**CBSE Previous Year Question Papers Class 10 Maths SA2 Delhi – 2011**

**Time allowed: 3 hours Maximum marks: 90**

**GENERAL INSTRUCTIONS:**

**All questions are compulsory.****The Question Taper consists of 31 questions divided into four Sections A, B. C. and D.****Section A contains 4 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.****Use of calculators is not permitted.**

**SET I**

**SECTION A**

**Questions number 1 to 4 carry 1 mark each.**

** Question.1 In Fig. 1, O is the centre of a circle, AB is a chord and AT is the tangent at A. If ∠AOB = 100°, then calculate ∠BAT.**

**Solution.**

**Question.2. In Fig. 2, PA and PB are tangents to the circle with centre O. If ∠APB = 60°, then calculate ∠OAB.**

** **

** Solution.**

**Question.3. 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 calculate the rise of water level (in cm).**

** Solution.**

**Question.4. In which quadrant the point P that divides the line segment joining the points A(2, – 5) and B(5, 2) in the ratio 2 : 3 lies?**

** Solution.**

**SECTION B**

**Questions number 5 to 10 carry 2 marks each.**

** Question.5. The angles of a triangle are in A.P., the least being half the greatest. Find the angles.**

** Solution.**

**Question.6. Three vertices of a parallelogram taken in order are (- 1, 0), (3, 1) and (2, 2) respectively. Find the coordinates of fourth vertex.**

** Solution.**

**Question.7. Find the value of p so that the quadratic equation p x(x – 3) + 9 = 0 has two equal roots.**

**Solution.**

**Question.8. Find whether – 150 is a term of the A.P. 17,12, 7, 2, … ?**

** Solution.**

**Question.9. Two concentric circles are of radii 7 cm and 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.**

** Solution.**

**Question.10. 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.**

** Solution.**

**SECTION C**

**Questions number 11 to 20 carry 3 marks each.**

** Question.11 In Fig. 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. [Use π = 22/7 ]**

**Or**

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

** [Use π = 22/7 ]**

** Solution.**

**Question.12 Two cubes, each of side 4 cm are joined end to end. Find the surface area of the resulting cuboid.**

** Solution.**

**Question.13 Find that value(s) of x for which the distance between the points P(x, 4) and Q(9,10) is 10 units.**

**Solution.**

**Question.14 A coin is tossed two times. Find the probability of getting at least one head.**

** Solution.**

**Question.15 Find the roots of the following quadratic equation: 2 √3 x^{2} – 5x + -√3 = 0**

**Solution.**

**Question.16 Find the value of the middle term of the following A.P.: – 6, – 2, 2, …, 58.**

** Or**

** Determine the A.P. whose fourth term is 18 and the difference of the ninth term from the fifteenth term is 30.**

** Solution.**

**Question.17 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 ^{2}, then find the lengths of sides AB and AC.**

**Solution.**

**Question.18 Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Then construct a triangle whose sides are y times the corresponding sides of ΔABC.**

** **

** Solution.**

**Question.19 Find the area of the major segment APB, in Fig. 5, of a circle of radius 35 cm and ∠AOB = 90°. [Use π = 22/7 ]**

**Solution.**

**Question.20 The radii of the circular ends of a bucket of height 15 cm are 14 cm and r cm (r< 14 cm)**

** the volume of bucket is 5390 cm3, then find the value of r. [Use π= 22/7 ]**

** Solution.**

**SECTION D**

**Questions number 21 to 31 carry 4 marks each.**

** Question.21 A survey has been done on 100 people out of which 20 use bicycles, 50 use motorbikes and 30 use cars to travel from one place to another. Find the probability of persons who use bicycles, motorbikes and cars respectively?**

** Which mode of transport do you think is better and why?**

** Solution.**

**Question.22 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.**

** Or**

** 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. [Use √3 = 1.73]**

** Solution.**

**The possible out comes on tossing a coin 3 times are,**

**Question.23 If (3, 3), (6, y), ( x, 7) and (5, 6) are the vertices of a parallelogram taken in order, find the values of x and y.**

