Quadratic Equations and Inequalities — JEE Main PYQs

148 previous year questions on Quadratic Equations and Inequalities (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let m and n be the numbers of real roots of the quadratic equations $x^{2} - 12 x + [x] + 31 = 0$ and $x^{2} - 5∣ x + 2∣ - 4 = 0$ respectively, where $[x]$ denotes the greatest integer $\leq x$ . Then $m^{2} + mn + n^{2}$ is equal to ______.
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The number of elements in the set ${n \in Z : n^{2} - 10 n + 19 < 6}$ is ____.
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Let $α, β$ be the roots of the quadratic equation $x^{2} + 6 x + 3 = 0$ . Then $\frac{α ^{23} + β ^{23} + α ^{14} + β ^{14}}{α ^{15} + β ^{15} + α ^{10} + β ^{10}}$ is equal to
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Let $α, β$ be the roots of the equation $x^{2} - 2 x + 2 = 0$ . Then $α^{14} + β^{14}$ is equal to
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If $a$ and $b$ are the roots of equation $x^{2} - 7 x - 1 = 0$ , then the value of $\frac{a ^{21} + b ^{21} + a ^{17} + b ^{17}}{a ^{19} + b ^{19}}$ is equal to
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The number of integral values of k, for which one root of the equation $2 x^{2} - 8 x + k = 0$ lies in the interval $(1, 2)$ and its other root lies in the interval $(2, 3)$ , is :
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Let $α_{1}, α_{2}, \dots, α_{7}$ be the roots of the equation $x^{7} + 3 x^{5} - 13 x^{3} - 15 x = 0$ and $∣ α_{1} ∣ \geq ∣ α_{2} ∣ \geq \dots \geq ∣ α_{7} ∣$ .

Then …
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Let $λ \neq = 0$ be a real number. Let $α, β$ be the roots of the equation $14 x^{2} - 31 x + 3 λ = 0$ and $α, γ$ be the roots of the equation $35 x^{2} - 53 x + 4 λ = 0$ . Then $\frac{3 α}{β}$ and …
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Let $a \in R$ and let $α, β$ be the roots of the equation $x^{2} + 6 0^{\frac{1}{4}} x + a = 0$ . If $α^{4} + β^{4} = - 30$ , then the product of all possible values of a is ______.
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The number of real solutions of the equation $3 (x^{2} + \frac{1}{x ^{2}}) - 2 (x + \frac{1}{x}) + 5 = 0$ , is
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The equation
$e^{4 x} + 8 e^{3 x} + 13 e^{2 x} - 8 e^{x} + 1 = 0, x \in R$ has:
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If the value of real number $a > 0$ for which $x^{2} - 5 a x + 1 = 0$ and $x^{2} - a x - 5 = 0$ have a common real root is $\frac{3}{2 β}$ then $β$ is equal to ____.
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The number of real roots of the equation $x^{2} - 4 x + 3 + x^{2} - 9 = 4 x^{2} - 14 x + 6$ , is:
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Let
$S = {x : x \in R and (3 + 2)^{x^{2} - 4} + (3 - 2)^{x^{2} - 4} = 10}$

Then $n (S)$ is equal to
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Let
$S = {α : lo g_{2} (9^{2 α - 4} + 13) - lo g_{2} (\frac{5}{2} \cdot 3^{2 α - 4} + 1) = 2} .$

Then the maximum value of $β$ for which the equation …
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For $p, q \in R$ , consider the real valued function $f (x) = (x - p)^{2} - q, x \in R$ and $q > 0$ . Let $a_{1}, a_{2}, a_{3}$ and $a_{4}$ be in an arithmetic progression with mean p …
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The sum of all real values of $x$ for which $\frac{3 x ^{2} - 9 x + 17}{x ^{2} + 3 x + 10} = \frac{5 x ^{2} - 7 x + 19}{3 x ^{2} + 5 x + 12}$ is equal to ______.
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Let $S = {x \in [- 6, 3] - {- 2, 2} : \frac{∣ x + 3∣ - 1}{∣ x ∣ - 2} \geq 0}$ and $T = {x \in Z : x^{2} - 7∣ x ∣ + 9 \leq 0}$ . Then the number of elements in $S \cap T$ is
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Let $α, β$ be the roots of the equation $x^{2} - 2 x + 6 = 0$ and $\frac{1}{α ^{2}} + 1, \frac{1}{β ^{2}} + 1$ be the roots of the equation $x^{2} + a x + b = 0$ . Then the roots of the equation …
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Let $f (x) = ax^{2} + bx + c$ be such that $f (1) = 3, f (- 2) = λ$ and $f (3) = 4$ . If $f (0) + f (1) + f (- 2) + f (3) = 14$ , then $λ$ is equal to
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Let $f (x) = 2 x^{2} - x - 1$ and $S = {n \in Z : ∣ f (n) ∣ \leq 800}$ . Then, the value of $\sum_{n \in S} f (n)$ is equal to ______.
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If the numbers appeared on the two throws of a fair six faced die are $α$ and $β$ , then the probability that $x^{2} + α x + β > 0$ , for all $x \in R$ , is:
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The minimum value of the sum of the squares of the roots of $x^{2} + (3 - a) x + 1 = 2 a$ is:
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If $α, β$ are the roots of the equation $x^{2} - (5 + 3^{l o g_{3} 5} - 5^{l o g_{5} 3}) x + 3 (3^{(l o g_{3} 5)^{1/3}} - 5^{(l o g_{5} 3)^{2/3}} - 1) = 0$ , then the equation whose roots are $α + \frac{1}{β}$ and $β + \frac{1}{α}$ is:
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Let $f (x)$ be a quadratic polynomial with leading coefficient 1 such that $f (0) = p, p \neq = 0$ and $f (1) = \frac{1}{3}$ . If the equation $f (x) = 0$ and $fofofof (x) = 0$ have a common real root, then …
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If for some $p, q, r \in R$ , not all have same sign, one of the roots of the equation $(p^{2} + q^{2}) x^{2} - 2 q (p + r) x + q^{2} + r^{2} = 0$ is also a root of the equation $x^{2} + 2 x - 8 = 0$ , then $\frac{q ^{2} + r ^{2}}{p ^{2}}$ is equal to-
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The number of distinct real roots of the equation $x^{5} (x^{3} - x^{2} - x + 1) + x (3 x^{3} - 4 x^{2} - 2 x + 4) - 1 = 0$ is
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Let $f (x)$ be a quadratic polynomial such that $f (- 2) + f (3) = 0$ . If one of the roots of $f (x) = 0$ is -1 , then the sum of the roots of $f (x) = 0$ is equal to :
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Let $α, β$ be the roots of the equation $x^{2} - 4 λ x + 5 = 0$ and $α, γ$ be the roots of the equation $x^{2} - (32 + 23) x + 7 + 3 λ 3 = 0$ .

