Solution of Quadratic Equations

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JEE Main 2016

The sum of all real values of $x$ satisfying the equation $(x^{2} - 5 x + 5)^{x^{2} + 4 x - 60} = 1$ is :

- 4

JEE Main 2005

LEVELJEE Main

A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of $2 cm/s^{2}$ and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then the lizard will catch the insect after

If $(1 - p)$ is a root of quadratic equation $x^{2} + p x + (1 - p) = 0$ then its root are

(A)

- 1, 2

(B)

- 1, 1

$x^{2} - 3∣ x ∣ + 2 = 0$ is

The set of all real numbers $x$ for which $x^{2} - ∣ x + 2∣ + x > 0$ , is

(A)

(- \infty, - 2) \cup (2, \infty)

(B)

(- \infty, - 2) \cup (2, \infty)

(C)

(- \infty, - 1) \cup (1, \infty)

(D)

(2, \infty)

JEE Advanced 1984

LEVELBoard

The equation $\frac{x}{x - 1} + \frac{2}{x - 1} = 1 - \frac{2}{x - 1}$ has

infinitely many roots

JEE Advanced 1982

LEVELBoard

The number of real solutions of the equation $∣ x ∣^{2} - 3∣ x ∣ + 2 = 0$ is

The value of $6 + lo g_{3/2} (\frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \dots)$ is

JEE Advanced 1978

LEVELBoard

Solve for $x : x + 1 - x - 1 = 1$ .

JEE Advanced 1978

LEVELJEE Main

Find all integers $x$ for which $(5 x - 1) < (x + 1)^{2} < (7 x - 3)$ .

JEE Advanced 1986

LEVELJEE Main

For $a \leq 0$ , determine all real roots of the equation $x^{2} - 2 a ∣ x - a ∣ - 3 a^{2} = 0$

JEE Advanced 1983

LEVELJEE Main

Find all real values of $x$ which satisfy $x^{2} - 3 x + 2 > 0$ and $x^{2} - 2 x - 4 \leq 0$

JEE Main 2018 (Paper 1)

LEVELJEE Main

Let $S = {x \in R : x \geq 0 and 2∣ x - 3∣ + x (x - 6) + 6 = 0}$ . Then $S$ :

(A)

contains exactly four elements

(B)

is an empty set

(C)

contains exactly one element

(D)

contains exactly two elements

JEE Main 2019 (10 January Shift 1)

LEVELBoard

Consider the statement : "P(n): $n^{2} - n + 41$ is prime." Then which one of the following is true?

(A)

(1) P(5) is false but P(3) is true

(B)

(2) Both P(3) and P(5) are false

(C)

(3) P(3) is false but P(5) is true

(D)

(4) Both P(3) and P(5) are true

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

The sum of the solutions of the equation $∣ x - 2∣ + x (x - 4) + 2 = 0, (x > 0)$ is equal to :

(A)

4

(B)

9

(C)

10

(D)

12

JEE Main 2019 (10 April Shift 2)

LEVELJEE Main

The number of real roots of the equation $5 + ∣ 2^{x} - 1∣ = 2^{x} (2^{x} - 2)$ is :

JEE Main 2020 (8 January Shift 2)

LEVELJEE Advanced

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2 = ∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣$ . Then $S$ :

contains at least four elements

(D)

contains exactly two elements.

JEE Main 2020 (8 January Shift 2)

LEVELJEE Main

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣ = 0$ . Then $S$ :

at least four elements

(D)

infinite elements

JEE Main 2021 (25 July Shift 1)

LEVELJEE Main

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 :

JEE Main 2021 (25 July Shift 2)

LEVELBoard

The number of real solutions of the equation, $x^{2} - ∣ x ∣ - 12 = 0$ is:

JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

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 :

(A)

56 \times 3^{25}

(B)

56 \times 3^{24}

(C)

52 \times 3^{24}

(D)

28 \times 3^{25}

JEE Main 2021 (24 February Shift 2)

LEVELJEE Main

The number of the real roots of the equation $(x + 1)^{2} + ∣ x - 5∣ = \frac{27}{4}$ is

JEE Main 2021 (March)

LEVELJEE Main

The value of $4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + ......\infty}}}}$

(A)

2 + \frac{30}{2}

(B)

2 + \frac{30}{5}

(C)

4 + \frac{30}{5}

(D)

5 + \frac{30}{2}

JEE Main 2021 (March)

LEVELJEE Main

The value of $3 + \frac{1}{4 + \frac{1}{3 + \frac{1}{4 + \frac{1}{3 + ...\infty}}}}$ is equal to

(A)

1.5 + 3

(B)

2 + 3

(C)

3 + 23

(D)

