Quadratic Equations jee mains questions | JEE Main Maths Quadratic Equations Previous Year Question

Q 11 :

The number of real solutions of the equation $x$ $| x + 3 |$ $+$ $| x - 1 |$ $- 2 = 0$ is: [2026]

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(1)

$(I)$

$x^{2} + 3 x + x - 1 - 2 = 0$

$x^{2} + 4 x - 3 = 0$

$x = - 2 + \sqrt{7} (rejected), - 2 - \sqrt{7} (rejected)$

$(II)$

$x^{2} + 3 x + 1 - x - 2 = 0$

$x^{2} + 2 x - 1 = 0$

$x = - 1 + \sqrt{2}, - 1 - \sqrt{2}$

$(III)$

$- x^{2} - 3 x + 1 - x - 2 = 0$

$x^{2} + 4 x + 1 = 0$

$x = - 2 - \sqrt{3}, - 2 + \sqrt{3} (rejected)$

Q 12 :

The number of distinct real solutions of the equation $x$ $| x + 4 |$ $+ 3$ $| x + 2 |$ $+ 10 = 0$ is [2026]

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(1)

$Case I x < - 4$

$x (- x + 4) + 3 (- x + 2) + 10 = 0$

$x^{2} + 7 x - 4 = 0$

$\Rightarrow x = - \frac{7 + \sqrt{65}}{2} or - \frac{7 - \sqrt{65}}{2}$

$Reject$ $Accept$

$Case II - 4 \leq x < - 2$

$x (x + 4) + 3 (- (x + 2)) + 10 = 0$

$x^{2} + x + 4 = 0$

$D < 0 No solution$

$Case III x \geq - 1$

$x (x + 4) + 3 (x + 2) + 10 = 0$

$x^{2} + 7 x + 16 = 0$

$D < 0 No solution$

$\Rightarrow No. of solutions = 1$

Q 13 :

The smallest positive integral value of $a$ , for which all the roots of $x^{4} - a x^{2} + 9 = 0$ are real and distinct, is equal to [2026]

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(1)

$x^{4} - a x^{2} + 9 = 0 . . . (1)$

$let x^{2} = t$

$t^{2} - a t + 9 = 0 . . . (2)$

$For roots of equation (1) to be real & distinct, roots of equation (2) must be positive & distinct.$

$(i) D > 0 \Rightarrow a^{2} - 36 > 0 \Rightarrow a \in (- \infty, - 6) \cup (6, \infty)$

$(ii) \frac{- b}{2 a} > 0 \Rightarrow \frac{a}{2} > 0 \Rightarrow a > 0$

$(iii) f (0) > 0 \Rightarrow 9 > 0 \Rightarrow a \in ℝ$

$By (i) \cap (ii) \cap (iii)$

$∴$ $a \in (6, \infty)$

$∴$ $least integral value of a = 7$

Q 14 :

The sum of all the real solutions of the equation

$\log_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 \log_{(6 x + 10)} (x^{2} + 6 x + 9)$ is equal to. [2026]

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(1)

$\log_{x + 3} [(x + 3) (6 x + 10)] = 5 - 2 \log_{6 x + 10} {(x + 3)}^{2}$

$1 + \log_{x + 3} (6 x + 10) = 5 - 4 \log_{6 x + 10} (x + 3)$

$Let \log_{x + 3} (6 x + 10) = A$

$\Rightarrow A + \frac{4}{A} = 4 or A = 2$

$\Rightarrow \log_{x + 3} (6 x + 10) = 2$

$\Rightarrow 6 x + 10 = {(x + 3)}^{2}$

$\Rightarrow 6 x + 10 = x^{2} + 9 + 6 x$

$\Rightarrow x^{2} = 1, x = \pm 1$

$So sum of roots = 0$

Q 15 :

Let $α and β$ be the roots of the equation $x^{2} + 2 a x + (3 a + 10) = 0$ such that $α < 1 < β$ . Then the set of all possible values of a is : [2026]

