Logical Reasoning — Chapter Test | Class 9 IMO

Q1

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.

Q2

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.

Q3

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

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°.

Q5

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.

Q6

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).

Q7

In a certain code, 'PARALLELOGRAM' is related to 'OPPOSITE_SIDES_EQUAL'. If we analyze the properties of a rhombus, which of the following logical pairs represents its unique diagonal property?

A rhombus is a special parallelogram whose diagonals bisect each other at right angles (90°). Thus, 'BISECT_AT_RIGHT_ANGLES' is the correct logical property.

Q8

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.

Q9

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.

Q10

Points A and B have coordinates that satisfy y = 3x − 5. If the x-coordinate of A increases by 4 units to become the x-coordinate of B, by how many units does the y-coordinate increase?

Let A be (x, 3x − 5). If x increases by 4, the new x-coordinate is x + 4. The new y-coordinate is 3(x + 4) − 5 = 3x + 12 − 5 = (3x − 5) + 12. Thus, the y-coordinate increases by exactly 12 units.

Q11

The sides of a triangle are 5 cm, 12 cm, and 13 cm. The altitude drawn on the longest side is:

Area = ½ × 5 × 12 = 30 cm². Also, ½ × 13 × altitude = 30 → altitude = 60/13 cm.

Q12

If a line segment AB is parallel to another line segment CD, and O is the midpoint of AD, then any line segment passing through O and terminating at AB and CD will be bisected at O. This logical step relies directly on which pair of angles?

To prove the triangles formed are congruent (which shows the segment is bisected), we use alternate interior angles (since AB || CD) and vertically opposite angles at O.

Q13

An angle tracking device notes that in a trapezium ABCD with AB parallel to CD, angle A = 50° and angle B = 70°. What are the values of angle D and angle C respectively?

Since AB || CD, adjacent angles on the same transverse side are supplementary. Angle A + angle D = 180° → 50° + angle D = 180° → angle D = 130°. Angle B + angle C = 180° → 70° + angle C = 180° → angle C = 110°.

Q14

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.

Q15

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.