Chapter 4 – Quadratic Equations

Class 10 Mathematics · 29 questions · 25 with answers

Solved examples

Ex. 1Multiple choice

Which one of the following is not a quadratic equation?

  • (A)(x + 2)2 = 2(x + 3)
  • (B)x2 + 3x = (–1) (1 – 3x)2
  • (C)(x + 2) (x – 1) = x2 – 2x – 3
  • (D)x3 – x2 + 2x + 1 = (x + 1)3
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Solution : Answer (C)

Ex. 2Short answer

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

Ex. 1Short answerExercise 4.1

Does (x – 1)2 + 2(x + 1) = 0 have a real root? Justify your answer.

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Solution : No, since the equation is simplified to x2 + 3 = 0 whose discriminant is –12.

Ex. 2True / FalseExercise 4.1

Is the following statement ‘True’ or ‘False’?Justify your answer. If in a quadratic equation the coefficient of x is zero, then the quadratic equation has no real roots.

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Solution : False, since the discriminant in this case is – 4ac which can still be non- negative if a and c are of opposite signs or if one of a or c is zero.

Ex. 1Short answerExercise 4.2

Find the roots of the quadratic equation 2x2 – 5x – 2 = 0 using the quadratic formula.

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Solution : b2 – 4ac = 5 – 4 × 2 × (–2) = 21 5 ± 21 5 + 21 5 − 21 Therefore, the roots are , i.e., and 4 4 4

Ex. 2Short answerExercise 4.2

Find the roots of 6x2– 2x – 2 = 0 by the factorisation of the corresponding quadratic polynomial.

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Solution : 6x2 – 2x – 2 = 6x2 – 3 2 x + 2 2 x – 2 = 3x (2x– 2 ) + 2 (2x – 2) = (3x + 2 ) (2x – 2) Now, 6x2 – 2x – 2 = 0 gives (3x + 2 ) (2x – 2 ) = 0, i.e., 3x + 2 = 0 or 2x – 2=0 2 2 So, the roots are − and . 3 2

Ex. 1Short answerExercise 4.3

Check whether the equation 6x2 – 7x + 2 = 0 has real roots, and if it has, find them by the method of completing the squares.

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Solution : The discriminant = b2 – 4ac = 49 – 4 × 6 × 2 = 1 > 0 So, the given equation has two distinct real roots. Now, 6x2 – 7x + 2 = 0 i.e., 36x2 – 42x + 12 = 0 7 49 i.e., 6x − + 12 – =0 2 4 2 2 7 1 2 2 7 1 i.e., 6x − – = 0 or 6 x – = 2 2 2 2 QUADRATIC EQUATIONS 41 7 1 The roots are given by 6 x − = ± 2 2 i.e., 6x = 4, 3 2 1 i.e., x = , . 3 2

Ex. 2Short answerExercise 4.3

Had Ajita scored 10 more marks in her mathematics test out of

Ex. 3Short answerExercise 4.3

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?

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Solution : Let its original average speed be x km/h. Therefore, 63 72 + =3 x x+6 7 8 3 1 i.e., + = = x x+6 9 3 7( x + 6) + 8 x 1 i.e., = x ( x + 6) 3 i.e., 21 (x + 6) + 24x = x (x + 6) i.e., 21x + 126 + 24x = x2 + 6x i.e., x2 – 39x – 126 = 0 i.e., (x + 3) (x – 42) = 0 i.e., x = – 3 or x = 42 Since x is the average speed of the train, x cannot be negative. Therefore, x = 42. So, the original average speed of the train is 42 km/h.

Questions

Q9Multiple choice

3 3 3

  • (A)
  • (B)
  • (C)
  • (D)16 16 4 4 Solution : Answer (B)
Q1Multiple choiceExercise 4.1

Which of the following is a quadratic equation? 2

  • (A)x2 + 2x + 1 = (4 – x)2 + 3
  • (B)–2x2 = (5 – x) 2 x − 5
  • (C)(k + 1)x2 + x = 7, where k = –1
  • (D)x3 – x2 = (x – 1)3
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(D) x3 – x2 = (x – 1)3

Q2Multiple choiceExercise 4.1

Which of the following is not a quadratic equation?

  • (A)2(x – 1)2 = 4x2 – 2x + 1
  • (B)2x – x2 = x2 + 5
  • (C)( 2 x + 3)2 + x 2 = 3x 2 − 5 x
  • (D)(x2 + 2x)2 = x4 + 3 + 4x3
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(C) ( 2 x + 3)2 + x 2 = 3x 2 − 5 x

Q3Multiple choiceExercise 4.1

Which of the following equations has 2 as a root?