$(- \infty, \frac{- 11}{5})$
$(- \infty, - 3)$
$(- \infty, - 2) \cup (5, \infty)$
$(- \infty, \frac{- 11}{5}) \cup (5, \infty)$

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(1)

$∵$ $α < 1 < β$

$f (1) < 0$

$\Rightarrow 1 + 2 a + (3 a + 10) < 0$

$\Rightarrow 5 a + 11 < 0$

$a < \frac{- 11}{5}$

$∴$ $a \in (- \infty, \frac{- 11}{5})$

Q 16 :

The positive integer n, for which the solutions of the equation

$x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3}$

are two consecutive even integers, is : [2026]

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(1)

$x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3}$

$\Rightarrow \sum_{r = 1}^{n} (x + 2 r - 2) (x + 2 r) = \frac{8 n}{3}$

$n x^{2} + 2 x \sum_{r = 1}^{n} (2 r - 1) + 4 \sum_{r = 1}^{n} r (r - 1) = \frac{8 n}{3}$

$n x^{2} + 2 x n^{2} + \frac{4 n (n^{2} - 1)}{3} - \frac{8 n}{3} = 0$

$x^{2} + 2 n x + \frac{4 (n^{2} - 1)}{3} - \frac{8}{3} = 0$

$∵$ $| α - β | = 2 \Rightarrow \frac{\sqrt{D}}{| a |} = 2 \Rightarrow D = 4$

$\Rightarrow 4 n^{2} - 4 (\frac{4 (n^{2} - 1)}{3} - \frac{8}{3}) = 4$

$\Rightarrow n^{2} - \frac{4 n^{2}}{3} = - 3$

$\Rightarrow n^{2} = 9$

$\Rightarrow n = 3$

Q 17 :

Let $α, β$ be the roots of the quadratic equation $12 x^{2} - 20 x + 3 λ = 0, λ \in Z .$ If $\frac{1}{2} \leq | β - α | \leq \frac{3}{2},$ then the sum of all possible values of $λ is$ . [2026]

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(2)

$\frac{1}{2} \leq | α - β | \leq \frac{3}{2}$

$\frac{1}{4} \leq | α - β |^{2} \leq \frac{9}{4}$

$\frac{1}{4} \leq {(α + β)}^{2} - 4 α β \leq \frac{9}{4}$

$\frac{1}{4} \leq \frac{25}{9} - 4 \times \frac{λ}{4} \leq \frac{9}{4}$

$- \frac{91}{36} \leq - λ \leq \frac{- 19}{36}$

$\frac{19}{36} \leq λ \leq \frac{91}{36}$

$λ = 1, 2$

$Sum = 3$

Q 18 :

If $α, β where α < β$ are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3},$ then the sum of all possible values of $λ$ is [2026]

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(3)

$\frac{β - α}{α β} = \frac{1}{3}, α + β = \frac{λ + 3}{λ}, α β = \frac{3}{λ}$

$β - α = \frac{α β}{3} = \frac{1}{λ}$

$on squaring$

$α^{2} + β^{2} - 2 α β = \frac{1}{λ^{2}} . . . (1)$

$α^{2} + β^{2} + 2 α β = \frac{{(λ + 3)}^{2}}{λ^{2}} . . . (2)$

(2) - (1) $4 α β = \frac{{(λ + 3)}^{2} - 1}{λ^{2}}$

$\frac{12}{λ} = \frac{λ^{2} + 6 λ + 8}{λ^{2}}$

$\Rightarrow λ^{2} - 6 λ + 8 λ = 0$

$\Rightarrow λ = 0, 2, 4$

$Sum of possible values of λ is 6$

Q 19 :

If the set of all solutions of $| x^{2} + x - 9 | = | x | + | x^{2} - 9 |$ is $[α, β] \cup [γ, \infty)$ , then $(α^{2} + β^{2} + γ^{2})$ is equal to : [2026]

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(2)

