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ICSE Class X Mid-term 2027 : Mathematics : Half yearly paper

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Rohit Gaming
The Bishop's Co - Ed School, Undri, Pune

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THE BISHOP S CO-ED SCHOOL, UNDRI FIRST TERMINAL EXAMINATION [2026-2027] MATHEMATICS [3 hours] CLASS: X MARKS: 80 Maximum Marks: 80 Time allowed: Three hours Answers to this Paper must be wri en on the paper provided separately. You will not be allowed write during rst 15 minutes. This me is to be spent reading the ques on paper. The me given at the head of this Paper is the me allowed for wri ng the answers. A empt all ques on from Sec on A and any four ques ons from Sec on 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 essen al working will result in loss of marks. The intended marks for ques ons or parts of ques on are given in brackets []. Mathema cal tables are provided. Instruc ons for the Supervising Examiner Kindly read aloud the Instruc ons given above to all the candidates present in the Examina on Hall. ___________________________________________________________________ SECTION A (A empt all ques ons from this sec on) QUESTION 1 [15M] Choose the correct answers to the ques ons from the given op ons. [Do not copy the ques ons. Write the answer only] i] The roots of Quadra c equa on: 2 3 = 2 2 + 21 a] -1 ii] If A = [ a] A + B b] 6 ] and B = c] Both a and b d] None of these then which of the following opera on is possible b] A B c] A B d] None of these iii] An integer is such that one-third of the next integer is at least 2 more than one-fourth of previous integer. The smallest integer is: a] 16 b] 15 c] 14 d] None of these iv] Asser on (A): The Interest on R.D account is calculated for Compound interest of 12 months. Reason(R): Compound Interest is interest on interest a] Both A and R are true are R is correct explana on of A b] Both A and R are true are R is not the correct explana on of A c] A is false and R is true d] both A and R are false v] A point ( , )is re ected successively in the X-axis, then the Y-axis, and nally through the origin. Its nal image is (4, 7).The value of is: a] 3 b] -3 c] 11 d] -11 vi] if (a-b), (b-c), (c-d) are in AP then what is the value of a: a] 2b 3c + d b] b + c d vii] The remainder when a] 0 c] 2b + 3c d 1 is divided by b] 1 d] 3b 3c + d 2 c] 2 d] 3 viii] Asser on(A): 0, 0, 0.. upto n terms are in both AP and GP Reason(R): The equa ons 2b = a + c and b2 = ac are sa s ed when a, b, c are 0. a] Both A and R are true are R is correct explana on of A b] Both A and R are true are R is not the correct explana on of A c] A is false and R is true d] both A and R are false ix] What number should be added to , , , to make them propor onal a] 1 b] 2 c] 0 d] none of these x] A man purchase 600 shares of face value 40 at par. If dividend was received of 1680 at the end of the year, nd the rate of dividend. a] 5% b] 7% c] 8% d] 6% xi] A retailer buys an ar cle at its listed price from a wholesaler and sells it to a consumer in the same state a er making up the price by 20%. The list price of ar cle is 2,500 and the rate of GST is 12%. What is tax liability of retailer to state government. a] 12 b] 15 c] 30 d] 60 xii] In what ra o does the line x y = 2 divide the line segment joining the points (3, -1) and (8, 9)? a] 3: 2 b] 1:2 c] 2: 1 d] 2: 3 xiii] The roots of Quadra c equa on 5 3 4 = 0 are 1.243 and -0.643. The roots correct upto three signi cant gures are: a] 1.24 and -0.64 b] 1.243 and -0.64 c] 1.24 and -0.643 d] none of these xiv] A number x is selected from the numbers 1, 2, 3 and second number y from 1, 4, 9. If the product xy of two numbers is less than 18 then probability will be: a] b] c] 0.77.. d] 0.88.. xv] The nature of roots of quadra c equa on: 8 + 1 = 0 a] Real and Equal b] Not Real c] Real and Ra onal d] None of these QUESTION 2 i] If a, b, c, d are in propor on prove that: + = ( ) [4] ii] The 3 terms of an AP is 3 , 14, 3 and when 4 is added to rst term, 11 to the second term and 98 to third term the numbers are in GP. Find the numbers in AP and GP respec vely [4] Page 2 of 5 iii] During the ba le, Ram had a certain number of arrows in his quiver. In the rst a ack, he discharged one-fourth of the arrows he ini ally possessed. From the arrows then remaining, he used a number equal to the square root of the number discharged in the rst a ack. He subsequently used six mes the number of arrows used in the second a ack. Of the arrows s ll remaining, he used them to kill the monster's assistant and the monster in the ra o 2: 3. The number of arrows allo ed to the monster exceeded those allo ed to his assistant by four fewer than ve mes the number used in the second a ack. Find the number of arrows Ram ini ally possessed. [4] QUESTION 3 i] The point ( , )divides the line segment joining ( , +3 3)and ( , 7 + 7)internally in the ra o 3: 7. If lies on the line + = 10, and the coordinates of sa sfy the equa ons 17 1382 + 3993 = 0and 17 1450 + 9317 = 0, nd the