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Class X Subject MATHEMATICS (Time Two hours and a half) Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during the first 15 minutes. This time is to be spent in reading the question paper. The time given at the head of this Paper is the time allowed for writing the answers. Attempt all questions from Section A and any four questions from Section B. All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. Omission of essential working will result in loss of marks. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables are provided. SECTION A (40 Marks) Attempt all questions from this Section Question 1. (a) What number must be subtracted from 2x3 5x2 + 5x so that the resulting polynomial has a factor 2x 3 ? [3] (b) D, E, F are mid points of the sides BC, CA and AB respectively of a ABC. Find the ratio of the areas of DEF and ABC. [3] (c) A man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end of the first year and Rs 4410 at the end of the second year. If the rate of compound interest is 5% per annum, find the sum borrowed. [4] Question 2. (a) The y-axis is a line of symmetry for the figure ACBD where A, B have co-ordinates (3, 6), ( 3, 4) respectively. (i) Find the co-ordinates of C and D. (ii) Name the figure ACBD and find its area. [3] (b) PAQ is a tangent at A to the circumcircle of ABC such that PAQ is parallel to BC, prove that ABC is an isosceles triangle. [3] (c) A rectangular piece of paper 30 cm long and 21 cm wide is taken. Find the area of the biggest circle that can be cut out from this paper. Also find the area of the paper left after cutting out the circle. [Take = 22/7] [4] Question 3. (a) Construct a 2 2 matrix whose elements aij are given by aij = i + j. [3] (b) The point P ( 4, 5) on reflection in y-axis is mapped on P . The point P on reflection in the origin is mapped on P . Find the co-ordinates of P and P . Write down the single transformation that maps P onto P . [3] (c) Let A = {1, 2, 3}, B = {1, 2, 3, 4} and R = {(x, y) : (x, y) A B, y = x + 1}, then (i) find A B (ii) write R in roster form (iii) write domain and range of R (iv) find R 1 in roster form (v) write R 1 in set builder form (vi) represent R and R 1 by arrow diagrams. Question 4. (a) Without using trigonometric table, evaluate : 7(sin 27 /cos 63 ) + 3(cos 21 /sin 69 ) 7(tan 36 /cot 54 ) [3] (b) Prabha saves Rs 1250 per month and invests in a cumulative deposit account giving interest at the rate of 8.5% per annum compounded quarterly. In order to have a total amount of nearly Rs 56000, how many instalments must she deposit ? [3] (c) In a class test, the marks obtained by 11 students are : 13, 17, 20, 5, 19, 7, 6, 11, 15, 17. Find (i) mean (ii) median (iii) upper quartile (iv) lower quartile. [4] SECTION B (40 marks) Attempt any four question from this section. Question 5. (a) Take any two different matrices A and B of order 2 2, and verify that AB BA. [3] (b) The diameter of the base of a right circular cylinder is 28 cm and its height is 21 cm. Find its (i) curved surface area (ii) total surface area (iii) volume. [3] (c) Which is better investment : 7% Rs 100 shares at Rs 120 or 8% Rs 10 shares at 13.50 ? [4] Question 6. (a) Solve the quadratic equation : 3 x2 x 7 = 0 and give your answer correct to two decimal places. [3] (b) An integer is chosen at random form 1 to 100. Find the probability that the number is : (i) is divisible by 5 (ii) is prime number (iii) is perfect cube. [3] (c) Find x from the following equations : {3x + (9x2 5)}/{3x (9x2 5)} = 5. [4] Question 7. (a) A road which is 7 m wide surrounds a circular park whose circumference is 352 m. Find the surface area of the road. [3] (b) Construct a ABC, in which AB = AC = 3 cm and BC = 2 cm. Using a ruler and compasses only, draw the reflection A BC of ABC, in BC. Draw lines of symmetry of the figure ABA C. [3] (c) An incomplete page from the passbook of a bank account is given below : Date Year 2012 Feb 17 Marc h 13 May 21 June 18 June 30 July 5 Particular s Withdrawa ls Deposi ts Balanc e By cash 8000 8000 By clearing To cheque 25000 33000 12000 By cash By interest To withdraw al By transfer By cash 21000 50000 71000 26000 Sep. 45000 23 Nov. 30000 18 Dec. To self 40000 17 If the rate of interest is 4% per annum and is credited at the end of June and December every year, find the interest up to 30th June and then after completing all the entries, find the balance of the account on 31st December, 2012. [4] Question 8. (a) Prove the following identities : cos A/(1 tan A) + sin A/(1 cot A) = sin A + cos A [3] (b) The centre of a circle of radius 13 units is the point (3, 6). P(7, 9) is a point inside the circle . APB is a chord of the circle such that AP = PB. Calculate the length of AB. [3] (c) The two vertices of a triangle are ( 1, 4) and (5, 2). If the centroid is (0, 3), find the third vertex. [4] Question 9. (a) A card is drawn at random from a pack of 52 playing cards. Find the probability of : (i) red card (ii) the card is an ace (iii) the card is a heart. [3] (b) A line passes through the point P(3, 2) and cuts off positive intercepts, on the x-axis and the y-axis in the ratio 3 : 4. Find the equation of the line. [3] (c) From the top of a cliff 90 m high, the angle of depression of the top and bottom of a tower are observed to be 30 and 60 respectively. Find the height of the tower. [4] Question 10. (a) 100 pupil in a school have heights as given below : Hei ght in cm 1 2 1 1 3 0 1 2 1 3 1 1 4 0 1 6 141 150 151 1 6 0 1 6 1 1 7 0 1 4 1 7 1 1 8 0 8 No. 30 2 of 0 pu pil Use a graph paper to draw an ogive and use this ogive to find : (i) the inter quartile range. (ii) the number of pupil whose height is more than 155 cm (iii) the number of pupil whose height is less than 148 cm. [6] (b) A manufacturer sold a refrigerator to a wholesaler for Rs 30000. The wholesaler sold it to retailer at a profit of Rs2000. If the retailer sold it to a customer at a profit of Rs3000 find : (i)the total VAT collected by the state government at the rate of 5% (ii) the amount the customer paid for the refrigerator. [4] Question 11. (a) The point (1, 3) and D(6, 8) are two opposite vertices of a square ABCD. Find the equation of the diagonal AC. [3] (b) AB is a diameter of a circle with centre O. CD is a chord equal to the radius of the circle. AC and BD produced meet at P. Prove that APB = 60 . [3] (c) Using ruler and compass only, construct a triangle ABC in which BC = 6 cm, BAC = 60 and , the foot of perpendicular drawn from A on BC, is 4 cm away from B. [4]
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