**Solution.**

**Question.24 If two vertices of an’equilateral triangle are (3, 0) and (6, 0), find the third vertex.**

** Or**

** Find the value of k, if the points P(5, 4), Q(7, k) and R(9, – 2) are collinear.**

** Solution**.

**Question.25 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 1/x+4- 1/x-7 = 11/30,x ≠ -4,7**

** Solution.**

**Question.26 If the sum of first 4 terms of an A.P. 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.**

** Solution.**

**Question.27 Prove that the lengths of tangents drawn from an external point to a circle are equal**

** Solution.**

**Question.28 In Fig. 6, arcs are drawn by taking vertices A, B and C of an equilateral triangle ABC of side 14 cm as centres to intersect the sides BC, CA and AB at their respective mid-points D, E and F. Find the area of the shaded region.**

** [Use π = 22/7 and√3 =1.73]**

**Solution.**

**Question.29 From a solid cylinder whose height is 15 cm and diameter 16 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. [Take π = 3.14] :**

** Solution.**

**Question.30 Two poles of equal heights are standing opposite to each other on either side of the road, which is 100 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles.**

** Solution.**

**Question.31 Two pipes running together can fill a cistern in 3 1/13 minutes. If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.**

** Solution.**

**SET II**

**Note: Except for the following questions, all the remaining questions have been asked in Set 1.**

**Question.5 Which term of the progression 4, 9,14,19,… is 109?**

** Solution.**

**Q.6 Find a relation between x and y such that the point P( x, y) is equidistant from the points A (2, 5) and B (-3, 7).**

**Solution.**

**Question.13 Find the value of k so that the quadratic equation k x (3x – 10) + 25 = 0, has two equal roots. **

**Solution.**

**Q.14. A coin is tossed two times. Find the probability of getting not more than one head.**

** Solution.**

**Question.23 Draw a triangle ABC with side BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a**

** triangle whose sides are 3/5 times the corresponding sides of AABC.**

** Solution.**

**Question.24 If P(2, 4) is equidistant from Q(7, 0) and R( x, 9), find the values of x. Also find the distance PQ**

**Solution.**

**Question.28 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.**

** Solution.**

**Question.29 Find the area of the shaded region in Fig. 7, 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.**

** [Use π = 22/7 ]**

**Solution.**

**Question.30 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]**

** Solution.**

**SET III**

**Note: Except for the following questions, all the remaining questions have been asked in Set I and Set II.**

**Question.5 Find the roots of 4***x*^{2}**+ 3 x + 5 = 0 by the method of completing the squares.**

**Solution.**

**Question.6 Determine the ratio in which the line 3 x + y – 9 = 0 divides the segment joining the points (1, 3) and (2, 7).**

**Solution.**

**Question.7 A coin is tossed two times. Find the probability of getting both heads or both tails.**

** Solution.**

**Question.8 Find the value of m so that the quadratic equation mx(5 x – 6) + 9 = 0 has two equal roots.**

**Solution.**

**Question.23 Draw a triangle PQR such that PQ = 5 cm, ∠P = 120° and PR = 6 cm. Construct another**

** triangle whose sides are 3/4 times the corresponding sides of APQR.**

** Solution.**

**Question.24 Find the point of y-axis which is equidistant from the points (- 5, – 2) and (3, 2).**

** Solution.**

**Question. 25 From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm**

** and radius 6 cm is hollowed out. Find the total surface area of the remaining solid.**

** [Use π = 22/7 ]**

** Solution.**

**Question.26 The length and breadth of a rectangular piece of paper are 28 cm and 14 cm respectively. A semi-circular portion is cut off from the breadth’s side and a semi-circular portion is added on length’s side, as shown in Fig. 8. Find the area of the shaded**

** region. [Use π = 22/7 ]**

**Solution.**

**Question.31 From the top of a 15 m high building, the angle of elevation of tire top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower.**

** Solution.**