Class 9 IMO — Full Syllabus Mock Test

Q1 Mathematical Reasoning

The factorisation of x² − 5x − 6 is:

Split the middle term: x² − 6x + x − 6 = x(x − 6) + 1(x − 6) = (x − 6)(x + 1).

Q2 Everyday Mathematics

A heap of wheat is in the form of a cone of diameter 10.5 m and height 3 m. Find the area of the canvas required to just cover the heap. (Use π = 22/7)

Radius r = 10.5/2 = 5.25 m and height h = 3 m. Slant height l = √(r² + h²) = √(27.5625 + 9) = √36.5625 ≈ 6.05 m. Canvas needed = curved surface area = πrl = (22/7) × 5.25 × 6.05 ≈ 99.83 m².

Q3 Mathematical Reasoning

The weights of 5 students in kg are 42, 45, 48, 51, 54. The range of their weights is:

Range = Maximum value − Minimum value = 54 − 42 = 12.

Q4 Mathematical Reasoning

The region between a chord and either of its arcs is called a:

The region bounded by a chord and an arc is called a segment of the circle.

Q5 Mathematical Reasoning

In ΔABC, the angle bisector of ∠A meets BC at D. If ∠B = 70° and ∠C = 30°, find ∠ADC.

Total sum of angles in ΔABC = 180° → ∠A = 180° − (70° + 30°) = 80°. Since AD bisects ∠A, ∠BAD = ∠CAD = 40°. In ΔADC, ∠ADC = 180° − (∠CAD + ∠C) = 180° − (40° + 30°) = 110°.

Q6 Logical Reasoning

The ratio of the volumes of two cones is 4:5 and the ratio of the radii of their bases is 2:3. Find the ratio of their vertical heights.

V₁/V₂ = [(1/3)πr₁²h₁] / [(1/3)πr₂²h₂] = (r₁/r₂)²(h₁/h₂). 4/5 = (2/3)² × (h₁/h₂) = 4/9 × (h₁/h₂). Therefore, h₁/h₂ = (4/5) × (9/4) = 9/5. Ratio is 9:5.

Q7 Mathematical Reasoning

ABCD is a parallelogram. If a line PQ is drawn parallel to diagonal AC cutting AB at P and BC at Q, then triangle APQ is equal in area to which of the following triangles?

Triangles APQ and CPQ share the same base PQ and lie between the same parallel lines PQ and AC. Therefore, their areas are equal.

Q8 Mathematical Reasoning

ABCD is a parallelogram and P is any point in its interior. Then Area(APB) + Area(PCD) is equal to:

Drawing a line through P parallel to the base shows that Area(APB) + Area(PCD) = ½ Area(ABCD).

Q9 Logical Reasoning

Assertion (A): Median of a triangle divides it into two triangles of equal areas. Reason (R): The centroid of a triangle divides the median in the ratio 2:1.

The median divides the base into two equal parts, so the two triangles have equal bases and the same height, giving equal areas. The centroid ratio (2:1) is true but doesn't explain the area division by the median.

Q10 Logical Reasoning

Which graph is most suitable to represent continuous grouped frequency distribution?

A histogram is used to represent a continuous grouped frequency distribution because there are no gaps between the class intervals.

Q11 Logical Reasoning

The probability of a sure (or certain) event is:

A sure event always happens. The number of favourable outcomes equals the total number of outcomes, making the probability 1.

Q12 Logical Reasoning

Statement A: A binomial can have degree 100. Statement B: A polynomial of degree 1 can have 3 terms. Which is true?

A is true (e.g. x¹⁰⁰ + 1 is a binomial of degree 100). B is false: a degree-1 polynomial has at most 2 terms (ax + b).

Q13 Achievers Section

In a school, 30 students obtained the following marks: 10 students got 20, 15 got 30, and 5 got 40. The mean score is:

Mean = Σ(fᵢxᵢ) / Σfᵢ = (10×20 + 15×30 + 5×40) / 30 = (200 + 450 + 200) / 30 = 850 / 30 ≈ 28.33.

Q14 Mathematical Reasoning

If a right circular cone has radius 5 cm and slant height 13 cm, what is its volume?

Radius r = 5, slant height l = 13. Height h = √(l² - r²) = √(169 - 25) = √144 = 12 cm. Volume = (1/3)πr²h = (1/3)π(25)(12) = 100π cm³.

Q15 Achievers Section

The value of k for which the linear equation kx − y = 2 has a solution where x is always 1 more than y, and passing through (3, 2), is:

The solution point given is (3, 2). Substitute x = 3 and y = 2 into the equation kx − y = 2: k(3) − 2 = 2 → 3k = 4 → k = 4/3.

Q16 Everyday Mathematics

Meena makes an umbrella by stitching 10 triangular pieces of cloth of two different colours. Each piece has sides 20 cm, 50 cm, and 50 cm. The total cloth required is:

s = 60 cm. Area of one piece = √(60 × 40 × 10 × 10) = 200√6 cm². Total cloth = 10 × 200√6 = 2000√6 cm².

