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ICSE Class X Question Bank 2024 : Mathematics : TRY TO SOLVE

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G.S.T 1) A shopkeeper buys an article whose printed price is Rs 4000 from a wholesaler at a discount of 20% and sells it to a consumer at the printed price. If the sales are intrastate and the rate of GST is 12%, find, (i) the price of the article inclusive of GST at which the shopkeeper bought it. (ii) the amount of tax (under GST) paid by the shopkeeper to the State Government. (iii) the amount of tax (under GST) received by the Central Government. (iv) the amount which the consumer pays for the article. Ans. (i)Rs 3584 (ii)Rs48 (iii)Rs240 (iv)Rs4480 2. The price of an article is Rs.5120 inclusive of GST, at the rate of 28% on its listed price. A customer asks the dealer for a discount on the listed price so that after charging the GST, the selling price will be same as the listed price. What is the amount of discount that the dealer must allow for the deal? Ans: 875 3. Ayush purchased a computer for 28320 which included 20% discount on the list price and 18% tax under GST on the remaining price. find the list price of the computer. Ans: 30000 4.The marked price of an article = Rs 9,000 and rate of GST on it = 18%. A shopkeeper buys this article at a reduced price and sells it at its marked price.If the shopkeeper paid Rs162 as CGST to the government, find the amount (inclusive of GST) paid by the shopkeeper. ANS-Rs 8496 5) Find the amount of bill for the following intra-state transaction of goods/services. The GST rate is 5%. Quantity (no. of items) 18 24 30 12 MRP of each item 150 240 100 120 Discount % 10 20 30 20 ANS-Rs. 10,804.5 6)Manufacturer A sells a washing machine to a dealer B for Rs12500. The dealer B sells it to a consumer at a profit of Rs 1500. If the sales are intra-State and the rate of GST is 12%, find (i) the amount of tax (under GST) paid by the dealer B to the Central Government. (ii) the amount of tax (under GST) received by the State Government. (iii) the amount that the consumer pays for the machine. Ans. (i) Rs 90 (ii) Rs 840 (iii) Rs 15680 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Banking THINGS TO REMEMBER Recurring Deposit SI = P n(n + 1) R 2 12 100 M. A. (Maturity Amount) = SI + P No. of months 7) Richard has a R. D. A/c. in a Post Office for 3 years, at 8% SI. If he gets Rs 1,998 as interest at the time of maturity, find a) The monthly installment. (b) The amount of maturity. Ans. : (a) Rs 450 (b) Rs 18,198 8) Mr. xyz opened a recurring deposit account in a bank and deposits Rs 2500 per month for 2 years .if he gets Rs 66250 at the time of maturity, find (i) The interest paid by the bank. (ii) The rate of interest. Ans. : (i) Rs 6250 (ii) 10% p.a. 9) Arav deposits 500 every month at 12% p.a.in a bank in a recurring deposit scheme. The bank pays 5275 on maturity. Find the time for which the account is held. Ans:10 months 10) ) Katrina opened a recurring deposit account with a Nationalised Bank for a period of 2 years. If the bank pays interest at the rate of 6% per annum and the monthly installment is 1,000, find the (i) maturity value (II) Interest earned in 2 years . RS. 1500 , RS .25500 ALGEBRA Linear Inequations THINGS TO REMEMBER N Natural Numbers {1, 2, 3, .} Not Zero ( 0 ) R Real Numbers { 1, 0, 1, 2, } W Whole Numbers {0, 1, ..} I Integers { 2, 3, ..} 11)IF x R, Solve the following and represent the solution set on the number line. 3x - 3 < 27 - 2x 3x + 7 Ans: { x : 4 x < 6, x R} GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 12)Solve the following inequations and represent the solution set on the number line. where x Ans: {-3,-2,-1,0,1,2,3,4,5} 13)A = {x : 11x 5 > 7x + 3, x R} and B = {x : 18x 9 15 + 12x, x R}. Find the range of set A B and represent it on a number line. Ans. {x : x 4, x R} Shares and Dividends THINGS TO REMEMBER The original value of a share is called nominal value or face value. The price at which the share can be sold any time is called market value. If the nominal value is Rs 100. a) If it is sold at premium 10% means 100 10 100 = 10 Rs 100 + 10 = Rs 110 b) If it is sold at discount; Rs 100 10 = Rs 90 c) If it sold at par than the M.V. will be same as of nominal value. This whole chapter can be solved using three formula is 1) Annual Income or Dividend = NOS ROD NV 2) Investment = NOS MV 3) Profit or % return = 1st 2nd 100% 14) A man invests Rs 16,800 in buying shares of nominal value of Rs 24 and selling at 12% premium. The dividend on the shares is 15% per annum. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 i) Calculate the number of shares he buys. ii) Calculate the dividend he receives annually Ans. : (i) 625 (ii) Rs 2,250 15) By purchasing 50 shares of a company for Rs 75 each Sonal gets 8% per annum on her investment. i) What rate per cent is the company paying? ii) What is her annual dividend, if she buys 350 shares? Ans. : (i) 12% (ii) Rs 2,100 16) Mukul invests Rs 9,000 in a company paying a dividend of 6% p.a. when a share of face value Rs 100 stands at Rs 150. What is his annual income? He sells 50% of his shares when the price rises to Rs 200. What is his gain on this transaction? Ans. : Rs 360, Rs 1,500 17)At what price should be a 6.25% Rs 50 share be quoted when the money is worth 10%. Ans. : Rs 31.25 18) A man invested Rs 45,000 in 15% Rs 100 shares quoted at Rs 125, when the market value of these shares rose to Rs 140, he sold some shares, just enough to raise Rs 8,400, calculate. i) His shares he still holds. ii) The dividend due to him on these shares. Ans. : (i) 300 (ii) Rs 4,500 19)Which is a better investment ? 7% at Rs 116 or 9% at Rs 145? Ans. : 9% at Rs 145 20) A man invests Rs 13,500, partly in 6% Rs 100 shares at Rs 140 and partly in 5% Rs 100 shares at Rs 125. If his total income is Rs 560, how much has he invested in each? Ans. : Rs 7,000 , Rs 6,500 21) Mrs. Sristi invested Rs 8000 in 7% Rs 100 shares at Rs 80. After a year he sold these shares at Rs 75 each and invested the proceeds (including his dividend) in 18% Rs 25 at Rs 41. Find : (i) his dividend for the first year. (ii) his annual income in the second year. (iii)The percentage increase in his return on his original investment. Ans. : (i)Rs 700 ,l(ii)Rs 900, (iii) 2.5% Quadratic Equations in One Variable 22) a x b + b x a =2 Ans. : a + b, GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 a+b 2 23) x (x 7) = 3 2 24) 1 + 1 = + Ans. : Ans. : + , 2, 9 2 + Solve up to 2 Decimal Places 25)x2 10x + 6 = 0 Ans. : 9.36, 0.64 Nature of roots of a quadratic equation: Equation ax2 + bx + c = 0 a 0 i) 2 distinct real roots if b2 4ac > 0 ii) 2 equal real roots if b2 4ac = 0 iii) No equal roots if b2 4ac < 0 26)Find the nature of roots of the following equations without solving: (i) 5x2 - 8x 12 = 0 (ii) 3x + (iii) 2x2 - 9x + 13 = 0 (iv) 7x2 - 7x 42 = 0 Ans: (i) irrational and unequal (ii) real and equal (iii) imaginary (iv) rational and unequal Equation has real & equal roots: 27) (3m + 1)x2 + 2(m + 1)x + m = 0 Ans. : 1, 1 28) Find the values of p for which the quadratic equation ( + 1) 2 6( + 1) + 3( + 9) = 0 Ans. : p = 3 p -1 has equal roots. Hence find the roots of the equation. 29) If x = 2 is one root of the equation (k - 3)x2 kx 8 = 0 ,find the value of k. also, find the other root of the equation. Ans: 10, - GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 30)A shopkeeper buys a certain number of books for Rs720. If the cost per book was Rs 5 less, the number of books that would be bought for Rs720 would be 2 more. Taking the original cost of each book to be Rs x, write the equation in x and solve. Hint: 720 x 5 720 x =2 Ans. : x = 45 31)A takes 10 days less than the time taken by B to finish a piece of work. If both A and B together can finish the work in 12 days, find the time taken by B to finish the work. Ans. :30 days 32) The speed of an express train is x km/hr. and the speed of ordinary train is 12 km/hr. less than of the express train. If ordinary train takes one hr. longer than the express train to cover a distance of 240 km, find the speed of the express train. Hint: Speed of ordinary train = (x 12) km/hr. Time taken by the express train to cover 240 km = 240 x hrs. Time taken by ordinary train = hrs. A.T.P. 240 x 12 240 x =1 Ans. : x = 60 33) In a two digit number, the unit s digit exceed its ten s digit by 2 and the product of given number and the sum of its digit is equal to 144. Find the number. Hint: Let 10 s digit be x, units digit = x + 2 10x + y 10x + x + 2 = 11x + 2 Sum of digits = x + x + 2 = 2x + 2 A.T.P. (11x + 2) (2x + 2) = 144 Ans. : x = 2 Required sum 11x + 2 = 11 2 + 2 = 24 34)The sum of the numerator and denominator of a fraction is 8. If 1 is added to both the numerator and denominator, the fraction is increased by Hint: Deno = x Fraction 1 . Find the Fraction. 