We have, $| x^{2} + x - 9 | = | x | + | x^{2} - 9 |$

If $| a + b | = | a | + | b |$ , then $a \cdot b \geq 0$

$∴ x \cdot (x^{2} - 9) \geq 0$

$\Rightarrow x (x + 3) (x - 3) \geq 0$

$\Rightarrow x \in [- 3, 0] \cup [3, \infty)$

Hence, $α = - 3, β = 0, γ = 3$

Thus, $α^{2} + β^{2} + γ^{2} = 18$

Q 20 :

The sum of squares of all the real solutions of the equation $\log_{(x + 1)} (2 x^{2} + 5 x + 3) = 4 - \log_{(2 x + 3)} (x^{2} + 2 x + 1)$ is equal to ________. [2026]

(2)

$\log_{(x + 1)} (2 x^{2} + 5 x + 3) = 4 - \log_{(2 x + 3)} (x^{2} + 2 x + 1)$

$\Rightarrow \log_{(x + 1)} [(x + 1) (2 x + 3)] = 4 - \log_{(2 x + 3)} [(x + 1) (x + 1)]$

$\Rightarrow \log_{(x + 1)} (x + 1) + \log_{(x + 1)} (2 x + 3) = 4 - \log_{(2 x + 3)} {(x + 1)}^{2}$

$\Rightarrow 1 + \log_{(x + 1)} (2 x + 3) = 4 - 2 \log_{(2 x + 3)} (x + 1)$

$\Rightarrow \log_{(x + 1)} (2 x + 3) + 2 \log_{(2 x + 3)} (x + 1) - 3 = 0$

Let $t = \log_{(x + 1)} (2 x + 3)$ , then $\log (2_{x} + 3) (x + 1) = \frac{1}{t}$

$\Rightarrow t + \frac{2}{t} - 3 = 0$

$\Rightarrow t^{2} - 3 t + 2 = 0 \Rightarrow (t - 1) (t - 2) = 0$

Case (i), t = 1;

$\log_{(x + 1)} (2 x + 3) = 1$

$\Rightarrow x + 1 = 2 x + 3 \Rightarrow x = - 2$

For $\log_{(x + 1)}, (x + 1) > 0$

Let x + 1 = –1, which is invalid.

Case (ii), t = 2;

$\log_{(x + 1)} (2 x + 3) = 2$

$\Rightarrow x^{2} + 2 x + 1 = 2 x + 3 \Rightarrow x^{2} = 2$

$\Rightarrow x = \pm \sqrt{2}$

$\Rightarrow x = \sqrt{2}$ (valid)

$\Rightarrow x = - \sqrt{2}$ (invalid)

$∴$ Sum of square of all real solution = ${(\sqrt{2})}^{2} = 2$

Topic Question Set

Q 11 :

The number of real solutions of the equation $x$ $| x + 3 |$ $+$ $| x - 1 |$ $- 2 = 0$ is: [2026]

Q 12 :

The number of distinct real solutions of the equation $x$ $| x + 4 |$ $+ 3$ $| x + 2 |$ $+ 10 = 0$ is [2026]

Q 13 :

The smallest positive integral value of $a$ , for which all the roots of $x^{4} - a x^{2} + 9 = 0$ are real and distinct, is equal to [2026]

Q 14 :

The sum of all the real solutions of the equation

$\log_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 \log_{(6 x + 10)} (x^{2} + 6 x + 9)$ is equal to. [2026]

Q 15 :

Let $α and β$ be the roots of the equation $x^{2} + 2 a x + (3 a + 10) = 0$ such that $α < 1 < β$ . Then the set of all possible values of a is : [2026]

Q 16 :

The positive integer n, for which the solutions of the equation

$x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3}$

are two consecutive even integers, is : [2026]

Q 17 :

Let $α, β$ be the roots of the quadratic equation $12 x^{2} - 20 x + 3 λ = 0, λ \in Z .$ If $\frac{1}{2} \leq | β - α | \leq \frac{3}{2},$ then the sum of all possible values of $λ is$ . [2026]

Q 18 :

If $α, β where α < β$ are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3},$ then the sum of all possible values of $λ$ is [2026]

Q 19 :

