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Half -Yearly Examination -2020-21 Class X Mathematics SECTION A (40 Marks) Attempt all questions from this Section TT:2 Hrs MM:80 Question 1 (a) Find the value of k if 4x3 2x2 + x + 5 leaves remainder -10 when divided by 2x + 1. (b) Amit deposits Rs.1600 per month in a bank for 18 months in a recurring deposit account. If he gets Rs. 31,080 at the time of maturity, what is the rate of interest per annum? (c) Construct a regular hexagon of side 4cm. Construct a circle circumscribing the hexagon. [3] [3] [4] Question 2 (a) Solve the following inequation and represent your solution on the real number line: (b) (c) 5 x 3x 3 x, x [3] Find the 16th term of the A.P. 7, 11, 15, 19 . Find the sum of the first 6 terms. [3] In the given figure CE is a tangent to the circle at point C. ABCD is a cyclic quadrilateral. If ABC = 930 and DCE = 350 Find:(i) Question 3 (a) (b) (c) ADC (ii) CAD (iii) ACD [4] Prove the following identity Find x and y if : [3] - = For what value of k will the following quadratic equation: ( + 1)2 4 2 + 9 = 0 have real and equal roots? Solve the equations. [3] [4] Question 4 (a) A box consists of 4 red, 5 black and 6 white balls. One ball is drawn out at random. Find the probability that the ball drawn is: (i) black (ii) red or white [3] (b) Calculate the median and mode for the following distribution: Weight (in kg) 35 47 52 56 60 No. of students 4 3 5 3 2 [3] (c) A solid cylinder of radius 7 cm and height 14 cm is melted and recast into solid spheres each of radius 3.5 cm. Find the number of spheres formed. [4] SECTION B (40 Marks) Attempt any four questions from this Section Question 5 (a) The 2nd and 45th term of an arithmetic progression are 10 and 96 respectively.Find the first term and the common difference and hence find the sum of the first 15 terms. [3] (b) If find matrix A = find matrix B,such that A2 2B = 3A + 5I where I is a 2 x 2 identity matrix. (c) With the help of a graph paper, taking 1cm=1unit along both x and y axis: (i) Plot points A (0, 3), B (2, 3), C (3, 0), D (2, -3), E (0, -3) (ii) Reflect points B, C and D on the y axis and name them as B', C' and D' respectively. (iii) Write the co-ordinates of B', C' and D'. (iv) Write the equation of line B' D'. (v) Name the figure BCDD'C'B'B Question 6 (a) In ABC and EDC, AB is parallel to ED. BD = (i) (iii) Prove that ABC ~ EDC. Find: (ii) [3] [4] BC and AB = 12.3 cm. Find DE [3] (b) Find the ratio in which the line joining (-2, 5) and (-5, -6) is divided by the line y = -3. Hence find the point of intersection. [3] (c)The given solid figure is a cylinder surmounted by a cone. The diameter of the base of the cylinder is 6 cm. The height of the cone is 4 cm and the total height of the solid is 25 cm. Take = 22 /7 Find the:(i) Volume of the solid (ii) Curved surface area of the solid Give your answers correct to the nearest whole number. [4] Question 7 (a) In the given figure, PAB is a secant and PT a tangent to the circle with centre O.If ATP = 40o, PA = 9 cm and AB = 7 cm. Find: (i) APT (ii) length of PT (b) The 1st and the 8th term of a AP are 4 and 512 respectively. Find: (i) the common ratio (ii) the sum of its first 5 terms. (c) [3] [3] The mean of the following distribution is 49. Find the missing frequency a . Class 0 20 20 40 40 60 60 80 80 100 Frequency 15 20 30 a 10 [4] Question 8 (a) Prove the following identity (sinA + cosecA)2 + (cosA + secA)2 = 5 + sec 2 A . cosec 2 A [3] (b) Find the equation of the perpendicular bisector of line segment joining A(4, 2) and B(-3, -5) (c) Using properties of proportion, find x : y if = [3] [4] Question 9 (a) The difference of the squares of two natural numbers is 84. The square of the larger number is 25 times the smaller number. Find the numbers. [4] (b) The following table shows the distribution of marks in Mathematics: Marks (less than) No. of students 10 7 20 28 30 54 40 71 50 84 60 105 70 147 80 180 With the help of a graph paper, taking 2 cm = 10 units along one axis and 2 cm= 20 units along the other axis, plot an ogive for the above distribution and use it to find the: (i)median. (ii) number of students who scored distinction marks (75% and above) (iii)number of students, who passed the examination if pass marks is 35%. [6] Question 10 (a) Prove that two tangents drawn from an external point to a circle are of equal length. [3] (b) From the given figure find the: (i) (ii) Coordinates of points P, Q, R. Equation of the line through P and parallel to QR. [3] (c) Sixteen cards are labelled as a, b, c, ., m, n, o, p. They are put in a box and shuffled. A boy is asked to draw a card from the box. What is the probability that the card drawn is: (i) a vowel (ii) a consonant (iii) none of the letters of the word median. [4]
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