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JEE • MATHEMATICS • QUADRATIC EQUATIONS

Quadratic equations practice · Easy

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Practice format and chapter quiz

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 511-528 of 528 questions

511 Easy

511 of 528

Application +4 | -1

If the roots of the given equation

\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0

are real, then

View answer and explanation

Correct option: C

p \in \left( {0,\;\pi } \right)

Option C (

p \in (0, \pi)

) is correct because the discriminant of the quadratic must be non-negative for real roots, and this condition is satisfied within the given interval.

512 Easy

512 of 528

Application +4 | -1

a + b + c = 0

, then the roots of the equation

4a{x^2} + 3bx + 2c = 0

are

View answer and explanation

Correct option: C

Real

Option C (Real) is correct because the discriminant of the quadratic equation, after substituting

b = -(a+c)

, simplifies to a form that is always non-negative, ensuring real roots.

513 Easy

513 of 528

Application +4 | -1

c > 0

and

4 a+c < 2 b

, then

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

has a root in the interval

View answer and explanation

Correct option: A

(0,2)

Option A (

(0,2)

) is correct because

f(0) > 0

and

f(2) < 0

, indicating a root exists within the interval (0,2) by the Intermediate Value Theorem.

514 Easy

514 of 528

Application +4 | -1

a, b \in\{1,2,3,4\}

the number of quadratic equation of the form

a x^2+b x+c=0

which have non-real complex roots is:

View answer and explanation

Correct option: C

Option C (52) is correct because we calculate the total number of possible quadratic equations and subtract the number of equations with real roots.

515 Easy

515 of 528

Analyzing +4 | -1

Let

S

denote the set of all real values of

a

for which the roots of the equation

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

lie between 5 and 10, then

S

equals

View answer and explanation

Correct option: D

\left( {6,\;9} \right)

Option D (

\left( {6,\;9} \right)

) is correct because the roots of the equation are a+1 and a-1, and for these roots to lie between 5 and 10, 'a' must lie between 6 and 9.

516 Easy

516 of 528

Analyzing +4 | -1

The inequality

\frac{2}{x} < 3

is true, when

x

belongs to

View answer and explanation

Correct option: C

(2 / 3, \infty) \cup(-\infty, 0)

Option C ((

2 / 3, \infty) \cup(-\infty, 0)

) is correct because solving the inequality

\frac{2}{x} < 3

leads to this set of values for

x

517 Easy

517 of 528

Application +4 | -1

{\log _{27}}\left( {{{\log }_2}x} \right) = \frac{1}{3}

, then the value of

x

View answer and explanation

Correct option: D

Option D (27) is correct because it is the value of x obtained by solving the given logarithmic equation.

518 Easy

518 of 528

Application +4 | -1

2+\sqrt{5} i

is a root of

x^2-p x+q=0

where

p

and

q

are real, then the ordered pair

(p, q)

is equal to

View answer and explanation

Correct option: A

(4,9)

Option A ((4,9)) is correct because since p and q are real, the other root is the conjugate, and we can find p and q using the sum and product of the roots.

519 Easy

519 of 528

Application +4 | -1

Rational roots of the equation

2{x^4} + {x^3} - 11{x^2} + x + 2 = 0

are

View answer and explanation

Correct option: A

\frac{1}{2}

and 2

Option A (

\frac{1}{2}

and 2) is correct because these are the only rational roots of the given quartic equation, found by reducing it to a quadratic.

520 Easy

520 of 528

Application +4 | -1

x_1

and

x_2

are the arithmetic and harmonic means of the roots of the equation

a x^2+b x+c=0

, the quadratic equation whose roots are

x_1

and

x_2

, is

View answer and explanation

Correct option: B

2 a b x^2+\left(b^2+4 a c\right) x+2 b c=0

Option B

2 a b x^2+\left(b^2+4 a c\right) x+2 b c=0

is correct because it is the quadratic equation formed using the arithmetic and harmonic means of the roots of the given equation.

521 Easy

521 of 528

Analyzing +4 | -1

The value of

x

in the given equation

{4^x} - {3^{x - \frac{1}{2}}} = {3^{x + \frac{1}{2}}} - {2^{2x - 1}}\;

View answer and explanation

Correct option: B

\frac{3}{2}

Option B (

\frac{3}{2}

) is correct because by simplifying and taking logarithms, we find that x =

\frac{3}{2}

satisfies the given equation.

522 Easy

522 of 528

Understanding +4 | -1

The number of real roots of the equation,

(x-2)^2+(x-3)^2+(x-4)^2=0

is -

View answer and explanation

Correct option: A

Option A (0) is correct because the sum of squares can only be zero if each term is zero, which is impossible in this case.

523 Easy

523 of 528

Analyzing +4 | -1

x = \frac{1}{2}\left( {\sqrt 3 + \frac{1}{{\sqrt 3 }}} \right)

, then

\frac{{\sqrt {{x^2} - 1} }}{{x - \sqrt {{x^2} - 1} }}

is equal to

View answer and explanation

Correct option: A

Option A (1) is correct because by simplifying the expression for x and substituting it into the given expression, the result is 1.

524 Easy

524 of 528

Application +4 | -1

If the sum of the roots of the equation

\lambda x^2+2 x+3 \lambda=0

be equal to their product, then

\lambda=

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because solving the equation under the given condition yields a value for

\lambda

that is not present in the provided options.

525 Easy

525 of 528

Application +4 | -1

\alpha \;\mathrm{and}\;\beta

be the roots of the equation

2{x^2} + 2\left( {a + b} \right)x + {a^2} + {b^2} = 0,

then the equation whose roots are

{\left( {\alpha + \beta } \right)^2}\;\mathrm{and}\;{\left( {\alpha - \beta } \right)^2}

, is

View answer and explanation

Correct option: B

{x^2} - 4abx - {\left( {{a^2} - {b^2}} \right)^2} = 0

Option B (

{x^2} - 4abx - {\left( {{a^2} - {b^2}} \right)^2} = 0

) is correct because it is the equation formed with roots

(\alpha + \beta)^2

and

(\alpha - \beta)^2

after using the relationships between roots and coefficients of the given quadratic equation.

Explore Other Chapters in MATHEMATICS

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526 Easy

526 of 528

Application +4 | -1

The real value of a for which the sum of the squares of the roots of the equation

x^2-(a-2) x-a-1=0

assume the least value is

View answer and explanation

Correct option: A

Option A (1) is correct because the sum of the squares of the roots is minimized when a = 1.

527 Easy

527 of 528

Analyzing +4 | -1

If the ratio of the roots of the equation

a x^2+b x+c=0

p: q

, then:

View answer and explanation

Correct option: B

p q b^2-(p+q)^2 a c=0

Option B (

p q b^2-(p+q)^2 a c=0

) is correct because it represents the relationship between the coefficients and the ratio of the roots of the quadratic equation.

528 Easy

528 of 528

Application +4 | -1

x^2+2 a x+10-3 a>0

for each

x \in R

then

View answer and explanation

Correct option: B

-5<\mathrm{a}<2

Option D (

-1 \leq x \leq 1

) is correct because the given equation simplifies to 0 for all x within the domain of the inverse trigonometric functions.

Revision notes and key takeaways

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