15 Num = 8 x 8 x 8 x+1 x x+1 8 x x = 1 15 Ans. : x = 5, Fraction 3 5 35)The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was 34. Find the ages of the son and the father. Hint: Father s age = x Son s age = 45 x GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Father s age 5 years ago = x 5 Son s age = 45 x 5 = 40 x Ans. : Father s age = 39 years Son s age = 6 years 36) in an auditorium seats were arranged In rows and columns. The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduces by 10, the total number of seats increased by 300. Find: (i) The number of rows in the original arrangement. (ii) The number of seats in the auditorium after re arrangement Hint: ( ) = + Ans. (i) = (ii) x = 1200 Factorization Tips : When without division: x 1 f(x) = 2x3 3x2 + 7x 8 x = 1 b(1) = 2(1)3 3(1)2 + 7(1) 8 =2 3+7 8 =9 3 8 = 2 When we are told to factorise completely: Then we will have to do long division. After getting the answer we have to do middle term. 37)What number should be added to 2x3 3x2 8x so that the resulting polynomial leaves the remainder 10 when divided by 2x + 1? Ans. : 7 38) When the two polynomials x3 - px2 + x + 6 and 2x3 - x2 - (p + 3)x - 6 are divided by (x - 3), the remainder is same. find the value of p. Ans: p = 1 39)Given that x + 2 and x + 3 are factors of 2x3 + ax2 + 7x b. Determine values of a and b. Ans. : a = 9, b = 6 40)If (x + a) is a common factor of expression ( ) = + + ( ) = + + ; show that : a= GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 41)If x 2 is a factor of 2 + + and a + b =1, find the values of a and b. Ans. : = , = 42) Given that (x + 2) is a factor of (3x + 4)3 - (5x + a)3 , find the value of a. Ans: a = 8 43) )Using the remainder theorem find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2.hence, find k if the sum of two remainders is 1. Ans = -2 44)Use factor theorem to factorise completely. x3 + x2 4x 4 Ans. : a) (x 2) (x + 1) (x + 2) 45) Show that 2x + 7 is a factor of 2x3 + 7x2 4x 14. Hence factorise 2x3 + 7x2 4x 14. Ans. : Yes, (2x + 7) (x + ) (x ) Ratio and Proportion FACTS : Compounded ratio of a : b and c : d is ac : bd Duplicate ratio of a : b = a2 : b2 Triplicate ratio of a : b = a3 : b3 Sub-duplicate ratio of a : b = a : b Sub- triplicate ratio of a : b = a : b Reciprocal ratio of a : b is b : a 3 a b 3 (1st term is antecedent) 2nd term is consequent) PROPORTION In a, b, c, d Fourth proportion is d first proportion is a Third proportion is c Mean Proportion is a b = b c b2 = ac. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Invertendo : a : b :: c : d b : a :: d : c Alternendo : Componendo : Dividendo : a : c :: b : d a+b b a b b Or c+d :: :: d c d d c d d b :: a b Componendo and dividend : a Componendo and dividend : If a b c = d = e then each ratio f a+b a : b :: c : d (a b) a+c+e b+d+f = :: a b :: c+d c d c (c d) sum of antecedents sum of consequents SUMS 46)Find the : i) Duplicate ratio of 7 : 9 ii) Triplicate ratio of 5 : 4 iii) Sub-duplicate ratio of 64 : 121 iv) Sub-triplicate ratio of 125 : 343 v) Reciprocal ratio of 9 : 13 Ans. : (i) 49 : 81 (ii) 125 : 64 (iii) 8 : 11 (iv) 5 : 7 (v) 13 : 9 47)Find compounded ratio of i) 5 : 6 and 9 : 10 Ans. : (i) 3 : 4 48) find the value of (i) x : y (ii) Ans: (i) 5:3 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 49)Use properties of proportion to solve for x: Ans: x=3 50)If x = 2mab a+b 51) If x = x+ ma find the value of x ma + x + mb Ans. x -mb 2 Ans -2 , find the value of 52) If a,b,c are in continued proportion, prove that (i) (a + b + c) (a b + c) = a2 + b2 + c2 +2 + a 2b 53)If x = a +2b a 2b , using properties of proportion show that bx2 ax + b = 0 CHAPTER 8 Matrices FACTS : 1) Order of matrix = Rows Columns 2) Equal Matrix A = B 3) 4) 5) 6) [ p a b ] = [ r c d q ] s [ a+p c+r b+q ] d+s [ a p c r b q ] d s [ p a b ] [ r c d Addition of Matrix Subtraction Multiplication a A=[ c b ] d a A = [ b ap + br q ] = [ s cp + dr aq + bs ] cq + ds c ] d GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 A = AT (Transpose) Row will be column and column will be Row. 7) Diagonal Matrix [ 8) Unit or Identity Matrix [ 9) 2 0 0 [0 8 0 ] 0 0 4 5 0 ] 0 9 1 0 0 1 0 ] Or [0 1 0] 0 1 0 0 1 Zero or Null Matrix 0 0 [ 0, 0], [ ] or [ 0 0 ORDER [1 2] 2 ] 3 5 54)If x + y = [ 1 1 ] 3 Ans. : [ Hint [ 3 4 ] 2 5 5 0 ] 0 5 show that 6x x2 = 9 I where I is unit matrix 2 ] find value of A , Hence Find A A 4 1 3 2 1 ] [ 4 2 Ans. : A = [ 58) Given [ 