Class 9 IMO — Full Syllabus Mock Test

Q1 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.

Q2 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.

Q3 Logical Reasoning

Which of the following conditions is NOT sufficient to prove that two lines intersected by a transversal are parallel?

The sum of ANY two random interior angles being 180° is meaningless unless they are specifically consecutive interior angles on the same side of the transversal.

Q4 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.

Q5 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.

Q6 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.

Q7 Achievers Section

If x = 1, y = 2 is a solution of the equation a²x + ay = 3, then the possible values of a are:

Substitute x = 1 and y = 2 into the equation: a²(1) + a(2) = 3 → a² + 2a − 3 = 0. Solving this quadratic equation: (a + 3)(a − 1) = 0 → a = 1 or a = −3.

Q8 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.

Q9 Logical Reasoning

Starting from the origin, a point moves 3 right, 2 up, 3 left, then 2 down. Where does it end?

(0,0) → (3,0) → (3,2) → (0,2) → (0,0): it returns to the origin.

Q10 Achievers Section

A capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

Radius r = 2.5 mm. Length of cylindrical part = Total length - 2(radius) = 14 - 2(2.5) = 14 - 5 = 9 mm. Total Surface Area = CSA of cylinder + 2 × CSA of hemisphere = 2πrh + 2(2πr²) = 2πr(h + 2r) = 2 × (22/7) × 2.5 × (9 + 5) = 2 × (22/7) × 2.5 × 14 = 220 mm².

Q11 Everyday Mathematics

A shopkeeper sells 5 shirts of size 38, 12 of size 40, 8 of size 42, and 2 of size 44. The modal shirt size sold is:

Size 40 has the highest frequency (12). Thus, the mode is 40.

Q12 Logical Reasoning

Look at the pattern: Term 1 = 5, Term 2 = 9, Term 3 = 13, Term 4 = 17. If y represents the value of the term and x represents the term number, which equation represents this rule?

Test y = 4x + 1. For x = 1: y = 4(1)+1 = 5. For x = 2: y = 4(2)+1 = 9. For x = 3: y = 4(3)+1 = 13. This linear equation perfectly matches the given values.

Q13 Mathematical Reasoning

In ΔABC, if AB = AC and ∠B = 50°, then find the value of ∠A.

Since AB = AC, the angles opposite to these sides are equal, so ∠C = ∠B = 50°. By angle sum property, ∠A + ∠B + ∠C = 180° → ∠A + 50° + 50° = 180° → ∠A = 80°.

Q14 Logical Reasoning

Assertion (A): Two triangles having the same base and equal areas will have their corresponding altitudes equal. Reason (R): Area of a triangle = ½ × base × altitude.

Since Area = ½ × b × h, if Area and base (b) are constant, the altitude (h) must also be equal. The reason correctly explains the assertion.

Q15 Everyday Mathematics

A farm has area 7^x square units and an adjacent plot 7^(x−2) square units. If their areas differ by 2352 square units, find x.

7^(x−2)(7² − 1) = 2352, so 7^(x−2) × 48 = 2352 and 7^(x−2) = 49 = 7². Thus x − 2 = 2 and x = 4.

Q16 Everyday Mathematics

Rohan sits at (3, 4). He is moved to a seat with the same abscissa but with the ordinate reduced by 5. His new position is:

Abscissa stays 3 and ordinate becomes 4 − 5 = −1, giving (3, −1).

Q17 Mathematical Reasoning

The diagonals of a rhombus are 16 cm and 12 cm long. Find the length of each side of the rhombus.

The diagonals of a rhombus bisect each other at right angles (90°). Half lengths of the diagonals are 8 cm and 6 cm. Using Pythagoras theorem for the right-angled triangle formed by the half-diagonals: side² = 8² + 6² = 64 + 36 = 100. Side = √100 = 10 cm.

Q18 Mathematical Reasoning

The area of a trapezium whose parallel sides are 9 cm and 16 cm and the distance between them is 8 cm is:

Area = ½ × (Sum of parallel sides) × height = ½ × (9 + 16) × 8 = ½ × 25 × 8 = 100 cm².

Q19 Logical Reasoning

Which of the following transformations changes the area of a triangle?

Translation, rotation, and reflection are rigid transformations that preserve shape and size (isometries). Scaling changes the dimensions and therefore the area.

Q20 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.

Q21 Logical Reasoning

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Which of the following is true?

Triangles ABD and ABC are on same base AB and between same parallels. Area(ABD) = Area(ABC). Subtracting Area(AOB) gives Area(AOD) = Area(BOC).

Q22 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).

Q23 Mathematical Reasoning

If a point X lies inside an angle ∠AOB, then the measure of ∠AOB is always ________ the measure of ∠AOX.

Since X lies inside the angle, ∠AOX is a part of the whole angle ∠AOB. According to Euclid's Fifth Axiom, the whole is greater than the part, so ∠AOB > ∠AOX.

Q24 Mathematical Reasoning

Which of these is the formula for class mark?

Class mark is the exact midpoint of the class interval, calculated as (Upper limit + Lower limit)/2.

