

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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• Q4

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

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• Q5

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

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• Q6

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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• 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

Draw a line segment of length 7 cm and divide it internally in the ratio 2 : 3.

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• Q11

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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• Q12

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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• Q13

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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• Q14

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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• Q15

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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• Q16

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

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• Q17

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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• Q18

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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• Q19

A metallic solid sphere of radius 10.5 cm is melted and recasted into smaller solid cones, each of radius 3.5 cm and height 3 cm. How many cones will be made?

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• Q20

From the top of a 7 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of its foot is 45°. Find the height of the tower.

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• Q21

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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• Q22

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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• Q23

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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• Q24

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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• 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

Draw a right triangle in which the sides (other than the hypotenuse) are of lengths 4 cm and 3 cm. Now construct another triangle whose sides are $\frac{3}{5}$ times the corresponding sides of the given triangle.

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• Q29

If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero.

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• Q30

Two points A and B are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60° and 45° respectively. If the height of the tower is 15 m, then find the distance between the points.

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• Q31

The height of a cone is 30 cm. From its topside a small cone is cut by a plane parallel to its base. If volume of smaller cone is $\frac{1}{27}$ of the given cone, then at what height it is cut from its base?

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