The equation $ (\cos p-1) x^{2}+\cos p x+\sin p=0 $ in $ x $ has real roots. Then the set of values of $ p $ is

$ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] $

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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 31-60 of 528 questions

31 Easy

31 of 528

Application +4 | -1

Sum of the roots of the equation is

4\left(x-\frac{1}{x}\right)^2-4\left(x-\frac{1}{x}\right)+1=0

View answer and explanation

Correct option: B

Option D (-1) is correct because the sum of the roots of the equation is 1.

32 Easy

32 of 528

Application +4 | -1

The roots of the equation

{\left| {2x - 1} \right|^2} - 3\left| {2x - 1} \right| + 2 = 0

are

View answer and explanation

Correct option: D

\left\{ { - \frac{1}{2},0,1,\frac{3}{2}} \right\}

Option D (

\left\{ { - \frac{{1}}{{2}},0,1,\frac{{3}}{{2}}} \right\}

) is correct because these are the solutions obtained by solving the equation after substituting

t = \left| {2x - 1} \right|

33 Easy

33 of 528

Application +4 | -1

\alpha

and

\beta

be the roots of equation

a x^2+b x+c=0

, then

\lim _{x \rightarrow \alpha}\left(1+a x^2+b x+c\right)^{\frac{1}{x-\alpha}}

is equal to

View answer and explanation

Correct option: C

e^{a(\alpha-\beta)}

Option C (

e^{a(\alpha-\beta)}

) is correct because it represents the limit of the given expression as x approaches α, using L'Hopital's rule and the properties of roots of a quadratic equation.

34 Easy

34 of 528

Analyzing +4 | -1

For any real

x

, the expression

2(k-x)\left[x+\sqrt{x^2+k^2}\right]

cannot exceed

View answer and explanation

Correct option: B

2 k^2

The correct answer is B (

2 k^2

) because it represents the maximum value that the expression can attain for any real x.

35 Easy

35 of 528

Application +4 | -1

The solution set of the inequation

0 < \left| {3x + 1} \right| < \frac{1}{3}\;,

View answer and explanation

Correct option: C

\left( { - 4/9\;, - 2/9} \right) - \left\{ { - 1/3} \right\}

The correct answer is C (

\left( { - 4/9\;, - 2/9} \right) - \left\{ { - 1/3} \right\}

) because it represents the interval where the absolute value of

3x+1

is greater than 0 but less than 1/3, excluding the point where

3x+1 = 0

36 Easy

36 of 528

Analyzing +4 | -1

Real roots of the equation

{x^2} + 5\left| x \right| + 4 = 0

are

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the equation has no real roots, as the left-hand side is always positive for real x.

37 Easy

37 of 528

Application +4 | -1

If the equation

{x^2} - 3kx + 2{e^{2\log k}} - 1 = 0

has real roots such that the product of roots is 7, then the value of

k

View answer and explanation

Correct option: B

Option B (2) is correct because solving the equation with the product of roots equal to 7 yields k = 2, considering k > 0.

38 Easy

38 of 528

Application +4 | -1

The quadratic equation with rational coefficients whose one root is

\frac{2+\sqrt{3}}{2-\sqrt{3}}

, is

View answer and explanation

Correct option: A

x^2-14 x+1=0

Option A (

x^2-14 x+1=0

) is correct because the quadratic equation is formed by finding the other root and then constructing the equation using the sum and product of the roots.

39 Easy

39 of 528

Application +4 | -1

p, q \in \mathbf{R}

and

2+\sqrt{3} i

a foot of

x^2+p x+

q=0

, then

View answer and explanation

Correct option: B

p=-4, q=7

Option B (p=-4, q=7) is correct because if

2 + \sqrt{3}i

is a root, then its conjugate

2 - \sqrt{3}i

is also a root, and we can find p and q using Vieta's formulas.

40 Easy

40 of 528

Application +4 | -1

If the roots of the equation

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

are imaginary and the sum of the roots is equal to their product, then

a =

View answer and explanation

Correct option: C

Option C (2) is correct because it satisfies both conditions: the roots are imaginary (discriminant < 0) and the sum of the roots equals their product.

