CBSE Class 10th Maths Important MCQs from Chapter 4 Quadratic Equations with Solutions 2023-24

1. The roots of quadratic equation 5x² – 4x + 5 = 0 are

(A) Real & Equal

(B) Real & Unequal

(C) Not real

(D) Non-real and equal

Answer:  (C)

Explanation: To find the nature, let us calculate b² – 4ac

b² – 4ac = 4² – 4 x 5 x 5

= 16 – 100

= -84 < 0

2. Equation (x+1)² – x² = 0 has _____ real root(s).

(A) 1

(B) 2

(C) 3

(D) 4

Answer:  (A)

Explanation:

Since (x + 1)² – x² = 0

⟹ x² + 1 + 2x – x² = 0

⟹ 1 + 2x = 0

⟹ x= -1/2

This gives only 1 real value of x.

3. Which constant should be added and subtracted to solve the quadratic equation 4x² − √3x + 5 = 0 by the method of completing the square?

(A) 9/16

(B) 3/16

(C) 3/4

(D) √3/4

Answer:  (B)

Explanation:

This can be written as

Hence the given equation can be solved by adding and subtracting 3/16.

4. If 1/2 is a root of the equation x² + kx – (5/4) = 0 then the value of k is

(A) 2

(B) – 2

(C) 3

(D) –3

Answer:  (A)

Explanation:

As one root of the equation x² + kx – (5/4) = 0 is 1/2

5. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.

(A) 3

(B) 8

(C) 4

(D) 7

Answer:  (B)

Explanation:

Let the number be x

Then according question,

x + 12 = 160/x

x² + 12x – 160 = 0

x² + 20x – 8x – 160 = 0

(x + 20) (x – 8) = 0

x = -20, 8

Since the number is natural, so we consider only positive value.

6. The product of two successive integral multiples of 5 is 300. Then the numbers are:

(A) 25, 30

(B) 10, 15

(C) 30, 35

(D) 15, 20

Answer:  (D)

Explanation:

Let the consecutive integral multiple be 5n and 5(n + 1) where n is a positive integer.

According to the question:

5n × 5(n + 1) = 300

⇒ n² + n – 12 = 0

⇒ (n – 3) (n + 4) = 0

⇒ n = 3 and n = – 4.

As n is a positive natural number so n = – 4 will be discarded.

Therefore the numbers are 15 and 20.

(A) 3.5

(B) 4

(C) 3

(D) – 3

Answer: (C)

Explanation:

Since y cannot be negative as negative square root is not real so y = 3.

8. If p²x² – q² = 0, then x =?

(A) ± q/p

(B) ±p/q

(C) p

(D) q

Answer:(A)

Explanation:

p²x² – q² = 0

⇒p²x² = q²

⇒x = ±p/q

(A) 3

(B)5

(C) 4

(D) 7

Answer:(B)

Explanation:

10. If x² (a² + b²) + 2x (ac + bd) + c² +d² = 0 has no real roots, then

(A) ad≠bc

(B) ad<bc

(C) ad>bc

(D) all of these

Answer: (D)

Explanation:

If equation has no real roots then discriminant of the equation must be less than zero.

11. If the one root of the equation 4x² – 2x + p – 4 = 0 be the reciprocal of other. Then value of p is

(A) 8

(B) – 8

(C) – 4

(D) 4

Answer: A

Explanation:

If one root is reciprocal of other, then product of roots is:

 

12. Rohini had scored 10 more marks in her mathematics test out of 30 marks, 9 times these marks would have been the square of her actual marks. How many marks did she get in the test?

(A) 14                                                   

(B) 16

(C) 15                    

(D) 18

Answer:  (C)

Explanation:

Let her actual marks be x

Therefore,

9 (x + 10) = x²

⇒x² – 9x – 90 = 0

⇒x² – 15x + 6x – 90 = 0

⇒x(x – 15) + 6 (x – 15) = 0

⇒(x + 6) (x – 15) = 0

Therefore  x = – 6 or x =15

Since x is the marks obtained, x ≠ – 6. Therefore, x = 15.

13. A train travels at a certain average speed for a distance of 63 km and then travels a distance of 72 km at an average speed of 6 km/h more than its original speed. If it takes 3 hours to complete the total journey, what is its original average speed?

(A) 42 km/hr

(B) 44 km/hr

(C) 46 km/hr

(D) 48 km/hr

Answer: (A)

Explanation:

Let the original speed be x,

Then according to question

This gives x = -3 and x = 42

Since speed cannot be negative, so we ignore –3,

Therefore original average speed is 42 km/hr.

14. Satvik observed that in a clock, the time needed by the minute hand of a clock to show 3 PM was found to be 3 min less than t²/4 minutes at t minutes past 2 PM. Then t is equal to

(a) 14                                                                    

(b) 15

(c) 16

(d) None of these

Answer: (A)

Explanation: We know that the time between 2 PM to 3 PM = 1 hr = 60 min

Given that at t minutes past 2 PM, the time needed by the minute’s hand of a clock to show 3 PM was found to be 3 minutes less than t²/4minutes

Therefore,

15. A takes 6 days less than B to finish a piece of work. If both A and B together can finish the work in 4 days, find the time taken by B to finish the work.

(A)12 days

(B) 12 ½ Days

(C) 13 days

(D) 15days

Answer: (A)

Explanation: Let B alone finish the work in x days.

Therefore, A alone can finish the work in (x – 6) days

A’s one day work = 1/x-6

B’s one day work = 1/x

Given that (A + B) can finish the work in 4 days.

Therefore, A’s one day work + B’s one day work = (A + B)’s one day work

As, x ≠ 2 , because if x = 2 , then A alone can finish work in (2 – 6) = – 4 days which is not possible.

Therefore we consider x = 12.

This implies B alone can finish work in 12 days and A alone will finish the work in 12 – 6 = 6 days.