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SECOND PRELIMINARY EXAMINATION Grade: X Subject: Mathematics Date: 27 January, 2021 Time: 2hours 30 minutes Max. Marks: 80 _______________________________________________________________________________ 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 ( ). ________________________________________________________________________________ SECTION A (40 Marks) Attempt all questions from this Section Question 1 sec60 cos90 cos90 cot45 2 (i) (a) If A = and B = , find A + BA. -3tan45 sin90 -4sin30 3sin90 (3) (b) Find the sum of all odd numbers between 100 and 150. (3) (c) If a, b, c, d are in continued proportion, prove that : (4) a 2 +b2 +c 2 b2 +c 2 + d2 = ab +bc +cd 2 Page 1 of 5 Question 2 (a) Using Remainder theorem, find the value of a if the division of x3 +5x2 - ax + 6 by (3) x -1 leaves the remainder 2a. (b) Determine the sum of the first 35 terms of an A.P. if t 2 = 2 and t 7 = 22 . (3) (c) A dealer supplied goods / services worth 5000 in Interstate transaction and worth (4) another 6000 in transactions within the state. The total value of his receipts of goods services within the state was 9000. Find the net tax payable by him, if the rate of GST is 12%. Question 3 (a) Find the values of k for which the following equation has equal roots: kx2 +kx +1= -4x2 - x (3) (b) In the given figure, (3) C ABCD is a cyclic quadrilateral whose side AB is a diameter of the circle. If ADC = 130 , find BAC (c) If x = D A . O B 2mab x +ma x +mb , find the value of + a +b x - ma x - mb (4) Question 4 (a) Calculate the ratio in which the line joining A(6, 5) and B(4, -3) is divided by the (3) line y = 2. (b) Determine the positive value of k for which both the equations x2 +kx + 64 = 0. (3) and x2 - 8x +k = 0. will have real roots. (c) The point P(3, 4) is reflected to P in the X axis and O is the image of (4) O (the origin) in the line PP . (i) Find the coordinates of P and O (ii) Find the length of the segments PP and OO . (iii) Find the perimeter of the quadrilateral, POP O . (iv) What is the special name of the quadrilateral POP O ? Page 2 of 5 SECTION B (40 Marks) Attempt any 4 out of 7 questions from this Section Question 5 (a) Solve the inequation given below and represent its solution set on a number line: (3) 1- x 2x 2x - 3 x + > ,x R 3 5 (b) Solve for x and give your answer correct to 2 decimal places: 3x2 - 32x +12 = 0 (3) (c) Shabana has a cumulative time deposit account in state bank of India. She (4) deposits Rs 500 per month for a period of 4 years. If at the time of maturity she gets Rs 28410, find the rate of interest. Question 6 (a) Prove that : (b) 1+ sin 1- sin + = 2sec 1- sin 1+ sin (3) In the figure, (3) P A PAT is a tangent to the circle at A, BC is a diameter of the circle and B ABC = 30 . Find PAB and TAC. 30 T . C O (c) If 4th term of an A.P. is equal to 3 times the first term and 7th term exceeds twice (4) the third term by 1. Find the first term and the common difference. Question 7 (a) Find the value of p and q if g(x) = x + 2 is a factor of f(x) = x3 - px + x + q and (3) f(2) = 4. (b) For what value of x, the inclination of the line passing through points P (3, 4) and (3) Q (x, 5) is 45 ? (c) Find the mode from the following distribution by drawing a histogram (4) Daily Wages (Rs) 31 36 37 42 43 48 49 54 55 60 61 66 No. of Workers 6 12 20 15 9 4 Page 3 of 5 Question 8 (a) The monthly income of a group of 320 employees in a company is given below: Monthly Income No. of Employees 6000 7000 20 7000 8000 45 8000 9000 65 9000 10000 95 10000 11000 60 11000 12000 30 12000 - 13000 5 (6) Draw an ogive of the given distribution on a graph sheet taking 2 cm = Rs 1000 on one axis and 2 cm = 50 employees on the other axis. From the graph determine (i) the median wage (ii) the number of employees whose income is below Rs 8500. (iii) If the salary of a senior employee is above 11,500, find the number of senior employees in the company. (iv) the upper quartile. (b) From the top of a hill, the angles of depression of two consecutive kilometer (4) stones, due east are found to be 30 and 45 respectively. Find the distance of the two stones from the foot of the hill. Question 9 (a) The line 4x - 3y +12 = 0 meets X axis at A. Write down the coordinates of A. (3) Determine the equation of the line passing through A and whose slope is reciprocal of slope of equation 4x - 3y +12 = 0 . (b) A train covers a distance of 300 km between two stations at a speed of x km per (3) hour. Another train covers the same distance at a speed of (x 5) km per hour. If the first train A train takes 3 hours less than the second train, finds the speed of each train. (c) A box contains 19 balls bearing numbers 1, 2, 3, 19. A ball is drawn at (4) random from the box. Find the probability that the number on the ball is (i) a prime number, (i) an odd number, ii) divisible by 3 or 5, (iv) neither divisible by 5 nor by 10. Page 4 of 5 Question 10 (a) In the given figure, PAT is tangent at A and BD is a diameter of the circle. If ABD = 280 and BDC = 520 . Find : BAD, PAB, CBD . (3) (3) A (b) ABC is a triangle and PQ is a straight line meeting AB at P and AC at Q. P Q If AP= 1 cm, PB = 3 cm, AQ = 1.5 cm, QC = 4.5 cm, B prove that area of APQ is one-sixteenth of the area of ABC C (c) A jewellery manufacturer in Surat (Gujarat) sold a bracelet to a dealer in Rajkot (4) (Gujarat) for 80000. This bracelet was then sold to a dealer in Cuttack (Odisha) for 95000. If the GST rate for gold jewellery is 3%, calculate (i) The net GST payable at Rajkot (ii) Input Tax Credit for the dealer in Cuttack. Question 11 (a) Prove the following identities: cos3 A + sin3 A cos3 A - sin3 A + =2 cosA + sinA cosA - sinA (b) In the given figure, AB is a diameter of a circle with centre O and chord ED is parallel to AB and EAB = 650. Calculate: (i) EBA (ii) BED (iii) BCD (3) (3) (c) The hotel bill for a number of people for overweight stay is Rs 14400. If there (4) were 4 more people, the bill each person had to pay would have reduced by Rs 600. Find the number of people staying overnight. End of Paper Page 5 of 5
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