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Villa Theresa High School STD: X Preliminary Examination DATE: 03/01/2024 SUBJECT: Mathematics MARKS: 80 WRITING TIME: 2 Hrs . Time allowed: Two and half hours Answers to this Paper must be written on the paper provided separately. You will not be allowed to write during 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 (Attempt all questions in this section) Question 1 Choose the correct answers to the questions from the given options. (Do not copy the question, write the correct answers only.) [15] i. Shilpa has 4 year recurring deposit account in HDFC bank and deposits Rs. 38,400 in 4 years. If she gets Rs. 48,200 in her account after 4 years then amount deposited per month is Rs.: a. 9800 b. 800 c. 400 d. 1600 ii. 0 1 2 If A = [ ] and B = [ ] then which of the following operations is possible 1 0 1 a. A + B b. A B c. AB d. BA iii. A retailer purchased an article for 2000 and sells it to the customer at 10% profit. If the GST paid by the customer is 220. The rate of GST is a. 22% b. 110% c. 11% d. 10% iv. The equation 2 2 6 + = 0 has real and distinct roots if values of is: 9 a. < 2 9 b. > 2 9 c. 2 9 d. 2 v. If , 2( + 1), 3 ( + 1) are in G.P. then the value of is a. 1 b. 4 c. 1 d. 4 vi. The table shows values of and . Where is proportional to 10 4 M 10 What a. b. c. d. vii. viii. N 15 are the values of M and N M = 25 and N= 5 M = 4 and N = 6 M= 25 and N= 6 M= 5 and N = 6 In the given figure ~ and AD and PM are the medians. If AD : PM = 3 : 5 then AB : PQ is a. b. c. d. The 6 : 10 3:5 9 : 25 5:3 maximum volume of the cone that can be carved out of a hemisphere of radius is a. 3 2 b. c. 3 3 2 3 d. 3 3 ix. A box contains 90 discs numbered 1 to 90. If one disc is drawn at random from the box, probability that it bears a prime number less than 23 is; a. b. c. d. 7 90 10 90 4 45 9 89 x. The length of the shadow is 3 times the length of the pole. Then the angle of elevation of the sun is: a. 30 b. 45 c. 60 d. 90 xi. The perimeter of triangle ABC is: a. b. c. d. 30 60 45 15 cm cm cm cm xii. Assertion: The co-ordinates of a mid-point P of a segment joining A ( 1 , 1 ) and B( 2 , 2 ) are ( xiii. xiv. xv. 1 + 2 1 + 2 2 , 2 ) Reason: The mid-point of line segment divides the line segment in the ratio 2 : 1 a. Both assertion and reason are true. b. Both assertion and reason are false. c. Assertion is true, but reason is false. d. Assertion is false, but reason is true. 25 shares of a company are selling at 20. If the company is paying a dividend of 12%. Then the rate of return is: a. 10% b. 12% c. 15% d. 18%. Three sides of a scalene triangle are equidistant from a fixed point. Then that point is a. Circumcentre of the triangle. b. Incentre of the triangle. c. Orthocentre of the triangle. d. Centroid of the triangle. Which of the following lines makes an acute angle with the axis a. = 9 b. 2 + 3 = 6 c. 2 3 = 8 d. = 5 Question 2 i. ii. iii. A toy in a shape of a right circular cylinder with hemisphere at one end and cone at the other. The height and radius of the cylindrical part is 13 cm and 5 cm respectively. The radii of the conical part and hemispherical part is same as the cylindrical part. Find the surface area of the toy if height of the conical part is 12 cm. [4] Richard has recurring deposit account in post office for 3 years at 7.5% p.a. simple interest. If he gets 8325 as interest at the time of maturity. Calculate: a. Monthly instalment b. Amount on maturity. [4] Prove that: cos (1 + ) +sin ( 1 + tan ) = sec + [4] Question 3 i. If a, b and c are in continued proportion prove that 2 + 2 + 2 ( + + )2 ii. + = + + [4] A in the figure below A, B, C are three points on a circle. The tangent at C meets BA produced at T. Given that ATC = 36 and ACT = 48 and AB = 7.8 cm and AT = 5 cm. Calculate: a. Angle subtended by AB with the centre of the circle b. PT [4] iii. The weights of 50 workers is given below Weight (in kg) No. of workers 50-60 60-70 70-80 80-90 90-100 100-110 110-120 4 7 11 14 6 5 3 Draw an ogive of the given information giving a graph sheet. Taking 2 cm as 10 kg on one axis and 2 cm = 5 workers on the other axis. Use the graph to estimate: a. Percentage of workers weighing 80 kg or more. b. The weight above which heaviest 30% of the workers fall? [5] Section B (Attempt Any 4 Questions from this Section) Question 