**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**The point P which divides the line segment joining the points A (2, −5) and B (5, 2) in the ratio 2:3 lies in the quadrant.

**A.**I**B.**II**C.**III

VIEW SOLUTION**D.**IV**Q2**A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36 cm, partly filled with water. If the sphere is completely submerged, then the water level rises (in cm) by

**A.**3**B.**4**C.**5

VIEW SOLUTION**D.**6**Q3**In figure 1, O is the centre of a circle, AB is a chord and AT is the tangent at A. If ∠AOB = 100°, then ∠BAT is equal to

**A.**100°**B.**40°**C.**50°

VIEW SOLUTION**D.**90°**Q4**The roots of the equation

*x*^{2}+*x*−*p*(*p*+ 1) = 0, where*p*is a constant, are**A.***p*,*p*+ 1**B.**−*p*,*p*+ 1**C.***p*, − (*p*+ 1)

VIEW SOLUTION**D.**−*p*, − (*p*+ 1)**Q5****Q6**The mid-point of segment AB is the point P (0, 4). If the coordinates of B are (−2, 3) then the coordinates of A are

**A.**(2, 5)**B.**(−2, −5)**C.**(2, 9)

VIEW SOLUTION**D.**(−2, 11)**Q7**In figure 2, PA and PB are tangents to the circle with centre O. If ∠APB = 60°, then ∠OAB is

**A.**30°**B.**60°**C.**90°

VIEW SOLUTION**D.**15°**Q8**The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 45°. The height of the tower (in metres) is

**A.**15**B.**30**C.**

VIEW SOLUTION**D.****Q9****Q10**If the perimeter and the area of a circle are numerically equal, then the radius of the circle is

**A.**2 units**B.**π units**C.**4 units

VIEW SOLUTION**D.**7 units**Q11**In figure 3, APB and CQD are semi-circles of diameter 7 cm each, while ARC and BSD are semi-circles of diameter 14 cm each. Find the perimeter of the shaded region.

**OR**Find the area of a quadrant of a circle, where the circumference of circle is 44 cm.

VIEW SOLUTION**Q12**Two concentric circles are of radii 7 cm and

VIEW SOLUTION*r*cm respectively, where*r*> 7. A chord of the larger circle, of length 48 cm, touches the smaller circle. Find the value of*r*.**Q13**Find the values(s) of

VIEW SOLUTION*x*for which the distance between the points P(*x*, 4) and Q(9, 10) is 10 units.**Q14****Q15**Two cubes, each of side 4 cm are joined end to end. Find the surface area of the resulting cuboid.

VIEW SOLUTION**Q16**Draw a line segment of length 6 cm. Using compasses and ruler, find a point P on it which divides it in the ratio 3:4.

VIEW SOLUTION**Q17**Find the value of

VIEW SOLUTION*k*so that the quadratic equation*kx*(3*x*− 10) + 25 = 0, has two equal roots.**Q18****Q19**In fig. 4, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 4 cm and 3 cm respectively. If area of ΔABC = 21 cm

VIEW SOLUTION^{2}, then find the lengths of sides AB and AC.**Q20**Two dice are rolled once. Find the probability of getting such numbers on two dice, whose product is a perfect square.

**OR**A game consists of tossing a coin 3 times and noting its outcome each time. Hanif wins if he gets three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

VIEW SOLUTION**Q21**If two vertices of an equilateral triangle are (3, 0) and (6, 0), find the third vertex.

**OR**Find the value of

VIEW SOLUTION*k*, if the points P(5, 4), Q(7,*k*) and R (9, −2) are collinear.**Q22****Q23**Find the value of the middle term of the following AP:

−6, −2, 2, ……, 58.

**OR**Determine the AP whose fourth term is 18 and the differences of the ninth term from the fifteenth term is 30.

VIEW SOLUTION**Q24**Find the area of the major segment APB, in Fig 5, of a circle of radius 35 cm and ∠AOB = 90°.

VIEW SOLUTION**Q25**From the top of a tower 100 m high, a man observes two cars on the opposite sides of the tower with angles of depression 30° and 45° respectively. Find the distance between the cars.

VIEW SOLUTION**Q26**The radii of the circular ends of a bucket of height 15 cm are 14 cm and

VIEW SOLUTION*r*cm (*r*< 14 cm). If the volume of bucket is 5390 cm^{3}, then find the value of*r*.**Q27**Draw a triangle ABC with side BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are times the corresponding sides of ΔABC.

VIEW SOLUTION**Q28**If P(2, 4) is equidistant from Q(7, 0) and R(

VIEW SOLUTION*x*, 9), find the values of*x*. Also find the distance PQ.**Q29**Prove that the lengths of tangents drawn from an external point to a circle are equal.

VIEW SOLUTION**Q30**A motor boat whose speed is 20 km/h in still water, takes 1 hour more to go 48 km upstream than to return downstream to the same spot. Find the speed of the stream.

**OR**Find the roots of the equation

VIEW SOLUTION**Q31**If the sum of first 4 terms of an AP is 40 and that of first 14 terms is 280, find the sum of its first

*n*terms.**OR**Find the sum of the first 30 positive integers divisible by 6.

VIEW SOLUTION**Q32**From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 10 m high building are 30° and 60° respectively. Find the height of the tower.

VIEW SOLUTION**Q33**Find the area of the shaded region in Fig. 6, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD, where the length of each side of square is 14 cm.

VIEW SOLUTION**Q34**A toy is in the shape of a solid cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 21 cm and 40 cm respectively, and the height of cone is 15 cm, then find the total surface area of the toy. [π = 3.14, be taken]

VIEW SOLUTION

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## Class X: Math, Board Paper 2011, Set-2