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AUXILIUM CONVENT SCHOOL, BANDEL PRE-BOARD EXAMINATION 2023-24 MATHEMATICS CLASS:X FULL MARKS: 80 Time allowed: Two and halfhours Answers to this Paper must be writen on the paper provided separately. You will not be allowed to write during the first 15 mimtes. 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 Aand any four questions from Section B. Al 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 Attempt all questions from this Section. [15| Question 1: Choose the correct answers to the questions from the given options. (Do not copy the questions, write the answers only) consecutive terms of an AP is: 5 are 3 he value of p, for which (2p+ 1), 10 and 5p + (c) 1 (a) -1 (b) -2 (ii) The 3rd proportion to 6- and 5is: (a) 4 (i) (d) 2 (b)7; (c) 3 Z0TA = 30 , then AT is a tangent to the circle with centre O such that OT =4 cm and AT is equal to: 4 cm 30 (a) 4 cm (iv) (b) 2 cm (c) 4v3 cm The co-ordinates of the point (5, 2) under reflection in X-axis is: (b) (-2, 5) (c) (5,-2) (a) (5, 2) (v) (d) 2v3 cm (d) (-5, -2) Ateacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over. When he increased the size of the square by one student, he found that he was short of 25 students. Then the total number of students is: (a) 500 (b) 600 (c) 700 (d) 800 P.T.O CL:X/Math/Pg.2 uopsan) (vi) If(2x+1) is a factor of 2x? + ax-3, then the value of ais: (b) -5 (a) 6 (vii) (Vi) (c) 4 (e) (d) -2 Asemicircular lamina of radius 35 cm is folded so that the two bounding radii are joined obtained is: together toform a cone. The lateral surface area of the cone thus (c) 1925 cm? (d) None of these (b) 962.5 cm (a) 3850 cm co-ordinates of its centroid is: If the vertices of atriangle are (1,3), (2,-4) and (-3,1), the (b) (0, 1) (a) (0, 0) (c) (1, 0) (d) (1, 1) (ix) If3 real numbers x, y and z form a geometric progression, then: (c) y-z = x (a) 2y =x+z (b) xz = y2 (x) From a solid circular cylinder with height 10 cm and radius 6 cm, a right circular cone of the same height and the same base is removed, then the volume of the remaining solid (d) x= y+z 1S: (a) 2807 Cm (xi) (b) 240ncm (d) 120 Cm3 7times the 7h term of an A.P. is equal to 11 times the 11th term, then the 18t term is: (a) 7 (xii) (c) 330rCm3 Tfthe lines 2x (b) 11 (c) 18 (d) 0 ay +3 = 0 and 4x -7y +1=0 are perpendicular to each other, then the value of'a' is: 8 (a) (b) (xii) The equation 2x2 + kx +3=0 has two real and equal roots, then the value of kis: (a) t6 (b) +4 (C)3V2 (d)22v6 (xiv) If the probability of a player winning a game is 0.56, the probability of his losing the game is: (a) 0.56 (xv) (b) 1 (c) 0.44 (d) 0 Assertion (A): The inclination of the line x-V3y + 2V3=0 is 30 Reason (R): Ify = mx +c represents the equation of a line, then the inclination of the line is given by m = tanb (a) A is false, R is true (c) A is true, R is false (b) both are true (d) both are false P.T.O CLXMath/Pg.3 Question 2: (a) The eross-section of a railway tunnel is a surmounted by a semi-circle as shown in the figure. The tunnel is 50 m long. Find square the cost of plastering the internal surface of the tunnel (excluding the floor) at the rate of Rs. 10 per m. [41 10.5 7m S0m (b) Each of Aand Bopened recurring deposit accounts in a bank. IfA deposited Rs. 1,200 per month for3 years and Bdeposited Rs. 1,500 per month for 2-years; find, on maturity, who Will get more amount and by how much? The ate of interest paid by the bank is 10 annum. Prove that: sinb - 2sin 0 2cos0- cos per (4] (4] tan Question 3: (a) - (b) Ifx = Va'+ b + vaz- b2 Va'+ b- Va'- h using properties of proportion, prove that b = 2a x x2+1 (4] In the given figure, PQ is atangent to the circle at A, AB and AD are bisectors of ZCAQ and ZPAC respectively. If 2BAQ = 30, prove that; ) BD is a diameter of the circle. [4] (i) ABC is an isosceles triangle B P (c) A Sourav invested Rs. 9600 in Rs. 100 shares at Rs. 20 premium paying 8% dividend. He sold the shares when the price rose to Rs. 160. He invested the proceeds in 10% Rs. 50 shares at 40. Find: (i) the original number of shares he had (ii) Number of new shares (ii) Change in the annual income P.T.O Que CL.XMath/Pg.4 SECTION B Question 4: () IfA= and B= (b) Solve: x-3 (x + 3) = 0. Give your answer correct to two significant figures. 