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CBSE Class 9  >  Class 9 Test  >  Mathematics Olympiad  >  Math Olympiad Test: Number System- 2 - Class 9 MCQ

Math Olympiad Test: Number System- 2 - Free MCQ with solutions Class 9


MCQ Practice Test & Solutions: Math Olympiad Test: Number System- 2 (15 Questions)

You can prepare effectively for Class 9 Mathematics Olympiad for Class 9 with this dedicated MCQ Practice Test (available with solutions) on the important topic of "Math Olympiad Test: Number System- 2". These 15 questions have been designed by the experts with the latest curriculum of Class 9 2026, to help you master the concept.

Test Highlights:

  • - Format: Multiple Choice Questions (MCQ)
  • - Duration: 15 minutes
  • - Number of Questions: 15

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Math Olympiad Test: Number System- 2 - Question 1
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If x - 3 is a factor of αx2 + 18 = 0, then the value of a is:

Detailed Solution: Question 1

x - 3 is a factor ⇒ x = 3 is a root of the polynomial f(x) = αx2 + 18.
Substituting the value of x in the given polynomial
f(3) = α·32 + 18 = 0
9α + 18 = 0 ⇒ 9α = −18 ⇒ α = −2.

Therefore α = −2, so option C is correct.

Math Olympiad Test: Number System- 2 - Question 2
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If  x - 1 is a factor of 4x3 + 3x2 - 4x + k , then k =

Detailed Solution: Question 2

If x = 1 is a factor of f(x), then f(1) = 0
⇒ = f(1) = 4 (1)3 + 3(1)2 - 4 (1) + k = 0
= 4 + 3 - 4 + k = 0
⇒ k = -3

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Math Olympiad Test: Number System- 2 - Question 3
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Factors of x2 + 6√2x + 10 will be

Detailed Solution: Question 3

x2 + 6√2 x + 10 can be factored using the identity:
x2 + (a + b)x + ab = (x + a)(x + b)

So we need two numbers a and b whose sum is 6√2 and product is 10.

Try 5√2 and √2:

5√2 + √2 = 6√2
(5√2)(√2) = 10

So they are the correct numbers.

Using the identity:

x2 + 6√2 x + 10 = (x + 5√2)(x + √2)

Correct answer: (x + 5√2) (x + √2)

Math Olympiad Test: Number System- 2 - Question 4
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If  x + 1/x = 3 then x4 + 1/x4 = 

Detailed Solution: Question 4

Given x + 1/x = 3

Math Olympiad Test: Number System- 2 - Question 5
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If α = b = c, then (a + b + c)2 - xα2 = 0, then x =

Detailed Solution: Question 5

If α = b = c, then
(α + b + c) = (α + α + α) = 3α
∴ (a + b + c)2 - xα2 = (3α)2 - xα2 = 0
⇒ xα2 = 9α2
⇒ x = 9

Math Olympiad Test: Number System- 2 - Question 6
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The value of 9α² + 4b² + 16c² + 12αb − 24αc − 16bc for α = 2, b = 1, c = −2 is

Detailed Solution: Question 6

The value of 9α² + 4b² + 16c² + 12αb − 24αc − 16bc for α = 2, b = 1, c = −2 is

Math Olympiad Test: Number System- 2 - Question 7
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If 3x + 2y = 13 and xy = 6, then 27x3 + 8y3 will be equal to

Detailed Solution: Question 7

Given 3x + 2y = 13, xy = 6
∵ 3x + 2y = 13
∴ (3x + 2y)3 = (13)3
⇒ 27x3 + 8y3 + 3 × 3x × 2y (3x + 2y)
= (169) × 13
⇒ 27x3 + 8y3 + 18xy (3x + 2y) = (169) × 13
⇒ 27x3 + 8y3 + 18 × 6 × (13) = (169) × 13
⇒ 27x3 + 8y3 = 61 × 13 = 793

Math Olympiad Test: Number System- 2 - Question 8
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The value of (x - 1) (x2 + 1 + x) (x6 + x3 + 1), will be equal to 

Detailed Solution: Question 8

We have (x - 1) (x2 + x + 1) = (x3 - 1)
Now,
Let x3 = p, then x6 = p2
∵ (x3 - 1) (x6 + x3 + 1)
= (p - 1)(p2 + p + 1) = (p3 - 1)
= (x3)3 - 1
= x9 - 1

Math Olympiad Test: Number System- 2 - Question 9
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For which of the following polynomials is (x2 - x + 1) a factor?

