## Class 10 Maths MCQs Chapter 4 Quadratic Equations

1. Which of the following is not a quadratic equation

(a) x² + 3x – 5 = 0

(b) x² + x3 + 2 = 0

(c) 3 + x + x² = 0

(d) x² – 9 = 0

**Answer/Explanation**

Answer: b

Explaination:Reason: Since it has degree 3.

2. The quadratic equation has degree

(a) 0

(b) 1

(c) 2

(d) 3

**Answer/Explanation**

Answer: c

Explaination:Reason: A quadratic equation has degree 2.

3. The cubic equation has degree

(a) 1

(b) 2

(c) 3

(d) 4

**Answer/Explanation**

Answer: c

Explaination:Reason: A cubic equation has degree 3.

4. A bi-quadratic equation has degree

(a) 1

(b) 2

(c) 3

(d) 4

**Answer/Explanation**

Answer: d

Explaination:Reason: A bi-quadratic equation has degree 4.

5. The polynomial equation x (x + 1) + 8 = (x + 2) {x – 2) is

(a) linear equation

(b) quadratic equation

(c) cubic equation

(d) bi-quadratic equation

**Answer/Explanation**

Answer: a

Explaination:Reason: We have x(x + 1) + 8 = (x + 2) (x – 2)

⇒ x² + x + 8 = x² – 4

⇒ x² + x + 8- x² + 4 = 0

⇒ x + 12 = 0, which is a linear equation.

6. The equation (x – 2)² + 1 = 2x – 3 is a

(a) linear equation

(b) quadratic equation

(c) cubic equation

(d) bi-quadratic equation

**Answer/Explanation**

Answer: b

Explaination:Reason: We have (x – 2)² + 1 = 2x – 3

⇒ x² + 4 – 2 × x × 2 + 1 = 2x – 3

⇒ x² – 4x + 5 – 2x + 3 = 0

∴ x² – 6x + 8 = 0, which is a quadratic equation.

7. The roots of the quadratic equation 6x² – x – 2 = 0 are

**Answer/Explanation**

Answer: c

Explaination:Reason: We have 6×2 – x – 2 = 0

⇒ 6x² + 3x-4x-2 = 0

⇒ 3x(2x + 1) -2(2x + 1) = 0

⇒ (2x + 1) (3x – 2) = 0

⇒ 2x + 1 = 0 or 3x – 2 = 0

∴ x =\(-\frac{1}{2}\), x = \(\frac{2}{3}\)

8. The quadratic equation whose roots are 1 and

(a) 2x² + x – 1 = 0

(b) 2x² – x – 1 = 0

(c) 2x² + x + 1 = 0

(d) 2x² – x + 1 = 0

**Answer/Explanation**

Answer: b

Explaination:Reason: Required quadratic equation is

9. The quadratic equation whose one rational root is 3 + √2 is

(a) x² – 7x + 5 = 0

(b) x² + 7x + 6 = 0

(c) x² – 7x + 6 = 0

(d) x² – 6x + 7 = 0

**Answer/Explanation**

Answer: d

Explaination:Reason: ∵ one root is 3 + √2

∴ other root is 3 – √2

∴ Sum of roots = 3 + √2 + 3 – √2 = 6

Product of roots = (3 + √2)(3 – √2) = (3)² – (√2)² = 9 – 2 = 7

∴ Required quadratic equation is x² – 6x + 7 = 0

10. The equation 2x² + kx + 3 = 0 has two equal roots, then the value of k is

(a) ±√6

(b) ± 4

(c) ±3√2

(d) ±2√6

**Answer/Explanation**

Answer: d

Explaination:Reason: Here a = 2, b = k, c = 3

Since the equation has two equal roots

∴ b² – 4AC = 0

⇒ (k)² – 4 × 2 × 3 = 0

⇒ k² = 24

⇒ k = ± √24

∴ k= ± \(\pm \sqrt{4 \times 6}\) = ± 2√6

11. The roots of the quadratic equation \(x+\frac{1}{x}=3\), x ≠ 0 are.

**Answer/Explanation**

Answer: c

Explaination:Reason: We have \(x+\frac{1}{x}=3\)

