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

Q 11 :
Consider the equation $x^{2} + 4 x - n = 0$ , where n $\in$ [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to [2025]

5
7
6
8

1

2

3

4

(3)

we have, $x^{2} + 4 x - n = 0$

$\Rightarrow x^{2} + 4 x + 4 = n + 4$

$\Rightarrow {(x + 2)}^{2} = n + 4$

$\Rightarrow x = - 2 \pm \sqrt{n + 4}$

But $20 \leq n \leq 100$

$\Rightarrow 24 \leq n + 4 \leq 104 \Rightarrow \sqrt{24} \leq \sqrt{n + 4} \leq \sqrt{104}$

$\Rightarrow \sqrt{n + 4} \in {5, 6, 7, 8, 9, 10}$ .

$∴$ 6 integral values of 'n' are possible.

Q 12 :
The product of all solutions of the equation $e^{5 {(\log_{e} x)}^{2} + 3} = x^{8}, x > 0$ , is: [2025]

$e^{2}$
e
$e^{8 / 5}$
$e^{6 / 5}$

1

2

3

4

(3)

We have, $e^{5 {(\log_{e} x)}^{2} + 3} = x^{8}, x > 0$

Taking log on both sides, we get

$5 {(\ln x)}^{2} + 3 = 8 \ln x \Rightarrow 5 {(\ln x)}^{2} + 3 - 8 \ln x = 0$

Put ln x = t

$\Rightarrow 5 t^{2} - 8 t + 3 = 0 \Rightarrow t_{1} + t_{2} = \frac{8}{5}$

$∴ \ln x_{1} + \ln x_{2} = 8 / 5 \Rightarrow \ln (x_{1} x_{2}) = 8 / 5 \Rightarrow x_{1} x_{2} = e^{8 / 5}$ .

Q 13 :
The product of all the rational roots of the equation ${(x^{2} - 9 x + 11)}^{2} - (x - 4) (x - 5) = 3$ , is equal to [2025]

28
21
7
14

1

2

3

4

(4)

We have, ${(x^{2} - 9 x + 11)}^{2} - (x - 4) (x - 5) = 3$

$\Rightarrow {(x^{2} - 9 x + 11)}^{2} - (x - 9 x + 20) = 3$

Let $x^{2} - 9 x = t$ , then

${(t + 11)}^{2} - (t + 20) = 3$

$\Rightarrow t^{2} + 121 + 22 t - t - 20 - 3 = 0$

$\Rightarrow t^{2} + 21 t + 98 = 0 \Rightarrow (t + 14) (t + 7) = 0$

$\Rightarrow t = - 7, - 14$

$\Rightarrow x^{2} - 9 x + 7 = 0 and x^{2} - 9 x + 14 = 0$

$\Rightarrow x = \frac{9 \pm \sqrt{53}}{2} and x = \frac{9 \pm 5}{2}$

$\Rightarrow$ Product of all rational roots = $(\frac{9 + 5}{2}) (\frac{9 - 5}{2}) = 7 \times 2 = 14$ .

Q 14 :
The number of real solution(s) of the equation $x^{2} + 3 x + 2 = \min {| x - 3 |, | x + 2 |}$ is : [2025]

2
3
1
0

1

2

3

4

(1)

$In (- \infty, \frac{1}{2})$ , |x + 2| is minimum and $In (\frac{1}{2}, \infty)$ , |x – 3| is minimum.

From the graph, only two solutions.

Q 15 :
The sum, of the squares of all the roots of the equation $x^{2} + | 2 x - 3 | - 4 = 0$ , is [2025]

$6 (2 - \sqrt{2})$
$3 (3 - \sqrt{2})$
$6 (3 - \sqrt{2})$
$3 (2 - \sqrt{2})$

1

2

3

4

(1)

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

Case I: $x^{2} + (2 x - 3) - 4 = 0 when x \geq \frac{3}{2}$

$\Rightarrow x^{2} + 2 x - 7 - 4 = 0 \Rightarrow x = 2 \sqrt{2} - 1$ $[∵ x \geq \frac{3}{2}]$

Case II : $x^{2} - (2 x - 3) - 4 = 0, x < \frac{3}{2}$

$\Rightarrow x^{2} - 2 x - 1 = 0$

$\Rightarrow x = \frac{2 \pm \sqrt{2}}{2} = 1 \pm \sqrt{2} \Rightarrow x = 1 - \sqrt{2}$ $[∵ x < \frac{3}{2}]$

$∴$ Required sum = ${(2 \sqrt{2} - 1)}^{2} + {(1 - \sqrt{2})}^{2}$

$= 12 - 6 \sqrt{2} = 6 (2 - \sqrt{2})$ .