Q17 Mathematical Reasoning

Two parallel chords of lengths 10 cm and 24 cm are on opposite sides of the centre of a circle of radius 13 cm. What is the distance between these chords?

For the 10 cm chord, half is 5 cm. Distance from centre = √(13² − 5²) = 12 cm. For the 24 cm chord, half is 12 cm. Distance from centre = √(13² − 12²) = 5 cm. Since they are on opposite sides, total distance = 12 + 5 = 17 cm.

Q18 Everyday Mathematics

For a circular ring, the area depends on (R² − r²). If R = 101 m and r = 99 m, find R² − r² without squaring.

R² − r² = (R − r)(R + r) = (101 − 99)(101 + 99) = 2 × 200 = 400.

Q19 Mathematical Reasoning

Two fair coins are tossed. What is the probability of getting at least one head?

Outcomes with at least one head = {HH, HT, TH}. Total outcomes = 4. Probability = 3/4.

Q20 Logical Reasoning

Complete the logic sequence: In triangle ABC, D and E are mid-points of AB and AC respectively. If DE = 4.5 cm, then the parallel side BC must logically be equal to:

By the Mid-point theorem, the line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. So, DE = BC/2 → BC = 2 × DE = 2 × 4.5 = 9 cm.

Q21 Logical Reasoning

Find the missing term in the sequence: 2, √5, √6, √7, 2√2, …

Writing each term under a root: √4, √5, √6, √7, √8. The next term is √9 = 3.

Q22 Everyday Mathematics

A circular cycling track has an inner radius of 21 m and an outer radius of 28 m. What is the width of the track?

The width of the circular track is the difference between the outer radius and the inner radius. Width = 28 m − 21 m = 7 m.

Q23 Achievers Section

If 2^x = 3^y = 6^(−z), then 1/x + 1/y + 1/z equals:

Let 2^x = 3^y = 6^(−z) = k. Then 2 = k^(1/x), 3 = k^(1/y), 6 = k^(−1/z). Since 2 × 3 = 6, k^(1/x + 1/y) = k^(−1/z), so 1/x + 1/y + 1/z = 0.

Q24 Logical Reasoning

If (a, b) lies in Quadrant II, then (−a, −b) lies in:

In Quadrant II, a < 0 and b > 0, so −a > 0 and −b < 0, which is Quadrant IV.

Q25 Mathematical Reasoning

Find the value of p if x = p and y = 2 is a solution of the equation 3x − 4y = 7.

Substitute x = p and y = 2 into 3x − 4y = 7: 3(p) − 4(2) = 7 → 3p − 8 = 7 → 3p = 15 → p = 5.

Q26 Logical Reasoning

In a histogram, the width of the rectangles is proportional to:

The width of a rectangle in a histogram corresponds to the size of the class interval on the x-axis.

Q27 Achievers Section

Three vertices of a square are (−1, 2), (3, 2) and (3, −2). The fourth vertex is:

Sides have length 4; the fourth vertex aligns with x = −1 and y = −2, giving (−1, −2).

Q28 Mathematical Reasoning

A cyclic parallelogram must be a:

Opposite angles of a parallelogram are equal (∠A = ∠C), and for a cyclic quadrilateral they sum to 180° (∠A + ∠C = 180°). Thus, 2∠A = 180°, so ∠A = 90°. A parallelogram with a right angle is a rectangle.

Q29 Mathematical Reasoning

An equilateral triangle has a perimeter of 60 cm. Its area is:

Side a = 60/3 = 20 cm. Area = (√3/4) × 20² = (√3/4) × 400 = 100√3 cm².

Q30 Mathematical Reasoning

Find the distance between C(−1, −2) and D(2, 2).

Horizontal change 3 and vertical change 4 give distance √(3² + 4²) = √25 = 5 units.

Q31 Logical Reasoning

Find the next term in the pattern of interior angles of a regular polygon with n sides, starting from n = 3, 4, 5... If a quadrilateral has three acute angles each equal to 75°, what is the logical value of its fourth angle?

Sum of angles of a quadrilateral is 360°. Sum of three angles = 75° + 75° + 75° = 225°. Fourth angle = 360° − 225° = 135°.

Q32 Everyday Mathematics

Anita is making a conical tent for a village fair in Gujarat. The tent is 10 m high and the radius of its base is 24 m. Find the cost of the canvas required to make the tent, if the cost of 1 m² canvas is ₹70.

h = 10 m, r = 24 m. l = √(24² + 10²) = √(576 + 100) = √676 = 26 m. CSA = πrl = (22/7) × 24 × 26. Area ≈ 1961.14 m². Cost = (22/7) × 24 × 26 × 70 = 22 × 24 × 26 × 10 = ₹1,37,280.

Q33 Everyday Mathematics

A metal sheet of dimensions (x + 5) by (x − 5) has area:

(x + 5)(x − 5) = x² − 25 by the identity (a + b)(a − b) = a² − b².

Q34 Mathematical Reasoning

Find the area of the triangle formed by O(0, 0), A(6, 0) and B(0, 5).

Right-angled at O with base 6 and height 5: area = ½ × 6 × 5 = 15 sq units.

