The solution of the inequation ${4^{ - x + 0.5}} - {7.2^{ - x}} < 4,x \in R$ IS

$\left( {2,\frac{7}{2}} \right)$

If $4 \le x \le 9,$ then

$\left( {x - 4} \right)\left( {x - 9} \right) \ge 0$

$\left( {x - 4} \right)\left( {x - 9} \right) < 0$

$\left( {x - 4} \right)\left( {x - 9} \right) > 0$

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JEE • MATHEMATICS • QUADRATIC EQUATIONS

Quadratic equations practice · Mixed difficulty

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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 151-180 of 914 questions

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151 Medium

151 of 914

Application +4 | -1

The number of real solutions of the equation

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

View answer and explanation

Correct option: C

Option C (4) is correct because solving the equation

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

yields four real solutions:

x= \pm 1, \pm 2

152 Easy

152 of 914

Application +4 | -1

The condition

f\left( x \right) = {x^3} + p{x^2} + qx + r\left( {x \in R} \right)

to have no extreme value, is

View answer and explanation

Correct option: A

{p^2} < 3q

Option A (

{p^2} < 3q

) is correct because it provides the condition for the given cubic function to have no extreme values, which occurs when its derivative has no real roots.

153 Easy

153 of 914

Application +4 | -1

\alpha ,\beta

are roots of the equation

6{x^2} - 5x + 1 = 0,

then the value of

{\tan ^{ - 1}}\alpha + {\tan ^{ - 1}}\beta

View answer and explanation

Correct option: B

\frac{\pi }{4}

Option B (

\frac{\pi}{4}

) is correct because by using the formula for the sum of two arctangents and substituting the sum and product of the roots, we obtain

\tan^{-1}(1) = \frac{\pi}{4}

154 Medium

154 of 914

Application +4 | -1

Let

a, b, c

be the lengths of the sides of a triangle (no two of them are equal) and

k \in R

. If the roots of the equation

x^2+2(a+b+c) x+6 k(a b+b c+c a)=0

are real, then:

View answer and explanation

Correct option: A

k<\frac{2}{3}

Option A (

k<\frac{2}{3}

) is correct because the discriminant of the quadratic equation must be non-negative for real roots, leading to the inequality

k < \frac{2}{3}

155 Medium

155 of 914

Analyzing +4 | -1

The values of

x

satisfying

\left| {{x^2} + 4x + 3} \right| + \left( {2x + 5} \right) = 0

are

View answer and explanation

Correct option: A

- 4,\; - 1 - \sqrt 3

Option A (

- 4,\; - 1 - \sqrt 3

) is correct because it involves solving the equation by considering cases based on the absolute value and checking the validity of the solutions.

156 Easy

156 of 914

Analyzing +4 | -1

\alpha

and

\beta

are the roots of the equation

{x^2} + px + 2 = 0

and

\frac{1}{\alpha }

and

\frac{1}{\beta }

are the roots of the equation

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

, then

\left( {\alpha - \frac{1}{\alpha }} \right)\left( {\beta - \frac{1}{\beta }} \right)\left( {\alpha + \frac{1}{\beta }} \right)\left( {\beta + \frac{1}{\alpha }} \right)

is equal to

View answer and explanation

Correct option: D

\frac{9}{4}(9 - {p^2})

Option D

\frac{9}{4}(9 - {p^2})

is correct because, after simplifying the expression using the relationships between roots and coefficients of the given quadratic equations, the result matches this option.

157 Easy

157 of 914

Analyzing +4 | -1

The equation

2{\cos ^2}\left( {\frac{x}{2}} \right){\sin ^2}x = {x^2} + \frac{1}{{{x^2}}},0 \le x \le \frac{\pi }{2}

has

View answer and explanation

Correct option: A

No real solution

Option A (No real solution) is correct because the maximum value of the LHS is 2, while the minimum value of the RHS is 2, but they don't attain these values at the same x.

158 Easy

158 of 914

Application +4 | -1

\alpha, \beta

are the roots of the equation

a x^2+

b x+c=0

, then the value of

\alpha^3+\beta^3

View answer and explanation

Correct option: C

\frac{3 a b c-b^3}{a^3}

Option C (

\frac{3 a b c-b^3}{a^3}

) is correct because it is derived from the formula for

\alpha^3 + \beta^3

using Vieta's formulas.

