Quadratic Equations Questions with Solutions

34 questions6 easy21 medium7 hard

A quadratic equation has the form ax^2 + bx + c = 0, where a is not zero. This topic covers solving quadratics by factorisation and by the quadratic formula, the nature of roots using the discriminant b^2 - 4ac, the sum (-b/a) and product (c/a) of roots, and forming an equation from its roots. In aptitude exams, a common question type gives two quadratic equations and asks you to compare their roots (whether x > y, x < y and so on). In engineering and MCA entrance exams, questions also test conditions on roots, common roots, and symmetric expressions such as the sum of squares of roots. Using the sum and product of roots is usually far faster than finding the roots themselves. Practise the quadratic equation questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q21.
Roots of equation $a x^{2} - 2 b x + c = 0$ are n and m, then the value of $\frac{b}{a n ^{2} + c} + \frac{b}{a m ^{2} + c}$ is
A. $\frac{c}{a}$ B. $\frac{b}{a}$ C. $\frac{a}{c}$ D. $\frac{b}{c}$
mediumAsked in NIMCET 2020View solution ›
Q22.
If $α \neq = β$ and $α^{2} = 5 α - 3$ , $β^{2} = 5 β - 3$ , then the equation whose roots are $\frac{α}{β}$ and $\frac{β}{α}$ is
A. $3 x^{2} - 25 x + 3 = 0$ B. $3 x^{2} + 5 x + 3 = 0$ C. $3 x^{2} - 5 x + 3 = 0$ D. $3 x^{2} - 19 x + 3 = 0$
mediumAsked in NIMCET 2021View solution ›
Q23.
If $α, β$ are the roots of $x^{2} - x - 1 = 0$ and $A_{n} = α^{n} + β^{n}$ , then the arithmetic mean of $A_{n - 1}$ and $A_{n}$ is
A. $2 A_{n} - 1$ B. $\frac{1}{2} A_{n + 1}$ C. $2 A_{n} - 2$ D. None of the above
mediumAsked in NIMCET 2022View solution ›
Q24.
If the roots of the quadratic equation $x^{2} + p x + q = 0$ are $t an 30^{\circ}$ and $t an 15^{\circ}$ respectively, then the value of $2 + p - q$ is
A. 3B. 0C. 1D. 2
mediumAsked in NIMCET 2022View solution ›
Q25.
If $α$ and $β$ are the roots of the equation $2 x^{2} + 2 p x + p^{2} = 0$ , where p is a non-zero real number, and $α^{4}$ and $β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the roots of $2 x^{2} - 4 p^{2} x + 4 p^{4} - 2 r = 0$ are:
A. Real and unequalB. Equal and zeroC. ImaginaryD. Equal and non-zero
hardAsked in NIMCET 2014View solution ›
Q26.
The value of $k$ for which the equation $(k - 2) x^{2} + 8 x + k + 4 = 0$ has both roots real, distinct and negative is
A. 0B. 2C. 3D. -4
mediumAsked in NIMCET 2015View solution ›
Q27.
If $α$ , $β$ are the roots of the equation $x^{2} - 2 x cos θ + 1 = 0$ , then the equation having roots $α^{n}$ and $β^{n}$ is
A. $x^{2} - (2 cos n θ) x + 1 = 0$ B. $x^{2} - (2 cos n θ) x - 1 = 0$ C. $x^{2} + (2 cos n θ) x + 1 = 0$ D. $x^{2} + (2 cos n θ) x - 1 = 0$
mediumAsked in NIMCET 2017View solution ›
Q28.
The equation $(x - a)^{3} + (x - b)^{3} + (x - c)^{3} = 0$ has
A. All three real rootsB. One real and two imaginary rootsC. Three real roots, namely x = a, x = b, x = cD. None of these
mediumAsked in NIMCET 2017View solution ›
Q29.
The quadratic equation whose roots are $s in^{2} 18^{\circ}$ and $cos^{2} 36^{\circ}$ is
A. $16 x^{2} - 12 x + 1 = 0$ B. $16 x^{2} + 12 x + 1 = 0$ C. $16 x^{2} - 12 x - 1 = 0$ D. None of these
mediumAsked in NIMCET 2018View solution ›
Q30.
Let $α$ and $β$ be the roots of the equation $x^{2} - p x + r = 0$ , and let $\frac{α}{2}$ and $2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of r is
A. $\frac{2}{9} (p - q) (2 q - p)$ B. $\frac{2}{9} (q - p) (2 q - p)$ C. $\frac{2}{9} (q - 2 p) (2 q - p)$ D. $\frac{2}{9} (2 p - q) (2 q - p)$
hardAsked in NIMCET 2018View solution ›
Q31.
If x is real, then the minimum value of $\frac{x ^{2} - x + 1}{x ^{2} + x + 1}$ is:
A. $\frac{1}{2}$ B. 2C. 3D. $\frac{1}{3}$
mediumAsked in NIMCET 2019View solution ›
Q32.
Some friends planned to contribute equally to jointly buy a CD player. However, two of them decided to withdraw at the last minute. As a result, each of the others had to shell out one rupee more than what they had planned for. If the price (in Rs.) of the CD player is an integer between 1000 and 1100, find the number of friends who actually contributed.
A. 44B. 23C. 21D. 46
mediumAsked in NIMCET 2019View solution ›
Q33.
If $α$ and $β$ are the two roots of the quadratic equation $x^{2} + a x + b = 0$ $(ab \neq = 0)$ , then the quadratic equation whose roots are $\frac{1}{α ^{3} + α}$ and $\frac{1}{β ^{3} + β}$ is
A. $b (b^{2} + 1 + a^{2} + 2 b) x^{2} - (a^{3} + a - 3 ab) x + 1 = 0$ B. $b (b^{2} + 1 + a^{2} - 2 b) x^{2} - (a^{3} + a - 3 ab) x + 1 = 0$ C. $b (b^{2} + 1 + a^{2} + 2 b) x^{2} + (a^{3} - a - 3 ab) x + 1 = 0$ D. $b (b^{2} + 1 + a^{2} - 2 b) x^{2} + (a^{3} + a - 3 ab) x + 1 = 0$
hardAsked in NIMCET 2025View solution ›
Q34.
At an IIM entrepreneurship summit, two young founders, Karan and Deepak, introduced their startup. In their quirky opening, they said: "The product of our ages is 240. And just like in our startup strategy, twice Deepak's age is 4 years more than Karan's age." How old was Deepak two years ago?
A. 6 yearsB. 8 yearsC. 15 yearsD. 10 years
easyAsked in NIMCET 2025View solution ›

‹ PreviousPage 2 of 2