4 + 3

JEE Main 2022 (24 June Shift 2)

LEVELBoard

The sum of all the real roots of the equation $(e^{2 x} - 4) (6 e^{2 x} - 5 e^{x} + 1) = 0$ is

(A)

lo g_{e} 3

(B)

- lo g_{e} 3

(C)

lo g_{e} 6

(D)

- lo g_{e} 6

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all real values of x satisfying the equation (x2−5x+5)x2+4x−60=1 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (1−p) is a root of quadratic equation x2+px+(1−p)=0 then its root are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x2−3∣x∣+2=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of all real numbers x for which x2−∣x+2∣+x>0, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation x−1x​+x−12​=1−x−12​ has

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real solutions of the equation ∣x∣2−3∣x∣+2=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 6+log3/2​(32​1​4−32​1​4−32​1​4−…​​​) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solve for x:x+1​−x−1​=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find all integers x for which (5x−1)<(x+1)2<(7x−3).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a≤0, determine all real roots of the equation x2−2a∣x−a∣−3a2=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find all real values of x which satisfy x2−3x+2>0 and x2−2x−4≤0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={x∈R:x≥0 and 2∣x​−3∣+x​(x​−6)+6=0}. Then S :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the statement : "P(n): n2−n+41 is prime." Then which one of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the solutions of the equation ∣x​−2∣+x​(x​−4)+2=0,(x>0) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real roots of the equation 5+∣2x−1∣=2x(2x−2) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S be the set of all real roots of the equation, 3x(3x−1)+2=∣3x−1∣+∣3x−2∣. Then S:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S be the set of all real roots of the equation, 3x(3x−1)+2∣3x−1∣+∣3x−2∣=0. Then S:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real roots of the equation e6x−e4x−2e3x−12e2x+ex+1=0 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real solutions of the equation, x2−∣x∣−12=0 is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β are the distinct roots of the equation x2+(3)1/4x+31/2=0, then the value of α96(α12−1)+β96(β12−1) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of the real roots of the equation (x+1)2+∣x−5∣=427​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 4+5+4+5+4+......∞1​1​1​1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 3+4+3+4+3+...∞1​1​1​1​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of all the real roots of the equation (e2x−4)(6e2x−5ex+1)=0 is

The sum of all real values of $x$ satisfying the equation $(x^{2} - 5 x + 5)^{x^{2} + 4 x - 60} = 1$ is :

If $(1 - p)$ is a root of quadratic equation $x^{2} + p x + (1 - p) = 0$ then its root are

$x^{2} - 3∣ x ∣ + 2 = 0$ is

The set of all real numbers $x$ for which $x^{2} - ∣ x + 2∣ + x > 0$ , is

The equation $\frac{x}{x - 1} + \frac{2}{x - 1} = 1 - \frac{2}{x - 1}$ has

The number of real solutions of the equation $∣ x ∣^{2} - 3∣ x ∣ + 2 = 0$ is

The value of $6 + lo g_{3/2} (\frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \dots)$ is

Solve for $x : x + 1 - x - 1 = 1$ .

Find all integers $x$ for which $(5 x - 1) < (x + 1)^{2} < (7 x - 3)$ .

For $a \leq 0$ , determine all real roots of the equation $x^{2} - 2 a ∣ x - a ∣ - 3 a^{2} = 0$

Find all real values of $x$ which satisfy $x^{2} - 3 x + 2 > 0$ and $x^{2} - 2 x - 4 \leq 0$

Let $S = {x \in R : x \geq 0 and 2∣ x - 3∣ + x (x - 6) + 6 = 0}$ . Then $S$ :

Consider the statement : "P(n): $n^{2} - n + 41$ is prime." Then which one of the following is true?

The sum of the solutions of the equation $∣ x - 2∣ + x (x - 4) + 2 = 0, (x > 0)$ is equal to :

The number of real roots of the equation $5 + ∣ 2^{x} - 1∣ = 2^{x} (2^{x} - 2)$ is :

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2 = ∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣$ . Then $S$ :

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣ = 0$ . Then $S$ :

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 :

The number of real solutions of the equation, $x^{2} - ∣ x ∣ - 12 = 0$ is:

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 :

The number of the real roots of the equation $(x + 1)^{2} + ∣ x - 5∣ = \frac{27}{4}$ is

The value of $4 + \frac{1}{5 + \frac{1}{4 + \frac{1}{5 + \frac{1}{4 + ......\infty}}}}$

The value of $3 + \frac{1}{4 + \frac{1}{3 + \frac{1}{4 + \frac{1}{3 + ...\infty}}}}$ is equal to

The sum of all the real roots of the equation $(e^{2 x} - 4) (6 e^{2 x} - 5 e^{x} + 1) = 0$ is