⇒ \(\frac{x^{2}+1}{x}=3\)

⇒ x² + 1 = 3x

On comparing with ax² + bx + c = 0

∴ a = 1, b = – 3, c = 1

⇒ D = b² – 4ac = (-3)² – 4 × (1) × (1) = 9 – 4 = 5

12. The roots of the quadratic equation 2x² – 2√2x + 1 = 0 are

**Answer/Explanation**

Answer: c

Explaination:Reason: Here a = 2, b = -2√2 , c = 1

∴ D = b² – 4ac = (-2√2 )² – 4 × 2 × 1 = 8 – 8 = 0

13. The sum of the roots of the quadratic equation 3×2 – 9x + 5 = 0 is

(a) 3

(b) 6

(c) -3

(d) 2

**Answer/Explanation**

Answer: c

Explaination:Reason: Here a = 3, b = -9, c = 5

∴ Sum of the roots \(=\frac{-b}{a}=-\frac{(-9)}{3}=3\)

14. If the roots of ax2 + bx + c = 0 are in the ratio m : n, then

(a) mna² = (m + n) c²

(b) mnb² = (m + n) ac

(c) mn b² = (m + n)² ac

(d) mnb² = (m – n)² ac

**Answer/Explanation**

Answer: c

Explaination:

15. If one root of the equation x² + px + 12 = 0 is 4, while the equation x² + px + q = 0 has equal roots, the value of q is

**Answer/Explanation**

Answer: a

Explaination:Reason: Since 4 is a root of x² + px + 12 = 0

∴ (4)² + p(4) + 12 = 0

⇒ p = -7

Also the roots of x² + px + q = 0 are equal, we have p² – 4 x 1 x q = 0

⇒ (-7)² -4q = 0

\(\therefore q=\frac{49}{4}\)

16. a and p are the roots of 4x² + 3x + 7 = 0, then the value of \(\frac{1}{\alpha}+\frac{1}{\beta}\) is

**Answer/Explanation**

Answer: b

Explaination:

17. If a, p are the roots of the equation (x – a) (x – b) + c = 0, then the roots of the equation (x – a) (x – P) = c are

(a) a, b

(b) a, c

(c) b, c

(d) none of these

**Answer/Explanation**

Answer: a

Explaination:Reason: By given condition, (x – a) (x – b) + c = (x – α) (x – β)

⇒ (x – α) (x – β) – c = (x – a) (x – b)

This shows that roots of (x – α) (x – β) – c are a and b

18. Mohan and Sohan solve an equation. In solving Mohan commits a mistake in constant term and finds the roots 8 and 2. Sohan commits a mistake in the coefficient of x. The correct roots are

(a) 9,1

(b) -9,1

(c) 9, -1

(d) -9, -1

**Answer/Explanation**

Answer: a

Explaination:Reason: Correct sum = 8 + 2 = 10 from Mohan

Correct product = -9 x -1 = 9 from Sohan

∴ x² – (10)x + 9 = 0

⇒ x² – 10x + 9 = 0

⇒ x² – 9x – x + 9

⇒ x(x – 9) – 1(x – 9) = 0

⇒ (x-9) (x-l) = 0 .

⇒ Correct roots are 9 and 1.

19. If a and p are the roots of the equation 2x² – 3x – 6 = 0. The equation whose roots are \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is

(a) 6x² – 3x + 2 = 0

(b) 6x² + 3x – 2 = 0

(c) 6x² – 3x – 2 = 0

(d) x² + 3x-2 = 0

**Answer/Explanation**

Answer: b

Explaination:

20. If the roots of px2 + qx + 2 = 0 are reciprocal of each other, then

(a) P = 0

(b) p = -2

(c) p = ±2

(d) p = 2

**Answer/Explanation**

Answer: d

Explaination:Reason: here α = \(\frac{1}{β}\)

