ML Aggarwal Quadratic Equations Exe-5.4 Class 10 ICSE Maths Solutions

ML Aggarwal Quadratic Equations Exe-5.4 Class 10 ICSE Maths Solutions . We Provide Step by Step Answer of Exe-5.4 ( discriminant ) Questions for Quadratic Equations in One Variable as council prescribe guideline for upcoming board exam.  Visit official Website  CISCE  for detail information about ICSE Board Class-10.

ML Aggarwal Ch-5 Quadratic Equations in one Variable Exercise- 5.4 Class 10 ICSE Maths Solutions

Board ICSE
Publications Avichal Publishig Company (APC)
Subject Maths
Class 10th
Chapter-5 Quadratic Equations in one Variable
Writer ML Aggarwal
Book Name Understanding
Topics Solution of Exe-5.4 ( discriminant ) Questions
Edition 2022-2023

 

 Quadratic Equations in one Variable Exe-5.4

(Class 10 ICSE ML Aggarwal Maths Solutions)

page-75

Question- 1 Find the discriminant of the following equations and hence find the nature of roots:
(i) 3x² – 5x – 2 = 0
(ii) 2x² – 3x + 5 = 0
(iii) 16x² – 40x + 25 = 0
(iv) 2x² + 15x + 30 = 0.

Answer -1

(i) 3x² – 5x – 2 = 0
Here a = 3, b = -5, c = -2

D = b2 – 4ac

= (-5)2 – 4(3) (-2)

= 25 + 24

= 49

Discriminate, D = 49

D > 0

Hence Roots are real and distinct.

(ii) 2x² – 3x + 5 = 0

a = 2, b = -3, c = 5

using the formula,

D = b2 – 4ac

= (-3)2 – 4(2) (5)

= 9 – 40

= -31

Discriminate, D = -31

D < 0

Hence Roots are not real.

(iii) 16x² – 40x + 25 = 0

a = 16, b = -40, c = 25

using the formula,

D = b2 – 4ac

= (-40)2 – 4(16) (25)

= 1600 – 1600

= 0

Discriminate, D = 0

D = 0

Hence Roots are real and equal.

(iv) 2x² + 15x + 30 = 0

Let us consider,

a = 2, b = 15, c = 30

using the formula,

D = b2 – 4ac

= (15)2 – 4(2) (30)

= 225 – 240

= – 15

Discriminate, D = -15

D < 0

Hence Roots are not real.

Question -2  Discuss the nature of the roots of the following quadratic equations :
(i) 3x² – 4√3x + 4 = 0
(ii) x² – x/2 + 4 = 0
(iii) – 2x² + x + 1 = 0
(iv) 2√3x² – 5x + √3 = 0

Answer -2

(i) 3x² – 4√3x + 4 = 0

a = 3, b = -4√3, c = 4

using the formula,

D = b2 – 4ac

= (-4√3)2 – 4(3) (4)

= 16(3) – 48

= 48 – 48

= 0

Discriminate, D = 0

D = 0

Hence Roots are real and equal.

(ii) x² – 1/2x + 4 = 0

a = 1, b = -1/2, c = 4

using the formula,

D = b2 – 4ac

= (-1/2)2 – 4(1) (4)

= 1/4 – 16

= -63/4

Discriminate, D = -63/4

D < 0

Hence  Roots are not real.

(iii) – 2x² + x + 1 = 0

a = -2, b = 1, c = 1

using the formula,

D = b2 – 4ac

= (1)2 – 4(-2) (1)

= 1 + 8

= 9

Discriminate, D = 9

D > 0

Hence Roots are real and distinct.

(iv) 2√3x² – 5x + √3 = 0

a = 2√3, b = -5, c = √3

using the formula,

D = b2 – 4ac

= (-5)2 – 4(2√3) (√3)

= 25 – 24

= 1

Discriminate, D = 1

D > 0

Hence Roots are real and distinct.

Question -3 Find the nature of the roots of the following quadratic equations:
(i) x² – x / 2 – 1/2 = 0
(ii) x² – 2√3x – 1 = 0 If real roots exist, find them.

Answer- 3

(i) x² – 1/2x – 1/2 = 0

a = 1, b = -1/2, c = -1/2

using the formula,

D = b2 – 4ac

= (-1/2)2 – 4(1) (-1/2)

= 1/4 + 2

= (1+8)/4

= 9/4

Discriminate, D = 9/4

D > 0

Hence Roots are real and unequal.

(ii) x² – 2√3x – 1 = 0

a = 1, b = 2√3, c = -1

using the formula,

D = b2 – 4ac

= (2√3)2 – 4(1) (-1)

= 12 + 4

= 16

Discriminate, D = 16

D > 0

Hence Roots are real and unequal.

