Chapter 4 · Quadratic Equations

NCERT Exemplar Class 10 Maths Exercise 4.2

All 7 questions of NCERT Exemplar Class 10 Maths Exercise 4.2 (Quadratic Equations), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   State ​whether the ​following ​quadratic equations have two ​distinct real ​roots. Justify ​your ​answer.

   1. (i)$x^2-3x+4=0$
   2. (ii)$2x^2+x-1=0$
   3. (iii)$2x^2-6x+\frac{9}{2}=0$
   4. (iv)$3x^2-4x+1=0$
   5. (v)$(x+4{)}^2-8x=0$
   6. (vi)$(x-\sqrt{2}{)}^2-2(x+1)=0$
   7. (vii)$\sqrt{2}{x}^2-\frac{3}{\sqrt{2}}x+\frac{1}{\sqrt{2}}=0$
   8. (viii)$x(1-x)-2=0$
   9. (ix)$(x-1)(x+2)+2=0$
   10. (x)$(x+1)(x-2)+x=0$
   Show answer

   Two distinct real roots: (ii), (iv), (vi), (vii), (ix), (x). No real roots: (i), (v), (viii). Equal (not distinct) roots: (iii).

   Step-by-step solution →
2. Q2
   Write whether the following ​statements are true or false. Justify your answers.

   1. (i)Every quadratic equation has exactly one root.
   2. (ii)Every quadratic equation has at least one real root.
   3. (iii)Every quadratic equation has at least two roots.
   4. (iv)Every quadratic equation ​has at most ​two roots.
   5. (v)If the coefficient of $x^2$ ​and the constant term ​of a quadratic equation have opposite ​signs, then the quadratic equation has real roots.
   6. (vi)If the coefficient of $x^2$ 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.
   Show answer

   (i) False -- it has two roots; (ii) False -- e.g. $x^2+1=0$ has no real root; (iii) False -- exactly two, not 'at least two' more; (iv) True; (v) True since $D=b^2-4ac>0$ when $ac<0$; (vi) True since $D=-4ac<0$ when $ac>0,b=0$.

   Step-by-step solution →
3. Q3
   A ​quadratic equation with ​integral ​coefficient ​has integral ​roots. Justify ​your ​answer.
   Show answer

   No, it is not true. For example $x^2-x-4=0$ has integral coefficients but roots $\frac{1\pm \sqrt{17}}{2}$, which are not integers.

   Step-by-step solution →
4. Q4
   Does ​there exist ​a ​quadratic ​equation whose coefficients are rational but ​both of its ​roots are irrational? ​Justify your answer.
   Show answer

   Yes. For example $x^2-4x+1=0$ has rational coefficients but irrational roots $2\pm \sqrt{3}$.

   Step-by-step solution →
5. Q5
   Does ​there exist ​a ​quadratic equation whose coefficients are ​all distinct irrationals ​but both the ​roots are ​rationals? Why?
   Show answer

   Yes. For example $\sqrt{2}{x}^2-3\sqrt{2}x+2\sqrt{2}=0$ has all-distinct-irrational coefficients but rational roots $1$ and $2$.

   Step-by-step solution →
6. Q6
   Is $0.2$ ​a ​root ​of ​the equation $x^2-0.4=0$? Justify.
   Show answer

   No, $0.2$ is not a root: $p(0.2)=0.04-0.4=-0.36\ne 0$. The actual roots are $x=\pm \sqrt{0.4}\approx \pm 0.632$.

   Step-by-step solution →
7. Q7
   If $b=0$, $c<0$, ​is ​it true that ​the ​roots of $x^2+bx+c=0$ are numerically ​equal and ​opposite in ​sign? ​Justify.
   Show answer

   Yes, it is true. With $b=0$ the equation is $x^2=-c$; since $c<0$, $-c>0$, so the roots are $+\sqrt{-c}$ and $-\sqrt{-c}$ - numerically equal and opposite in sign.

   Step-by-step solution →