Q35 Logical Reasoning

Two cubes each of volume 64 cm³ are joined end to end. The surface area of the resulting cuboid is:

Volume of one cube = 64 cm³ → side a = 4 cm. When joined, length of cuboid l = 4+4 = 8 cm, breadth b = 4 cm, height h = 4 cm. TSA = 2(lb + bh + hl) = 2(32 + 16 + 32) = 2(80) = 160 cm².

Q36 Logical Reasoning

The algebraic sum of the deviations of a set of values from their mean is:

By definition, the sum of deviations of all observations from their mean is always 0. Σ(xᵢ − x̄) = 0.

Q37 Mathematical Reasoning

In a parallelogram ABCD, diagonal AC and BD intersect at O. If AO = 3 cm and BD = 8 cm, find the length of OC and OD respectively.

Diagonals of a parallelogram bisect each other. So, OC = AO = 3 cm. OD = BD / 2 = 8 / 2 = 4 cm.

Q38 Logical Reasoning

In a logical sequence of figures, the first circle is divided into 2 sectors, the second into 4 sectors, and the third into 8 sectors. Following this pattern, how many sectors will the fifth circle have?

The number of sectors follows the pattern 2¹, 2², 2³. The fourth circle will have 2⁴ = 16. The fifth circle will have 2⁵ = 32.

Q39 Everyday Mathematics

An express train track switches onto a loop line. The main line and the loop line form an obtuse angle of 135° at the point of divergence. What is the measure of the linear pair partner angle formed by this track layout?

The straight track line implies a linear pair layout. 180° − 135° = 45°.

Q40 Logical Reasoning

If a triangle has vertices with coordinates that follow a specific pattern: (1,1), (2,3), (3,5), then these points are:

Slope of line joining (1,1) and (2,3) is (3-1)/(2-1) = 2. Slope joining (2,3) and (3,5) is (5-3)/(3-2) = 2. Since slopes are equal, points are collinear and do not form a triangle.

Q41 Everyday Mathematics

A triangular plot with sides 18 m, 24 m, and 30 m needs clearing. If the contractor charges ₹10 per m², what is the total bill?

The sides form a right triangle (18, 24, 30 are a multiple of 3, 4, 5). Area = ½ × 18 × 24 = 216 m². Bill = 216 × 10 = ₹2160.

Q42 Mathematical Reasoning

If AB = CD, then which of the following expressions is true when we add BC to both sides?

Given AB = CD. Adding BC to both sides (Euclid's Second Axiom): AB + BC = CD + BC. Since B lies between A and C, AB + BC = AC. Since C lies between B and D, CD + BC = BD. Therefore, AC = BD.

Q43 Mathematical Reasoning

Which of the following is an irrational number?

√(4/9) = 2/3, √12/√3 = √4 = 2, and 4.333… = 13/3 are rational. √7 cannot be simplified to a rational, so it is irrational.

Q44 Logical Reasoning

Rays OC and OE lie above a straight line AB through point O, dividing the straight angle into three parts: ∠AOC = 3y, ∠COE = y and ∠EOB = 2y. Find the value of y.

The three angles on the straight line AB add up to 180°: 3y + y + 2y = 180° → 6y = 180° → y = 30°.

Q45 Mathematical Reasoning

If the exterior angle of a triangle is 115° and one of the interior opposite angles is 45°, then the other interior opposite angle is:

By the exterior angle property, the exterior angle is equal to the sum of the two interior opposite angles. So, 115° = 45° + x → x = 115° − 45° = 70°.

Q46 Mathematical Reasoning

The lateral surface area of a cube is 256 m². The volume of the cube is:

LSA of cube = 4a² = 256. Therefore, a² = 64 → a = 8 m. Volume = a³ = 8³ = 512 m³.

Q47 Everyday Mathematics

A taxi driver in Delhi earns ₹500, ₹450, ₹600, ₹400, and ₹550 in 5 days. What is the median earning?

Arranging in ascending order: 400, 450, 500, 550, 600. The median (middle value) is ₹500.

Q48 Achievers Section

An equilateral triangle can be constructed on any given line segment. This proposition was proven by Euclid using which of his tools?

In Proposition 1 of Book 1, Euclid constructed an equilateral triangle on a segment AB by drawing two circles centered at A and B with radius AB (Postulate 3), finding their intersection point via lines (Postulate 1), and equating the sides using Axiom 1.

Q49 Everyday Mathematics

The cost of a notebook is twice the cost of a pen. Writing this statement as a linear equation in two variables (where x is cost of notebook and y is cost of pen) gives:

Let cost of notebook = x and cost of pen = y. According to the given statement: Cost of notebook = 2 × Cost of pen → x = 2y → x − 2y = 0.

Q50 Mathematical Reasoning

In ΔABC, AD is the perpendicular bisector of BC. Then ΔABC is a/an:

In ΔABD and ΔACD: BD = CD (since AD bisects BC), ∠ADB = ∠ADC = 90° (perpendicular), and AD = AD (common). By SAS, ΔABD ≅ ΔACD, which means AB = AC. Hence, it is an isosceles triangle.