Mathematics — Suggested Paper 3
Three hours80 marks Questions from 2017–2026
A suggested paper: every question is a real ICSE board question, chosen from different years and arranged in the current pattern. Start the timed test to have your written answers marked.
Instructions to Candidates: 1. Answers to this Paper must be written on the paper provided separately. 2. You will not be allowed to write during first 15 minutes. 3. This time is to be spent in reading the question paper. 4. The time given at the head of this Paper is the time allowed for writing the answers. 5. Attempt all questions from Section A and any four questions from Section B. 6. All working, including rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. 7. Omission of essential working will result in loss of marks. 8. The intended marks for questions or parts of questions are given in brackets [ ]. 9. Mathematical tables and graph papers are to be provided by the school.
Section A (40 marks)
Attempt all questions from this Section.
1. Choose the correct answers to the questions from the given options. (Do not copy the questions, write the correct answers only.)
1(i). The factor common to the two polynomials x² - 4 and x³ - x² - 4x + 4
[1 mark]- (a)(x + 1)
- (b)(x - 1)
- (c)(x - 2)
- (d)(x - 4)
1(ii). the values of x and y respectively are :
[1 mark]
- (a)1, -2
- (b)-2, 1
- (c)1, 2
- (d)-2, -1
1(iii). The roots of the quadratic equation px² - qx + r = 0 are real and equal if:
[1 mark]- (a)q² = 4pr
- (b)q² = 4pr
- (c)-q² = 4pr
- (d)p² > 4qr
1(iv). In the given diagram ΔABC ∼ ΔEFG. If ∠ABC = ∠EFG = 60°, then the length of the side FG is:
[1 mark]
- (a)15 cm
- (b)20 cm
- (c)25 cm
- (d)30 cm
1(v). If the volume of two spheres is in the ratio 27: 64, then the ratio of their radii is:
[1 mark]- (a)3 : 4
- (b)4 : 3
- (c)9 : 16
- (d)16 : 9
1(vi). Which of the following cannot be determined graphically for a grouped frequency distribution ?
[1 mark]- (a)Median
- (b)Mode
- (c)Quartiles
- (d)Mean
1(vii). The selling price of a shirt excluding GST is 800. lf the rate of GST is 12% then the total price of the shirt is
[1 mark]- (a)704
- (b)96
- (c)896
- (d)848
1(viii). In an Arithmetic Progression (A.P.) if first term is 5, common difference is -3 and the nth term is -7, then n is equal to
[1 mark]- (a)5
- (b)17
- (c)-13
- (d)7
1(ix). Assertion (A): If a die is rolled, the probability of getting a number greater than 6 is 1/6 Reason (R): There are six possible outcomes when rolling a die, {1, 2, 3, 4, 5, 6}.
[1 mark]- (a)(A) is true and (R) is false.
- (b)(A) is false and (R) is true.
- (c)Both (A) and (R) are true and (R) is the correct explanation of (A).
- (d)Both (A) and (R) are true but (R) is not the correct explanation of (A).
1(x). The solution set of the inequation x - 3 ≥ - 5 , x ∈ R is:
[1 mark]- (a){x : x > - 2 , x ∈ R}
- (b){x : x ≤ - 2 , x ∈ R}
- (c){x : x ≥ - 2 , x ∈ R}
- (d){-2, -1, 0, 1, 2}
1(xi). (x + 3), 1, (3x − 7) and −5 are in proportion. The value of x is :
[1 mark]- (a)−1
- (b)1
- (c)−5
- (d)5
1(xii). Naveen deposits ₹ 800 every month in a recurring deposit account for 6 months. If he receives ₹ 4884 at the time of maturity, then the interest he earns is :
[1 mark]- (a)₹ 84
- (b)₹ 42
- (c)₹ 24
- (d)₹ 284
1(xiii). If a, b, c, and d are proportional then (a + b)/(a - b) is equal to
[1 mark]- (a)c/d
- (b)(b - d)/(c + d)
- (c)d/c
- (d)(c + d)/(c - d)
1(xiv). Find the equation of a line whose y-intercept is 6 and is parallel to x-axis.
[1 mark]- (a)y = 6
- (b)x = 6
- (c)x + y = 6
- (d)y − x = 6
1(xv). ABCD is a cyclic quadrilateral. If ∠BAD = (2x+5)° and ∠BCD = (x + 10)° then x is equal to:
[1 mark]
- (a)65°
- (b)45°
- (c)55°
- (d)5°
2(i). Draw a Histogram for the given data, using a graph paper: Estimate the mode from the graph.
[4 marks]
2(ii). AB and CD are two parallel chords of a circle such that AB = 24 cm and CD = 10 cm. If the radius of the circle is 13 cm. find the distance between the two chords.
[4 marks]
2(iii). Find the values of ‘a’ and ‘b’.
[4 marks]
3(i). In what ratio is the line joining P(5, 3) and Q(– 5, 3) divided by the y-axis ? Also find the coordinates of the point of intersection.
[4 marks]3(ii). Jaya borrowed Rs. 50,000 for 2 years. The rates of interest for two successive years are 12% and 15% respectively. She repays 33,000 at the end of the first year. Find the amount she must pay at the end of the second year to clear her debt.
[4 marks]3(iii). In the given graph ABCD is a parallelogram.
3(iii). Using the graph, answer the following:
3(iii)(a). write down the coordinates of A, B, C and D.
[2 marks]3(iii)(b). calculate the coordinates of ‘P’, the point of intersection of the diagonals AC and BD.