31 (c) In the figure given below, AB || CR and LM || QR. 41 Prove that (i1) find a, b, cwhere A+B= A.B BM AL MC LQ (31 A B Calculate LM : QR, given that BM : MC = 1:2 M R Question 5: (a) A survey regarding height (in cm) of 60 students belonging to class 10 of a school was conducted. The following data was recorded. Height in 140-145 145-150 150-155 155-160 160-165 165-170 170-175 Cm No. of 4 8 20 14 7 6 1 boys Find the mean of the above distribution by step deviation method. [3] (b) The price of abicycle is Rs. 3136 inclusive of tax (under GST) at the rate of 10% on its listed price. Abuyer asks for a discount on the listed price so that after charging GST, the selling price becomes equal to the listed price. Find the amount of discount which the seller has to allow for the deal. [3] (c) In the given figure, '0' is the centre of the circle and AB is a tangent to the circle at B. If ZPQB = 55 , (a) find the value of the angles x, yand z [4) (b) Prove that RB is parallel to PQ 55 B P.T.O gA CL.XIMath/Pg.5 Question 6: b-P, (a) Ifph, qlh and rth term ofa GP. are a, band Icrespectively, prove that a9-T. (b) The table given bclowshows the marks Marks 20-30 No. of students [31 in an examination. Scored by a set of students 30-40 40-50 50-60 60-70 14 9 4 8 Use graph paper for this question. cP-4 =1 70-80 2 students axis and 2 cm =2 one along marks 10 = Take 2cm distribution. Hence find the along the other axis. Draw a histogram representing the above (3] modal marks of the students. of its top, which 11 cm and the radius is height Its cone. is (c) Avessel is in the form of an inverted some lead shots, each of which When rim. the upto water with Find the IS open, is 2.5 cm. It is filled of the water flows out. vessel, the into dropped 5 cm. are [4] a sphere of radius 0.25 number of lead shots dropped into the vessel. 7: A Time passes (4 meets the through the point P (2, 3) and coordinates axes at the points A figure. If 2PA =3 PB, find: Bas shown in the adjoining () (1) the coordinates of A and B and [4] B the equation of the line AB P(2,3) X y' a point P' on the ground which is 12Om (b) When a building under construction was observed from fter its completion when it was A away from its base, the angle of elevation of the top was 30 . higher was the building observed from the same point, the angle changed to 60. How much [6] raised, from the time it was first observed? 8: [3] (a) Solve the following inequation and represent the solution set on the number line: 8x 11 -4+xs+1s,+2r, xEI P.T.0 CLX/Math/Pg.6 (b) On a map drawn to a scale of 1:50,000, a rectangular plot of land ABCD has the following dimensions. AB = 6cm, BC=8 cm, find: The actual length of the diagonal AC of the plot in km (1) The actual area of the plot in sq. km [31 (0) ABO. Use graph paper for this question. A(0, 3), B(3, -2) and O(0,0) are the vertices of triangle Plot the triangle on a graph sheet taking 1 cm = 1 unit on both axes Plot D, the reflection of B, in the Y-axis and write its coordinates. () (im) Name two invariant points under reflection in the Y-axis. (iv) Give the geometrical name of the figure ABOD. [4] Qv stion 9: (a) The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 years more than three times the age of his son. Find their present ages. [4] (b) Use graph paper for this question. The marks obtained by 120 students ofatest are given below: Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90 100 No. of 16 22 26 18 6 4 3 students Draw the ogive and hence estimate: ( The median marks () (iii) The number of students who did not pass the test if the pass percentage was 50 The upper quartile marks [6] gdestion 10: (a) Sixteen cards are labelled as a,b,c,....n,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: (i1) (ii) A vowel A consonant None of the letters of the word 'median' [31 (b) Show that (x-3) is a factor of x' - 7x+ 15x 9. Hence factorize x 7x + 15x -9. (3] (c) Using ruler and compass only construct a semi-circle with diameter BC =7 cm. Locate a point 'A' on the circumference of the semi-circle such that A' is equidistant from Band C. Complete the cyclic quadrilateral ABCD, such that Dis equidistant from AB and BC. Measure zADC and write it down. [4] ********
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