Number Systems — Chapter Test | Class 9 IMO

Q1

If a > b and b > c, then a > c. This property is called:

A relation that carries from a–b and b–c to a–c is the transitive property.

Q2

Container A has (√50 + √18) litres of milk and Container B has √32 litres. Poured together, the total milk is:

A = 5√2 + 3√2 = 8√2 and B = 4√2, so total = 12√2 litres.

Q3

Rationalise the denominator of 6 / (3√2 − 2√3):

Multiply by (3√2 + 2√3): 6(3√2 + 2√3) / ((3√2)² − (2√3)²) = 6(3√2 + 2√3)/(18 − 12) = 3√2 + 2√3.

Q4

Point P is at √2, Q at √8 and R at √18 from zero. If the pattern continues, the next point S is at:

√2, √8 = 2√2, √18 = 3√2, so the next is 4√2 = √32.

Q5

The value of (64/125)^(−2/3) is:

(64/125)^(−2/3) = (125/64)^(2/3) = ((5/4)³)^(2/3) = (5/4)² = 25/16.

Q6

If 4^x − 4^(x−1) = 24, then the value of (2x)² is:

4^(x−1)(4 − 1) = 24, so 4^(x−1) = 8 = 2³, giving 2x − 2 = 3 and x = 5/2. Then (2x)² = 5² = 25.

Q7

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.

Q8

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.

Q9

Simplify: [ { (625)^(−1/2) }^(−1/4) ]²

Multiply the powers: (−1/2) × (−1/4) × 2 = 1/4. Then (625)^(1/4) = (5⁴)^(1/4) = 5.

Q10

If x = 9 − 4√5, then √x is:

9 − 4√5 = (√5)² + 2² − 2(2)(√5) = (√5 − 2)², so √x = √5 − 2.

Q11

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.

Q12

What is the value of (2⁰ + 7¹ + 5³) / (11⁰ × 2³)?

Numerator = 1 + 7 + 125 = 133; denominator = 1 × 8 = 8; value = 133/8.

Q13

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.

Q14

A rectangular rug has length 4√3 m and width 2√3 m. Its area is:

Area = 4√3 × 2√3 = 8 × 3 = 24 square metres.

Q15

In the sequence P₁ = √3, P₂ = √12, P₃ = √27, P₄ = √48, the next term P₅ is:

The terms are 1√3, 2√3, 3√3, 4√3, so the next is 5√3 = √(25 × 3) = √75.