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 Mathematical Reasoning

If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 2:3, then the greater of the two angles is:

Co-interior angles are supplementary. Let the angles be 2x and 3x. 2x + 3x = 180° → 5x = 180° → x = 36°. The greater angle is 3x = 3 × 36° = 108°.

Q27 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.

Q28 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.

Q29 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.

Q30 Logical Reasoning

Which of the following is not a measure of central tendency?

Mean, median, and mode measure central tendency. Range measures dispersion.

Q31 Mathematical Reasoning

Find the length of the line segment joining the mid-points of the non-parallel sides of a trapezium whose parallel sides are 12 cm and 18 cm long.

The length of the line segment joining the mid-points of the non-parallel sides of a trapezium is equal to half the sum of the parallel sides. Length = (12 + 18) / 2 = 30 / 2 = 15 cm.

Q32 Mathematical Reasoning

Find the distance between P(−3, 2) and Q(5, 2).

Same y-coordinate, so the distance is |5 − (−3)| = 8 units.

Q33 Logical Reasoning

If two lines are intersected by a transversal such that the bisectors of a pair of corresponding angles are parallel, then the two lines are:

When the bisectors of corresponding angles are parallel, the corresponding angles themselves must be equal because they are exactly double the angles formed by the parallel bisectors. Since corresponding angles are equal, the original lines are parallel.

Q34 Achievers Section

If a regular hexagon is inscribed in a circle of radius r, what is the perimeter of the hexagon?

A regular hexagon can be divided into 6 equilateral triangles from the centre. Each side of the hexagon is equal to the radius r. Perimeter = 6 × side = 6r.

Q35 Everyday Mathematics

A circular garden has radius √7 m. If the radius is increased to 3√7 m, how many times does the area increase?

Area is proportional to radius². The radius becomes 3 times, so the area becomes 3² = 9 times.

Q36 Mathematical Reasoning

In a parallelogram ABCD, if angle A is a right angle, then the parallelogram is specifically classified as a:

By definition, a parallelogram with one right angle is a rectangle (since opposite and adjacent angles will also become 90°).

Q37 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.

Q38 Everyday Mathematics

Harish is measuring a piece of land that is a parallelogram ABCD. He knows that the fence lengths are AB = 40 m and BC = 30 m. If he walks from corner A directly to corner C along a straight path of length 50 m, what is the angle ABC between the two fences?

In triangle ABC, the sides are 40 m, 30 m, and 50 m. Since 30² + 40² = 900 + 1600 = 2500 = 50², triangle ABC is a right-angled triangle with the right angle opposite to the side of 50 m (AC). Thus, angle ABC = 90°.

Q39 Everyday Mathematics

To find the area of a quadrilateral field ABCD, it is divided by diagonal AC which is 24 m. Perpendicular offsets from B and D are 8 m and 12 m. The area of the field is:

Area = ½ × d × (h₁ + h₂) = ½ × 24 × (8 + 12) = 12 × 20 = 240 m².

Q40 Achievers Section

If both (x + 1) and (x − 1) are factors of ax³ + x² − 2x + b, find a and b.

p(−1) = −a + 1 + 2 + b = 0 → −a + b = −3; p(1) = a + 1 − 2 + b = 0 → a + b = 1. Solving gives b = −1 and a = 2.

Q41 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°.

Q42 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.

Q43 Mathematical Reasoning

Two fair dice are rolled together. What is the total number of possible outcomes?

Each die has 6 outcomes. For two dice, total outcomes = 6 × 6 = 36.

Q44 Mathematical Reasoning

The linear equation 2x − 5y = 7 has:

A linear equation in two variables represents a straight line. Since a line contains infinitely many points, the equation has infinitely many solutions.

Q45 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).

Q46 Logical Reasoning

If 5 points are marked on the circumference of a circle, what is the total number of unique chords that can be drawn joining these points?

Choosing any 2 points out of 5 creates a chord. This is combinations: 5C2 = (5 × 4) / 2 = 10.

Q47 Achievers Section

Mean of 20 observations is 15.5. Later it was found that 24 was misread as 42. The corrected mean is:

Difference = 24 − 42 = −18. Correct sum = (20 × 15.5) − 18 = 310 − 18 = 292. Correct mean = 292/20 = 14.6.

Q48 Mathematical Reasoning

In a rectangle ABCD, diagonal AC = 25 cm and side AB = 24 cm. Find the perimeter of the rectangle.

In a rectangle, angle B = 90°. In right-angled triangle ABC, BC = √(AC² − AB²) = √(25² − 24²) = √(625 − 576) = √49 = 7 cm. Perimeter of the rectangle = 2(length + width) = 2(24 + 7) = 2 × 31 = 62 cm.

Q49 Logical Reasoning

Find the point on the graph of 3x + 4y = 24 whose x-coordinate is double its y-coordinate.

Since the x-coordinate is double the y-coordinate, let x = 2y. Substitute x = 2y into 3x + 4y = 24: 3(2y) + 4y = 24 → 6y + 4y = 24 → 10y = 24 → y = 2.4. Then x = 2(2.4) = 4.8. The point is (4.8, 2.4).

Q50 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.