2 6 y= [ ] 3 2 evaluate A2 4A. 1 ] 2 4 1 1 57)If A = [ 3 [2 2] 1 10 x y= [ ] find x and y 5 7 Ans. : x = [ 56)If x = [ [2 1] A = A = I A = ***** ***** 1 55)If A = [ 8 0 ] 0 3 ] 4 1 3 ] 2 4 [ 0 1 ] 1 0 7 2 1 ] = [ ] , : 3 4 6 (i) The order of the matrix x (ii) the matrix x GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 (ii) [ ] Ans. (i) 2 59) construct a matrix 2x2 aij = 2i-j 60)Let Q= and R = . find the matrix P if PQ = R. Ans: Arithmetic Progressions / G.P If an AP is 11, 15, 19, 23, then first term is a = 11, d = difference 15 11 = 4 or 19 15 = 4. It will be always same. Formula: an = a + (n 1) d, where an is the general term. When last term is given L = a + (m 1) d m is the terms. Numbers in an A.P. Always difference is equal b a = c b * 2b = a + c Three numbers are taken as a d, a, a + d. Four numbers are taken as a 3d, a d, a + d, a + 3d. 61) The nth term of sequence is (2n - 3), find its fifteenth term Ans. : 27 62) Find the value of p, if x, 2x + p and 3x + 6 are in AP. Ans. : p=3 63)Find the sum of 28 terms of an A.P. whose nth term is 8n 5. Ans . 3108 64) Which term of the AP: 23,44,65,86, ., is 212? Ans: 10 65)If Sn denotes the sum of first n terms of an AP; prove that S12 = 3(S8 S4) 66)Find the sum of integers between 100 and 200 that are divisible by 9 Ans. : 1683 67)The sum of first 7 terms of an AP is 49 and that of first 17 terms of it is 289. Find the sum of first n terms GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Ans. : a = 1, d = 2 , term is -17 and the sum of n terms is 132. Find n and the common difference. Ans. : = , = 68)The sum of n natural numbers is 5n2+ 4n. Find its 8th term Ans. : 79 69) Solve the equation: 4 + ( 1) + 2+. . . + = 437 Ans. : x = 50 70) In an AP,18th term is 48 and 32nd term is 104. find (i) the first term and the common difference. (ii) the sum of first 50 terms. Ans: (i) -20 (ii) 3900 71) Find three numbers in A.P. whose sum is 24 and whose product is 440. Ans 5,8,11 or 11,8,5 . find the 25th term. 72) The sum of first n terms of an AP is Ans: 76 G. P. FACTS General term an = arn 1 a = is first term r = ratio 2nd term / 1st term n = number of terms. L = arn 1 Three numbers a, b, c are in G. P. = * b2 = ac If the product of the numbers is given, then is G. P. a Three numbers are taken as Four numbers are taken as Five numbers are taken as as r a r3 , a, ar. a , , ar, ar3. r a r2 a , , a, ar, ar2. r GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 1 1 nth term from the end of given G. P. = l( ) r SUMS 73)Find the 20th term and the nth term of the list of number Ans. : , 5 5 5 , , 2 4 8 , .. 74) Find the value of x such that , + 3, + 9 . . Ans. : 3 75) The 5th term ,8th and 11th terms of G.P. are p,q and s respectively. Show that q2 = ps CO-ORDINATE GEOMETRY Reflection HINTS : Co-ordinates of the origin O are Ans. : (0, 0) Co-ordinates of a point on x axis are Ans. : (x, 0) Co-ordinates of a point on y axis are Ans. : (0, y) Co-ordinates (x, y) on invariant means no change The line x = a is parallel to y axis at a distance a from it The line y = b is parallel to x axis at a distance b from it. Reflection of p(x, y) : a) On x axis is p (x, y) b) On y axis is p ( x, y) c) On origin is p ( x, y) d) y = a is p (x, y + 2a) e) x = a is p ( x + 2a, y) 76)Plot A(2,3) and B(6,3) (i) Reflect A in the origin to get the image D. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 (ii) Reflect A in the x-axis to get the image C. (iii)write the coordinates of C and D. (iv)What kind of figure is ABCD? find its area. (v) What is the reflection of C in the y-axis? (vi) Name two points from the figure which are invariant on reflection in y-axis. Ans: (iii) C(2,-3),D(-2,-3) (IV) parallelogram, 24 sq.units v) D(-2,-3) (vi) (0,0),(0,-3) 77)Plot P(2,4), Q(-2,1),R(5,0) . Reflect points P and Q in x-axis to get P and Q . (i) Write the coordinates. (ii) Give a geometrical name to the figure formed by joining the points PQQ P R. Find its area and perimeter. (iii)Name its axis of symmetry and write its equation. (iv) Name two points from the figure which are invariant on reflection in x-axis. Ans: (ii) pentagon, 32 sq.units,22 iii) x axis, y=0 iv)(5,0), (-2,0) Section Formula Section Formula : The co-ordinates of the point which divides internally, the line segment joining the points P(x1 y1) and Q(x2 y2) in ratio m1 : m2 are Mid-point formula P(x1 y1) Q(x2 y2) = x1 + x2 2 , m1 x2 + m2 x1 m1 +m2 , m1 y 2 + m2 y 1 m1 + m2 y1 + y2 2 In unknown ratio we will take k : 1 Centroid Formula : x1 + x2 +x3 y1 + y2 +y3 , 3 3 78)In the given figure, M(3,-2) is the midpoint of AB. if A and B are on x-axis and y-axis . find the coordinates of A and B. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Ans: A(6,0) , B (0,-4) 79) Find the co-ordinates of the centroid of a triangle whose vertices are A( 1, 3) B(1, 1) C(5, 1). Ans. : ( , 1) 80) In the given figure, the line segment AB meets X-axis at A and Y-axis at B. The point P(-3,4) on AB divides it in the ratio 2:3. Find the coordinates of A and B. Ans: A(-5,0), B(0,10) 81) P divides the line segment joining A(1,-6) and B(6,4) in the ratio 2:3. find the coordinates of P. Ans: (3,-2) 82)Find the ratio in which C(p, 1) divides the join of A( 4, 4) and B(6, 1). Hence find the value of p. Ans. : 3 : 2; p = 2 83)If the line joining the points A(4, 5) and B(4, 5) is divided by the point P such that AP AB 2 5 = find the co- ordinates of P. HINT. AP AB = 2 5 5AP = 2AB 5AP = 2 (AP + PB) AP : PB = 2 : 3 84) The line segment joining the points (2, 1) and (5, 8) is trisected at the points P and Q. If the point P lies on the line 2x y + k = 0, find value of k. Ans. : k = 8 Equation of a Straight Line Slope m = tan Slope of a line through points A(x1 y1) & B(x2 y2) will be m = y2 y1 x2 x1 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 y intercept is denoted by c. Equation of a line parallel to x-axis is y = b Equation of x-axis y = 0 Equation of a line parallel to y-axis is x = a. Equation of the y-axis is x = 0. Equation of a line with slope m and y-intercept c is y = mx + c Equation of a line through the point A(x1y1) with slope m is y y1 = m(x x1). Equation of a line passing through two points A(x1y1) & B(x2y2) is y y1 = y 2 y1 x2 x1 (x x1) Condition of parallel m1 = m2 condition of perpendicular m1m2 = 1 Reflection of the point P(x1 y1) in line B = x is the point P(y, x). SUMS 85)If A (-3,5), B (2,-1), C(a,11) are collinear ,find the value a. Ans: -8 86)Given A = (k,4), B = (6,-3),C = (1,8), D = (5,-6).find the value of k if (ii) AB CD. (i) AB || CD Ans: (i) 4 (ii) 87)Write down the equation of the line whose gradient is 3 2 and which passes through p, where P divides the line segment joining A( 2, 6) and B(3, 4) are in ratio 2 : 3. Ans. : 3x 2y + 4 = 0 88)The equation of a straight line is 3x 3y 7 = 0, Find (i) Gradient (ii) inclination (iii) y-intercept Ans. : (i) m = 1 (ii) = 450 (iii) c = 89)Points A and B co-ordinates (7, 3) and (1, 9). Find (i) Slope of AB. (ii) The equation of the perpendicular bisector of the line segment AB. (iii) The value of P if ( 2, P) lies on it. Ans. : (i) 2 (ii) x 2y + 2 = 0 (iii) 0 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 90)A(2, 4) B(3, 3) C( 1, 5) are the vertices of triangle ABC. Find equation of : (i) The median of the triangle through A. (ii) The altitude of the triangle through in B. Ans. : (i) 8x + y 12 = 0 (ii) x 3y + 6 = 0 91)Point A(a, 0) B(0, b) C(x, y) are collinear prove using slope : x a + y b =1 Hint. : AB = BC y 92) A and B are two points on the x axis and y axis respectively, P(2, 3) is the mid-point of AB, Find the : (i) Co-ordinates of A and B (ii) Slope of line AB. (iii) Equation of line AB. x A O B P(2, 3) y ANS- A (4,0) B(0,-6) 3/2, 3x-2y=12 GEOMETRY Similarity 93)In ABC, PQ || BC and AP : PB = 2:3.If AQ = x + 1 QC = x + 5 ,find the value of x. Ans: 7 94)In the given figure, ABC and AMP are right angled at B and M. given AC = 10 cm, AP = 15 cm and PM = 12 cm (i) prove that ABC ~ AMP. (ii) find AB and BC. Ans: 6 cm and 8 cm 95)In ABC, PQ || BC. IF AP : PB=2:3 ,find i) the length of PQ , IF BC = 7.5 cm. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 x Ans: 3 cm 96)AB and ED are perpendiculars to BD. AE meets BD at C. if AB = 16 cm, BC = 12 cm, CD = 3 cm, find: the lengths of DE and CE. Ans: 4 cm, 5 cm 97)In the given figure, AB = 9 cm ,BC = 12 cm and AC = 15 cm . BP AC. (i) what is the measure of ABC? (ii) Prove that APB ~ ABC. (iii) Find the lengths of BP and AP. Ans : 90 , 7.2 cm, 5.4 cm 98)In the given figure, AB || CD || EF , AB = 5 cm, AC = 4 cm, EF = 7.5 cm , CF = x and CD = y. (i) prove that FEC ~ ABC. 99) (ii) solve for x and y. Ans: 6 cm , 3 cm GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Ans- EC =4.5 cm AC =8.25 cm 100) Ans- ii) 144 iii)9/4 101) I)2km ii)4km sq iii)5cmsq LOCUS 102) GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 103) 104) GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Circles Property I Double Angle P P B A O O B A AOB = 2 APB Reflex AOB = 2 APB Property II Angles in the same segment of a circle are equal. C D C A ACB = ADB O A DAB = DCB O ADC = ABC B D B B Property III 5 cm 4 cm O BAD + BCD = 1800 C A D ADC + ABC = 1800 The opposite angles of a cyclic quadrilateral inscribed in a circle are supplementary. C Property IV O A The angle in a semi-circle is a right angle. B ACB = 900 105)Find x and y. C x = 600 y= 1200 Hence find the value of 6x 2y B y0 A 1200 O x0 D GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Ans. : x = 1200. 