**General instructions:**

1. All questions are compulsory.

2. The question paper consists of 34 questions divided into four sections A, B, C and

D.

3. Section A contains 10 questions of 1 mark each, which are multiple choices type

questions, Section B contains 8 questions of 2 marks each, Section C contains 10

questions of 3 marks each, Section D contains 6 questions of 4 marks each.

4. There is no overall choice in the paper. However, internal choice is provided in one

question of 2 marks, 3 questions of 3 marks each and two questions of 4 marks each.

5. Use of calculators is not permitted.

**Q1**If 1 is a root of the equations

*ay*^{2}+*ay*+ 3 = 0 and*y*^{2}+*y*+*b*= 0 then*ab*equals:A. 3

B.

C. 6

D. −3

**Q2****Q3**In Fig. 1, the sides AB, BC and CA of a triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then the length of BC (in cm) is:

A. 11

B. 10

C. 14

D. 15

**Q4**In Fig 2, a circle touches the side DF of ΔEDF at H and touches ED and EF produced at

K and M respectively. If EK = 9 cm, then the perimeter of ΔEDF (in cm) is:

A. 18

B. 13.5

C. 12

D. 9

**Q5**If the radius of the base of a right circular cylinder is halved, keeping the height the same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is:

A. 1 : 2

B. 2 : 1

C. 1 : 4

D. 4 : 1

**Q6**If the area of a circle is equal to sum of the areas of two circles of diameters 10 cm and 24 cm, then the diameter of the larger circle (in cm) is:

A. 34

B. 26

C. 17

D. 14

VIEW SOLUTION**Q7**The length of shadow of a tower on the plane ground is times the height of the tower.

The angle of elevation of sun is:

A. 45°

B. 30°

C. 60°

D. 90°

**Q8**If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (−2, 5), then the coordinates of the other end of the diameter are:

A. (−6, 7)

B. (6, −7)

C. (6, 7)

D. (−6, −7)

VIEW SOLUTION**Q9**The coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1 are:

A. (2, 4)

B. (3, 5)

C. (4, 2)

D. (5, 3)

**Q10**Two dice are thrown together. The probability of getting the same number on both dice is:

A.

B.

C.

D.

**Q11**Find the value(s) of

*k*so that the quadratic equation*x*^{2}− 4*kx*+*k*= 0 has equal roots.**Q12****Q13**Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in Fig. 3. If AP = 15 cm, then find the length of BP.

**Q14**In Fig. 4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC.

**OR**In Fig. 5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.

**Q15****Q16**In Fig. 6, OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region.

**Q17**If a point A (0, 2) is equidistant from the points B (3,

*p*) and C (*p*, 5), then find the value of*p*.**Q18**A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.

**Q19****Q20**Prove that the parallelogram circumscribing a circle is a rhombus.

**OR**Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

VIEW SOLUTION**Q21**Construct a right triangle in which the sides, (other than the hypotenuse) are of length 6 cm and 8 cm. Then construct another triangle, whose sides are times the corresponding sides of the given triangle.

**Q22**In Fig. 7, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and

3.5 cm and centre O. If ∠POQ = 30°, then find the area of the shaded region.

VIEW SOLUTION**Q23**From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid.

**OR**A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.

VIEW SOLUTION**Q24**The angles of depression of two ships from the top of a light house and on the same side of it are found to be 45° and 30°. If the ships are 200 m apart, find the height of the light house.

**Q25**A point P divides the line segment joining the points A (3, −5) and B (−4, 8) such that

. If P lies on the line

*x*+*y*= 0, then find the value of K.**Q26**If the vertices of a triangle are (1, −3), (4,

*p*) and (−9, 7) and its area is 15 sq. units, find the value(s) of*p*.**Q27**A box contains 100 red cards, 200 yellow cards and 50 blue cards. If a card is drawn at random from the box, then find the probability that it will be (i) a blue card (ii) not a yellow card (iii) neither yellow nor a blue card.

**Q28**The 17

^{th}term of an AP is 5 more than twice its 8^{th}term. If the 11^{th}term of the AP is 43, then find its*n*^{th}term.**Q29**A shopkeeper buys some books for Rs 80. If he had bought 4 more books for the same amount, each book would have cost Rs 1 less. Find the number of books he bought.

**OR**The sum of two numbers is 9 and the sum of their reciprocals is. Find the numbers.

**Q30**Sum of the first 14 terms of an AP is 1505, and its first term is 10. Find its 25

^{th}term.**Q31**Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

**OR**A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

**Q32**A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them being 3.5 cm and the total height of solid is 9.5 cm. Find the volume of the solid.

VIEW SOLUTION**Q33**A bucket is in the form of a frustum of a cone and it can hold 28.49 litres of water. If the radii of its circular ends are 28 cm and 21 cm, find the height of the bucket.

VIEW SOLUTION**Q34**The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of depression from the top of the tower to the foot of the hill is 30°. If the tower is 50 m high, find the height of the hill.

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## Class X: Math, Board Paper 2012, Set-1