coordinates of , , and . [4] ii] A polynomial ( ) of order 3 has (3 + 7) as a factor. The remaining factors form a quadratic equation which when divided by ( 1) and ( 5) leaves remainder of 8. If 4 is subtracted from the quadratic equation its discriminant becomes zero. Find the quadratic equation and the polynomial ( ). [4] iii] Use a graph paper for this ques on (Take 1 cm = 1 unit on both X and Y axis). [5] Plot the points A (2,2), and B (6, -2) in the graph and answer the following: (a) Re ect points A in origin to point D and write the co-ordinates of point D. (b) Re ect points A in line y = - 2 to point C and write the co-ordinates of points C. (c) Find a point P on CD which is invariant under re ec on in x = 0, write its coordinates. (d) Write the geometrical name of the closed gure ABCD. (e) Write the co-ordinates of the point of intersec on of the diagonals of ABCD. SECTION B (A empt any 4 Ques ons from this sec on.) QUESTION 4 i] One-third of a bamboo pole is buried in mud, one-sixth of it is in water and the part above the water is greater than or equal to 3 metres. Find the length of the shortest pole. [3] ii] Amit deposited 600 per month in a recurring deposit account. The bank pays a simple interest of 12% p.a. Calculate the: [3] (a) number of monthly instalments Amit deposits to get a maturity amount of 11826? (b) total interest paid by the bank. (c) total amount deposited by him. iii] A man invests 20304 partly in company A in 9% 50 shares at 8% premium and in company B at 8% 25 shares at 8% discount, the annual income from both the investments are equal. Find the investment in Company A and B respec vely. [4] QUESTION 5 i] Show that the equa on 2( + ) + 2( + ) + 1 = 0 has no real roots when . [3] Page 3 of 5 ii] The probability of selec ng a blue marble and a red marble from a bag containing red, blue and green marbles is and respec vely. If the bag contains 14 green marbles, then nd: [3] (a) number of red marbles. (b) total number of marbles in the bag. iii] Solve the equa on upto 3 signi cant gures: 0.73 4.61 + 2.17 = 0 [4] QUESTION 6 i] Let Matrix M be such that 2 4 1331 5 = . 1728 2 [3] a] Find the order of Matrix M b] Find the Matrix M ii] The marked price of a toy is same as the percentage of GST that is charged. The price of the toy is 24 including GST. Taking the marked price as x, form an equa on and solve it to nd x. [3] iii] If a, b, c are in con nued propor on, prove that: = ( ) ( ) [4] QUESTION 7 i] Solve: 10 + 34 + 142 upto n terms. [4] ii] A dealer purchases two ar cles. The total cost price of the two ar cles is 2,80,000. He marks both ar cles 50% above their cost prices. He gives a discount of x% on the rst ar cle and (x+10)% on the second ar cle. GST is charged at 18% on both ar cles. It is further known that: The total bill (including GST) is 3,68,160. The GST collected on the second ar cle exceeds the GST collected on the rst ar cle by 4,320. [6] Find: a] The cost price of each ar cle. b] The taxable value of each ar cle. c] The value of x. d] The GST collected on each ar cle. e] The net GST deposited if the dealer had paid input GST at 18% while purchasing the goods. QUESTION 8 2 2 1 and B = 1 1 2 If the sum of all the elements of is 3 mes the sum of all the elements of , nd . i] A = Hence, nd . [3] ii] In the given gure, P (10, a), R (b, 5) and Q (c, 3) are collinear. If PQ: RQ = 5: 2. Find the value of a, b, c. [3] iii] Given ( ) = + + 2 and ( ) = + + 1. If ( 2) is a factor of ( ) but leaves remainder -15 when it divides ( ), nd the values of and . With these values of and , factorise the expression ( ) + ( ) + 4 + 7 [4] Page 4 of 5 QUESTION 9 i] Find coordinates of centroid of triangle of ver ces A (3, -3), B (2, -5) and C (6, -7) without using the centroid formula. [3] ii] a] Find the probability of 2 people having same birthday in a year. b] Find the probability of choosing a random point in a square of area which is not present in the biggest circle in that square. [3] iii] Rajiv Bhardwaj has a R.D account in a bank. If he deposits 600 per month the bank pays him a simple interest rate of 7% p.a and he receives 15450 as maturity amount, nd the total e for which account was held. [4] QUESTION 10 i] a] Point P (2, -3) on re ec on becomes P (2, 3). Name the line of refec on (say L1). b] Point P is re ected to P along line L2 which is perpendicular to line L1 and passes through the point, which is invariant along the both axes. Write coordinates of P . c] Point P is re ected to P on re ec on through the line named in answer of part (a) of this ques on. Write coordinates of P . Comment on loca on of points P and P . [3] ii] A person invests 4368 and buys certain 100 shares at 91. He sells out the shares worth 2400 when they have risen to 95 and the remainder when they have fallen to 85. Find the gain or loss on the total transac on. [3] iii] If is A.M of and and the G.M and the two geometric means are and , then prove that + = 2 [4] ******************************************* Page 5 of 5

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