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ICSE Class X Question Bank 2023 : Mathematics : MOCK TEST

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G.S.T 1. A shopkeeper buys certain quantity of cashew nuts for 7200 and sells it to a consumer at a profit of 25%. If the rate of GST is 5%, find the GST paid by the shopkeeper to the Central and State Governments. Ans: CGST = 45 SGST = 45 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 2023 BY -ASHISH SARAF -9903312612 7)Rajiv visits a department stores and buys the following articles: Medicine costing 1250 , GST @ 5%; packet of sweets 450 , GST @ 5%, oil 500, GST @ 18%, a chess board 300 with a discount of 10%, GST @ 12% calculate: (1) the total amount of GST paid (2) the total bill amount including GST paid by Sachin. Ans: (i) 207.40 (ii) 2677.40 Banking THINGS TO REMEMBER Recurring Deposit SI = P n(n + 1) R 2 12 100 M. A. (Maturity Amount) = SI + P No. of months 8) Ranjana opens a recurring deposit scheme and deposits 800 per month for a period of 2 years . if the rate of interest is 9% p.a. find the amount payable at the end of 2 years. Ans: 21000 9) Rita deposits 800 per month in a cumulative deposit account for 3 years. if the amount payable at the time of maturity is 31464; calculate the rate of interest. Ans: 6% 10) 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 11)Peter has a recurring deposit account in Punjab National Bank at Sadar Bazar, Delhi for 4 years at 10% p.a. He will get 6,370 as interest on maturity. Find : (i)installment (ii) the maturity value of the account. RS 650 ,RS 37570 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 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, ..} 12)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} 13)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} 14)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} Quadratic Equations in One Variable 15) x x+1 + x+1 =2 1 12 Ans. : 3, 4 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 16) a x b + b x a =2 Ans. : a + b, 17) x (x 7) = 3 2 18) 1 + 1 = + 19) Solve the equation: x- 2 Ans. : 2, 9 Ans. : 18 a+b + , 2 + =6. Give your answer correct to two significant figures. Ans: 8.2,-2.2 Solve up to 2 Decimal Places 20)x2 10x + 6 = 0 Ans. : 9.36, 0.64 21)2(x 1) (x 5) = 5 Ans. : 5.55, 0.45 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 22)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: 23) (3m + 1)x2 + 2(m + 1)x + m = 0 Ans. : 1, 1 24) 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. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 25) 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, 26)300 apples are distributed equally among a certain number of students. Had there been 10 more students, each would have received one apple less. Find the number of students. Ans. : 50 27)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 28) 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 29) 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 30)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 31)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 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 Son s age = 45 x Father s age 5 years ago = x 5 Son s age = 45 x 5 = 40 x Ans. : Ans. : Son s age Father s age = 39 years = 6 years 32) 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. 33)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 34) 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 35)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 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 36)If (x + a) is a common factor of expression ( ) = + + ( ) = + + ; show that : a= 37)If x 2 is a factor of 2 + + and a + b =1, find the values of a and b. Ans. : = , = 38) Given that (x + 2) is a factor of (3x + 4)3 - (5x + a)3 , find the value of a. Ans: a = 8 39) )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 40)Use factor theorem to factorise completely. a) x3 + x2 4x 4 Ans. : a) (x 2) (x + 1) (x + 2) 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 = 3 a : b Reciprocal ratio of a : b is b : a 3 a b (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 2023 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 41)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 42)Find compounded ratio of i) 5 : 6 and 9 : 10 Ans. : (i) 3 : 4 43) find the value of (i) x : y (ii) Ans: (i) 5:3 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 44)Use properties of proportion to solve for x: Ans: x=3 45)If x = 2mab a+b find the value of 3 x+ ma x ma + x + mb Ans. x -mb 2 3 a + 1 + a 1 46) If x = 3 prove that : x3 3ax2 + 3x a = 0 3 a + 1 a 1 47) If x = Ans -2 , find the value of 48) If a,b,c are in continued proportion, prove that (i) (a + b + c) (a b + c) = a2 + b2 + c2 +2 + a 2b 49)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 [a c 3) 4) 5) p b] = [ r d q ] s Addition of Matrix a+p [ c+r b+q ] d+s a p [ c r b q ] d s Subtraction Multiplication [a c b] [p r d ap + br q ] = [ s cp + dr aq + bs ] cq + ds GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 6) a A=[ c b] d A = [ a c ] b d A = AT (Transpose) Row will be column and column will be Row. 7) Diagonal Matrix [5 0 8) 0 0 8 0] 0 4 1 0] Or [0 1 0 0 0 1 0] 0 1 Unit or Identity Matrix [1 0 9) 2 [0 0 0] 9 Zero or Null Matrix 0 0 [ 0, 0], [ ] or [ 0 0 ORDER [1 2] [2 1] 5 2 ] 1 3 x y= [ 1 10 ] find x and y 5 7 3 4] Ans. : x = [ 2 5 51)If A = [ 1 8 1 ] 3 1 ] 2 4 52)If x = [ 1 1 3 2 y= [ 3 6 ] 2 evaluate A2 4A. 5 Ans. : [ 0 53)If A = [ [2 2] A = A = I A = ***** ***** 50)If x + y = [ 0] 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 Hint [ 3 2 1 ] [ 4 2 1 Ans. : A = [ 2 3] 4 3 ] 4 [0 1] 1 0 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 54) Given [ 2 1 7 ] = [ ] , : 3 4 6 (i) The order of the matrix x (ii) the matrix x Ans. (i) 2 (ii) [ ] 55) construct a matrix 2x2 aij = 2i-j 56) Let Q= and R = . find the matrix P if PQ = R. Ans: Arithmetic Progressions 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. 