If $β + γ = 32$ , then $(α + 2 β + γ)^{2}$ is equal to :
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The number of real solutions of the equation $e^{4 x} + 4 e^{3 x} - 58 e^{2 x} + 4 e^{x} + 1 = 0$ is ____ .
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Let p and q be two real numbers such that $p + q = 3$ and $p^{4} + q^{4} = 369$ . Then $(\frac{1}{p} + \frac{1}{q})^{- 2}$ is equal to ______.
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Let $α$ and $β$ be the roots of the equation $x^{2} + (2 i - 1) = 0$ . Then, the value of $α^{8} + β^{8}$ is equal to :
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The sum of the cubes of all the roots of the equation $x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 10$ is ______.
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If the sum of all the roots of the equation $e^{2 x} - 11 e^{x} - 45 e^{- x} + \frac{81}{2} = 0$ is $lo g_{e} P$ , then p is equal to ____.
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If the sum of the squares of the reciprocals of the roots $α$ and $β$ of the equation $3 x^{2} + λ x - 1 = 0$ is 15, then $6 (α^{3} + β^{3})^{2}$ is equal to :
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The sum of all the real roots of the equation $(e^{2 x} - 4) (6 e^{2 x} - 5 e^{x} + 1) = 0$ is
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Let $a, b \in R$ be such that the equation $ax^{2} - 2 bx + 15 = 0$ has a repeated root $α$ . If $α$ and $β$ are the roots of the equation $x^{2} - 2 b x + 21 = 0$ , then $α^{2} + β^{2}$ is equal to:
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The sum of all integral values of $k (k \neq = 0)$ for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is ______.
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Let $p$ and $q$ be two positive numbers such that $p + q = 2$ and $p^{4} + q^{4} = 272$ . Then $p$ and $q$ are roots of the equation:
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Let $λ \neq = 0$ be in R . If $α$ and $β$ are the roots of the equation $x^{2} - x + 2 λ = 0$ , and $α$ and $γ$ are the roots of the equation $3 x^{2} - 10 x + 27 λ = 0$ , then $\frac{β γ}{λ}$ is equal to ______.
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The probability of selecting integers $a \in [- 5, 30]$ such that $x^{2} + 2 (a + 4) x - 5 a + 64 > 0$ , for all $x \in R$ is :
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If $α$ and $β$ are the distinct roots of the equation $x^{2} + (3)^{1/4} x + 3^{1/2} = 0$ , then the value of $α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)$ is equal to :
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The set of all values of $k > - 1$ , for which the equation $(3 x^{2} + 4 x + 3)^{2} - (k + 1) (3 x^{2} + 4 x + 3) (3 x^{2} + 4 x + 2) + k (3 x^{2} + 4 x + 2)^{2} = 0$ has real roots is:
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The integer ' $k$ ', for which the inequality $x^{2} - 2 (3 k - 1) x + 8 k^{2} - 7 > 0$ is valid for every $x$ in $R$ , is :
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The sum of $16 2^{th}$ power of the roots of the equation $x^{3} - 2 x^{2} + 2 x - 1 = 0$ is ______
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Let $a$ be an integer such that all the real roots of the polynomial $2 x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 10 x + 10$ lie in the interval $(a, a + 1)$ . Then $∣ a ∣$ is equal to ____
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The number of real roots of the equation $e^{4 x} + 2 e^{3 x} - e^{x} - 6 = 0$ is:
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Let $α, β$ be two roots of the equation $x^{2} + (20)^{\frac{1}{4}} x + (5)^{\frac{1}{2}} = 0$ Then $α^{8} + β^{8}$ is equal to :
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The number of real roots of the equation $e^{6 x} - e^{4 x} - 2 e^{3 x} - 12 e^{2 x} + e^{x} + 1 = 0$ is :
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If $α, β$ are roots of the equation $x^{2} + 5 (2) x + 10 = 0, α > β$ and $P_{n} = α^{n} - β^{n}$ for each positive integer $n$ , then the value of $(\frac{P _{17} P _{20} + 5 2 P _{17} P _{19}}{P _{18} P _{19} + 5 2 P _{18}^{2}})$ is e …
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