**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 If 1 is a root of the equations ay ^{2} + ay + 3 = 0 and y^{2} + y + b = 0, then find the value of ab.**

**Solution.**

**Question.2 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, find the length of BC (in cm).**

**Solution.**

**Question.3 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, calculate the perimeter of ΔEDF (in cm).**

**Solution.**

**Question.4 If the area of a circle is equal to sum of the areas of two circles of diameters 10 cm and 24 cm, calculate the diameter of the larger circle (in cm).**

** Solution.**

**SECTION B**

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

** Question.5 Find the sum of the first 25 terms of an A.P. whose n ^{th} term is given by _{ }t_{n}= 2 – 3n.**

**Solution.**

**Question.6 In a simultaneous toss of two coins, find the probability of getting:**

** (i) exactly one head, (ii) atmost one head.**

** Solution.**

**Question.7 Find the value(s) of k so that the quadratic equation x ^{2} – 4kx + k = 0 has equal roots.**

**Solution.**

**Question.8 Find the sum of all three digit natural numbers, which are multiples of 11.**

** Solution.**

**Question.9 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.**

**Solution.**

**Question.10 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.**

**Solution.**

**SECTION C**

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

** Question.11 A milkman was serving his customers using two types of mugs A and B of inner diameter 5 cm to serve the customers. The height of the mugs is 10 cm.**

** He decided to serve the customers in ‘B’ type of mug.**

** (a) Find the volume of the mugs of both types.**

** (b) Which mathematical concept is used in the above problem?**

** (c) By chasing the mug of type ‘B’, which value is being depicted MUg ‘A’ bv the milkman?**

**Solution.**

**Question.12 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.**

** [Use π = 22/7 ]**

**Solution.**

**Question.13 If a point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), then find the value of p,**

** Solution.**

**Question.14 A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.**

** Solution.**

**Question.15 Solve for x: 4×2 – 4ax + (a ^{2} – b^{2}) = 0**

**Or**

**Solve for x: 3x**

^{2}– 2√6x + 2 = 0**Solution.**

**Question.16 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.**

** Solution.**

**Question.17 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 3/5 times the corresponding sides of the given triangle.**

** Solution.**

**Question.18 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. [Use π= 22/7 ]**

**Solution.**

**Question.19 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. [Use π = 22/7 ]**

** 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.**

** Solution.**

**Question.20 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.**

** Solution.**

**SECTION D**

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

** Question.21 A point P divides the line segment joining the points A(3, -5) and B(-4, 8) such that AP K**

** pg = Y . If P lies on the line x + y = 0, then find the value of K.**

** Solution.**

**Question.22 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.**

** Solution.**

**Question.23 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.**

** Solution.**

**Question.24 The 17th term of an AP is 5 more than twice its 8th term. If the 11th term of the AP is 43, then find its nth term.**

** Solution.**

**Question. 25 A shopkeeper buys some books for Rs80. 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 1/2. Find the numbers.**

** Solution.**

**Question.26 Sum of the first 14 terms of an AP is 1505 and its first term is 10. Find its 25 ^{th} term.**

**Solution.**

**Question.27 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.**

** Solution.**

**Question.28 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.**

** [Use π = 22/7 ]**

** Solution.**

**Question.29 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.**

** [Use π = 22/7 ]**

** Solution.**

**Question.30 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 of the foot of the hill is 30°. If the tower is 50 m high, find the height of the hill.**

** Solution.**

**Question.31 From the top of a hill, the angles of depression of two consecutive kilometre stones due east are found to be 30° and 45°. Find the height of the hill.**

** Solution.**

**SET II**

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

** Question.13 Find the value(s) of k so that the quadratic equation 2x ^{2} + kx + 3 = 0 has equal roots.**

**Solution.**

**Question.14 Find the sum of all three digit natural numbers, which are multiples of 9.**

** Solution.**

**Question. 21 A box contains 35 blue, 25 white and 40 red marbles. If a marble is drawn at random from the box, find the probability that the drawn marble is**

** (i) white**

** (ii) not blue**

** (iii) neither white nor blue.**

** Solution.**

**Question.22 The 15th term of an AP is 3 more than twice its 7th term. If the 10th term of the AP is 41, then find its n ^{th }term.**

**Solution.**

**Question.23 Draw a triangle ABC with side BC = 7 cm, ∠ABC = 60° and AB = 6 cm. Then construct**

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

** Solution.**

**Question.24 The shadow of a tower standing on a level ground is found to be 20 m longer when the Sun’s altitude is 45° than when it is 60°. Find the height of the tower.**

** Solution.**

**Question.29 The sum of the first 15 terms of an AP is 750 and its first term is 15. Find its 20th term.**

** Solution.**

**Question.30 A container shaped like a right circular cylinder having base radius 6 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled into cones of height 12 cm and radius 3 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.**

** Solution.**

**SET III**

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

** Question.13 Find the sum of all three digit natural numbers, which are multiples of 7.**

** Solution.**

**Question.14 Find the value(s) of k so that the quadratic equation 3x ^{2}– 2kx + 12 = 0 has equal roots.**

**Solution.**

**Question.21 A kite is flying at a height of 45 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string assuming that there is no slack in the string.**

** Solution.**

**Question.22 Draw a triangle ABC with side BC = 6 cm, ∠C = 30° and ∠A = 105°. Then construct another**

** triangle whose sides are 2/3 times the corresponding sides of ΔABC.**

** Solution.**

**Question.23 The 16th term of an AP is 1 more than twice its 8th term. If the 12th term of the AP is 47, then find its nth term.**

** Solution.**

**Question.24 A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting**

** (i) a king of red colour**

** (ii) a face card**

** (iii) the queen of diamonds.**

** Solution.**

**Question.29 Sum of the first 20 terms of an AP is -240, and its first term is 7. Find its 24th term.**

** Solution.**

**Question.30 A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid.**

** [Use π = 22/7]**

** Solution.**