Class 9 IMO — Full Syllabus Mock Test

Q1 Logical Reasoning

If every side of a triangle is doubled, the ratio of the area of the original triangle to the new triangle is:

When sides are doubled, the semi-perimeter doubles, making each (s-a) term double. √(2s × 2(s-a) × 2(s-b) × 2(s-c)) = √16 × Area = 4 × Area. Ratio is 1:4.

Q2 Everyday Mathematics

Marks of 90 students: 0-20(7), 20-40(10), 40-60(10), 60-70(20), 70-100(43). The probability of a student getting more than 70% is:

Students scoring 70-100 are 43. Total students = 90. Probability = 43/90.

Q3 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.

Q4 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.

Q5 Mathematical Reasoning

One card is drawn from a well-shuffled deck of 52 cards. What is the probability of getting a face card?

Face cards are King, Queen, Jack in 4 suits (total 12). Probability = 12/52 = 3/13.

Q6 Everyday Mathematics

The total weight of a box containing x identical cricket balls (each weighing 150 grams) and a container weighing 200 grams is given by y grams. Write the linear expression.

The total weight y is the sum of the combined weight of x balls (150 grams each) and the constant weight of the empty container (200 grams). Therefore, y = 150x + 200.

Q7 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.

Q8 Achievers Section

ABCD is a rhombus with side length 5 cm. If diagonal AC = 6 cm, find the area of the rhombus.

The diagonals of a rhombus bisect each other at right angles. Half of AC is 3 cm. Let half of BD be x. Using Pythagoras theorem: 3² + x² = 5² → 9 + x² = 25 → x² = 16 → x = 4 cm. So, the full diagonal BD = 2 × 4 = 8 cm. Area of rhombus = ½ × d1 × d2 = ½ × 6 × 8 = 24 cm².

Q9 Achievers Section

Find the length of the altitude corresponding to the smallest side of a triangle whose sides are 17 cm, 25 cm, and 26 cm.

s = 34. Area = √(34 × 17 × 9 × 8) = 204 cm². The smallest side is 17 cm. ½ × 17 × h = 204 → h = 408 / 17 = 24 cm.

Q10 Mathematical Reasoning

If a² + b² + c² = 250 and ab + bc + ca = 3, find |a + b + c|.

(a + b + c)² = 250 + 2(3) = 256, so |a + b + c| = √256 = 16.

Q11 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.

Q12 Mathematical Reasoning

If a regular hexagon is divided into 6 equilateral triangles by its diagonals, and the area of one triangle is 10 cm², the area of the hexagon is:

A regular hexagon is made of 6 identical equilateral triangles. Area = 6 × 10 = 60 cm².

Q13 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.

Q14 Achievers Section

The cost of painting the total outside surface of a closed cylindrical oil tank at ₹60 per sq. m is ₹237.60. The height of the tank is 6 times the radius of the base of the tank. Find its volume.

Total surface area = cost ÷ rate = 237.60 ÷ 60 = 3.96 m². With h = 6r, TSA = 2πr(r + h) = 2πr(7r) = 14πr² = 44r², so 44r² = 3.96, r² = 0.09 and r = 0.3 m (h = 1.8 m). Volume = πr²h = (22/7) × 0.09 × 1.8 ≈ 0.509 m³.

Q15 Mathematical Reasoning

If ABCD is a parallelogram, which of the following expressions correctly represents the relationship between its consecutive interior angles?

In a parallelogram, consecutive interior angles are supplementary because the opposite sides are parallel. Therefore, angle A + angle B = 180°.

Q16 Everyday Mathematics

A class consists of 15 boys and 10 girls. The average weight of boys is 40 kg and that of girls is 35 kg. The average weight of the whole class is:

Total weight = (15 × 40) + (10 × 35) = 600 + 350 = 950. Total students = 25. Average = 950/25 = 38 kg.

Q17 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.

Q18 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).

