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Mathematics · Quadratic Equations · NCERT Exercises

Exercise 4.3

Complete, independently verified solutions for NCERT Exercise 4.3.

Mathematics · Chapter 4 · NCERT Exercises

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Verified NCERT questions and solutionsSolutions are checked independently and follow the supplied NCERT chapter edition.

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Question

Question 1

discriminant · nature of roots · quadratic formula

Find the nature of the roots; if real roots exist, find them.

  1. 2x2−3x+5=02x^2-3x+5=0
  2. 3x2−43x+4=03x^2-4\sqrt3x+4=0
  3. 2x2−6x+3=02x^2-6x+3=0

Solution

  1. D=(−3)2−4(2)(5)=−31<0D=(-3)^2-4(2)(5)=-31<0

    No real roots.

  2. D=(−43)2−4(3)(4)=0D=(-4\sqrt3)^2-4(3)(4)=0
    x=436=233x=\frac{4\sqrt3}{6}=\frac{2\sqrt3}{3}

    Two equal real roots.

  3. D=(−6)2−4(2)(3)=12>0D=(-6)^2-4(2)(3)=12>0
    x=6±124=3±32x=\frac{6\pm\sqrt{12}}4=\frac{3\pm\sqrt3}{2}

    Two distinct real roots.

Question

Question 2

discriminant · equal roots

Find k so that each equation has two equal roots.

  1. 2x2+kx+3=02x^2+kx+3=0
  2. kx(x−2)+6=0kx(x-2)+6=0

Solution

  1. D=k2−24=0⇒k=±26D=k^2-24=0\Rightarrow k=\pm2\sqrt6
  2. kx2−2kx+6=0kx^2-2kx+6=0
    D=4k2−24k=4k(k−6)=0D=4k^2-24k=4k(k-6)=0

    k = 0 would make the equation non-quadratic, so k = 6.

Question

Question 3

nature of roots · word problem · rectangle

Can a rectangular mango grove have length twice its breadth and area 800 m²? If so, find its dimensions.

Solution

Let breadth = x m and length = 2x m.

2x2=800⇒x2−400=02x^2=800\Rightarrow x^2-400=0
x=20or−20x=20 \text{or} -20

A breadth cannot be negative, so x = 20 m; length = 40 m. Their product is 800 m².

Final answer

Yes; breadth 20 m and length 40 m.

Question

Question 4

nature of roots · word problem · ages

Two friends’ present ages sum to 20 years. Four years ago, the product of their ages was 48. Is this possible?

Solution

Let one present age be x years; the other is 20 − x.

(x−4)(16−x)=48⇒x2−20x+112=0(x-4)(16-x)=48\Rightarrow x^2-20x+112=0
D=(−20)2−4(1)(112)=−48<0D=(-20)^2-4(1)(112)=-48<0

There are no real ages satisfying the conditions, so the situation is not possible.

Final answer

No, the stated situation is not possible.

Question

Question 5

nature of roots · word problem · rectangle

Can a rectangular park have perimeter 80 m and area 400 m²? If so, find its dimensions.

Solution

Let one side be x m. Since length + breadth = 40 m, the other side is 40 − x.

x(40−x)=400⇒x2−40x+400=0x(40-x)=400\Rightarrow x^2-40x+400=0
D=(−40)2−4(1)(400)=0D=(-40)^2-4(1)(400)=0
x=402=20x=\frac{40}{2}=20

The equal roots give a 20 m by 20 m square; its perimeter and area verify both conditions.

Final answer

Yes; 20 m × 20 m.