106)In the figure given below O is the centre of the circle. If OAD = 500; find the values of x and y. B Ans. : x = 1300, y= 1000. O y x C 500 D A 107)In the figure AC is a diameter of the circle AB = BC and AED = 1180, Calculate. i) DEC D E (ii) DAB 1180 A C Ans. : (i) DEC = 280 (ii) DAB = 730. B 108)If DAB = 52 , CBD = 28 , find ABD, BCD, BDC. Ans: 38 , 128 , 24 109)ABCD is a cyclic quadrilateral in the circle with centre O. ADC = 50 and AOB = 68 , Find ACB, ABO, BOC. Ans: 34 , 56 , 32 110) Side CB of cyclic quadrilateral BCDE is extended to a point A. ABE = 110 , BCE = 35 CED = 25 .Find DCE AND CMD. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 , Ans: 45 , 60 111)In triangle PQR, PQ = 24 cm, QR = 7 cm and PQR = 900. Find radius of the P inscribed circle. Ans. : x = 3 cm. A O x c Q R 112)In the given figure 1, PA and PB are tangents to the circle, CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of PEC. E A FIG 1 P D Ans. : 30 cm. C B 113)In the circle with centre O, TP and TQ are tangents from T. If PTQ =70 , find POQ, OPQ and TPQ. Ans: 110 , 35 , 55 114)In the circle with centre O, chords AB and CD intersect externally at P and PT is a tangent to the circle at T. If PB = 12 cm, AB = 15 cm and DP = 6 cm , find the lengths of : (i) PT (ii) radius GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Ans: 18 cm , 24 cm 115)Chords AB and CD intersect at P. AP = 3 cm, CD = 10 cm and PB = 8 cm. find the length of CP. Ans: 4 cm or 6 cm 116) Chords BA and CD are extended to meet at P. If AB = 5 cm, AP = 3 cm, DP = 4 cm, find CD. Ans: 2 cm 117) in the given figure , AB is a diameter and AC is a chord of a circle such that angle BAC = 30 . The tangent at C intersects AB produced at D . prove that BC = BD. 0 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 118) Ans x=36 y=48 z= 60 119) Ans- i)105, ii)15,iii)75 120) Ans- i)27ii)153 CONSTRUCTION 121)CONSTRUCT the inscribed circle of a given triangle ABC with BC = 6.4 cm CA = 5.8 cm B = 600. 122)CONSTRUCT a circle with regular hexagon; with 5 cm. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Mensuration FORMULAE 1) Cuboid :V = L b h Area of 4 walls = 2h(l + b) TSA = 2(lb + bh + Lh) 2) Cube : VOL = a3 3) Right Circular Cylinder : V = r2h TSA = 6a2 Diag = 3 a CSA = 2 rh (Lateral) TSA = 2 r (h + r) 4) Right Circular Cone CSA = rL V= 1 3 r Sphere V= 4 3 7) r3 4 r2 TSA or SA 6) h TSA = r (r + L) L2 = r2 + h2 5) L r2h 2 r3 Hemisphere V= CSA = 2 r2 TSA = 3 r2 3 Spherical shell with outer radius & inner radius. V= 4 3 (R3 r3) SUMS 123)The surface area of a solid metallic sphere is 616 cm2. It is melted and recast into smaller spheres of diameter 3.5 cm. How many such spheres can be obtained? Ans. : 64 124)A hemispherical bowl of diameter 7.2 cm is completely filled with chocolate sauce. This sauce is poured into an inverted cone of radius 4.8 cm. Find the height of the cone. 125)The volume of a conical tent is 1232 m3 and area of the base floor is 154 m2. Calculate the : i) Radius. ii) Height of the tent GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 iii) Length of the canvas required to cover conical tent if its width is 2 m. Ans. : (i) r = 7 m (ii) 24 m (iii) 275 m 126)A solid sphere of radius 15 cm is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Find the number of cones recast. Ans. : 270 127) An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85 m and height of the cylindrical part is 50 m. If the diameter of the base is 168 m,find the quantity of canvas required to make the tent. Allow 20% extra for folding and for stitching.Give your answer to the nearest m2. Ans: 60509 m2 128) In the given figure a hemisphere is surmounted by a conical block of wood .The diameter of their bases is 6 cm each and the slant height is 5 cm . Find : (i) height. (ii)the volume. Ans: 4 cm, 94.2 cm3 129) The following figures represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. their common radius is 7 cm. the height of the cylinder and cone are each of 4 cm. find the volume of the solid. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Ans: 1540 cm3 130)A hemispherical and a conical hole is scooped out of a solid wooden cylinder. find the volume of the remaining solid where the measurement are as follows: the height of the solid cylinder is 7 cm, radius of each of hemisphere, cone and cylinder is 3 cm, height of cone is 3 cm. give your answer correct to the nearest whole number . Ans: 113 cm3 Trigonometric Identities A PHYTO HEIGHT PERPENDICULAR AB2 + BC2 = AC2 * SIN PERPEN HYPOTEN SOME PEOPLE HAVE COS BASC HYPOTEN CURLY BROWN HAIR TAN Perpen BASE TURN Permanently BLACK Sin = 1 Cosec Cosec = 1 Sin HYPOTENUSE P SIN = H COS = BASE P Tan = B Tan = C B B H 1 Cot GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 1 1 * Cos = Sec = * Cot = * Sin . Cosec = 1 *Cos . Sec = 1 Sec Cos 1 OR Tan Cot = Tan = Sin Cos Cos Sin *tan . Cot = 1 GOLDEN IDENTITIES : * Sin2 + Cos2 = 1 * 1 + Tan2 = Sec2 * 1 + Cot2 = Cosec2 131) 132) 1 + sin 1 sin = sec + tan tanA + secA 1 tanA secA + 1 = 133)tan2A tan2B = 1 + sinA cosA sin2 A sin2 B cos2 A cos2 B = sec2A sec2B 134)If sin +sin2 =1 then prove that cos2 + cos4 = 1 135) show that (1- + )2 = 2(1+ ) (1 ). 136) tan (1 cot ) + cot (1 tan ) = (1 + sec cosec ) 137)sin4 + cos4 = 1 2 sin2 cos2 138) + cos 1 - cos = cosec + cot Heights and Distances 139)If the length of a shadow cast by a pole is 3 times the length of the pole, find angle of elevation of the sun. Ans. : 300 140) A man on the top of a tower observes a car moving at a uniform speed towards it. if it takes 12 minutes for the angle of depression to change from 30 to 45 , how soon will the car reach the tower? give the answer correct to nearest second. Ans: 16 minutes 24 seconds GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 141)A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 600. When he moves 50 m away from the bank he finds that the angle of elevation is to be 300. Calculate. (i) The width of the river (ii) height of the tree Ans. : (i) 25 m, (ii) 43.3 m 142)The shadow of a vertical tower on a level ground increases by 10 m, when the altitude of the sun changes from 450 to 300. Find the height of the tower, correct to two decimal places. Ans. : 13.66 m 143)The angles of elevation of the top of a tower from two points P & Q at distances a & b respectively, from the base and in the same straight line with it, are complementary. Prove that the height of the tower is ab . Ans. : n = ab 144)From the top of a light house 100 m high the angles of depression of two ships on opposite sides of it are 480 and 360 respectively. Find the distances between the two ships to nearest metre. Ans. : 227.68 m 145)The length of a shadow of a tower standing on a level plane is found to be 2x metres longer when Sun s altitude is 300 then it is 450. Prove that the height of the tower is x ( 3 + 1) m. 146)The angle of elevation of a Jet Plane from a point A on the ground is 600. After a flight of 15 seconds, the angle of elevation changes to 300. If the Jet is flying at a constant height of 1500 3 m. Find the speed of the Jet Plane in km/hr. Ans. : Speed = 720 km/hr. 600 300 1500 3 147) Ans- 228 m (approx) GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 1500 3 Measures of Central Tendency * x Mean ungrouped = n For discrete data Fx * Direct Method * Short cut Method = A + * Step deviation method = A + f f.d. f f.d. f C * Note : The answer will be same of Mean in all cases whether we use any of the 3 methods. * Median ungrouped First arrange in ascending order : n+1 th ( If n is odd; then 2 n+1 th Median = ( ) 2 ) value. If n is even; then Median = * 1 2 n+1 th [( 2 n th ) observation+ (2 + 1) observation] Mode for ungrouped : Which occurs maximum number of times 2, 3, 3, 4, 5, 3, 3, 7, 6 Ans. : = 3 (Since it has occurred maximum times). * Relationship between Mean, Median, Mode Mode = 3(Median) 2 Mean Empirical Formula. * Quartiles : First arrange in ascending order: n+1 th Lower Quartile Q1 = ( n th =( 4 ) 4 ) [n is odd] (When n is even] * Middle Quartile is also known as Median = Q2. * Upper Quartile 3(n+1)th 4 when n is odd. n 3 th when n is even. 4 * Interquartile Range = Q3 Q1 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 Semi Interquartile Range = Q3 