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CBSE Class 10  >  Class 10 Test  >  Mathematics (Maths)   >  Test: Nature Of Roots - Class 10 MCQ

Nature Of Roots - Free MCQ Practice Test with solutions, Class 10 Maths


MCQ Practice Test & Solutions: Test: Nature Of Roots (25 Questions)

You can prepare effectively for Class 10 Mathematics (Maths) Class 10 with this dedicated MCQ Practice Test (available with solutions) on the important topic of "Test: Nature Of Roots". These 25 questions have been designed by the experts with the latest curriculum of Class 10 2026, to help you master the concept.

Test Highlights:

  • - Format: Multiple Choice Questions (MCQ)
  • - Duration: 25 minutes
  • - Number of Questions: 25

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Test: Nature Of Roots - Question 1
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The quadratic equation 2x2 – 3x + 5 = 0 has​

Test: Nature Of Roots - Question 2
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The real roots of a quadratic equation ax2 + bx + c = 0 are given by 

Detailed Solution: Question 2

When b2 - 4ac=0 then we have the root as -b/2a.
When  b2-4ac < 0,then the root is a complex root ,since we have a negative number ,for which real square root is not possible.
When b2 - 4ac > 0 then we have a square root of it and hence we have real roots. So the correct answer is b2 - 4ac > 0

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Test: Nature Of Roots - Question 3
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The solutions of the quadratic equation   are​

Detailed Solution: Question 3

Multiply by LCM of denominators (x-1) (x+2) on both sides

Left-hand side (LHS):

  1. (x + 1)(x + 2) = x² + 2x + x + 2 = x² + 3x + 2

  2. (x − 2)(x − 1) = x² − x − 2x + 2 = x² − 3x + 2

Now add both parts:

x² + 3x + 2 + x² − 3x + 2 = 2x² + 4

LHS = 2x² + 4

Right-hand side (RHS):

3(x − 1)(x + 2)
= 3[x² + 2x − x − 2] = 3[x² + x − 2]
= 3x² + 3x − 6

RHS = 3x² + 3x − 6

Comparing RHS and LHS, we get the quadratic equation

x² + 3x − 10 = 0

Now, on solving the equation, we get two solutions:

x = 2; x = -5

Test: Nature Of Roots - Question 4
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The value of b2 - 4ac  for equation 3x2 - 7x - 2 = 0 is

Test: Nature Of Roots - Question 5
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Which of the following is not a quadratic equation ?

Detailed Solution: Question 5

 A quadratic equation in one variable takes the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. Option a) 5x + 3y² = 0 includes a y² term, meaning it involves two different variables. This does not satisfy the definition of a quadratic equation in a single variable.

Test: Nature Of Roots - Question 6
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For what value(s) of k will the equation kx2-5x+k = 0 have a repeated root?​

Test: Nature Of Roots - Question 7
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A real number α is said to be a root of ax2 + bx + c = 0 if

Detailed Solution: Question 7

The correct answer is:

d) aα² + bα + c = 0

Explanation:

A real number α is a root of the quadratic equation ax² + bx + c = 0 if plugging α into the equation makes it true.

In other words, when you substitute x = α into the equation, you should get:
aα² + bα + c = 0

Why not the other options?

  • a) a(-α)² + b(-α) + c = 0 → This would mean -α is a root, not necessarily α.

  • b) aα² + bα - c = 0 → The sign of c is wrong. It must be +c, not -c.

  • c) a(α)² + b(-α) + c = 0 → This is mixing signs and treats -α as part of the linear term, which changes the meaning.

So, the only correct condition for α being a root is option d.

Test: Nature Of Roots - Question 8
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-3 is a root of the quadratic equation 2x2 +px – 15 = 0. For what value of q, the equation p(x2 + x ) + q = 0 has equal roots?​

Test: Nature Of Roots - Question 9
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The roots of x2 – 8x + 12 = 0, are

Test: Nature Of Roots - Question 10
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If 1/2 is a root of the equation x2+kx – (5/4) = 0, then the other root of the quadratic equation is​

Test: Nature Of Roots - Question 11
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The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. Find their present ages

Test: Nature Of Roots - Question 12
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The nature of the roots of following quadratic equation 

Test: Nature Of Roots - Question 13
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If the equation px2 – 6x – 2 = 0 has real roots then, ________

Test: Nature Of Roots - Question 14
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The positive value of k, for which both equations: x2 + kx + 64 = 0 and x2 – 8x + k = 0 have real roots is_______

Test: Nature Of Roots - Question 15
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If one root of a Quadratic equation is m +√n, then the other root is​

Detailed Solution: Question 15

In a quadratic equation with rational coefficients has an irrational root  α + √β, then it has a conjugate root α - √β.
So if the root is m+ √n the other root will be m- √n

Test: Nature Of Roots - Question 16
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If the value of the Discriminant function of a quadratic equation is D = 27, then its roots are

Detailed Solution: Question 16

Given:

Discriminant D = 27

What is the Discriminant?

For a quadratic equation ax² + bx + c = 0, the discriminant is:

D = b² - 4ac

Now interpret the value of D = 27:

  • D > 0 → Roots are real and distinct

  • But 27 is not a perfect square, so the roots are irrational

Therefore, the roots are:

Real, Distinct, and Irrational

Correct Answer:

c) Distinct, Irrational

Test: Nature Of Roots - Question 17
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Comment on the nature of the roots of the equation 7x - 3x2 - 2 = 0​

Test: Nature Of Roots - Question 18
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The roots of the equation x2 – 3x – m (m + 3) = 0, where m is a constant, are​

Detailed Solution: Question 18

X2 - 3x - m(m+3) = 0

=> x2 + mx - (m+3)x- m(m+3) = 0

=> x(x+m) - (m+3)(x+m) = 0

=> (x+m) (x-m-3) = 0

x = - m and m+3

Roots are - m and m+3

Test: Nature Of Roots - Question 19
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If the equation 9x2 – 6x + c = 0 has equal roots then, the value of c is 

Test: Nature Of Roots - Question 20
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If -3 is a root of both the Quadratic equations 2x2 + px – 15 = 0, and p(x2 + x) + q = 0, then the value of q if the equation containing it has equal roots is

Detailed Solution: Question 20

-3 is root of both

2(9) -3p -15 = 0

p = 1

and

second equation has two equal roots so, 

b2-4ac = 0

p2 = 4pq

p = 4q

q = 1/4

Test: Nature Of Roots - Question 21
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If a,b,c are real and b2-4ac >0 then roots of equation are​

Test: Nature Of Roots - Question 22
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The quadratic equation whose roots are real and equal is:​

Test: Nature Of Roots - Question 23
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Which of the following equations has two distinct real roots?​

Test: Nature Of Roots - Question 24
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If the equation 2x2 + 2(c - 20) x + (c - 20) = 0 has equal roots then, the value of c is

Test: Nature Of Roots - Question 25
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Which of the following is a solution of the quadratic equation x2 - b2 = a (2x - a)

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About this Class 10 Practice Test

This test contains 25 MCQs on Nature Of Roots with detailed solutions for each question. It is part of the Mathematics (Maths) Class 10 course on EduRev, designed to align with the current Class 10 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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