Class 10: Quadratic Equations – Very Short Answer Type Questions

$\displaystyle \textbf{Question 1: }\text{Find the value(s) of }k\text{ so that the quadratic equation } \\ 4x^2+kx+1=0\text{ has real and equal roots.}\hfill\text{[CBSE 2025]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{For real and equal roots, }D=0.$
$\displaystyle \therefore k^2-4(4)(1)=0\Rightarrow k^2=16\Rightarrow \boxed{k=\pm4}$
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$\displaystyle \textbf{Question 2: }\text{Write the value of }k\text{ for which the quadratic equation } \\ x^2-kx+4=0\text{ has equal roots.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{For equal roots, }D=0.$
$\displaystyle \therefore (-k)^2-4(1)(4)=0\Rightarrow k^2=16\Rightarrow \boxed{k=\pm4}$
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$\displaystyle \textbf{Question 3: }\text{What is the nature of roots of the quadratic equation } \\ 4x^2-12x-9=0?$
$\displaystyle \text{Answer:}$
$\displaystyle D=(-12)^2-4(4)(-9)=144+144=288>0$
$\displaystyle \therefore \boxed{\text{The roots are real and distinct.}}$
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$\displaystyle \textbf{Question 4: }\text{Write a quadratic polynomial, sum of whose zeros is } \\ 2\sqrt{3}\text{ and their product is }2.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Required polynomial}=x^2-(\text{sum of zeros})x+(\text{product of zeros})$
$\displaystyle \therefore \boxed{x^2-2\sqrt{3}x+2}$
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$\displaystyle \textbf{Question 5: }\text{Show that }x=-3\text{ is a solution of }x^2+6x+9=0.\hfill\text{[CBSE 2008]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Putting }x=-3,\quad (-3)^2+6(-3)+9=9-18+9=0$
$\displaystyle \therefore \boxed{x=-3\text{ is a solution of the given equation.}}$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Show that }x=-2\text{ is a solution of }3x^2+13x+14=0.\hfill\text{[CBSE 2008]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Putting }x=-2,\quad 3(-2)^2+13(-2)+14=12-26+14=0$
$\displaystyle \therefore \boxed{x=-2\text{ is a solution of the given equation.}}$
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$\displaystyle \textbf{Question 7: }\text{Find the discriminant of the quadratic equation } \\ 3\sqrt{3}\,x^2+10x+\sqrt{3}=0.\hfill\text{[CBSE 2009]}$
$\displaystyle \text{Answer:}$
$\displaystyle D=b^2-4ac=10^2-4(3\sqrt{3})(\sqrt{3})=100-36=\boxed{64}$
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$\displaystyle \textbf{Question 8: }\text{The sides of a right triangle are such that the longest side is }$
$\displaystyle 4\text{ m more than the shortest side} \ \text{and the third side is }2\text{ m less than the longest side.}$
$\displaystyle \text{Find the length of each side of the triangle. Also, find the difference between the numerical}$
$\displaystyle \text{values of the area and the perimeter of the given triangle.}\hfill\text{[CBSE 2025]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the shortest side be }x\text{ m. Then, the other sides are }(x+2)\text{ m and }(x+4)\text{ m.}$
$\displaystyle \text{By Pythagoras theorem, }x^2+(x+2)^2=(x+4)^2$
$\displaystyle \Rightarrow x^2-4x-12=0\Rightarrow (x-6)(x+2)=0\Rightarrow x=6$
$\displaystyle \therefore \boxed{\text{The sides of the triangle are }6\text{ m, }8\text{ m and }10\text{ m.}}$
$\displaystyle \text{Area}=\frac{1}{2}\times6\times8=24\text{ m}^2,\quad \text{Perimeter}=6+8+10=24\text{ m}$
$\displaystyle \therefore \boxed{\text{Difference between their numerical values}=24-24=0}$
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$\displaystyle \textbf{Question 9: }\text{If }x=-\frac{1}{2}\text{ is a solution of the quadratic equation } \\ 3x^2+2kx-3=0,\text{ find the value of }k.\hfill\text{[CBSE 2015]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Putting }x=-\frac{1}{2},\quad 3\left(-\frac{1}{2}\right)^2+2k\left(-\frac{1}{2}\right)-3=0$
$\displaystyle \Rightarrow \frac{3}{4}-k-3=0\Rightarrow -k=\frac{9}{4}\Rightarrow \boxed{k=-\frac{9}{4}}$
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$\displaystyle \textbf{Question 10: }\text{If }x=3\text{ is one root of the quadratic equation } \\ x^2-2kx-6=0,\text{ then find the value of }k.\hfill\text{[CBSE 2018]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Putting }x=3,\quad 3^2-2k(3)-6=0$
$\displaystyle \Rightarrow 9-6k-6=0\Rightarrow 3-6k=0\Rightarrow \boxed{k=\frac{1}{2}}$
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$\displaystyle \textbf{Question 11: }\text{For what value of }k,\text{ the roots of the equation } \\ x^2+4x+k=0\text{ are real?}\hfill\text{[CBSE 2019]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{For real roots, }D\geq0.$
