Mathematics — Suggested Paper 4
Three hours80 marks Questions from 2016–2026
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). What must be added to x³ + 7x² + 3x + 2 so that the result is completely divisible by (x + 2)?
[1 mark]- (a)−40
- (b)−16
- (c)16
- (d)40
1(ii). Question 3 The product AB of two matrices A and B is possible if:
[1 mark]- (a)A and B have the same number of rows.
- (b)The number of columns of A is equal to the number of rows of B.
- (c)The number of rows of A is equal to the number of columns of B.
- (d)A and B have the same number of columns
1(iii). If 3 is a root of the quadratic equation x² - px + 3 = 0 then p is equal to :
[1 mark]- (a)4
- (b)3
- (c)5
- (d)2
1(iv). In the given diagram, ∆ABC ∼ ∆PQR. If AD and PS are bisectors of ∠BAC and ∠QPR respectively then:
[1 mark]
- (a)∆ABC ∼ ∆PQS
- (b)∆ABD ∼ ∆PQS
- (c)∆ABD ∼ ∆PSR
- (d)∆ABC ∼ ∆PSR
1(v). The total surface area of a solid sphere (S1) and a solid hemisphere (S2), as shown in the diagram, are equal. The ratio of radii R and r is :
[1 mark]
- (a)1 : 1
- (b)2 : 1
- (c)√3 : 2
- (d)2 : √3
1(vi). Assertion (A): The mean of first 9 natural numbers is 4.5. Statement 2: Mean = Sum of all observations / Total number of observations
[1 mark]- (a)(A) is true, (R) is false.
- (b)(A) is false, (R) is true.
- (c)Both (A) and (R) are true, and (R) is the correct reason for (A).
- (d)Both (A) and (R) are true, and (R) is the incorrect reason for (A).
1(vii). The first four terms of an Arithmetic Progression (A. P.) whose first term is 4 and common difference is -6, are
[1 mark]- (a)4, -10, -16, -22
- (b)4, 10, 16, 22
- (c)4, -2, -8, -14
- (d)4, 2, 8, 14
1(viii). The probability of getting a number divisible by 3 in throwing a dice is:
[1 mark]- (a)1/6
- (b)1/3
- (c)1/2
- (d)2/3
1(ix). A(1, 4), B (4, 1) and C (x, 4) are the vertices of ΔABC If the centroid of the triangle is G (4, 3) then x is equal to
[1 mark]- (a)2
- (b)1
- (c)7
- (d)4
1(x). In the given diagram, chords AC and BC are equal. If ∠ACD = 120°, then ∠AEC is:
[1 mark]
- (a)30°
- (b)60°
- (c)90°
- (d)120°
1(xi). The marked price of an article is ₹1,375. If the CGST is charged at a rate of 4%, then the price of the article including GST is:
[1 mark]- (a)₹55
- (b)₹110
- (c)₹1,430
- (d)₹1,485
1(xii). Statement (i) : sin² θ + cos² θ = 1 Statement (ii) : cosec² θ + cot² θ = 1 Which of the following is valid ?
[1 mark]- (a)only (i)
- (b)only (ii)
- (c)both (i) and (ii)
- (d)neither (i) nor (ii)
1(xiii). If x, 5.4, 5, 9 are in proportion then x is
[1 mark]- (a)3
- (b)9.72
- (c)25
- (d)25/3
1(xiv). Mohit opened a Recurring deposit account in a bank for 2 years. He deposits ₹1000 every month and receives ₹25500 on maturity. The interest he earned in 2 years is
[1 mark]- (a)₹13500
- (b)₹3000
- (c)₹24000
- (d)₹1500
1(xv). The median class for the given distribution is:
[1 mark]
- (a)0-10
- (b)10-20
- (c)20-30
- (d)30-40
2(i). In the given figure O, is the centre of the circle. CE is a tangent to the circle at A. If ∠ABD = 26° , then find:
2(i)(a). ∠BDA
[1 mark]2(i)(b). ∠BAD
[1 mark]2(i)(c). ∠CAD
[1 mark]2(i)(d). ∠ODB
[1 mark]2(ii). Prove the following identity : (sin² θ - 1)(tan² θ + 1) + 1 = 0
[4 marks]2(iii). Use graph paper for this question. (Take 2 cm = 1 unit along both x and y axis.) Plot the points O(0, 0), A(-4, 4), B(-3, 0) and C(0, -3)
2(iii)(i). Reflect points A and B on the y-axis and name them A’ and B’ respectively. Write down their coordinates.
[2 marks]2(iii)(ii). Name the figure OABCB’A’
[1 mark]2(iii)(iii). State the line of symmetry of this figure
[1 mark]3(i). A solid metallic cylinder is cut into two identical halves along its height. The diameter of the cylinder is 7 cm and the height is 10 cm. Find :
3(i)(a). The total surface area (both the halves).
[2 marks]3(i)(b). Calculate the total cost of painting the two halves at the rate of ₹ 30 per cm². (use π = 22/7)
[2 marks]3(ii). Solve the following inequation and represent the solution set on a number line.
[4 marks]
3(iii). Use graph sheet for this question.
3(iii)(a). Plot A(0, 3), B(2, 1) and C(4, -1).
