Transformatio...

Transformation of Roots: Linear Combination of Roots

Trending Questions

Q. If

α, β

are the roots of quadratic equation

x^{2} + 7 x + 10 = 0

. Then the quadratic equation with roots as

α + 2

and

β + 2

$x^{2} + 7 x + 18 = 0$
$x^{2} + 3 x + 10 = 0$
$x^{2} + 3 x = 0$
$x^{2} + 11 x + 18 = 0$

View Solution

Q. If the equation

a x^{2} + b x + c = 0

is not altered when each of the coefficient is increased by same quantity, then the value of

x^{2} + x + 1

View Solution

Q. If

α, β

are the roots of

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

, then the equation whose roots are

- \frac{α}{3}, - \frac{β}{3}

$9 x^{2} + 6 x + 1 = 0$
$27 x^{2} + 6 x + 1 = 0$
$27 x^{2} + 6 x - 1 = 0$
$27 x^{2} - 6 x + 1 = 0$

View Solution

Q. If

α, β

are the roots of the equation

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

, then the equation whose roots are

3 α - 5, 3 β - 5

$x^{2} + 22 x + 76 = 0$
$x^{2} + 2 x - 44 = 0$
$9 x^{2} - 18 x + 4 = 0$
$x^{2} + 22 x - 76 = 0$

View Solution

Q. If

α, β, γ

are the roots of

x^{3} - 3 x + 7 = 0

, then the equation whose roots are

α + β - γ, β + γ - α, γ + α - β

$x^{3} + 12 x - 56 = 0$
$x^{3} - 12 x + 56 = 0$
$x^{3} - 12 x - 56 = 0$
$x^{3} + 12 x + 56 = 0$

View Solution

Q. If

α, β

are the roots of the equation

x^{2} - (3 + \sqrt{2^{5 {log}_{2} 4} - {log}_{2} ({log}_{3} ({log}_{6} 216)}) x - 2 (3^{{log}_{3} 2} - 2^{{log}_{2} 3}) = 0

, then the equation whose roots are

- α, - β

$x^{2} + 35 x - 2 = 0$
$x^{2} + 29 x + 2 = 0$
$x^{2} + 19 x + 2 = 0$
$x^{2} + 35 x + 2 = 0$

View Solution

Q. If roots of the equation

(x - 12)^{2} + (2 x - 24) (x + 12) + (x + 12)^{2} = 0

are decreased by

12

, then the roots of the transformed equation are

rational and equal roots
non real roots
irrational roots
rational and distinct roots

View Solution

Q. If $\alpha, \beta$ are the roots of $2x^2+5x+1=0$, then the equation whose roots are $2\alpha +1, 2\beta +1$ is

View Solution

Q. If roots of the equation

a x^{2} + b x + c = 0

are

α, β

, then the equation whose roots are

\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β},

where

α \neq 1, β \neq 1

$a (x^{2} - 1) + b (x - 1)^{2} + c (x + 1)^{2} = 0$
$a (x + 1)^{2} + b (x^{2} - 1) + c (x + 1) = 0$
$a (x - 1)^{2} + b (x^{2} - 1) + c (x - 1) = 0$
$a (x - 1)^{2} + b (x^{2} - 1) + c (x + 1)^{2} = 0$

View Solution

Q. If

α, β

are zeroes of a polynomial

6 x^{2} + x - 2

, then the polynomial whose zeroes are

2 α + 3 β

and

3 α + 2 β

, is .

$6 x^{2} + 5 x + 1$
$6 x^{2} - 5 x + 1$
$6 x^{2} + 5 x - 1$
$6 x^{2} - 5 x - 1$

View Solution

Q. If

α, β

are the roots of the quadratic equation

a x^{2} + b x + c = 0, a \neq 0

. The equation with roots as

k α & k β \forall k \in R

is:

$a (x - k)^{2} + b (x - k) + c = 0$
$a (k x)^{2} + b (k x) + c = 0$
$k a x^{2} + b k x + c = 0$
$a {(\frac{x}{k})}^{2} + b (\frac{x}{k}) + c = 0$

View Solution

Q. If one root of

4 x^{2} - 2 x + (λ - 4) = 0

is reciprocal of the other, then value of

λ

View Solution

Q. If roots of the equation

a x^{2} + b x + c = 0

are

α, β

, then the equation whose roots are

\frac{1 + α}{1 - α}, \frac{1 + β}{1 - β},

where

α \neq 1, β \neq 1

$a (x + 1)^{2} + b (x^{2} - 1) + c (x + 1) = 0$
$a (x - 1)^{2} + b (x^{2} - 1) + c (x - 1) = 0$
$a (x - 1)^{2} + b (x^{2} - 1) + c (x + 1)^{2} = 0$
$a (x^{2} - 1) + b (x - 1)^{2} + c (x + 1)^{2} = 0$

View Solution

Q. If

α, β

are the roots of the quadratic equation

a x^{2} + b x + c = 0, a \neq 0

. Then the equation with roots

α + k & β + k \forall k \in R

is:

$a (x + k)^{2} + b (x + k) + c = 0$
$a (x - k)^{2} + b (x - k) - c = 0$
$a (x - k)^{2} - b (x - k) + c = 0$
$a (x - k)^{2} + b (x - k) + c = 0$

View Solution

If $α, β$ are the roots of $a x^{2} + b x + c = 0$ , the correct statements about the equation a ${(x - 2)}^{2} + b (x - 2) + c = 0$ is

$2 α, 2 β$ are the roots
$α - 2, β - 2$ are the roots
$α + 2, β + 2$ are the roots
$\frac{α}{2}, \frac{β}{2}$ are the roots

