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JEE • MATHEMATICS • THEORY OF EQUATION

Theory of equation practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 31-48 of 48 questions

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31 Medium

31 of 48

Analyzing +4 | -1

Difference between the corresponding roots of

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

and

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

is same and

a \neq b

, then

View answer and explanation

Correct option: A

a+b+4=0

Option A (

a+b+4=0

) is correct because by equating the difference between the roots of the two equations, we can derive the relationship between a and b.

32 Medium

32 of 48

Application +4 | -1

\alpha ,\;\beta ,\;\gamma

are the roots of

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

, then

{\alpha ^{ - 2}} + {\beta ^{ - 2}} + {\gamma ^{ - 2}}

is equal to

View answer and explanation

Correct option: B

Option B (13) is correct because it is the value of the expression

{\alpha ^{ - 2}} + {\beta ^{ - 2}} + {\gamma ^{ - 2}}

calculated using Vieta's formulas.

33 Medium

33 of 48

Analyzing +4 | -1

In the complex plane, let

A

be the set of solutions to

z^{3}-8=0

and let

B

be the set of solution to

z^{3}-8 z^{2}-8 z+64=0

, then the maximum distance between a point of

A

and a point of

B

View answer and explanation

Correct option: D

2 \sqrt{21}

The correct answer is

2 \sqrt{21}

because the maximum distance is found by identifying the points in sets A and B that are farthest apart in the complex plane.

34 Medium

34 of 48

Analyzing +4 | -1

\alpha ,\;\beta ,\;\gamma

are the roots of the equation

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

, then

\frac{1}{{{\alpha ^4}}} + \frac{1}{{{\beta ^4}}} + \frac{1}{{{\gamma ^4}}}

View answer and explanation

Correct option: B

\frac{3}{7}

Option B (

\frac{3}{7}

) is correct because it involves using Vieta's formulas and algebraic manipulation to find the sum of the reciprocals of the fourth powers of the roots.

35 Medium

35 of 48

Analyzing +4 | -1

If the functions

f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}

and

g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b

have a common extreme point, then

a+2 b+7

is equal to

View answer and explanation

Correct option: C

The correct answer is 6 because the common extreme point implies that the derivatives of the two functions have a common root, which allows us to solve for a + 2b + 7.

36 Medium

36 of 48

Analyzing +4 | -1

{x^2} + ax + 1

is a factor of

a{x^3} + bx + c

, then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the conditions derived from polynomial division do not match any of the given options.

37 Medium

37 of 48

Analyzing +4 | -1

If the roots of the equation

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

from a non-decreasing H.P., then

View answer and explanation

Correct option: B

b = - 3

Option B (

b = - 3

) is correct because the roots form a non-decreasing H.P., which leads to a specific value for b.

38 Medium

38 of 48

Analyzing +4 | -1

If the sum of the roots of the equation

a x^2+b x+c=0

equal to the sum of the reciprocals of their squares. Then

\frac{a}{c}, \frac{b}{a}, \frac{c}{b}

are in

View answer and explanation

Correct option: C

H.P

Option C (H.P) is correct because after manipulating the given equation and simplifying, the resulting terms are in Arithmetic Progression, which implies the reciprocals are in Harmonic Progression.

39 Medium

39 of 48

Analyzing +4 | -1

x

and

y

are positive integers such that

x y+x+y=71, x^2 y+x y^2=880

, then

x^2+y^2

is equal to

View answer and explanation

Correct option: C

146

Option C (146) is correct because it is the result of solving the system of equations and calculating x² + y².

40 Medium

40 of 48

Analyzing +4 | -1

The sum of the roots of the equation

2^{33 x-2}+2^{11 x+2}=2^{22 x+1}+1

View answer and explanation

Correct option: B

\frac{2}{11}

Option B (2/11) is correct because the sum of the roots can be found by transforming the equation and using properties of exponents.

41 Medium

41 of 48

Analyzing +4 | -1

If the equation

{a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + \ldots + {a_1}x = 0

has a positive root

\alpha

, then the equation

n\;{a_n}{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + \ldots + {a_1} = 0

has

View answer and explanation

Correct option: A

A positive root less than

\alpha

Option A is correct because by applying Rolle's theorem to a related function, we can show that the given equation has a positive root less than α.

42 Medium

42 of 48

Application +4 | -1

x = 2 + {2^{2/3}} + {2^{1/3}},

then the value of

{x^3} - 6{x^2} + 6x

View answer and explanation

Correct option: B

Option B (2) is correct because it involves algebraic manipulation to simplify the given expression.

43 Medium

43 of 48

Analyzing +4 | -1

f(x)

is a polynomial of degree

4, f^{\prime}(1)=f(2)=f(3)=f(4)=0

, then root of

f(x)=0

View answer and explanation

Correct option: A

\frac{5}{11}

Option C (

\frac{5}{11}

) is correct because it is derived from the condition

f'(1) = 0

using the given information about the polynomial

f(x)

44 Medium

44 of 48

Application +4 | -1

\alpha, \beta, \gamma

are the roots of

2 x^3-x+1=0

then the value of

\alpha^3+\beta^3+\gamma^3

View answer and explanation

Correct option: C

-\frac{3}{2}

Option C (-

\frac{3}{2}

) is correct because by using Vieta's formulas and the given cubic equation, we can find the sum of the cubes of the roots.

45 Medium

45 of 48

Understanding +4 | -1

x = c

is a root of order 2 of a polynomial

f\left( x \right),

then

x = c

is also a root of the polynomial

View answer and explanation

Correct option: A

f'\left( x \right)

Option A (

f'(x)

) is correct because if

x=c

is a root of order 2 of

f(x)

, then

f(x) = (x-c)^2 g(x)

, and

f'(x)

will also have

(x-c)

as a factor, making

x=c

a root of

f'(x)

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46 Medium

46 of 48

Application +4 | -1

\alpha, \beta, \gamma

and

\sigma

are the roots of the equation

x^4+4 x^3-6 x^2+7 x-9=0

, then the value of

\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\gamma^2\right)\left(1+\sigma^2\right)

View answer and explanation

Correct option: C

Option C (13) is correct because substituting

x=i

and

x=-i

into the given polynomial and multiplying the results yields 13.

47 Medium

47 of 48

Application +4 | -1

Let

\left(1-x-2 x^2\right)^6=1+a_1 x+a_2 x^2+\ldots . .+a_{12} x^{12}

. Then

\frac{a_2}{2^2}+\frac{a_4}{2^4}+\frac{a_6}{2^6}+\ldots .+\frac{a_{12}}{2^{12}}

is equal to

View answer and explanation

Correct option: B

-1 / 2

Option B (-1/2) is correct because by substituting x = 1/2 and x = -1/2 into the given equation and adding the resulting equations, we can isolate the desired sum.

48 Medium

48 of 48

Application +4 | -1

\alpha, \beta, \gamma

are roots of equation

x^3-2 x^2-1=0

and

T_n=\alpha^n+\beta^n+\gamma^n

then the value of

\frac{T_{11}-T_8}{T_{10}}

is equal to

View answer and explanation

Correct option: B

Option B (2) is correct because by using the recurrence relation derived from the equation

x^3 - 2x^2 - 1 = 0

, we can express

T_{11}

in terms of

T_{10}

and

T_8

, and then simplify the given expression.

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