41 Easy

41 of 528

Application +4 | -1

The number of solutions of

|x+2|=2(3-x)

View answer and explanation

Correct option: A

Option A (1) is correct because there is only one solution to the equation, which is x = 4/3.

42 Easy

42 of 528

Application +4 | -1

a, b, c

are rational and

a x^2+b x+c=0

and

3 x^2+

x-5=0

have a common root, then

3 a+b+2 c=

View answer and explanation

Correct option: A

Option A (0) is correct because the equations must be the same, leading to a relationship where 3a + b + 2c = 0.

43 Easy

43 of 528

Application +4 | -1

\sin \theta + \cos \theta = h

, then the quadratic equation having

\sin \theta

and

\cos \theta

as its roots, is

View answer and explanation

Correct option: B

2{x^2} - 2hx + \left( {{h^2} - 1} \right) = 0

Option B (

2{x^2} - 2hx + \left( {{{h^2} - 1}} \right) = 0

) is correct because it is the quadratic equation formed using the given sum of roots and calculated product of roots.

44 Easy

44 of 528

Application +4 | -1

r

be the ratio of the roots of the equation

a x^2+b x+c=0

, then

\frac{(r+1)^2}{r}=

View answer and explanation

Correct option: B

\frac{b^2}{c a}

Option B (

\frac{b^2}{c a}

) is correct because it is derived from the relationship between the roots and coefficients of the quadratic equation.

45 Easy

45 of 528

Application +4 | -1

The solutions of the inequation

2 x^2+3 x-9 \leq 0

are given by

View answer and explanation

Correct option: B

-3 \leq x \leq \frac{3}{2}

Option B (

-3 \leq x \leq \frac{3}{2}

) is correct because it represents the solution to the quadratic inequality obtained by factoring and finding the interval where the expression is less than or equal to zero.

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46 Easy

46 of 528

Analyzing +4 | -1

x

is real, then expression

\frac{{x + 2}}{{2{x^2} + 3x + 6}}

takes all values in the interval

View answer and explanation

Correct option: B

\left[ { - \frac{1}{{13}},\frac{1}{3}} \right]

Option B (

\left[ { - \frac{1}{{13}},\frac{1}{3}} \right]

) is correct because it represents the range of the given expression when x is real, found by analyzing the discriminant of the quadratic equation.

47 Easy

47 of 528

Analyzing +4 | -1

If one root of the equation

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

is reciprocal of the one root of the equation

{a_1}{x^2} + {b_1}x + {c_1} = 0,

then

View answer and explanation

Correct option: A

{\left( {a{a_1} - c{c_1}} \right)^2} = \left( {b{c_1} - {b_1}a} \right)\left( {{b_1}c - {a_1}b} \right)

Option A (

{\left( {a{a_1} - c{c_1}} \right)^2} = \left( {b{c_1} - {b_1}a} \right)\left( {{b_1}c - {a_1}b} \right)

) is correct because it represents the condition derived when one root of the first quadratic equation is the reciprocal of a root of the second quadratic equation.

48 Easy

48 of 528

Analyzing +4 | -1

\alpha ,\beta

are the roots of

a{x^2} + c = bx,

then the equation

{\left( {a + cy} \right)^2} = {b^2}y

y

has the roots

View answer and explanation

Correct option: B

{\alpha ^2},{\beta ^2}

Option B (α², β²) is correct because substituting y = x² into the transformed equation yields the roots as the squares of the original roots.

49 Easy

49 of 528

Analyzing +4 | -1

The number of real solutions of the equation

\left(\frac{9}{10}\right)^x=-3+x-x^2

View answer and explanation

Correct option: A

Option A (0) is correct because the left-hand side of the equation is always positive, while the right-hand side is always negative, resulting in no real solutions.

50 Easy

50 of 528

Application +4 | -1

\tan \theta

and

\cot \theta

are roots of

x^2+2 a x+b=0

, then least value of

|a|

View answer and explanation

Correct option: B

Option A (1) is correct. Using the AM-GM inequality, we can find the minimum value of |a| to be 1.