∴ αβ = 1

⇒ \(\frac{2}{p}\) = 1

∴ p = 2

21. If one root of the quadratic equation 2x² + kx – 6 = 0 is 2, the value of k is

(a) 1

(b) -1

(c) 2

(d) -2

**Answer/Explanation**

Answer: b

Explaination:Reason: Scice x = 2 is a root of the equation 2x² + kx -6 = 0

∴ 2(2)² +k(2) – 6 = 0

⇒ 8 + 2k – 6 = 0

⇒ 2k = -2

∴ k = -1

22. The roots of the quadratic equation

(a) a, b

(b) -a, b

(c) a, -b

(d) -a, -b

**Answer/Explanation**

Answer: d

Explaination:

23. The roots of the equation 7x² + x – 1 = 0 are

(a) real and distinct

(b) real and equal

(c) not real

(d) none of these

**Answer/Explanation**

Answer: a

Explaination:Reason: Here a = 2, b = 1, c = -1

∴ D = b² – 4ac = (1)² – 4 × 2 × (-1) = 1 + 8 = 9 > 0

∴ Roots of the given equation are real and distinct.

24. The equation 12x² + 4kx + 3 = 0 has real and equal roots, if

(a) k = ±3

(b) k = ±9

(c) k = 4

(d) k = ±2

**Answer/Explanation**

Answer: a

Explaination:Reason: Here a = 12, b = 4k, c = 3

Since the given equation has real and equal roots

∴ b² – 4ac = 0

⇒ (4k)² – 4 × 12 × 3 = 0

⇒ 16k² – 144 = 0

⇒ k² = 9

⇒ k = ±3

25. If -5 is a root of the quadratic equation 2x² + px – 15 = 0, then

(a) p = 3

(b) p = 5

(c) p = 7

(d) p = 1

**Answer/Explanation**

Answer: c

Explaination:Reason: Since – 5 is a root of the equation 2x² + px -15 = 0

∴ 2(-5)² + p (-5) – 15 = 0

⇒ 50 – 5p -15 = 0

⇒ 5p = 35

⇒ p = 7

26. If the roots of the equations ax² + 2bx + c = 0 and bx² – 2√ac x + b = 0 are simultaneously real, then

(a) b = ac

(b) b2 = ac

(c) a2 = be

(d) c2 = ab

**Answer/Explanation**

Answer: b

Explaination:Reason: Given equations have real roots, then

D_{1} ≥ 0 and D_{2} ≥ 0

(2b)² – 4ac > 0 and (-2√ac)² – 4b.b ≥ 0

4b² – 4ac ≥ 0 and 4ac – 4b2 > 0

b² ≥ ac and ac ≥ b²

⇒ b² = ac

27. The roots of the equation (b – c) x² + (c – a) x + (a – b) = 0 are equal, then

(a) 2a = b + c

(b) 2c = a + b

(c) b = a + c

(d) 2b = a + c

**Answer/Explanation**

Answer: d

Explaination:Reason: Since roots are equal

∴ D = 0 => b² – 4ac = 0

⇒ (c – a)² -4(b – c) (a – b) = 0

⇒ c² – b² – 2ac -4(ab -b² + bc) = 0 =>c + a-2b = 0 => c + a = 2b

⇒ c² + a² – 2ca – 4ab + 4b² + 4ac – 4bc = 0

⇒ c² + a² + 4b² + 2ca – 4ab – 4bc = 0

⇒ (c + a – 2b)² = 0

⇒ c + a – 2b = 0

⇒ c + a = 2b

28. A chess board contains 64 equal squares and the area of each square is 6.25 cm². A border round the board is 2 cm wide. The length of the side of the chess board is

(a) 8 cm

(b) 12 cm

(c) 24 cm

(d) 36 cm

**Answer**

Answer: c

29. One year ago, a man was 8 times as old as his son. Now his age is equal to the square of his son’s age. Their present ages are

(a) 7 years, 49 years

(b) 5 years, 25 years

(c) 1 years, 50 years

(d) 6 years, 49 years

**Answer**

Answer: a

30. The sum of the squares of two consecutive natural numbers is 313. The numbers are

(a) 12, 13

(b) 13,14

(c) 11,12

(d) 14,15

**Answer**

Answer: a

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