[1 mark]3(iii)(b). Reflect point B and C in y-axis and name their images as B' and C' respectively. Plot and write coordinates of the points B' and C'.
[1 mark]3(iii)(c). Reflect point A in the line BB' and name its images as A'.
[1 mark]3(iii)(d). Plot and write coordinates of point A'.
[1 mark]3(iii)(e). Join the points ABA'B' and give the geometrical name of the closed figure so formed.
[1 mark]Section B (40 marks)
Attempt any four questions from this Section.
4(i). Kabir bought 120 shares of a company with nominal value ₹100, available at a premium of ₹25. Find :
4(i)(a). the money invested by Kabir in buying these shares.
[1 mark]4(i)(b). the rate of dividend, if he received ₹ 1,080 as dividend from these shares after one year.
[1 mark]4(i)(c). his rate of return.
[1 mark]4(ii). Find the mean of the following frequency distribution using step-deviation method. Take assumed mean = 28
[3 marks]
4(iii). The difference of two natural numbers is 5 and sum of their reciprocals is 3/10 . Find the two numbers.
[4 marks]5(a). A model of a high rise building is made to a scale of 1 : 50.
5(a)(i). If the height of the model is 0.8 m, find the height of the actual building.
[1.5 marks]5(a)(ii). If the floor area of a flat in the building is 20 m2, find the floor area of that in the model.
[1.5 marks]5(b). From a solid wooden cylinder of height 28 cm and diameter 6 cm, two conical cavities are hollowed out. The diameters of the cones are also of 6 cm and height 10.5 cm. Taking π = 22/7 find the volume of the remaining solid.
[3 marks]
5(c). Prove the identity ((1 - tan θ) / (1 - cot θ))² = tan² θ
[4 marks]6(a). If x/a = y/b = z/c, show that (x³ / a³) + (y³ / b³) + (z³ / c³) = 3xyz/abc
[3 marks]6(b). Draw a line AB = 5 cm. Mark a point C on AB such that AC = 3 cm. Using a ruler and a compass only, construct :
6(b)(i). A circle of radius 2.5 cm, passing through A and C.
[2 marks]6(b)(ii). Construct two tangents to the circle from the external point B. Measure and record the length of the tangents.
[2 marks]6(c). A line AB meets X – axis at A and Y –axis at B. P (4, -1) divides AB in the ratio 1 : 2.
6(c)(i). Find the coordinates of A and B.
[2 marks]6(c)(ii). Find the equation of the line through P and perpendicular to AB.
[1 mark]7(a). In the figure given, O is the centre of the circle. ∠DAE = 70º, Find giving suitable reasons the measure of:
7(a)(i). ∠BCD
[1 mark]7(a)(ii). ∠BOD
[1 mark]7(a)(iii). ∠OBD
[1 mark]7(b). A(-1, 3), B(4, 2) and C(3, -2) are the vertices of a triangle.
7(b)(i). Find the coordinates of the centroid G of the triangle
[1.5 marks]7(b)(ii). Find the equation of the line through G and parallel to AC
[1.5 marks]7(c). Prove that (sin θ - 2sin³ θ) / (2cos³ θ - cos θ) = tan θ
[4 marks]8(a). Construct a regular hexagon of side 5 cm. Hence construct all its lines of symmetry and name them.
[3 marks]8(b). In the given figure PQRS is a cyclic quadrilateral PQ and SR produced meet at T.
8(b)(i). Prove ΔTPS ~ ΔTRQ
[1 mark]8(b)(ii). Find SP if TP = 18 cm, RQ = 4 cm and TR = 6 cm.
[1 mark]8(b)(iii). Find area of quadrilateral PQRS if area of ΔPTS = 27 cm².
[2 marks]8(c). Given matrix
8(c). If AX = B
8(c)(i). Write the order of matrix X.
[1 mark]8(c)(ii). Find the matrix 'X'.
[2 marks]9(i). The polynomial 3x³ + 8x² - 15x + k has (x - 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.
[4 marks]9(iii). Oil is stored in a spherical vessel occupying 3/4 of its full capacity. Radius of this spherical vessel is 28 cm. This oil is then poured into a cylindrical vessel with a radius of 21 cm. Find the height of the oil in the cylindrical vessel (correct to the nearest cm). Take π = 22/7
[3 marks]
9(ii). The following letters A, D, M, N, O, S, U, Y of the English alphabet are written on separate cards and put in a box. The cards are well shuffled and one card is drawn at random. What is the probability that the card drawn is a letter of the word,
9(ii)(a). MONDAY?
[1 mark]9(ii)(b). which does not appear in MONDAY?
[1 mark]9(ii)(c). which appears both in SUNDAY and MONDAY?
[1 mark]10(i). The given graph with a histogram represents the number of plants of different heights grown in a school campus. Study the graph carefully and answer the following questions :
10(i)(a). Make a frequency table with respect to the class boundaries and their corresponding frequencies.
[2 marks]10(i)(b). State the modal class.
[1 mark]10(i)(c). Identify and note down the mode of the distribution.
[1 mark]10(i)(d). Find the number of plants whose height range is between 80 cm to 90 cm.
[1 mark]10(ii). The angle of elevation of the top of a 100 m high tree from two points A and B on the opposite side of the tree are 52° and 45° respectively. Find the distance AB, to the nearest metre.
[5 marks]