Q19 Everyday Mathematics

A milkman sells milk at ₹50, ₹55, ₹60, and ₹65 per litre in four different localities. If he sells equal quantities in each locality, the mean price is:

Mean = (50 + 55 + 60 + 65) / 4 = 230 / 4 = ₹57.5.

Q20 Logical Reasoning

The area of a triangle is calculated using base = 12 cm and height = 8 cm. If calculated correctly using Heron's formula with its three side lengths, the result will be:

Both methods yield the exact same area. Area = ½ × base × height = ½ × 12 × 8 = 48 cm².

Q21 Everyday Mathematics

To find 249² − 248² quickly, which identity helps, and what is the value?

Using a² − b² = (a − b)(a + b): 249² − 248² = (249 − 248)(249 + 248) = 1 × 497 = 497.

Q22 Logical Reasoning

A card is selected from a standard deck. What is the probability it is a red king?

There are 2 red kings (King of Hearts, King of Diamonds). Probability = 2/52 = 1/26.

Q23 Logical Reasoning

How many total lines of symmetry does a perfect circle have?

Any straight line passing through the centre of a circle divides it into two identical semicircles. Since there are infinite such lines (diameters), a circle has infinite lines of symmetry.

Q24 Logical Reasoning

Point A is to the North of Point B. Point B is to the West of Point C. If AB = BC = 6 km, then what is the direction of Point A with respect to Point C, and what is the shortest distance between them?

ABC forms a right-angled isosceles triangle with right angle at B. Distance AC = √(6² + 6²) = √72 = 6√2 km. Direction of A from C is North-West.

Q25 Mathematical Reasoning

For a triangle whose semi-perimeter s and sides a, b, c satisfy s − a = 5 cm, s − b = 10 cm and s − c = 1 cm, what is the area of the triangle?

s = 16. Area = √(s(s−a)(s−b)(s−c)) = √(16 × 5 × 10 × 1) = √800 = 20√2 cm².

Q26 Mathematical Reasoning

A chord of length 16 cm is drawn in a circle of radius 10 cm. Find the distance of the chord from the centre of the circle.

Half of the chord = 8 cm. Using Pythagoras theorem in the right triangle formed by the radius, distance to chord, and half-chord: distance = √(10² − 8²) = √(100 − 64) = √36 = 6 cm.

Q27 Mathematical Reasoning

Three coins are tossed. The probability of getting no tails is:

No tails means all heads {HHH}. Probability = 1/8.

Q28 Logical Reasoning

P is any point on the median AD of triangle ABC. The ratio of Area(ABP) to Area(ACP) is:

AD is the median, so triangles ABD and ACD have equal areas. PD is the median of PBC, so PBD and PCD have equal areas. Subtracting gives Area(ABP) = Area(ACP). Ratio is 1:1.

Q29 Logical Reasoning

Euclid belonged to which ancient country and town?

Euclid was a Greek mathematician who lived and taught mathematics at Alexandria in Egypt.

Q30 Mathematical Reasoning

A point P is 4 units from the x-axis and 3 units from the y-axis, in Quadrant III. Its coordinates are:

|y| = 4 and |x| = 3; in Quadrant III both are negative, so (−3, −4).

Q31 Mathematical Reasoning

Two dice are rolled simultaneously. What is the probability of getting a sum of exactly 12?

The only outcome giving a sum of 12 is (6,6). Probability = 1/36.

Q32 Mathematical Reasoning

The total surface area of a solid cylinder of radius 14 cm and height 20 cm is:

TSA = 2πr(h + r) = 2 × (22/7) × 14 × (20 + 14) = 88 × 34 = 2992 cm².

Q33 Logical Reasoning

Thales, a famous Greek philosopher, is credited with giving the first known proof in geometry. What was this proof about?

Thales is famous for providing the first geometric proof, which showed that a circle is bisected (cut into two equal halves) by its diameter.

Q34 Everyday Mathematics

Meena makes a circular rangoli design in her courtyard. She wants to divide the boundary into equal arcs using 8 dots. What will be the measure of the angle subtended by each arc at the centre?