Detailed Solution: Question 9

To determine if (x2 - x + 1) is a factor of the given polynomials, we can use polynomial division or check if the polynomial equals zero when x2 - x + 1 is equal to zero.

1. For the first polynomial, x2 + x + 1, substituting the roots of x2 - x + 1 does not yield zero.

2. The second polynomial, x4 - 7x2 + 2, does not give zero when evaluated with the roots of x2 - x + 1.

3. Evaluating the third polynomial, x4 + x2 + 1, also does not result in zero.

4. The fourth polynomial, x4 + 2x2 + 1, similarly fails to yield zero when substituting the roots.

Thus, after testing all options, we find that only the third polynomial does not contradict our factor check, confirming that (x2 - x + 1) is indeed a factor.

Math Olympiad Test: Number System- 2 - Question 10
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If x + y = 4, xy = 4, then 2x3 + y3 will be equal to

Detailed Solution: Question 10

Given x + y = 4, xy = 4
⇒ (x + y)2 = (x - y)2 + 4xy
⇒ (4)2 = (x - y)2 + 4 × 4
⇒ (x - y)2 = 16 - 16 = 0
⇒ x = y
∴ if x + y = 4
⇒ y = x = 2
Hence, 2x3 + y3 = 3x3 = 3(2)3 = 24

Math Olympiad Test: Number System- 2 - Question 11
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If the polynomials αx3 + 4x2 + 3x - 4 and x3 - 4x + α equal to

Detailed Solution: Question 11

f(3) will be remained in both cases.
∴ α(3)3 + 4(3)3 + 3(3) - 4 = (3)3 - 4(3) + α
⇒ 27α + 36 + 9 - 4 = 27 - 12 + a
⇒ 26α = 15 - 5 - 36
⇒ 26α = -26
⇒ α = -1

Math Olympiad Test: Number System- 2 - Question 12
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If α + b + c = 15, and α2 + b2 + c2 = 83, then the value of α3 + b3 + c3 - 3αbc will be  

Detailed Solution: Question 12

α3 + b3 + c3 - 3αbc = (α + b + c) (α2 + b2 + c2 - αb - bc - cα) = (15) (83 - 71)
= 15 (12)
= 180

Math Olympiad Test: Number System- 2 - Question 13
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If α + b + c = 15 and α2 + b2 + c2 = 83 then the value of (αb + bc + cα) will be equal to

Detailed Solution: Question 13

Here (α + b + c)2 = (15)2
⇒ (α2 + b2 + c2) = (15)2 - 2(αb + bc + cα)
⇒ (83) - 225/2 = - (αb + bc + cα)
⇒ - 71 = - (αb + bc + cα) ...(i)
⇒ αb + bc + cα = 71 ...(i)

Math Olympiad Test: Number System- 2 - Question 14
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The expression (α - b)3 + (b - c)3 + (c - α)3 can be factorised as 

Detailed Solution: Question 14

Let α - b = A, b - c = B,  c - α = C
Then, A + B + C = (α - b) + (b - c) + (c - α) = 0
∴ A3 + B3 + C3 = 3ABC
⇒ 3(α - b) (b - c) (c - α)

Math Olympiad Test: Number System- 2 - Question 15
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If α + b + c = 6 and α3 + b3 + c3 = , then αb + bc + cα will have the value equal to

Detailed Solution: Question 15

Given α + b + c = 6 , and, α3 + b3 + c3 = 18 + 3αbc
⇒ a3 + b3 + c3 - 3αbc = 18

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About this Class 9 Practice Test

This test contains 15 MCQs on Math Olympiad Test: Number System- 2 with detailed solutions for each question. It is part of the Mathematics Olympiad for Class 9 course on EduRev, designed to align with the current Class 9 syllabus. Each solution explains not just the correct answer but also gives you an understanding of the topic — not just to attempt the question but also help you prepare for the exam with practice.

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