Phrases Quadratic Equations Previous Year Question Paper and Solution with PDF

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Phrases Previous Year Questions (PYQs)

Phrases Quadratic Equations PYQ

Qus : 1

If the equation

$|x^2 – 6x + 8| = a$ has four real solution then find the value of

$a$ ?

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Solution

Qus : 2

Between any two real roots of the equation

$e^x sin x = 1$ , the equation

$e^x cos x = –1$ has

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Solution

Number of Roots

Given:

$e^x \sin x = 1$ has two real roots → say $x_1$ and $x_2$

Apply Rolle’s Theorem:

Since $f(x) = e^x \sin x$ is continuous and differentiable, and $f(x_1) = f(x_2)$ , ⇒ There exists $c \in (x_1, x_2)$ such that $f'(c) = 0$

Compute:

$f'(x) = e^x(\sin x + \cos x) = 0 \Rightarrow \tan x = -1$ At this point,

$e^x \cos x = -1$

$\boxed{\text{At least one root}}$

Qus : 3

Not Available

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Solution

Qus : 4

Let C denote the set of all tuples (x,y) which satisfy

$x^2 -2^y=0$ where x and y are natural numbers. What is the cardinality of C?

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Solution

Qus : 5

$\alpha , \beta$ are the roots of

$x^2-x-1=0$ and

$A_n=\alpha^n+\beta^n$ , the Arithmetic mean of

$A_{n-1}$ and

$A_n$ is

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Solution

Qus : 6

If the roots of the quadratic equation

$x^2+px+q=0$ are tan 30° and tan 15° respectively, then the value of 2 + p - q is

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Solution

Qus : 7

The quadratic equation whose roots are

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Solution

Qus : 8

For what value of p, the polynomial

$x^4-3x^3+2px^2-6$ is exactly divisible by

$(x-1)$

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Solution

Qus : 9

α, β are the roots of the an equation

$x^2- 2x cosθ + 1 = 0$ , then the equation having roots αn and βn is

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Solution

Qus : 10

The equation (x-a)³+(x-b)³+(x-c)³ = 0 has

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Solution

Let f(x) = (x – a)³ + (x – b)³ + (x – c)³.

Then f'(x) = 3{(x – a)² + (x – b)² + (x –c)²}

clearly , f'(x) > 0 for all x.

so, f'(x) = 0 has no real roots.

Hence, f(x) = 0 has two imaginary and one real root

Qus : 11

Let

and

be the roots f the equation

and

are the roots of the equation

, then the value of r,

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Solution

Qus : 12

$\alpha$ and

$\beta$ are the two roots of the quadratic equation

$x^2 + ax + b = 0, (ab \ne 0)$ then the quadratic roots whose roots

$\frac{1}{\alpha^3+\alpha}$ and

$\frac{1}{\beta^3+\beta}$ is

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Solution

Qus : 13

If α≠β and

$\alpha^2=5\alpha-3,\beta^2=5\beta-3$ , then the equation whose roots are

$\frac{\alpha}{\beta}$ and

$\frac{\beta}{\alpha}$ is

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Solution

Qus : 14

Given the equation

$x+y =1$ ,

$x^2+y^2 =2$ ,

$x^5 +y^5 =A$ . Let N be the number of solution pairs (x,y) to this system of equations. Then AN is equal to

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Solution

Qus : 15

$\alpha$ and

$\beta$ are the roots of the equation

$2x^{2}+ 2px + p^{2} = 0$ , where

$p$ is a non-zero real number, and

$\alpha^{4}$ and

$\beta^{4}$ are the roots of

$x^{2} - rx + s = 0$ , then the roots of

$2x^{2} - 4p^{2}x + 4p^{4} - 2r = 0$ are:

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Solution

Qus : 16

If x and y are positive real numbers satisfying the system of equations

$x^{2}+y\sqrt{xy}=336$ and

$y^{2}+x\sqrt{xy}=112$ , then x + y is:

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Solution

Qus : 17

The value of k for which the equation

$(k-2)x^{2}+8x+k+4=0$ has both real, distinct and negative roots is

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Solution

Qus : 18

Roots of equation are

$ax^2-2bx+c=0$ are n and m , then the value of

$\frac{b}{an^2+c}+\frac{b}{am^2+c}$ is

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Solution

Qus : 19

If a + b + c = 0, then the value of

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Solution

Qus : 20

$a, b, c$ are positive integers such that

$a^{2}+2b^{2}-2bc=100$ and

$2ab-c^{2}=100$ . Then the value of

$\frac{a+b}{c}$ is

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Solution

Qus : 21

$x^{2} + 2ax + 10 - 3a > 0$ for all x ∈ R, then

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Solution

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Phrases Quadratic Equations PYQ

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