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F.M 80 Tim e 2 1 2 Hrs HALF-YEARLY EXAMINATION 2018-2019 SUBJECT: MATHEMATICS CLASS- X 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 same sheet as 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. Question1 a) Find the value of a if the division of ax3+9x2+4x-10 by (x-3) leaves a remainder 5. [3] b) Manoj opened a recurring deposit account in a bank and deposited `500 per month for 3 years. The bank paid him `20220 on maturity. Find the rate of interest paid by the bank. [3] c) Solve the following quadratic equation for x and give your answer correct to three significant figures: 2x2-4x-3=0 [4] Question 2 a) Solve the following inequation and represent the solution set on the number line. 2 x 2 <1+ , x R 3 3 3 b) If sum of 1st 7 terms of an A.P. is 49 and that of 1st 17 terms is 289, find sum of 1st n terms. [ 2 3 5 6 [4] c) Let Q= ] and R= [ 8 15 19 15 ] [3] [3] . Find the matrix P, if PQ=R Question 3 a) A game of numbers has cards numbered 10, 11, 12, 13, ., 49, 50. A card is drawn at random, find the probability that the number on the card is (i) A perfect square (ii) Has one of the digit as 3 (iii) A number divisible by 3 and 4. [3] 1 b) Prove that: cot A 1 cot A = 2 2 sec A 1+ tan A [3] c) Calculate the median and mode for the following distribution: Weight(kg) 10 11 12 13 14 No. of Children 1 4 7 5 9 [4] 15 3 16 5 Question 4 a) A train travels a distance of 300km at a constant speed. If the speed of the train is increased by 5km/hr, the journey would have taken 2 hour less. Find the speed of train. [3] 0 0 0 0 2 0 b) Find x, if sin 47 sec 43 + cos 43 cosec 47 xcos 45 =0 [3] c) Draw a circle of radius 3.2cm. Take a point N at a distance of 5.7cm from the centre of the circle. Construct the tangents to the circle from the point N. [4] SECTION B (40 Marks) Attempt any four questions from this Section. Question 5 a) Find the sum of 1st 51 terms of an A.P. whose 2nd and 3rd terms are 14 and 18 respectively. b) Let A= [ 2 1 0 2 ] , B= [ 4 1 3 2 ] and C= [ 3 2 1 4 ] [3] . Find A2 + AC 5B. [3] c) With the help of a graph paper, taking 1cm =1unit along both axes. (i) Plot points A(0,3), B(2,3), C(3,0), D(2,-3) and E(0,-3). (ii) Reflect points B, C and D on the y-axis and name them as B , C and D . (iii) Write the co-ordinates of B , C and D . (iv) Name the figure BCDD C B . [4] Question 6 a) ABCD is trapezium in which OA=2OC and AB||DC. If AC and BD intersect at O and AB=10.8cm. (i) Prove that AOB COD (ii) Find DC. areaof COD (iii) Find area of AOB [4] b) Find the ratio in which the line joining (-3,-1) and (5,7) is divided by the line x=2. Also, find the coordinates of the point of intersection. [3] 3 3 tan cot + =sec cosec 2sin cos c) Prove that [3] 2 1+ tan 1+cot 2 Question 7 a) The 1st and 8th term of a G.P. are 5 and 640 respectively. Find (i) The common ratio (ii) Sum of its first 6 terms b) The mean of the following distribution is 52, find f. 2 [3] [4] Class-interval 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Frequency 5 3 f 7 2 6 13 2 c) Find the value of m , for which the given equation x +m(2x+1) 2x+5=0 has real and equal roots. [3] Question 8 a) A piece of cloth costs `200. If the piece was 5m longer, and each meter of cloth costs `2 less, the cost of the piece would have remain unchanged. How long is the piece and what is the original rate of cloth per metre? [4] b) An airplane at an altitude of 250m, observes the angles of depression of two boats on the opposite banks of a river to be 450 and 600 respectively. Find the width of the river. [3] c) A man invests a sum of money in `100 shares, paying 10% dividend and quoted at 20% premium. If his annual dividend from these shares is `560. Calculate: [3] (i) His total investment. (ii) Rate of return on his investment. Question 9 a) (i) Construct BCP given BC=5cm, BP=4cm and PBC=450. (ii) Complete the rectangle ABCD such that (1) P is equidistant from AB and BC. (2) P is equidistant from C and D. (iii) Measure and record the length of AB. [4] b) The weight of 50 workers is given below: Weight(kg) 50-60 60-70 70-80 80-90 90-100 100-110 110-120 No. of workers 4 7 11 14 6 5 3 Draw an ogive of the given distribution using a graph paper. Take 2cm=10kg on one axis and 2cm=5workers along the other axis. Use the ogive drawn to estimate the following: (i) Median (ii) The upper and lower quartiles (iii) If weighing 95kg and above is considered overweight, find the number of workers who are overweight. [6] Question 10 a) All the three face cards of spades are removed from a well-shuffled deck of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting: [3] (i) A black face card (ii) A queen (iii) A black card 2 1 1 3 55 + is b) Sum of how many terms of the GP . [3] 9 3 2 4 72 c) Construct an inscribing circle of a regular hexagon of side 5cm and measure its radius. [4] Question 11 a) From a window, 15m high above the ground in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are 30 0 and 450 3 respectively. Find the height of the opposite tower. (Take 3 =1.732) [3] b) Mr. Ghosh invested `8000 in 8% `100 shares at `80. After a year, he sold these shares at `75 each and invested the proceeds (including his dividend) in 18% `25 shares at `41. Find: [4] st (i) His dividend for the 1 year (ii) His annual income for the 2nd year (iii) Percentage increase in his return on his original investment. c) Factorize x3-7x2+15x-9, if it is given that (x 3) is a factor of it. [3] 4
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