Q 16 :
If $α + i β$ and $γ + i δ$ are the roots $x^{2} - (3 - 2 i) x - (2 i - 2) = 0, i = \sqrt{- 1}$ , then $α γ + β δ$ is equal to : [2025]

–6
6
–2
2

1

2

3

4

(4)

We have, $x^{2} - (3 - 2 i) x - (2 i - 2) = 0$

$x = \frac{(3 - 2 i) \pm \sqrt{{(3 - 2 i)}^{2} + 4 (1) (2 i - 2)}}{2}$

$= \frac{(3 - 2 i) \pm \sqrt{9 - 4 - 12 i + 8 i - 8}}{2} = \frac{(3 - 2 i) \pm \sqrt{- 3 - 4 i}}{2}$

$= \frac{(3 - 2 i) \pm \sqrt{{{(1)}^{2} - (2 i)}^{2} - 2 (1) (2 i)}}{2}$

$= \frac{(3 - 2 i) \pm \sqrt{{(1 - 2 i)}^{2}}}{2} = \frac{(3 - 2 i) \pm (1 - 2 i)}{2}$

$∴ x = \frac{3 - 2 i + 1 - 2 i}{2} or \frac{(3 - 2 i - 1 - 2 i)}{2}$

$∴ x = 2 - 2 i or 1 + 0 i$

So, $α γ + β δ = 2 (1) + (- 2) (0) = 2$ .

Q 17 :
The number of solutions of the equation $(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2) (\frac{7}{x} - \frac{7}{\sqrt{x}} + 3) = 0$ is : [2025]

1
3
4
2

1

2

3

4

(3)

We have, $(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2) (\frac{7}{x} - \frac{7}{\sqrt{x}} + 3) = 0$ ... (i)

Put in (i), we get

$(9 y^{2} - 9 y + 2) (2 y^{2} - 7 y + 3) = 0$

$\Rightarrow (3 y - 2) (3 y - 1) (2 y - 1) (y - 3) = 0$

$\Rightarrow y = \frac{2}{3}, \frac{1}{3}, \frac{1}{2}, 3 \Rightarrow x = \frac{9}{4}, 9, 4, \frac{1}{9}$

$∴$ Number of solutions = 4.

Q 18 :
If the set of all $a \in R$ , for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z : α < x < β}$ , then $\sum_{x \in X} x^{2}$ is equal to : [2025]

2129
2119
2109
2139

1

2

3

4

(4)

Since, the given equation has no real root

$∴ b^{2} - a c < 0 \Rightarrow {(a - 5)}^{2} - 4 (2) (15 - 3 a) < 0$

$\Rightarrow {(a - 5)}^{2} - 8 (15 - 3 a) < 0$

$\Rightarrow a^{2} - 10 a + 25 - 120 + 24 a < 0$

$\Rightarrow a^{2} - 14 a - 95 < 0 \Rightarrow a \in (- 5, 19)$

$∴ \sum_{x \in X} x^{2} = (1^{2} + 2^{2} + 3^{2} + 4^{2}) + o^{2} + {(1^{2} + 2^{2} + . . . + 18)}^{2}$

$= \frac{4 (5) (9)}{6} + \frac{18 (19) (37)}{6} = \frac{12834}{6} = 2139$ .

Q 19 :
Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $I m (β) < I m (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 5 \sqrt{7} i, P_{4} = - 3 \sqrt{7} i, P_{5} = 11 \sqrt{7} i and P_{6} = 45 \sqrt{7} i$ , then $| α^{4} + β^{4} |$ is equal to __________. [2025]

0%

31

We have, $α + β = a and α β = - b$

$∵ P_{6} = a P_{5} + b P_{4}$

$\Rightarrow 45 \sqrt{7} i = a \times 11 \sqrt{7} i + b (- 3 \sqrt{7}) i$

$\Rightarrow 45 = 11 a - 3 b$ ... (i)

and $P_{5} = a P_{4} + b P_{3}$

$\Rightarrow 11 \sqrt{7} i = a (- 3 \sqrt{7} i) + b (- 5 \sqrt{7} i)$

$\Rightarrow 11 = - 3 a - 5 b$ ... (ii)

On solving equations (i) and (ii), we get a = 3, b = –4

$∴ | α^{4} + β^{4} | = \sqrt{{(α^{4} - β^{4})}^{2} + 4 α^{4} β^{4}}$

$= \sqrt{- 63 + 4 . 4^{4}} = \sqrt{- 63 + 1024} = \sqrt{961} = 31$ .