51 Easy

51 of 528

Understanding +4 | -1

The number of real roots of

f\left( x \right) = 0

, where

f\left( x \right) = \left( {x - 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)\left( {x - 4} \right)

lying in the interval

\left( {1,3} \right)

View answer and explanation

Correct option: A

Option A (1) is correct because only the root x=2 lies within the interval (1,3).

52 Easy

52 of 528

Application +4 | -1

The roots of the equation

i x^2-4 x-4 i=0

are:

View answer and explanation

Correct option: C

-2 i,-2 i

Option C (

-2i, -2i

) is correct because the quadratic equation has a repeated root of -2i.

53 Easy

53 of 528

Analyzing +4 | -1

The minimum value of

{e^{\left( {{x^4} - {x^3} + {x^2}} \right)}}

, is

View answer and explanation

Correct option: C

Option C (1) is correct because the minimum value of the exponent is 0, resulting in a minimum value of 1 for the exponential function.

54 Easy

54 of 528

Application +4 | -1

The equation

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

x

has real roots. Then the set of values of

p

View answer and explanation

Correct option: D

[0, \pi]

Option D (

[0, \pi]

) is correct because the discriminant of the quadratic equation must be non-negative for real roots, which leads to the condition that

\sin p

must be non-negative, restricting p to the interval

[0, \pi]

55 Easy

55 of 528

Application +4 | -1

If the sum of the roots of a quadratic equation is 3 and the sum of their cubes is 7 , then the equation is

View answer and explanation

Correct option: D

9 x^2-27 x+20=0

Option D (

9 x^2-27 x+20=0

) is correct because it is obtained by using the given sum of roots and sum of their cubes to find the product of roots and then constructing the quadratic equation.

56 Easy

56 of 528

Application +4 | -1

Let

\alpha

and

\beta

be the roots of

x^2-6 x-2=0

, with

\alpha>\beta

. If

a_n=\alpha^n-\beta^n

for

n \geq 1

, the value of

\frac{a_{10}-2 a_8}{2 a_9}

View answer and explanation

Correct option: C

Option C (3) is correct because by substituting the roots into the equation and simplifying the expression, we find that the value is 3.

57 Easy

57 of 528

Application +4 | -1

Let

x

be an integer and

x^2+x+1

is divisible by 3. When

x

is divided by 3 , it leaves remainder

View answer and explanation

Correct option: B

Option B (1) is correct because when x is divided by 3, the remainder must be 1 for x² + x + 1 to be divisible by 3.

58 Easy

58 of 528

Application +4 | -1

x = \sqrt {1 + \sqrt {1 + \sqrt {1 + \ldots \infty } } } ,\;

then

x

is equal to

View answer and explanation

Correct option: A

\frac{{1 + \sqrt 5 }}{2}

The correct answer is A (

\frac{{1 + \sqrt 5 }}{2}

) because it is the positive solution to the equation

x = \sqrt {1 + x}

which arises from the given infinite nested radical.

59 Easy

59 of 528

Application +4 | -1

a, b, c

are integers and

b^2=4\left(a c+5 d^2\right), d \in N

, then roots of the quadratic equation

a x^2+b x+c=0

are

View answer and explanation

Correct option: A

Irrational

Option A (Irrational) is correct because the discriminant

D = b^2 - 4ac = 20d^2

, and since

d \in N

\sqrt{D} = 2d\sqrt{5}

which is irrational.

60 Easy

60 of 528

Application +4 | -1

Let

p,q \in \left\{ {1,\;2,\;3,\;4} \right\}.

The number of equations of the form

p{x^2} + qx + 1 = 0

having real roots, is

View answer and explanation

Correct option: C

Option C (7) is correct because by checking each possible value of p from the set {1, 2, 3, 4}, we can determine the number of valid values of q that satisfy the condition for real roots.

Revision notes and key takeaways

Review QUADRATIC EQUATIONS in your textbook or class notes, then use this set to check your understanding.

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