4 (i) X =[ 3 5 5 ][ ], Find a and b if XY=I 1 2 1 [3] (ii) Solve the following equation and calculate the answer correct to 3 significant figures: ( 2)2 2 5 = 0 [3] (iii) In , = , AP= 6 cm, AQ= 5cm, PB= 4 cm [4] (a) Prove ~ (b) Find the length of QC (c) Find the area of (d) Find the area of : Question 5 (i) In an AP, the fourth and sixth terms are 8 and 14 respectively. Find: (a) first term (b) common difference (c) sum of first 20 terms [3] (ii) Use graph for this question. In a school the weekly pocket money of 50 students is as follows: Weekly 40-50 50-60 60-70 70-80 80-90 pocket money(Rs) No. of 2 8 12 14 8 Students Draw a histogram and find the mode from the same graph. 90-100 6 [3] (iii) A cylindrical can of radius 9 cm and height 12 cm is full of ice-cream. The can was emptied completely when each child who attended a party was given a cone full of ice cream with a hemispherical topping. How many children attended the party if the base radius of the cone is 3 cm and height 6 cm? [4] Question 6 (i) The following are the marks obtained by 70 boys in a class test Marks 30-40 40-50 50-60 60-70 No. of 10 12 14 12 boys Find the mean by step deviation method 70-80 9 80-90 7 (ii) The following bill shows the GST and the marked price of the articles 90-100 6 [3] BILL: COMPUTERS Articles Marked price Graphic card Rs. 15500.00 Laptop Adapter Rs 1900.00 Find the final amount to be paid. Rate of GST 18% 28% [3] (iii) In the figure given below, O is the centre of the circle. SP is a tangent. If = 65 , find the values of x, y and z. Also, prove TQ || SU. [4] U Question 7 (i) Find the geometric progression whose 5th term is 48 and 8th term is 384. [3] (ii) Gopal has some ` 100 shares of a company A paying 10% dividend. He sells certain number of shares at a discount of 20% and invests the proceeds in ` 100 shares at ` 60 of company B paying 20% dividend. If his income from the shares sold increases by ` 18,000, find the number of shares sold by Gopal [3] (iii) An observer standing on the top of a light house 150m above the sea level watches a ship sailing away. As he observes, the angle of depression of the ship changes from 50 30 . Determine the distance travelled by the ship during the period of observation. Give your answer correct to the nearest metre. (Use mathematical table) [4] Question 8 2 + 2 + 2 2 (i) Given = 2 2 2 + 2 2 2 Use componendo and dividendo to prove that 2 = 2 +1 (ii) Some grass flower vases were bought for Rs 6000. Ten were damaged during transportation, the remaining were sold for a total profit of Rs 1200 by selling each for Rs 60 more than what was paid for. Form a quadratic equation to find the number of vases. [3] [4] (iii) Use ruler and compasses only for the following question. All construction lines and arcs must be clearly shown. a. b. c. d. Construct ABC in which BC = 6.5 cm ABC = 60 and AB = 5 cm. Construct the locus of points at a distance of 3.5 cm from A. Construct the locus of points equidistant from AC and BC. Mark two points X and Y which are at a distance of 3.5 cm from A and also equidistant from AC and BC. Measure XY. Question 9 (i) Raju wakes up in the morning and notice that his digital clock reads 07: 25 [4] Afternoon, he looks at the clock again. What is the probability that (a) the number in column A is a 4? (b) the number in column B is an 8? (c) the number in column B is greater than 5? (ii) Using remainder theorem factorise 4 3 + 7 2 36 63 completely (iii) Use graph sheet for this. Take 2cm = 1 unit along the axes. [3] [3] [4] a) Plot A (0,3) , B (1,1) , C(3,1) , D(2,-1), E(3,-3), F(0,-2) b) Reflect B, C, D, E on y- axis to get B , C , D and E respectively. Write the co-ordinate of B , C , D and E c) Join A, B, C, D, E, F, E , D , C , B and A to form a closed figure and give geometrical name for the figure. Question 10 (i) Solve the following inequation. Write the solution set and represent it on 14 4 the number line: 3 + 3 > 3 2 2 4 , [3] (ii) In the given figure , O is the centre. = 70 , = 40 [3] Calculate a) b) c) (iii) A (2,5), B (-1,2), C (4,9) are the vertices of a . M is a point on AB such that AM: MB= 1 : 2 , N is the midpoint of AC. a) Find coordinates of M b) coordinates of N c) equation of the line passing through N parallel to BC ******************************* [4]
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