Question -4 Without solving the following quadratic equation, find the value of ‘p’ for which the given equations have real and equal roots:
(i) px² – 4x + 3 = 0
(ii) x² + (p – 3) x + p = 0.

Answer -4

(i) px² – 4x + 3 = 0
Here a = p, b = -4, c = 3

using the formula,

D = b2 – 4ac

= (-4)2 – 4(p) (3)

= 16 – 12p

Hence, roots are real.

16 – 12p = 0

16 = 12p

p = 16/12

= 4/3

∴ p = 4/3

(ii) x² + (p – 3)x + p = 0

a = 1, b = (p – 3), c = p

using the formula,

D = b2 – 4ac

= (p – 3)2 – 4(1) (p)

= p2 – 32 – 2(3) (p) – 4p

= p2 + 9 – 6p – 4p

= p2 – 10p + 9

roots are real and have equal roots.

p2 – 10p + 9 = 0

p2 – 9p – p + 9 = 0

p(p – 9) – 1 (p – 9) = 0

(p – 9) (p – 1) = 0

(p – 9) = 0 or (p – 1) = 0

p = 9 or p = 1

∴ p = 1, 9

Question 5 Find the value (s) of k for which each of the following quadratic equation has equal roots :
(i) x2 + 4kx + (k– k + 2) =0

(ii) (k – 4) x² + 2(k – 4) x + 4 = 0

Answer -5

(i) x² + 4kx + (k2 – k + 2) = 0

a = 1, b = 4k, c = k2 – k + 2

using the formula,

D = b2 – 4ac

= (4k)2 – 4(1) (k2 – k + 2)

= 16k2 – 4k2 + 4k – 8

= 12k2 + 4k – 8

As, roots are equal, D = 0

12k2 + 4k – 8 = 0

3k2 + k – 2 = 0

3k2 + 3k – k – 2 = 0

3k(k + 1) – 1(k + 2) = 0

(3k – 1) (k + 2) = 0

(3k – 1) = 0 or (k + 2) = 0

k = 1/3 or k = -2

∴ k = 1/3, -2

(ii) (k – 4)x² + 2(k – 4)x + 4 = 0

a = (k – 4), b = 2(k – 4), c = 4

using the formula,

D = b2 – 4ac

= (2(k – 4))2 – 4(k – 4) (4)

= (4(k2 + 16 – 8k)) – 16(k – 4)

= 4(k2 – 8k + 16) – 16k + 64

= 4 [k2 – 8k + 16 – 4k + 16]

= 4 [k2 – 12k + 32]

Since, roots are equal.

4 [k2 – 12k + 32] = 0

k2 – 12k + 32 = 0

k2 – 8k – 4k + 32 = 0

k(k – 8) – 4 (k – 8) = 0

(k – 8) (k – 4) = 0

(k – 8) = 0 or (k – 4)  0

k = 8 or k  4

∴ k = 8

Question -6 Find the value(s) of m for which each of the following quadratic equation has real and equal roots:
(i) (3m + 1)x² + 2(m + 1)x + m = 0
(ii) x² + 2(m – 1) x + (m + 5) = 0

Answer- 6

(i) (3m + 1)x² + 2(m + 1)x + m = 0
Here a = 3m + 1, b = 2(m + 1), c = m

Let a = (3m + 1), and b = 2(m + 1), c = m

using the formula,

D = b2 – 4ac

= (2(m + 1))2 – 4 (3m + 1) (m)

= 4(m2 + 1 + 2m) – 4m(3m + 1)

= 4(m2 + 2m + 1) – 12m2 – 4m

= 4m2 + 8m + 4 – 12m2 – 4m

= -8m2 + 4m + 4

, roots are equal.

D = 0

-8m2 + 4m + 4 = 0

-2m2 + m + 1 = 0

2m2 – m – 1 = 0

2m2 – 2m + m – 1 = 0

2m(m – 1) +1 (m – 1) = 0

(m – 1) (2m + 1) = 0

(m – 1) = 0 or (2m + 1) = 0

m = 1 or 2m = -1

m = 1 or m = -1/2

∴ m = 1, -1/2

(ii) x² + 2(m – 1) x + (m + 5) = 0

a = 1, b = 2(m – 1), c = (m + 5)

using the formula,

D = b2 – 4ac

= (2(m – 1))2 – 4 (1) (m + 5)

= [4(m2 + 1 – 2m)] – 4m – 20

= 4m2 – 8m + 4 – 4m – 20

= 4m2 – 12m – 16

Hence, roots are equal.

