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ICSE Class X Board Specimen 2024 : Mathematics

12 pages, 64 questions, 16 questions with responses, 16 total responses,

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Indian Certificate of Secondary Education (ICSE), New Delhi

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ICSE 2024 EXAMINATION SPECIMEN QUESTION PAPER MATHEMATICS Maximum Marks: 80 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 from this Section.) Question 1 Choose the correct answers to the questions from the given options. [15] (Do not copy the question, write the correct answers only.) (i) 1 = [ 1 2] = [ 0 2 ] 3 Which of the following operation is possible? (a) A B (b) A+B (c) AB (d) BA T24 511 SPECIMEN 1 of 12 (ii) (iii) If 2 + + 6 = ( 2 )( 3) for all values of x, then the value of k is: (a) 5 (b) 3 (c) 2 (d) 5 A retailer purchased an item for 1500 from a wholesaler and sells it to a customer at 10% profit. The sales are intra-state and the rate of GST is 10%. The amount of GST paid by the customer: (iv) (v) (a) 15 (b) 30 (c) 150 (d) 165 If the roots of equation 2 6 + = 0 are real and distinct, then value of k is: (a) > 9 (b) > 6 (c) <6 (d) <9 Which of the following is/are an Arithmetic Progression (A.P.)? 1. 1, 4, 9, 16, . 2. 3, 2 3, 3 3, 4 3, 3. 8, 6, 4, 2, (a) only 1. (b) only 2. (c) only 2. and 3. (d) all 1., 2. and 3. T24 511 SPECIMEN 2 of 12 (vi) The table shows the values of x and y, where x is proportional to y. x 6 12 N y M 18 6 What are the values of M and N? (vii) (a) M = 4, N = 9 (b) M = 9, N = 3 (c) M = 9, N = 4 (d) M = 12, N =0 In the given diagram, ABC ~ PQR and 3 = 8. The value of AB : PQ is: P A B (viii) (a) 8:3 (b) 3:5 (c) 3:8 (d) 5:8 D CQ S R A right angle triangle shaped piece of hard board is rotated completely about its hypotenuse, as shown in the diagram. The solid so formed is always: 1. a single cone 2. a double cone Which of the statement is valid? (a) only 1. (b) only 2. (c) both 1. and 2. (d) neither 1. nor 2. T24 511 SPECIMEN 3 of 12 (ix) Event A: The sun will rise from east tomorrow. Event B: It will rain on Monday. Event C: February month has 29 days in a leap year. Which of the above event(s) has probability equal to 1? (x) (a) all events A, B and C (b) both events A and B (c) both events B and C (d) both events A and C The three vertices of a scalene triangle are always equidistant from a fixed point. The point is: (xi) (a) Orthocentre of the triangle. (b) Incentre of the triangle. (c) Circumcentre of the triangle. (d) Centroid of the triangle. In a circle with radius R, the shortest distance between two parallel tangents is equal to (xii) (a) R (b) 2R (c) 2 R (d) R An observer at point E, which is at a certain distance from the lamp post AB, finds the angle of elevation of top of lamp post from positions C, D and E as , and . It is given that B, C, D and E are along a straight line. Which of the following condition is satisfied? (a) tan > tan (b) tan < tan (c) tan > tan (d) tan < tan A B T24 511 SPECIMEN C D E 4 of 12 (xiii) 1. Shares of company A, paying 12%, 100 shares are at 80. 2. Shares of company B, paying 12%, 100 shares at 100. 3. Shares of company C, paying 12%, 100 shares are at 120. Shares of which company are at premium? (xiv) (xv) (a) Company A (b) Company B (c) Company C (d) Company A and C Which of the following equation represent a line passing through origin? (a) 3x 2y + 5 = 0 (b) 2x 3y = 0 (c) x=5 (d) y = 6 For the given 25 variables: , , . Assertion (A): To find median of the given data, the variate needs to be arranged in ascending or descending order. Reason (R): The median is the central most term of the arranged data. (a) A is true, R is false (b) A is false, R is true (c) both A and R are true (d) both A and R are false Question 2 (i) Shown below is a horizontal water tank composed of a cylinder and two [4] hemispheres. The tank is filled up to a height of 7 m. Find the surface area of the tank in contact with water. Use = T24 511 SPECIMEN 22 7 . 5 of 12 (ii) In a recurring deposit account for 2 years, the total amount deposited by a person is [4] 9600. If the interest earned by him is one-twelfth of his total deposit, then find: (iii) (a) the interest he earns. (b) his monthly deposit. (c) the rate of interest. Find: [4] (a) (sin + )2 (b) ( + )2 Using the above results prove the following trigonometry identity. (sin + )2 + (cos + sec )2 = 7 + 2 + 2 Question 3 (i) If a, b and c are in continued proportion, then prove that: [4] 3 2 + 5 + 7 2 = 3 2 + 5 + 7 2 (ii) In the given diagram, O is the centre of circle circumscribing the ABC. [4] CD is perpendicular to chord AB. OAC=32 . Find each of the unknown angles x, y and z. T24 511 SPECIMEN 6 of 12 (iii) Study the graph and answer each of the following: (a) Name the curve plotted (b) Total number of students (c) The median marks (d) Number of students scoring between 50 and 80 marks T24 511 SPECIMEN [5] 7 of 12 SECTION B (Attempt any four questions from this Section.) Question 4 (i) (ii) If = [ [3] 4 4 ] , find 2 . If 2 = , then find the value of . 