**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****Q2**In Fig. 1, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA ⊥ PB, then the length of each tangent is:

(A) 3 cm

(B) 4 cm

(C) 5 cm

(D) 6 cm

VIEW SOLUTION**Q3**In Fig.2, a circle with centre O is inscribed in a quadrilateral ABCD such that, it touches the sides BC, AB, AD and CD at points P, Q, R and S respectively, If AB = 29 cm, AD = 23 cm, ∠B = 90° and DS = 5 cm, then the radius of the circle (in cm.) is:

(A) 11

(B) 18

(C) 6

(D) 15

VIEW SOLUTION**Q4**The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30°. The distance of the car from the base of the tower (in m.) is:

(A)

(B)

(C)

(D) 150

VIEW SOLUTION**Q5**The probability of getting an even number, when a die is thrown once, is :

(A)

(B)

(C)

(D)

VIEW SOLUTION**Q6**A box contains 90 discs, numbered from 1 to 90. If one disc is drawn at random from the box, the probability that it bears a prime-number less than 23, is :

(A)

(B)

(C)

(D)

VIEW SOLUTION**Q7****Q8**If the difference between the circumference and the radius of a circle is 37 cm, then using, the circumference (in cm) of the circle is:

(A) 154

(B) 44

(C) 14

(D) 7

VIEW SOLUTION

**Q9****Q10****Q11**In Fig. 4, a circle inscribed in triangle ABC touches its sides AB, BC and AC at points D, E and F respectively. If AB = 12 cm, BC = 8 cm and AC = 10 cm, then find the lengths of AD, BE and CF.

VIEW SOLUTION**Q12****Q13**A card is drawn at random from a well shuffled pack of 52 playing cards.

Find the probability that the drawn card is neither a king nor a queen.

VIEW SOLUTION**Q14**Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions 14 cm × 7 cm. Find the area of the remaining card board.

VIEW SOLUTION

**Q15**For what value of

VIEW SOLUTION*k*, are the roots of the quadratic equation*kx*(*x*− 2) + 6 = 0 equal?**Q16****Q17**Construct a triangle with sides 5 cm, 4 cm and 6 cm. Then construct another triangle whose sides are times the corresponding sides of first triangle.

VIEW SOLUTION**Q18**The horizontal distance between two poles is 15 m. The angle of depression of the top of first pole as seen from the top of second pole is 30°. If the height of the second pole is 24 m, find the height of the first pole.

VIEW SOLUTION**Q19**Prove that the points (7, 10), (−2, 5) and (3, −4) are the vertices of an isosceles right triangle.

VIEW SOLUTION**Q20**Find the ratio in which the

VIEW SOLUTION*y*-axis divides the line segment joining the points (−4, − 6) and (10, 12). Also find the coordinates of the point of division.**Q21**In Fig.5, AB and CD are two diameters of a circle with centre O, which are perpendicular to each other. OB is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

VIEW SOLUTION**Q22**A vessel is in the form of hemispherical bowl surmounted by a hollow cylinder of same diameter. The diameter of the hemispherical bowl is 14 cm and the total height of the vessel is 13 cm. Find the total surface area of the vessel.

VIEW SOLUTION**Q23**A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy.

VIEW SOLUTION**Q24**In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc (ii) area of the sector formed by the arc.

VIEW SOLUTION

**Q25****Q26**Sum of the areas of two squares is 400 cm

VIEW SOLUTION^{2}. If the difference of their perimeters is 16 cm, find the sides of the two squares.**Q27**If the sum of first 7 terms of an AP is 49 and that of first 17 terms is 289, find the sum of its first

VIEW SOLUTION*n*terms.**Q28**Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

VIEW SOLUTION**Q29**In fig. 6,

VIEW SOLUTION*l*and*m*are two parallel tangents to a circle with centre O, touching the circle at A and B respectively. Another tangent at C intersects the line*l*at D and*m*at E. Prove that ∠DOE = 90°.**Q30**The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building.

VIEW SOLUTION**Q31**A Group consists of 12 persons, of which 3 are extremely patient, other 6 are extremely honest and rest are extremely kind. A person from the group is selected at random. Assuming that each person is equally likely to be selected, find the probability of selecting a person who is (i) extremely patient (ii) extremely kind or honest.

Which of the above values you prefer more?

VIEW SOLUTION**Q32**The three vertices of a parallelogram ABCD are A(3, −4), B(−1, −3) and C(−6, 2). Find the coordinates of vertex D and find the area of ABCD.

VIEW SOLUTION**Q33**Water is flowing through a cylindrical pipe, of internal diameter 2 cm, into a cylindrical tank of base radius 40 cm, at the rate of 0.4 m/s. Determine the rise in level of water in the tank in half an hour.

VIEW SOLUTION**Q34**A bucket open at the top, and made up of a metal sheet is in the form of a frustum of a cone. The depth of the bucket is 24 cm and the diameters of its upper and lower circular ends are 30 cm and 10 cm respectively. Find the cost of metal sheet used in it at the rate of Rs 10 per 100 cm

VIEW SOLUTION^{2}. [Use π = 3.14]

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## Class X: Math, Board Paper 2013, Set-1