D = 0

4m2 – 12m – 16 = 0

Divide by 4, we get.

m2 – 3m – 4 = 0

m2 – 4m + m – 4 = 0

m(m – 4) + 1 (m – 4) = 0

(m – 4) (m + 1) = 0

(m – 4) = 0 or (m + 1) = 0

m = 4 or m = -1

∴ m = 4, -1

Question -7 Find the values of k for which each of the following quadratic equation has equal roots:
(i) 9x² + k x + 1 = 0
(ii) x² – 2kx + 7k – 12 = 0
Also, find the roots for those values of k in each case.

Answer -7

(i) 9x² + k x + 1 = 0
Here a = 9, b = k, c = 1

using the formula,

D = b2 – 4ac

= (k)2 – 4 (9) (1)

= k2 – 36

roots are equal. given so

D = 0

k2 – 36 = 0

(k + 6) (k – 6) = 0

(k + 6) = 0 or (k – 6) = 0

k = -6 or k = 6

∴ k = 6, -6

When k = 6, then

9x² + kx + 1 = 0

9x2 + 6x + 1 = 0

(3x)2 + 2(3x)(1) + 12 = 0

(3x + 1)2 = 0

3x + 1 = 0

3x = -1

x = -1/3, -1/3

When k = -6, then

9x² + kx + 1 = 0

9x2 – 6x + 1 = 0

(3x)2 – 2(3x)(1) + 12 = 0

(3x – 1)2 = 0

3x – 1 = 0

3x = 1

x = 1/3, 1/3

(ii) x² – 2kx + 7k – 12 = 0

a = 1, b = -2k, c = (7k – 12)

using the formula,

D = b2 – 4ac

= (-2k)2 – 4 (1) (7k – 12)

= 4k2 – 28k + 48

Since, roots are equal.

D = 0

4k2 – 28k + 48 = 0

k2 – 7k + 12 = 0

k2 – 3k – 4k + 12 = 0

k(k – 3) – 4 (k – 3) = 0

(k – 3) (k – 4) = 0

(k – 3) = 0 or (k – 4) = 0

k = 3 or k = 4

∴ k = 3, 4

Now, let us substitute in the equation

When k = 3, then

using the formula,

= [-(-2k) ± √0] / 2(1)

= [2(3)]/2

= 3

x = 3, 3

When k = 4, then

using the formula,

= [-(-2k) ± √0] / 2(1)

= [2(4)] / 2

= 8/2

= 4

x = 4, 4

Question -8  Find the value(s) of p for which the quadratic equation (2p + 1)x² – (7p + 2)x + (7p – 3) = 0 has equal roots. Also find these roots.

Answer -8

The quadratic equation given is (2p + 1)x² – (7p + 2)x + (7p – 3) = 0
Comparing with ax² + bx + c = 0,

a = (2p + 1), b = – (7p + 2), c = (7p – 3)

using the formula,

D = b2 – 4ac

0 = (– (7p + 2))2 – 4 (2p + 1) (7p – 3)

= 49p2 + 4 + 28p – 4[14p2 – 6p + 7p – 3]

= 49p2 + 4 + 28p – 56p2 – 4p + 12

= -7p2 + 24p + 16

-7p2 + 28p – 4p + 16 = 0

-7p(p – 4) – 4 (p – 4) = 0

(p – 4) (-7p – 4) = 0

(p – 4) = 0 or (-7p – 4) = 0

p = 4 or -7p = 4

p = 4 or p = -4/7


page-76

Question-9  Find the value(s) of p for which the equation 2x² + 3x + p = 0 has real roots.

Answer -9

2x² + 3x + p = 0
Here, a = 2, b = 3, c = p

using the formula,

D = b2 – 4ac

= (3)2 – 4 (2) (p)

= 9 – 8p

Since, roots are real.

9 – 8p ≥ 0

9 ≥ 8p

8p ≤ 9

p ≤ 9/8

Question-10 Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots.

Answer -10

x² + kx + 4 = 0
Here, a = 1, b = k, c = 4

using the formula,

D = b2 – 4ac

= (k)2 – 4 (1) (4)

= k2 – 16

, roots are real and positive.

k2 – 16 ≥ 0

k2 ≥ 16

k ≥ 4

k = 4

Question-11 Find the values of p for which the equation 3x² – px + 5 = 0 has real roots.

Answer-11

3x² – px + 5 = 0
Here, a = 3, b = -p, c = 5

By using the formula,

D = b2 – 4ac

= (-p)2 – 4 (3) (5)

= p2 – 60

Since, roots are real.

p2 – 60 ≥ 0

p2 ≥ 60

p ≥ ± √60

p ≥ ± 215

p ≥ 215 or p ≤ -215

—  : End of ML Aggarwal Quadratic Equations Exe-5.4 Class 10 ICSE Maths Solutions : –

Return to :- ML Aggarwal Solutions for ICSE Class-10

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