4 4 Solve the given equation 2 4 2 = 0 and express your answer correct to two places [3] of decimal. (You may use mathematical tables for this question). (iii) In the given diagram, ABC is right angled at B. BDFE is a rectangle. AD = 6 cm, CE = 4 cm and BC = 12 cm (a) prove that ADF ~ FEC (b) prove that ADF ~ ABC (c) find the length of FE (d) find area ADF : area ABC [4] C 4 cm E F A 6 cm D B Question 5 (i) Shown below is a table illustrating the monthly income distribution of a company with [3] 100 employees. Monthly Income (in 10, 000) Number of employees 0-4 4-8 8 - 12 12 - 16 16 - 20 20 - 24 55 15 06 08 12 4 Using step- deviation method, find the mean monthly income of an employee. (ii) The following bill shows the GST rate and the marked price of articles: [3] Vidhyut Electronics S. No. Item Marked Price Quantity Rate of GST (a) LED TV set 12000 01 28% (b) MP4 player 5000 01 18% Find the total amount to be paid (including GST) for the above bill. T24 511 SPECIMEN 8 of 12 (iii) In the given figure, O is the centre of the circle and AB is a tangent to the circle at B. If PQB=55o. [4] P (a) find the value of the angles x, y and z. (b) prove that RB is parallel to PQ. R O S 55 A Q B Question 6 (i) There are three positive numbers in a Geometric Progression (G.P.) such that: (a) [3] their product is 3375 (b) the result of the product of first and second number added to the product of second and third number is 750. Find the numbers. (ii) The table given below shows the ages of members of a society. Age (in years) Number of Members of the Society 25 35 05 35 45 32 45 55 69 55 - 65 80 65 75 61 75 - 85 13 [3] Use graph sheet for this question. Take 2cm = 10 years along one axis and 2cm=10 members along the other axis. (a) Draw a histogram representing the above distribution. (b) Hence find the modal age of the members. T24 511 SPECIMEN 9 of 12 (iii) A tent is in the shape of a cylinder surmounted by a conical top. If the height and radius of [4] the cylindrical part are 7 m each and the total height of the tent is 14 m. Find the: (a) quantity of air contained inside the tent. (b) radius of a sphere whose volume is equal to the quantity of air inside the tent. Use = 22 7 Question 7 (i) The line segment joining A(2,-3) and B(-3, 2) is intercepted by the -axis at the point M [5] and the y axis at the point N. PQ is perpendicular to AB produced at R and meets the y- axis at a distance of 6 units from the origin O, as shown in the diagram, at S. Find the: (a) P coordinates of M and N (b) coordinates of S (c) R slope of AB. (d) equation of line PQ. Q -6 -5 (ii) 6 S 5 4 3 B (-3, 2) 2 M 1 -4 -3 -2 -1 O 1 -1 N -2 -3 -4 2 3 A (2, -3) The angle of depression of two ships A and B on opposite sides of a light house of height [5] 100m are respectively 42o and 54o. The line joining the two ships passes through the foot of the lighthouse. (a) Find the distance between the two ships A and B. (b) Give your final answer correct to the nearest whole number. (Use mathematical tables for this question) T24 511 SPECIMEN 10 of 12 Question 8 (i) Solve the following inequation write the solution set and represent it on the real number [3] line. 3 2 + (ii) 1 2 > , 3 5 ABCD is a cyclic quadrilateral in which BC = CD and EF is a tangent at A. [3] CBD = 43 and ADB = 62 . Find: (a) ADC (b) ABD (c) FAD F A D 62 E 43 B C (iii) A (a, b), B(-4, 3) and C(8,-6)are the vertices of a ABC. Point D is on BC such that [4] BD : DC is 2 : 1 and M (6, 0) is mid point of AD. Find: (a) coordinates of point D. (b) coordinates of point A. (c) equation of a line passing through M and parallel to line BC. Question 9 (i) Using componendo and dividend, find the value of x, when: [3] 3 + 3 14 = 3 2 + 1 13 (ii) The total expense of a trip for certain number of people is 18000. If three more people join [3] them, then the share of each reduces by 3000. Taking x to be the original number of people, form a quadratic equation in x and solve it to find the value of x. T24 511 SPECIMEN 11 of 12 (iii) Using ruler and compass only construct ABC = 60 , AB = 6 cm and BC = 5 cm. (a) [4] construct the locus of points equidistant from AB and BC. (b) construct the locus of points equidistant from A and B. (c) Mark the point which satisfies both the conditions (a) and (b) as P. Hence, construct a circle with centre P and passing through A and B. Question 10 (i) Using remainder and factor theorem, factorize completely, the given polynomial: [3] 2 3 9 2 + 7 + 6 (ii) Each of the letter of the word HOUSEWARMING " is written on cards and put in a bag. [3] If a card is drawn at random from the bag after shuffling, what is the probability that the letter on the card is (a) a vowel (b) one of the letters of the word SEWING. (c) not a letter from the word WEAR. (iii) Use graph sheet for this question. Take 2 cm = 1 unit along the axes. (a) Plot A (1, 2), B(1, 1)and C (2, 1) (b) Reflect A, B and C about y-axis and name them as A , B and C . (c) Reflect A, B, C, A , B and C about x-axis and name them as A , B , C , A , B [4] and C respectively. (d) Join A, B, C, C , B , A , A , B , C , C , B , A and A to form a closed figure. T24 511 SPECIMEN 12 of 12

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