View Solution

Q. If

α, β

are the roots of

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

, then the equation whose roots are

2 α, 2 β

$4 x^{2} + 3 x + 4 = 0$
$x^{2} + 3 x + 8 = 0$
$4 x^{2} + 3 x + 2 = 0$
$x^{2} + 3 x + 4 = 0$

View Solution

Q. Choose the correct pair of quadratic equations with roots with opposite sign.

$- x^{2} - 5 x + 6 = 0$
$- x^{2} + 5 x = 0$
$- x^{2} - 5 x = 0$
$- x^{2} + 5 x + 6 = 0$
$x^{2} + 5 x + 6 = 0$
$x^{2} - 5 x + 6 = 0$

View Solution

Q. If

α, β

are the roots of the quadratic equation

a x^{2} + b x + c = 0, a \neq 0

. Find the equation with roots

α - k

and

β - k \forall k \in R

$a (x - k)^{2} + b (x - k) - c = 0$
$a (x - k)^{2} + b (x - k) + c = 0$
None of the above
$a (x + k)^{2} + b (x + k) + c = 0$

View Solution

Find the equation whose roots are equal in magnitude but opposite in sign to the roots of the equation $x^{5}$ - 3 $x^{3}$ + 2 $x^{2}$ + 4x + 1 = 0.

$x^{5}$ - 3 $x^{3}$ + 2 $x^{2}$ + 4x + 1 = 0
$x^{5}$ + 2 $x^{3}$ + 3 $x^{2}$ + 5x - 1 = 0
$x^{5}$ + $x^{3}$ + $x^{2}$ + x + 1 = 0
$x^{5}$ - 3 $x^{3}$ - 2 $x^{2}$ + 4x - 1 = 0

View Solution

If $α, β$ are the roots of the quadratic equation $x^{2} + 3 x + 6 = 0$ , then find the equation whose roots are $\frac{1 + α}{1 - α}$ , $\frac{1 + β}{1 - β}$

$5 y^{2} + 5 y + 2 = 0$
$5 y^{2} + 2 y + 5 = 0$
$2 y^{2} + 5 y + 5 = 0$
$5 y^{2} - 2 y + 5 = 0$

View Solution

Q. If the roots of

a x^{2} - b x - c = 0

are increased by same quantity, then which of the following expressions does not change?

$\frac{b^{2} + 4 a c}{a^{2}}$
$\frac{b^{2} - 4 a c}{a^{2}}$
$\frac{b^{2} - 4 a c}{a}$
$\frac{b^{2} + 4 a c}{a}$

View Solution

Q. The value of

a

for which one root of the quadratic equation

(a^{2} - 5 a + 3) x^{2} + (3 a - 1) x + 2 = 0

is twice as large as the other is:

$\frac{2}{3}$
$\frac{- 2}{3}$
$\frac{- 3}{2}$
$\frac{3}{2}$

View Solution

Q. If

α, β

are the roots of the quadratic equation

x^{2} - 7 x + 12 = 0

. Then the equation with roots

- α, - β

$x^{2} + 7 x - 12 = 0$
$x^{2} - 7 x - 12 = 0$
$x^{2} + 7 x + 12 = 0$
$x^{2} - 7 x + 12 = 0$

View Solution

Q. If roots of the equation

(x - 12)^{2} + (2 x - 24) (x + 12) + (x + 12)^{2} = 0

are decreased by

12

, then the roots of the transformed equation are

irrational roots
rational and equal roots
non real roots
rational and distinct roots

View Solution

Q. If

α, β

are the roots of

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

, then the equation whose roots are

2 α, 2 β

$x^{2} + 3 x + 4 = 0$
$4 x^{2} + 3 x + 4 = 0$
$x^{2} + 3 x + 8 = 0$
$4 x^{2} + 3 x + 2 = 0$

View Solution

If $α, β$ are the roots of the quadratic equation $x^{2} + 3 x + 6 = 0$ , then find the equation whose roots are $\frac{1 + α}{1 - α}$ , $\frac{1 + β}{1 - β}$

$5 y^{2} + 2 y + 5 = 0$
$5 y^{2} + 5 y + 2 = 0$
$2 y^{2} + 5 y + 5 = 0$
$5 y^{2} - 2 y + 5 = 0$

View Solution

Q. If

α, β

are the roots of the quadratic equation

a x^{2} + b x + c = 0, a \neq 0

. Then the equation with roots

α + k & β + k \forall k \in R

is:

$a (x - k)^{2} - b (x - k) + c = 0$
$a (x - k)^{2} + b (x - k) + c = 0$
$a (x - k)^{2} + b (x - k) - c = 0$
$a (x + k)^{2} + b (x + k) + c = 0$

View Solution

Q. The value of

k

for which the equation

3 x^{2} + 2 x (k^{2} + 1) + k^{2} - 3 k + 2 = 0

has roots of opposite signs, lies in the interval

$(1, 2)$
$(\frac{3}{2}, 2)$
$(- \infty, 0)$
$(- \infty, - 1)$

View Solution

Q. If

α, β

are the roots of

x^{2} + 7 x + 5 = 0

, then the equation whose roots are

α - 1, β - 1

$x^{2} + 5 x - 1 = 0$
$x^{2} + 5 x + 1 = 0$
$x^{2} + 7 x + 13 = 0$
$x^{2} + 9 x + 13 = 0$

View Solution

Q. If

α, β

are the roots of

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

, then the equation whose roots are

- \frac{α}{3}, - \frac{β}{3}

$27 x^{2} + 6 x + 1 = 0$
$27 x^{2} + 6 x - 1 = 0$
$27 x^{2} - 6 x + 1 = 0$
$9 x^{2} + 6 x + 1 = 0$

View Solution