  • (A)x2 – 4x + 5 = 0
  • (B)x2 + 3x – 12 = 0
  • (C)2x2 – 7x + 6 = 0
  • (D)3x2 – 6x – 2 = 0 QUADRATIC EQUATIONS 37
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(C) 2x2 – 7x + 6 = 0

Q1Short answerExercise 4.1

5

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(D)

Q4Multiple choiceExercise 4.1

If is a root of the equation x2 + kx – = 0, then the value of k is 2 4 1 1

  • (A)2
  • (B)– 2
  • (C)
  • (D)4 2
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(A) 2

Q5Multiple choiceExercise 4.1

Which of the following equations has the sum of its roots as 3?

  • (A)2x2 – 3x + 6 = 0
  • (B)–x2 + 3x – 3 = 0
  • (C)2 x2 − x +1 = 0
  • (D)3x2 – 3x + 3 = 0
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(B) –x2 + 3x – 3 = 0

Q6Multiple choiceExercise 4.1

Values of k for which the quadratic equation 2x2 – kx + k = 0 has equal roots is

  • (A)0 only
  • (B)4
  • (C)8 only
  • (D)0, 8
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(D) 0, 8

Q7Short answerExercise 4.1

Which constant must be added and subtracted to solve the quadratic equation 9x2 + x – 2 = 0 by the method of completing the square?

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(B)

Q1Multiple choiceExercise 4.1

1 1 9

  • (A)
  • (B)
  • (C)
  • (D)8 64 4 64 8. The quadratic equation 2x2 – 5x + 1 = 0 has (A) two distinct real roots (B) two equal real roots (C) no real roots (D) more than 2 real roots 9. Which of the following equations has two distinct real roots? (A) 2x2 – 3 2 x + =0 (B) x2 + x – 5 = 0 (C) x2 + 3x + 2 2 = 0 (D) 5x2 – 3x + 1 = 0 10. Which of the following equations has no real roots? (A) x2 – 4x + 3 2 = 0 (B) x2 + 4x – 3 2 = 0 (C) x2 – 4x – 3 2 = 0 (D) 3x2 + 4 3 x + 4 = 0 11. (x2 + 1)2 – x2 = 0 has (A) four real roots (B) two real roots (C) no real roots (D) one real root.
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(D) 8 64 4 64 8. The quadratic equation 2x2 – 5x + 1 = 0 has (A) two distinct real roots (B) two equal real roots (C) no real roots (D) more than 2 real roots 9. Which of the following equations has two distinct real roots? (A) 2x2 – 3 2 x + =0 (B) x2 + x – 5 = 0 (C) x2 + 3x + 2 2 = 0 (D) 5x2 – 3x + 1 = 0 10. Which of the following equations has no real roots? (A) x2 – 4x + 3 2 = 0 (B) x2 + 4x – 3 2 = 0 (C) x2 – 4x – 3 2 = 0 (D) 3x2 + 4 3 x + 4 = 0 11. (x2 + 1)2 – x2 = 0 has (A) four real roots (B) two real roots (C) no real roots (D) one real root.

Q1Multiple choiceExercise 4.2

State whether the following quadratic equations have two distinct real roots. Justify your answer.

  • (i)x2 – 3x + 4 = 0
  • (ii)2x2 + x – 1 = 0
  • (iii)2x2 – 6x + =0
  • (iv)3x2 – 4x + 1 = 0 (v) (x + 4)2 – 8x = 0 (vi) (x – 2 2 ) – 2(x + 1) = 0
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(i) x2 – 3x + 4 = 0

(ii) 2x2 + x – 1 = 0

(iii) 2x2 – 6x + =0

(iv) 3x2 – 4x + 1 = 0 (v) (x + 4)2 – 8x = 0 (vi) (x – 2 2 ) – 2(x + 1) = 0

Q3Multiple choiceExercise 4.2

1 (vii) 2 x2 – x + = 0 (viii) x (1 – x) – 2 = 0 2 2 (ix) (x – 1) (x + 2) + 2 = 0 (x) (x + 1) (x – 2) + x = 0 2. Write whether the following statements are true or false. Justify your answers.

  • (i)Every quadratic equation has exactly one root.
  • (ii)Every quadratic equation has at least one real root.
  • (iii)Every quadratic equation has at least two roots.
  • (iv)Every quadratic equations has at most two roots. (v) If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots. QUADRATIC EQUATIONS 39 (vi) If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots. 3. A quadratic equation with integral coefficient has integral roots. Justify your answer.
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x2–3x + 1 = 0 is an equation with integral coefficients but its roots are not integers.

