Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 3

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In a triangle $P Q R, ∠ R = \frac{π}{2}$ . If $\tan (\frac{P}{2})$ and $\tan (\frac{Q}{2})$ are the roots of $a x^{2} + b x + c = 0; a \neq 0$ then

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Important Questions on Quadratic Equations

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 5

Let

p

and

q

be real numbers such that

p \neq 0, p^{3} \neq q

and

p^{3} \neq - q

. If

α

and

β

are non zero complex numbers satisfying

α + β = - p

and

α^{3} + β^{3} = q

, then a quadratic equation having

\frac{α}{β}

and

\frac{β}{α}

as its roots is

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 6

Let

α

and

β

be the roots of

x^{2} - 6 x - 2 = 0

, with

α > β

. If

a_{n} = α^{n} - β^{n}

for

n \geq 1

, then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 7

The quadratic equation

p (x) = 0

with real coefficients has purely imaginary roots. Then the equation

p (p (x)) = 0

has?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 8

Let

S

be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1}

and

x_{2}

satisfying the inequality

|x_{1} - x_{2}| < 1

. Which of the following intervals is (are) a subset(s) of

S

?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 9

Let

- \frac{π}{6} < θ < - \frac{π}{12} .

Suppose

α_{1}

and

β_{1}

are the roots of the equation

x^{2} - 2 x \sec θ + 1 = 0

and

α_{2}

and

β_{2}

are the roots of the equation

x^{2} + 2 x \tan θ - 1 = 0 .

If

α_{1} > β_{1}

and

α_{2} > β_{2},

then

α_{1} + β_{2}

equals?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 10

A value of $b$ for which the equations $x^{2} + b x - 1 = 0$ and $x^{2} + x + b = 0$ have one root in common is?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 12

Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots.

(4, 3) .

Rahul made a mistake in writing down the coefficient of

x

to get roots

(3, 2) .

The correct roots of the equation are?

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Alpha Question Bank for Engineering: Mathematics>Chapter 4 - Quadratic Equations>Exercise-3>Q 13

Let for

a \neq a_{1} \neq 0

,

f (x) = a x^{2} + b x + c

,

g (x) = a_{1} x^{2} + b_{1} x + c_{1}

and

p (x) = f (x) - g (x) .

If

p (x) = 0

only for

x = - 1

and

p (- 2) = 2,

then the value of

p (2)

is?