**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.

**Q1**If two different dice are rolled together, the probability of getting an even number on both dice, is:

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(A) $\frac{1}{36}$

(B) $\frac{1}{2}$

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

(D) $\frac{1}{4}$**Q2**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:

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(A) $\frac{4}{\sqrt{3}}$

(B) $4\sqrt{3}$

(C) $2\sqrt{2}$

(D) 4**Q3**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 :

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(A) 3.8

(B) 7.6

(C) 5.7

(D) 1.9**Q4**In Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals:

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(A) 67°

(B) 134°

(C) 44°

(D) 46°**Q5**The first three terms of an AP respectively are 3

VIEW SOLUTION*y*– 1, 3*y*+ 5 and 5*y**y*equals:

(A) –3

(B) 4

(C) 5

(D) 2**Q6**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:

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(A) 3

(B) 5

(C) 4

(D) 6**Q7**A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:

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(A) $\frac{2}{3}$

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

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

(D) $\frac{11}{30}$**Q8**If the points A(

VIEW SOLUTION*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

**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.

VIEW SOLUTION**Q10**Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

VIEW SOLUTION**Q11**Solve the quadratic equation 2

VIEW SOLUTION*x*^{2}+*ax*−*a*^{2}= 0 for*x*.**Q12**Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

VIEW SOLUTION**Q13**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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**Q14**The first and the last terms of an AP are 8 and 65 respectively. If the sum of all its terms is 730, 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**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 $\sqrt{3}=1.73$]

VIEW SOLUTION**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?

VIEW SOLUTION**Q18**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.

VIEW SOLUTION**Q19**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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**Q20**If the seventh term of an AP is $\frac{1}{9}$ and its ninth term is $\frac{1}{7}$, find its 63

VIEW SOLUTION^{rd}term.**Q21**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.

VIEW SOLUTION**Q22**If the points A(−1, −4), B(

VIEW SOLUTION*b*,*c*) and C(5, −1) are collinear and 2*b*+*c*= 4, find the values of*b*and*c*.**Q23**If the point P(2, 2) is equidistant from the points A(−2,

VIEW SOLUTION*k*) and B(−2*k*, −3), find*k*. Also find the length of AP.**Q24**Solve the equation $\frac{14}{x+3}-1=\frac{5}{x+1};x\ne -3,-1,$for

VIEW SOLUTION*x*.

**Q25**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?

VIEW SOLUTION**Q26**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.

VIEW SOLUTION**Q27**Prove that a parallelogram circumscribing a circle is a rhombus.

VIEW SOLUTION**Q28**Find the ratio in which the point P(

VIEW SOLUTION*x*, 2) divides the line segment joining the points A(12, 5) and B(4, – 3). Also find the value of*x*.**Q29**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.

VIEW SOLUTION**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. $\left[\mathrm{Take}\pi =\frac{22}{7}\right]$

VIEW SOLUTION**Q31**Find the value of

VIEW SOLUTION*p*for which the quadratic equation (2*p*+ 1)*x*^{2}− (7*p*+ 2)x + (7*p*− 3) = 0 has equal roots. Also find these roots.**Q32**Cards numbered from 11 to 60 are kept in a box. If a card is drawn at random from the box, find the probability that the number on the drawn card is :

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(i) an odd number

(ii) a perfect square number

(iii) divisible by 5

(iv) a prime number less than 20**Q33**Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

VIEW SOLUTION**Q34**The difference of two natural numbers is 5 and the difference of their reciprocals is $\frac{5}{14}$. Find the numbers.

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## Class X: Math, Board Paper 2014, Set-3