57)If m times the m th term of an AP is equal to n times its n th term; then find its (m + n)th term. Ans. : (m + n)th = 0 58) The nth term of sequence is (2n - 3), find its fifteenth term Ans. : 27 59) Find the value of p, if x, 2x + p and 3x + 6 are in AP. Ans. : p=3 60)Find the sum of 28 terms of an A.P. whose nth term is 8n 5. Ans . 3108 61) Which term of the AP: 23,44,65,86, ., is 212? Ans: 10 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 62)The ratio of the 11th term to the 18th term of an A.P is 2 : 3. Find the ratio of the 5th term to 21st term and also the ratio of the sum of first five terms to the sum of first 21 terms. Ans. : , 63)If Sn denotes the sum of first n terms of an AP; prove that S12 = 3(S8 S4) 64)Find the sum of integers between 100 and 200 that are divisible by 9 Ans. : 1683 65)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 Ans. : a = 1, d = 2 , 66) in an AP, the first term is 25, nth term is -17 and the sum of n terms is 132. Find n and the common difference. Ans. : = , = 67)The sum of n natural numbers is 5n2+ 4n. Find its 8th term Ans. : 79 68) Solve the equation: 4 + ( 1) + 2+. . . + = 437 Ans. : x = 50 69) 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 70) Find three numbers in A.P. whose sum is 24 and whose product is 440. Ans 5,8,11 or 11,8,5 71) The sum of first n terms of an AP is . find the 25th term. Ans: 76 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 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) 72)Plot A(2,3) and B(6,3) (i) Reflect A in the origin to get the image D. (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) 73)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. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 (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 , m 1 x2 + m 2 x1 m1 +m2 , m1 y2 + m2 y1 m1 + m2 y1 + y2 2 In unknown ratio we will take k : 1 Centroid Formula : x1 + x2 +x3 y1 + y2+y3 3 , 3 74)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. Ans: A(6,0) , B (0,-4) 75) Find the co-ordinates of the centroid of a triangle whose vertices are A( 1, 3) B(1, 1) C(5, 1). Ans. : ( , 1) 76) 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) 77) 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) GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 78)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 AP 2 79)If the line joining the points A(4, 5) and B(4, 5) is divided by the point P such that AB = 5 find the coordinates of P. HINT. AP AB = 2 5 5AP = 2AB 5AP = 2 (AP + PB) AP : PB = 2 : 3 80) 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 = 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(x 1y1) & B(x2y2) is y y1 = y2 y1 x2 x1 y2 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 81)If A (-3,5), B (2,-1), C(a,11) are collinear ,find the value a. Ans: -8 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 82)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) 83)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 84)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 = 85)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 86)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 87)Point A(a, 0) B(0, b) C(x, y) are collinear prove using slope : x a + y b =1 Hint. : AB = BC 88) A and B are two points on the x axis and y axis respectively, P(2, 3) is the mid-point of AB, Find y the : (i) Co-ordinates of A and B (ii) Slope of line AB. (iii) Equation of line AB. x A O P(2, 3) B ANS- A (4,0) B(0,-6) 3/2, 3x-2y=12 y GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 x GEOMETRY Similarity 89)In ABC, PQ || BC and AP : PB = 2:3.If AQ = x + 1 QC = x + 5 ,find the value of x. Ans: 7 90)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 91)In ABC, PQ || BC. IF AP : PB=2:3 ,find i) the length of PQ , IF BC = 7.5 cm. Ans: 3 cm 92)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 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 93)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 94)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. (ii) solve for x and y. Ans: 6 cm , 3 cm GUESS QUESTIONS ICSE 2023 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 95)Find x and y. C x = 600 B y = 120 0 Hence find the value of 6x 2y y0 A 1200 O x0 D GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 Ans. : x = 1200. 96)In the figure given below O is the centre of the circle. If OAD = 500; find the values of x and y. B 0 0 Ans. : x = 130 , y = 100 . O y x C 500 D A 97)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 98)If DAB = 52 , CBD = 28 , find ABD, BCD, BDC. Ans: 38 , 128 , 24 99)ABCD is a cyclic quadrilateral in the circle with centre O. ADC = 50 and AOB = 68 , Find ACB, ABO, BOC. Ans: 34 , 56 , 32 100) 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 2023 BY -ASHISH SARAF -9903312612 , Ans: 45 , 60 P 101)In triangle PQR, PQ = 24 cm, QR = 7 cm and PQR = 900. Find radius of the inscribed circle. A O Ans. : x = 3 cm. x c Q R 102)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. FIG 1 E A P D Ans. : 30 cm. C B 103)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 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 104)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 Ans: 18 cm , 24 cm 105)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 106) 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 107) in the given figure , AB is a diameter and AC is a chord of a circle such that angle BAC = 300. The tangent at C intersects AB produced at D . prove that BC = BD. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 CONSTRUCTION 108)CONSTRUCT the inscribed circle of a given triangle ABC with BC = 6.4 cm CA = 5.8 cm B = 600. HINT : 109)CONSTRUCT a circle with regular hexagon; with 5 cm. HINT : CONSTRUCT a hexagon side = 5 cm, ABCDEF draw A & B bisectors such that they meet O & From O draw ON perpendicular to ED. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 CHAPTER 16 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 7) h r Sphere V= 4 3 r3 4 r2 TSA or SA 6) r h TSA = r (r + L) L2 = r 2 + h 2 5) L 2 2 r3 Hemisphere V= CSA = 2 r2 TSA = 3 r2 3 Spherical shell with outer radius & inner radius. V= 4 3 (R3 r3) SUMS 110)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. : 64 111)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. 112)The volume of a conical tent is 1232 m3 and area of the base floor is 154 m2. Calculate the : i) Radius. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 ii) Height of the tent 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 113)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. : 270 114) 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 115) 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 116) 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 2023 BY -ASHISH SARAF -9903312612 Ans: 1540 cm3 117)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 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 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 1 * Sin = Cosec = * Cos = * Cot = * Sin . Cosec = 1 *Cos . Sec = 1 Cosec 1 Sec = Sec 1 OR Tan 1 Sin 1 Cos Cot = HYPOTENUSE P SIN = H B Tan = BASE P B Tan = Tan = 1 Cot Sin Cos Cos Sin *tan . Cot = 1 GOLDEN IDENTITIES : * Sin2 + Cos2 = 1 * 1 + Tan2 = Sec2 * 1 + Cot2 = Cosec2 118) 119) 1 + sin 1 sin = sec + tan tanA + secA 1 tanA secA + 1 = 1 + sinA cosA 2 120)tan A tan B = 2 2 2 sin A sin B cos2 A cos2 B = sec2A sec2B 121)If sin +sin2 =1 then prove that cos2 + cos4 = 1 122) show that (1- + )2 = 2(1+ ) (1 ). 123) tan (1 cot ) + cot (1 tan ) C B COS = H = (1 + sec cosec ) GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 124)sin4 + cos4 = 1 2 sin2 cos2 125) + cos 1 - cos = cosec + cot Heights and Distances 126)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. : 300 127) 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 128)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 129)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 130)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 131)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 132)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. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 133)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 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+ ( + 1) observation] 2 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: Lower Quartile Q1 = ( =( n th 4 ) n+1 th 4 ) [n is odd] (When n is even] GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 * 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 Semi Interquartile Range = Q3 Q1 1 2 134)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 135)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 = 3 136)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 = 24 137)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) 11 138)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. GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 Ans. : (i) 930 (ii) 7.23% (approx) (iii) 20 (iv) 220. 139) 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 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. SUMS : 140)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 Ans. : (i) 1 4 (ii) 1 2 (iii) 3 4 (iv) 3 4 (v) 1 4 141)An unbiased die is thrown once. Find the probability of getting: (i) A prime number (ii) An even number GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 0 7 =0 4 (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 3 142)Anushka and Virat are friends. They both were born is 1975. What is the probability that they have (i) Same birthdays (ii) Different birthdays. Ans. : (i) 1 365 (ii) 364 365 143)Find the probability of having 53 Wednesday in a (i) a non-leap year, (ii) a leap year. Ans. : (i) 1 7 (ii) 2 7 144)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 = 3 145)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) 146) 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 , 5 147)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 GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 148)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: 149) There are 30 blue balls and x red balls in a bag. A ball is drawn at random from the bag. (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: , 35 150) A card is drawn from a well shuffled pack of 52 cards. Find the probability that the card drawn is: (i) a spade (ii) a red card (iii) a face card (iv) 5 of heart or diamond Ans. : (i) ( ) ( ) ( ) (v) Jack or queen (vi) ace and king (vii) a red and a king (viii) a red or a king ( ) ( ) ( ) GUESS QUESTIONS ICSE 2023 BY -ASHISH SARAF -9903312612 ( )

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