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# Class X: Math, Board Paper 2014, Set-2

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.
(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.
(iv) Use of calculators is not permitted.
Question 1
• Q1

A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:
(A) $\frac{4}{\sqrt{3}}$

(B) $4\sqrt{3}$

(C) $2\sqrt{2}$

(D) 4

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• Q2

If two different dice are rolled together, the probability of getting an even number on both dice, is:
(A) $\frac{1}{36}$
(B) $\frac{1}{2}$
(C) $\frac{1}{6}$
(D) $\frac{1}{4}$

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• Q3

A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:
(A) $\frac{2}{3}$

(B) $\frac{1}{6}$

(C) $\frac{1}{3}$

(D) $\frac{11}{30}$

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• Q4

If the points A(x, 2), B(−3, −4) and C(7, 5) are collinear, then the value of x is:
(A) −63
(B) 63
(C) 60
(D) −60

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• Q5

In Fig. 1, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is :

(A) 3.8
(B) 7.6
(C) 5.7
(D) 1.9

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• Q6

In Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals:

(A) 67°
(B) 134°
(C) 44°
(D) 46°

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• Q7

The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:
(A) 3
(B) 5
(C) 4
(D) 6

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• Q8

The first three terms of an AP respectively are 3y – 1, 3y + 5 and 5y + 1. Then y equals:
(A) –3
(B) 4
(C) 5
(D) 2

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• Q9

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.

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• Q10

Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

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• Q11

In fig. 3, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)

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• Q12

Solve the quadratic equation 2x2 + ax a2 = 0 for x.

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• Q13

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

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• Q14

The first and the last terms of an AP are 7 and 49 respectively. If sum of all its terms is 420, find its common difference.

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• Q15

In  Fig 4, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region. [Use π = 3.14 and $\sqrt{3}=1.73$]

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• Q16

In Fig.5, PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use π = 3.14]

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• Q17

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely?

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• Q18

A solid metallic right circular cone 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{12}\phantom{\rule{0ex}{0ex}}$ cm, find the length of the wire.

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• Q19

If the seventh term of an AP is $\frac{1}{9}$ and its ninth term is $\frac{1}{7}$, find its 63rd term.

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• Q20

Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.

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• Q21

Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. [Use $\sqrt{3}=1.73$]

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• Q22

Solve the equation , for x.

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• Q23

Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle.

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• Q24

If the points P(–3, 9), Q(a, b) and R(4, – 5) are collinear and a + b = 1, find the values of a and b.

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• Q25

The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them.

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• Q26

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, – 3). Also find the value of x.

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• Q27

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

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• Q28

Prove that a parallelogram circumscribing a circle is a rhombus.

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• Q29

Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which $\frac{2}{5}$th of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?

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• Q30

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid.

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• Q31

The difference of two natural numbers is 3 and the difference of their reciprocals is $\frac{3}{28}$ . Find the numbers.

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• Q32

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

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• Q33

All the black face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a:
(i) face card
(ii) red card
(iii) black card
(iv) king

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• Q34

Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots.

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