$\displaystyle \therefore 4^2-4(1)(k)\geq0\Rightarrow16-4k\geq0\Rightarrow \boxed{k\leq4}$
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$\displaystyle \textbf{Question 12: }\text{Find the value of }k\text{ for which the roots of the equation } \\ 3x^2-10x+k=0\text{ are reciprocal of each other.}\hfill\text{[CBSE 2019]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{For reciprocal roots, }\alpha\beta=1.$
$\displaystyle \therefore \frac{c}{a}=1\Rightarrow\frac{k}{3}=1\Rightarrow \boxed{k=3}$
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$\displaystyle \textbf{Question 13: }\text{Solve the quadratic equation }2x^2-5x-1=0\text{ for }x.\hfill\text{[CBSE 2022]}$
$\displaystyle \text{Answer:}$
$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{5\pm\sqrt{25+8}}{4}$
$\displaystyle \therefore \boxed{x=\frac{5+\sqrt{33}}{4}\text{ or }x=\frac{5-\sqrt{33}}{4}}$
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$\displaystyle \textbf{Question 14(a): }\text{Find the nature of the roots of the quadratic equation } \\ x^2-5x+9=0.$
$\displaystyle \text{Answer:}$
$\displaystyle D=(-5)^2-4(1)(9)=25-36=-11<0$
$\displaystyle \therefore \boxed{\text{The equation has no real roots.}}$
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$\displaystyle \textbf{Question 14(b): }\text{Write a quadratic equation with roots }-3\text{ and }5.\hfill\text{[CBSE 2022]}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Required equation}=(x+3)(x-5)=0$
$\displaystyle \therefore \boxed{x^2-2x-15=0}$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Write the set of values of }a\text{ for which the equation } \\ x^2+ax-1=0\text{ has real roots.}$
$\displaystyle \text{Answer:}$
$\displaystyle D=a^2-4(1)(-1)=a^2+4>0\text{ for all real }a.$
$\displaystyle \therefore \boxed{a\in\mathbb{R}}$
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$\displaystyle \textbf{Question 16: }\text{Is there any real value of }a\text{ for which the equation } \\ x^2+2x+(a^2+1)=0\text{ has real roots?}$
$\displaystyle \text{Answer:}$
$\displaystyle D=2^2-4(1)(a^2+1)=-4a^2$
$\displaystyle \text{For real roots, }D\geq0\Rightarrow -4a^2\geq0\Rightarrow a=0.$
$\displaystyle \therefore \boxed{\text{Yes, for }a=0.}$
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$\displaystyle \textbf{Question 17: }\text{Write the set of values of }k\text{ for which the quadratic equation } \\ 2x^2+kx-8=0\text{ has real roots.}$
$\displaystyle \text{Answer:}$
$\displaystyle D=k^2-4(2)(-8)=k^2+64>0\text{ for all real }k.$
$\displaystyle \therefore \boxed{k\in\mathbb{R}}$
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$\displaystyle \textbf{Question 18: }\text{If }1+\sqrt{2}\text{ is a root of a quadratic equation with rational} \\ \text{coefficients, write its other root.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since irrational roots occur in conjugate pairs, the other root is }\boxed{1-\sqrt{2}}.$
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$\displaystyle \textbf{Question 19: }\text{Write the number of real roots of the equation } \\ x^2+3|x|+2=0.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Since }x^2\geq0,\ 3|x|\geq0,\text{ we have }x^2+3|x|+2>0\text{ for all real }x.$
$\displaystyle \therefore \boxed{\text{Number of real roots}=0}$
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$\displaystyle \textbf{Question 20: }\text{Write the sum of real roots of the equation }x^2+|x|-6=0.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }y=|x|.\text{ Then }y^2+y-6=0\Rightarrow(y+3)(y-2)=0.$
$\displaystyle \text{Since }|x|\geq0,\ |x|=2\Rightarrow x=\pm2.$
$\displaystyle \therefore \boxed{\text{Sum of real roots}=2+(-2)=0}$
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$\displaystyle \textbf{Question 21: }\text{Write the value of }\lambda\text{ for which }x^2+4x+\lambda\text{ is a perfect square.}$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+4x+\lambda=(x+2)^2\Rightarrow \boxed{\lambda=4}$
$\displaystyle \\$

$\displaystyle \textbf{Question 22: }\text{Write the condition to be satisfied for which equations }$
$\displaystyle ax^2+2bx+c=0\text{ and} \ bx^2-2\sqrt{ac}\,x+b=0\text{ have equal roots.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{For }ax^2+2bx+c=0,\quad D=(2b)^2-4ac=0\Rightarrow b^2=ac.$
$\displaystyle \text{For }bx^2-2\sqrt{ac}\,x+b=0,\quad D=4ac-4b^2=0\Rightarrow ac=b^2.$
$\displaystyle \therefore \boxed{b^2=ac}$
$\displaystyle \\$