Total angle at the centre is 360°. Since there are 8 equal arcs, each arc subtends 360° / 8 = 45°.

Q35 Mathematical Reasoning

If p(x) = x + 4, then p(x) + p(−x) is equal to:

p(x) = x + 4 and p(−x) = −x + 4, so their sum is (x + 4) + (−x + 4) = 8.

Q36 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°.

Q37 Mathematical Reasoning

A box has 90 discs (1 to 90). The probability of picking a number divisible by 5 is:

Numbers divisible by 5 up to 90 are 90 ÷ 5 = 18 numbers. Probability = 18/90 = 1/5.

Q38 Mathematical Reasoning

The perimeter of a rhombus is 32 cm and its corresponding height is 5 cm. Its area is:

Side of the rhombus = 32 / 4 = 8 cm. Area = base × height = 8 × 5 = 40 cm².

Q39 Mathematical Reasoning

An equilateral triangle ABC is inscribed in a circle with centre O. The measure of ∠BOC is:

In an equilateral triangle, each angle is 60°. Thus ∠A = 60°. The angle subtended by arc BC at the centre O is twice the angle subtended at the remaining circumference. ∠BOC = 2 × ∠A = 2 × 60° = 120°.

Q40 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.

Q41 Logical Reasoning

The coordinates of a point on the line x + 2y = 8 whose y-coordinate is half its x-coordinate is:

Given y-coordinate is half of x-coordinate, so x = 2y. Substitute x = 2y into x + 2y = 8: 2y + 2y = 8 → 4y = 8 → y = 2. Then x = 2(2) = 4. The point is (4, 2).

Q42 Everyday Mathematics

A bag contains 4 red balls and 5 black balls. A ball is drawn at random. What is the probability of getting a red ball?

Total balls = 4 + 5 = 9. Red balls = 4. Probability = 4/9.

Q43 Mathematical Reasoning

The class mark of the class interval 120 − 150 is:

Class mark = (Lower limit + Upper limit)/2 = (120 + 150)/2 = 270/2 = 135.

Q44 Logical Reasoning

If a sphere and a cube have the same surface area, then the ratio of the volume of the sphere to the volume of the cube is:

Surface areas equal: 4πr² = 6a² → a² = (4π/6)r² = (2π/3)r² → a = r√(2π/3). Ratio of volumes = (4/3)πr³ / a³ = (4/3)πr³ / (r³(2π/3)√(2π/3)) = (4/3)π / ((2π/3)√(2π/3)) = 2 / √(2π/3) = √6 / √π = √6 : √π.

Q45 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.

Q46 Everyday Mathematics

A floral design is made from 16 identical triangular tiles and has a total area of 576√6 cm². If polishing costs 50 paise per cm², what is the total polishing cost?

Total area = 576√6 cm². Cost in Rupees = 576√6 × 0.50 = ₹288√6.

Q47 Everyday Mathematics

A school decides to plant trees in a triangular area of the playground. The base is 30 m and altitude is 10 m. If 1 tree needs 3 m² of space, how many trees can be planted?

Area of triangular area = ½ × 30 × 10 = 150 m². Number of trees = 150 / 3 = 50.

Q48 Logical Reasoning

In ancient India, altars with combinations of shapes like rectangles, triangles, and trapeziums were required for:

In the Vedic period, square and circular altars were used for household rituals, while altars made of combinations of rectangles, triangles, and trapeziums were required for public worship and grand sacrifices.

Q49 Mathematical Reasoning

In the given figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE.

Since AB and CD intersect at O, ∠AOC = ∠BOD = 40° (vertically opposite angles). Given ∠AOC + ∠BOE = 70°, we get 40° + ∠BOE = 70° → ∠BOE = 30°. Also, ∠AOC + ∠COE + ∠BOE = 180° (angles on a straight line AB) → 70° + ∠COE = 180° → ∠COE = 110°. Thus, reflex ∠COE = 360° − 110° = 250°.

Q50 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.