HARD

Earn 100

If $a, b, c$ are in $G P,$ then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, if $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in.

50% studentsanswered this correctly

Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

Two distinct polynomials

f (x)

and

g (x)

are defined as follows:

f (x) = x^{2} + a x + 2; g (x) = x^{2} + 2 x + a

If the equations

f (x) = 0

and

g (x) = 0

have a common root then the sum of roots of the equation

f (x) + g (x) = 0

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

f (x) = x^{2} + a x + 2 = 0

and

g (x) = x^{2} + 2 x + a = 0

have only one real common root, then sum of the roots of

f (x) + g (x) = 0

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

Let

b_{i} > 1

for

i = 1, 2, \dots ., 101 .

Suppose

\log_{e} b_{1}, \log_{e} b_{2}, \dots . ., \log_{e} b_{101}

are in Arithmetic Progression (A.P.) with the common difference

\log_{e} 2 .

Suppose

a_{1}, a_{2}, \dots ., a_{101}

are in A.P. such that

a_{1} = b_{1}

and

a_{51} = b_{51} .

t = b_{1} + b_{2} + \dots + b_{51}

and

s = a_{1} + a_{2} + \dots + a_{51}

, then

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

If the equations

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

and

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

a, b, c \in R,

have a common root, then

a : b : c

is:

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

The quadratic equations

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

and

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

have one root in common. The other roots of the first and second equations are integers in the ratio

4 : 3

. Then, the common root is

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

The number of real solutions of the equation

\sin^{- 1} (\sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} {(\frac{x}{2})}^{i}) = \frac{π}{2} - \cos^{- 1} (\sum_{i = 1}^{\infty} {(- \frac{x}{2})}^{i} - \sum_{i = 1}^{\infty} {(- x)}^{i})

lying in the interval

(- \frac{1}{2}, \frac{1}{2})

is____.

(Here, the inverse trigonometric functions

\sin^{- 1} x & \cos^{- 1} x

assume values in

[- \frac{π}{2}, \frac{π}{2}] & [0, π]

respectively.)

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

If $m$ is the $A . M .$ of two distinct real numbers $I$ and $n$ $(I, n > 1)$ and $G_{1}, G_{2} and G_{3}$ are three geometric means between $I and n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

Suppose

p, q, r

are real numbers such that

q = p (4 - p), r = q (4 - q), p = r (4 - r) .

The maximum possible value of

p + q + r

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

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

and

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

have a common root where

a, b, c \in N

(set of natural numbers), the least value of

a + b + c

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

The number of pairs

(a, b)

of real numbers, such that whenever

α

is a root of the equation

x^{2} + a x + b = 0, α^{2} - 2

is also a root of this equation, is :

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

The sum of first

20

terms of the sequence

0.7, 0.77, 0.777, . . . . . .,

is :

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

If the equations

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

and

x^{2} + x + b = 0

have a common root different from

- 1,

then

|b|

is equal to :

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

The sum of the

3^{r d}

and the

4^{t h}

terms of a

G . P .

60

and the product of its first three terms is

1000.

If the first term of this

G . P .

is positive, then its

7^{t h}

term is:

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

Let

a, b

and

c

be the

7^{t h}, 11^{t h}

and

13^{t h}

terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then

\frac{a}{c}

is equal to:

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

Three positive numbers form an increasing

G . P .

If the middle term in this

G . P .

is doubled, the new numbers are in

A . P .

Then the common ratio of the

G . P .

is :

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

Let

x, y, z

be three non-negative integers such that

x + y + z = 10

. The maximum possible value of

x y z + x y + y z + z x

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

The quotient when

1 + x^{2} + x^{4} + . . . + x^{34}

is divided by

1 + x + x^{2} + . . . + x^{17}

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

For

a \neq b

, if the equation

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

and

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

have a common root, then the value of

a + b =

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

Let

λ \neq 0

be in

R

. If

α

and

β

are the roots of the equation,

x^{2} - x + 2 λ = 0

and

α

and

γ

are the roots of the equation,

3 x^{2} - 10 x + 27 λ = 0,

then

\frac{β γ}{λ}

is equal to:

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

If the equations

x^{2} + a x + 1 = 0

and

x^{2} - x - a = 0

have a real common root

b

,

then the value of

b

is equal to

If a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root, if da, eb, fc are in.

Important Questions on Theory of Equation

Learn and Prepare for any exam you want !

If $a, b, c$ are in $G P,$ then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root, if $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in.