Q1 1 2 148)Weight of 50 eggs were recorded as given below : Weight in gm 80-84 85-89 90-94 95-99 100-104 105-109 110-114 5 10 12 12 8 2 1 No. of eggs Ans. : 94 gm 149)Calculate the mean, the median and the mode of the following nos. 3, 1, 5, 6, 3, 4, 5, 3, 7, 2 Ans. : Mean = 3.9, Median = 3.5, Mode = 3 150)The mean of the following frequency distribution is 50 and the sum of all frequencies is 120. Find the values of p and q. Class Intervals Frequency 0-20 20-40 40-60 60-80 80-100 17 p 32 q 19 Ans. : p = 28, q = 24 151)In a class test, the marks scored by 11 students are, 13, 17, 20, 5, 3, 19, 7, 6, 11, 15, 17 find i) Median, (ii) Lower Quartile, (iii) Upper Quartile, (iv) Inter Quartile range Ans. : (i) 13 (ii) 6 (iii) 17 (iv) 11 152)Draw ogive for the following : Monthly Income 600-700 700-800 800-900 900-1000 1000-1100 1100-1200 1200-1300 No. of Employees 40 68 86 120 90 40 26 Calculate : i) The median. ii) % of employees whose income exceeds 1180. iii) The lower and upper quartiles. iv) The inter-quartile range. Ans. : (i) 930 (ii) 7.23% (approx) (iii) 20 (iv) 220. 153) The marks obtained by 100 students in a test are given below: Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90100 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 No. of students 3 7 12 17 23 14 9 6 5 Draw an ogive for the given distribution on a graph sheet. Use a scale of 2 cm= 10 units on both axis. Estimate from the ogive. i)The median ii)The lower quartile iii)The number of students who obtained more than 85% marks in the test. iv)The number of students who did not pass in the test if the pass percentage was 35. Ans: 45,32,6,29 Probability 1) SURE EVENT is 1 2) Impossible EVENT is 0 For eg. What is the probability of getting 7 in a single throw of a die = 3) A pack of playing cards has in all 52 cards : 13 each of (i) Spades (ii) Clubs (iii) Hearts (iv) Diamonds. 4) Face cards are Kings, Queens & Jacks = 12 cards. 5) Honour cards are Ace, Kings, Queens, Jacks = 16 cards 6) There are 26 black cards and 26 are red cards. 0 7 =0 SUMS : 154)Two unbiased coins are tossed simultaneously, find the probability of getting (i) Two heads (ii) One head (iii) At least one head iv)At most one head, (v) No head (ii) 1 2 (iii) 3 4 (iv) 3 4 (v) 1 4 Ans. : (i) 1 4 155)An unbiased die is thrown once. Find the probability of getting: (i) A prime number (ii) An even number (iii) The number 5 (iv) A number greater than 4 (v) A number less than 3 (vi) A number greater between 3 and 6. Ans. : (i) 1 2 (ii) 1 2 (iii) 1 6 (iv) 1 3 (v) 1 3 (vi) 1 3 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 4 156)Find the probability of having 53 Wednesday in a (i) a non-leap year, (ii) a leap year. Ans. : (i) 1 7 (ii) 2 7 157)A bag contains 12 balls out of which x balls are black. (i) If a ball is drawn at random, what is the probability of drawing a black ball? (ii) If 6 more black balls are put in the bag, the probability of drawing a black ball will be double than that of (i). Find the value of x. Ans. : (i) x 12 (ii) x = 3 158)Cards numbered from 2 to 25 are put in a box and mixed thoroughly. One card is drawn at random. Find the probability that the card drawn bears: (i) (ii) (iii) (iv) (v) An odd number Not a prime number A perfect square A number greater than 17 A number divisible by 2 and 3 both Ans. (i) (ii) (iii) (iv) (v) 159) If an office works for 5 days in a week (Monday to Friday) and if two employees of the office remain absent in the same week, what is the probability that it is (i) the same day (ii) different days 1 4 Ans: 5 , 5 160)The probability of selecting a red ball at random from a jar that contains only red, blue and orange 1 1 balls is 4 .the probability of selecting a blue ball at random from the same jar is 3. If the jar contains 10 orange balls find the total number of balls in the jar. Ans. :24 balls 161)Two coins are tossed together 100 times and the results are as follows: No. of heads 0 1 2 frequency 28 52 20 What is the probability of (i) One head (ii) at most 1 head Ans: 162) there are 30 blue balls and x red balls in a bag. A ball is drawn at random from the bag. GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612 (i) write down, in terms of x, an expression for the probability that the ball drawn is red. (ii) given that the probability is , find x. Ans: , 35 GUESS QUESTIONS ICSE 2024 BY -ASHISH SARAF -9903312612

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