[1 mark]3(iii)(c). find the slope of sides CB and DA and verify that they represent parallel lines.
[1 mark]3(iii)(d). find the equation of the diagonal AC.
[1 mark]Section B (40 marks)
Attempt any four questions from this Section.
4(a). In the given figure AC is a tangent to the circle with centre O. If ∠ADB = 55°, find x and y. Give reasons for your answers.
[3 marks]
4(b). The model of a building is constructed with the scale factor 1 : 30.
4(b)(i). If the height of the model is 80 cm, find the actual height of the building in meters.
[1.5 marks]4(b)(ii). If the actual volume of a tank at the top of the building is 27 m³, find the volume of the tank on the top of the model.
[1.5 marks]4(c). Given [[4, 2], [-1, 1]] M = 6I, where M is a matrix and I is unit matrix of order 2×2.
4(c)(i). State the order of matrix M.
[1 mark]4(c)(ii). Find the matrix M.
[3 marks]5(a). The data on the number of patients attending a hospital in a month are given below. Find the average (mean) number of patients attending the hospital in a month by using the shortcut method. Take the assumed mean as 45. Give your answer correct to 2 decimal places.
[3 marks]
5(b). Using properties of proportion solve for x, given: (√(5x) + √(2x - 6)) / (√(5x) - √(2x - 6)) = 4
[3 marks]5(c). Sachin invests ₹ 8500 in 10%, ₹ 100 shares at ₹ 170. He sells the shares when the price of each share rises by ₹ 30. He invests the proceeds in 12%, ₹ 100 shares at ₹ 125. Find:
5(c)(i). the sale proceeds.
[2 marks]5(c)(ii). the number of ₹ 125 shares he buys
[1 mark]5(c)(iii). the change in his annual income.
[1 mark]6(a). ₹7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received ₹100 more. Find the original number of children.
[3 marks]6(b). If the mean of the following distribution is 24, find the value of ‘a’.
[3 marks]
6(c). Using ruler and compass only, construct a ΔABC such that BC = 5 cm and AB = 6.5 cm and ∠ABC = 120°
6(c)(i). Construct a circum-circle of ∆ABC
[2 marks]6(c)(ii). Construct a cyclic quadrilateral ABCD, such that D is equidistant from AB and BC.
[2 marks]7(a). Given matrix B = [[1, 1], [8, 3]], find the matrix X if X = B² - 4B. Hence solve for a and b given X * [[a], [b]] = [[5], [50]].
[3 marks]7(b). How much should a man invest in Rs. 50 shares selling at Rs. 60 to obtain an income of Rs. 450, if the rate of dividend declared is 10%. Also find his yield percent, to the nearest whole number.
[3 marks]7(c). Sixteen cards are labeled as a, b, c, ……………. m, n, o, p. They are put in a box and shuffled. A boy is asked to draw a card from the box. What is the probability that the card drawn is:
7(c)(i). a vowel
[2 marks]7(c)(ii). a consonant
[1 mark]7(c)(iii). none of the letters of the word median
[1 mark]8(i). In the given diagram, an isosceles ∆ABC is inscribed in a circle with centre O. PQ is a tangent to the circle at C. OM is perpendicular to chord AC and ∠COM = 65°. Find :
8(i)(a). ∠ABC
[1 mark]8(i)(b). ∠BAC
[1 mark]8(i)(c). ∠BCQ
[1 mark]8. In the given diagram, an isosceles ∆ABC is inscribed in a circle with centre O. PQ is a tangent to the circle at C. OM is perpendicular to chord AC and ∠COM = 65°. Find :
8(ii). Solve the following inequation, write down the solution set and represent it on the real number line. -3 + x ≤ 7x/2 + 2 < 8 + 2x, x ∈ I.
[3 marks]8(iii). In the given diagram, ABC is a triangle, where B(4, -4) and C(-4, -2). D is a point on AC.
8(iii)(a). Write down the coordinates of A and D.
[1 mark]8(iii)(b). Find the coordinates of the centroid of ∆ABC.
[1 mark]8(iii)(c). If D divides AC in the ratio k : 1, find the value of k.
[1 mark]8(iii)(d). Find the equation of the line BD.
[1 mark]9(i). The following table gives the marks scored by a set of students in an examination. Calculate the mean of the distribution by using the short cut method.
[3 marks]
9(ii). What number must be added to each of the number 4, 6, 8, 11 in order to get the four numbers in proportion ?
[3 marks]9(iii). Using ruler and compass construct a triangle ABC in which AB = 6 cm. ∠BAC = 120° and AC = 5 cm. Construct a circle passing through A, B and C. Measure and write down the radius of the circle.
[4 marks]10(a). Using a ruler and a compass construct a triangle ABC in which AB = 7 cm, ∠CAB = 60° and AC = 5 cm. Construct the locus of
10(a)(i). points equidistant from AB and AC
[1 mark]10(a)(ii). points equidistant from BA and BC Hence construct a circle touching the three sides of the triangle internally.
[2 marks]10(b). A conical tent is to accommodate 77 persons. Each person must have 16 m³ of air to breathe. Given the radius of the tent as 7 m, find the height of the tent and also its curved surface area
[3 marks]10(c). If (7m + 2n) / (7m - 2n) = 5 / 3, use properties of proportion to find
10(c)(i). m : n
[2 marks]10(c)(ii). (m² + n²) / (m² - n²)
[2 marks]