Topic Question Set

Q 11 :
Consider the equation $x^{2} + 4 x - n = 0$ , where n $\in$ [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to [2025]

Q 12 :
The product of all solutions of the equation $e^{5 {(\log_{e} x)}^{2} + 3} = x^{8}, x > 0$ , is: [2025]

Q 13 :
The product of all the rational roots of the equation ${(x^{2} - 9 x + 11)}^{2} - (x - 4) (x - 5) = 3$ , is equal to [2025]

Q 14 :
The number of real solution(s) of the equation $x^{2} + 3 x + 2 = \min {| x - 3 |, | x + 2 |}$ is : [2025]

Q 15 :
The sum, of the squares of all the roots of the equation $x^{2} + | 2 x - 3 | - 4 = 0$ , is [2025]

Q 16 :
If $α + i β$ and $γ + i δ$ are the roots $x^{2} - (3 - 2 i) x - (2 i - 2) = 0, i = \sqrt{- 1}$ , then $α γ + β δ$ is equal to : [2025]

Q 17 :
The number of solutions of the equation $(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2) (\frac{7}{x} - \frac{7}{\sqrt{x}} + 3) = 0$ is : [2025]

Q 18 :
If the set of all $a \in R$ , for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z : α < x < β}$ , then $\sum_{x \in X} x^{2}$ is equal to : [2025]

Q 19 :
Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $I m (β) < I m (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 5 \sqrt{7} i, P_{4} = - 3 \sqrt{7} i, P_{5} = 11 \sqrt{7} i and P_{6} = 45 \sqrt{7} i$ , then $| α^{4} + β^{4} |$ is equal to __________. [2025]

0%

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Topic Question Set

Q 11 : Consider the equation x2+4x–n=0, where n ∈ [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to [2025]

Q 12 : The product of all solutions of the equation e5(logex)2+3=x8, x>0, is: [2025]

Q 13 : The product of all the rational roots of the equation (x2–9x+11)2–(x–4)(x–5)=3, is equal to [2025]

Q 14 : The number of real solution(s) of the equation x2+3x+2=min{|x–3|,|x+2|} is : [2025]

Q 15 : The sum, of the squares of all the roots of the equation x2+|2x–3|–4=0, is [2025]

Q 16 : If α+iβ and γ+iδ are the roots x2–(3–2i)x–(2i–2)=0, i=–1, then αγ+βδ is equal to : [2025]

Q 17 : The number of solutions of the equation (9x–9x+2)(7x–7x+3)=0 is : [2025]

Q 18 : If the set of all a∈R, for which the equation 2x2+(a–5)x+15=3a has no real root, is the interval (α,β), and X={x∈Z : α<x<β}, then ∑x∈Xx2 is equal to : [2025]

Q 19 : Let α, β be the roots of the equation x2–ax–b=0 with Im(β)<Im(β). Let Pn=αn–βn. If P3=–57i, P4=–37i, P5=117i and P6=457i, then |α4+β4| is equal to __________. [2025]

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Q 11 :
Consider the equation $x^{2} + 4 x - n = 0$ , where n $\in$ [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to [2025]

Q 12 :
The product of all solutions of the equation $e^{5 {(\log_{e} x)}^{2} + 3} = x^{8}, x > 0$ , is: [2025]

Q 13 :
The product of all the rational roots of the equation ${(x^{2} - 9 x + 11)}^{2} - (x - 4) (x - 5) = 3$ , is equal to [2025]

Q 14 :
The number of real solution(s) of the equation $x^{2} + 3 x + 2 = \min {| x - 3 |, | x + 2 |}$ is : [2025]

Q 15 :
The sum, of the squares of all the roots of the equation $x^{2} + | 2 x - 3 | - 4 = 0$ , is [2025]

Q 16 :
If $α + i β$ and $γ + i δ$ are the roots $x^{2} - (3 - 2 i) x - (2 i - 2) = 0, i = \sqrt{- 1}$ , then $α γ + β δ$ is equal to : [2025]

Q 17 :
The number of solutions of the equation $(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2) (\frac{7}{x} - \frac{7}{\sqrt{x}} + 3) = 0$ is : [2025]

Q 18 :
If the set of all $a \in R$ , for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z : α < x < β}$ , then $\sum_{x \in X} x^{2}$ is equal to : [2025]

Q 19 :
Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $I m (β) < I m (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 5 \sqrt{7} i, P_{4} = - 3 \sqrt{7} i, P_{5} = 11 \sqrt{7} i and P_{6} = 45 \sqrt{7} i$ , then $| α^{4} + β^{4} |$ is equal to __________. [2025]