If the set of all solutions of $| x^{2} + x - 9 | = | x | + | x^{2} - 9 |$ is $[α, β] \cup [γ, \infty)$ , then $(α^{2} + β^{2} + γ^{2})$ is equal to : [2026]

Q 20 :

The sum of squares of all the real solutions of the equation $\log_{(x + 1)} (2 x^{2} + 5 x + 3) = 4 - \log_{(2 x + 3)} (x^{2} + 2 x + 1)$ is equal to ________. [2026]

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Topic Question Set

Q 11 : The number of real solutions of the equation x|x+3|+|x-1|-2=0 is: [2026]

Q 12 : The number of distinct real solutions of the equation x|x+4|+3|x+2|+10=0 is [2026]

Q 13 : The smallest positive integral value of a, for which all the roots of x4-ax2+9=0 are real and distinct, is equal to [2026]

Q 14 : The sum of all the real solutions of the equation log(x+3)(6x2+28x+30)=5-2log(6x+10)(x2+6x+9) is equal to. [2026]

Q 15 : Let α and β be the roots of the equation x2+2ax+(3a+10)=0 such that α<1<β. Then the set of all possible values of a is : [2026]

Q 16 : The positive integer n, for which the solutions of the equation x(x+2)+(x+2)(x+4)+⋯+(x+2n-2)(x+2n)=8n3 are two consecutive even integers, is : [2026]

Q 17 : Let α, β be the roots of the quadratic equation 12x2-20x+3λ=0, λ∈Z. If 12≤|β-α|≤32, then the sum of all possible values of λ is. [2026]

Q 18 : If α,β where α<β are the roots of the equation λx2-(λ+3)x+3=0 such that 1α-1β=13, then the sum of all possible values of λ is [2026]

Q 19 : If the set of all solutions of |x2+x–9|=|x|+|x2–9| is [α,β]∪[γ,∞), then (α2+β2+γ2) is equal to : [2026]

Q 20 : The sum of squares of all the real solutions of the equation log(x+1)(2x2+5x+3)=4–log(2x+3)(x2+2x+1) is equal to ________. [2026]

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Q 11 :

The number of real solutions of the equation $x$ $| x + 3 |$ $+$ $| x - 1 |$ $- 2 = 0$ is: [2026]

Q 12 :

The number of distinct real solutions of the equation $x$ $| x + 4 |$ $+ 3$ $| x + 2 |$ $+ 10 = 0$ is [2026]

Q 13 :

The smallest positive integral value of $a$ , for which all the roots of $x^{4} - a x^{2} + 9 = 0$ are real and distinct, is equal to [2026]

Q 14 :

The sum of all the real solutions of the equation

$\log_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 \log_{(6 x + 10)} (x^{2} + 6 x + 9)$ is equal to. [2026]

Q 15 :

Let $α and β$ be the roots of the equation $x^{2} + 2 a x + (3 a + 10) = 0$ such that $α < 1 < β$ . Then the set of all possible values of a is : [2026]

Q 16 :

The positive integer n, for which the solutions of the equation

$x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3}$

are two consecutive even integers, is : [2026]

Q 17 :

Let $α, β$ be the roots of the quadratic equation $12 x^{2} - 20 x + 3 λ = 0, λ \in Z .$ If $\frac{1}{2} \leq | β - α | \leq \frac{3}{2},$ then the sum of all possible values of $λ is$ . [2026]

Q 18 :

If $α, β where α < β$ are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3},$ then the sum of all possible values of $λ$ is [2026]

Q 19 :

If the set of all solutions of $| x^{2} + x - 9 | = | x | + | x^{2} - 9 |$ is $[α, β] \cup [γ, \infty)$ , then $(α^{2} + β^{2} + γ^{2})$ is equal to : [2026]

Q 20 :

The sum of squares of all the real solutions of the equation $\log_{(x + 1)} (2 x^{2} + 5 x + 3) = 4 - \log_{(2 x + 3)} (x^{2} + 2 x + 1)$ is equal to ________. [2026]