Q4Short answerExercise 4.2

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

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x 2 – 6 x 7 0 , which has roots 3 2, 3– 2

Q5Short answerExercise 4.2

Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals? Why?

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Yes. 3x 2 – 7 3 x 12 3 0, which has roots 3, 4

Q6Short answerExercise 4.2

Is 0.2 a root of the equation x2 – 0.4 = 0? Justify.

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No.

Q7Short answerExercise 4.2

If b = 0, c < 0, is it true that the roots of x2 + bx + c = 0 are numerically equal and opposite in sign? Justify.

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Yes

Q1Multiple choiceExercise 4.3

Find the roots of the quadratic equations by using the quadratic formula in each of the following:

  • (i)2x2 – 3x – 5 = 0
  • (ii)5x2 + 13x + 8 = 0
  • (iii)–3x2 + 5x + 12 = 0
  • (iv)–x2 + 7x – 10 = 0 (v) x2 + 2 2 x – 6 = 0 (vi) x2 – 3 5 x + 10 = 0
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(i) 2x2 – 3x – 5 = 0

(ii) 5x2 + 13x + 8 = 0

(iii) –3x2 + 5x + 12 = 0

(iv) –x2 + 7x – 10 = 0 (v) x2 + 2 2 x – 6 = 0 (vi) x2 – 3 5 x + 10 = 0

Q1Short answerExercise 4.3

2 (vii) x – 11 x + 1 = 0

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(i) , –1 (ii) –1, – (iii) – , 3 (iv) 5, 2 2 5 3 (v) –3 2, 2 (vi) 5, 2 5 (vii) 11 3 , 11 –3 3 2 1 2

Q2Short answerExercise 4.3

Find the roots of the following quadratic equations by the factorisation method:

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(i) – , (ii) – ,3 (iii) 2, – 2 3 2 6 pu T 5 ,–2 5 1 1 , (iv) (v) 3 21 21 1. (i) Real roots exist; roots are

Q5Multiple choiceExercise 4.3

2 2 3

  • (i)2x2 + x –2=0
  • (ii)x –x– =0 3 5 5
  • (iii)3 2x 2 – 5x – 2 =0
  • (iv)3x2 + 5 5x – 10 = 0 (v) 21x2 – 2x + =0
Q30Long answerExercise 4.3

marks, 9 times these marks would have been the square of her actual marks. How many marks did she get in the test? Solution : Let her actual marks be x Therefore, 9 (x +10) = x2 i.e., x2 – 9x – 90 = 0 i.e., x2 – 15x + 6x – 90 = 0 i.e., x(x – 15) + 6(x –15) = 0 i.e., (x + 6) (x –15) = 0 Therefore, x=–6 or x =15 Since x is the marks obtained, x ≠ – 6. Therefore, x = 15. So, Ajita got 15 marks in her mathematics test.

Q1Multiple choiceExercise 4.4

Find whether the following equations have real roots. If real roots exist, find them.

  • (i)8x2 + 2x – 3 = 0
  • (ii)–2x2 + 3x + 2 = 0 1 1 3
  • (iii)5x2 – 2x – 10 = 0
  • (iv)+ =1, x ≠ ,5 2x −3 x −5 2 (v) x2 + 5 5 x – 70 = 0
Q2Short answerExercise 4.4

Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

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The natural number is 12

Q3Short answerExercise 4.4

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

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The natural number is 8

Q4Short answerExercise 4.4

A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.

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Original speed of the train is 45 km/h

Q5Short answerExercise 4.4

If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?

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Zeba’s age now is 14 years

Q6Short answerExercise 4.4

At present Asha’s age (in years) is 2 more than the square of her daughter Nisha’s age. When Nisha grows to her mother’s present age, Asha’s age would be one year less than 10 times the present age of Nisha. Find the present ages of both Asha and Nisha. QUADRATIC EQUATIONS 43

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Nisha’s age is 5 years and Asha’s age is 27 years

Q7Short answerExercise 4.4

In the centre of a rectangular lawn of dimensions 50 m × 40 m, a rectangular pond has to be constructed so that the area of the grass surrounding the pond would be 1184 m2 [see Fig. 4.1]. Find the length and breadth of the pond.

This question refers to a figure in the original PDF.

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Length of the pond is 34 m and breadth is 24 m

Q8Short answerExercise 4.4

At t minutes past 2 pm, the time needed by the minutes hand of a clock to show t2 3 pm was found to be 3 minutes less than minutes. Find t.

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14 ANSWERS 189