159 Easy

159 of 914

Application +4 | -1

The real values of

x

for which

3^{72}\left(\frac{1}{3}\right)^x\left(\frac{1}{3}\right)^{\sqrt{x}}>1

, are

View answer and explanation

Correct option: C

x \in[0,64)

Option C (

x \in[0,64)

) is correct because it represents the range of real values of x for which the given inequality holds true, considering the domain of the square root function.

160 Medium

160 of 914

Application +4 | -1

\log _2 x+\log _x 2=\frac{10}{3}=\log _2 y+\log _y 2

and

x \neq y

, then

(x+y)=?

View answer and explanation

Correct option: D

None of these

The correct answer is D (None of these) because solving the logarithmic equation leads to

x=8

and

y=2^{1/3}

, so

x+y = 8 + 2^{1/3}

161 Medium

161 of 914

Application +4 | -1

a, b

and

c

are real numbers such that

a^2+b^2+c^2=1

, then

a b+b c+c a

lies in the interval

View answer and explanation

Correct option: B

\left[-\frac{1}{2}, 1\right]

The correct answer is B ([

\frac{-1}{2}, 1

]) because using

(a+b+c)^2 \geq 0

and

(a-b)^2+(b-c)^2+(c-a)^2 \geq 0

allows us to find the interval for

ab+bc+ca

162 Medium

162 of 914

Application +4 | -1

The value(s) of

y

for which the equation

4 \sin x+3 \cos x=y^2-6 y+14

has a real solution, is (are)

View answer and explanation

Correct option: A

Option A (3) is correct because the equation

4 \sin x + 3 \cos x = y^2 - 6y + 14

has a real solution only when

y=3

, which makes the right-hand side equal to 5, the maximum value of the left-hand side.

163 Easy

163 of 914

Application +4 | -1

The set of values of

p

for which the roots of the equation

3 x^2+2 x+p(p-1)=0

are of opposite sign, is

View answer and explanation

Correct option: B

(0,1)

The correct answer is B ((0,1)) because the roots are of opposite sign when the product of the roots is less than 0, which leads to

p(p-1) < 0

164 Easy

164 of 914

Application +4 | -1

The solution of

6 + x - {x^2} > 0,

View answer and explanation

Correct option: B

- 2 < x < 3

Option B (

- 2 < x < 3

) is correct because it represents the interval where the quadratic expression is greater than zero, found by factoring and analyzing the inequality.

165 Easy

165 of 914

Understanding +4 | -1

The expression

y=a x^2+b x+c

has always the same sign as of 'a' if :

View answer and explanation

Correct option: B

4 a c>b^2

Option B (

4 a c>b^2

) is correct because the quadratic expression has the same sign as 'a' when the discriminant is negative, ensuring no real roots and thus no sign change.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

166 Easy

166 of 914

Application +4 | -1

If the quadratic equation

2 x^2+a x+b=0

and

2 x^2

+b x+a=0,(a \neq b)

have a common root, the value of

a+b

View answer and explanation

Correct option: B

-2

Option B (-2) is correct because subtracting the two equations leads to x=1 as a common root, which can then be used to solve for a+b.

167 Easy

167 of 914

Analyzing +4 | -1

The number of real solution of the equation

\left( {\frac{9}{{10}}} \right) = - 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, so there are no real solutions.

168 Medium

168 of 914

Application +4 | -1

Let

f\left( x \right)

be a quadratic expression which is positive for all real

x

and

g\left( x \right) = f\left( x \right) + f'\left( x \right) + f''\left( x \right),

then for any real

x

View answer and explanation

Correct option: B

g\left( x \right) > 0

Option B (

g(x) > 0

) is correct because if f(x) is always positive, then g(x), which is a combination of f(x) and its derivatives, is also always positive.

169 Easy

169 of 914

Application +4 | -1

The value of

\sqrt {42 + \sqrt {42 + \sqrt {42 + ...} } }

is equal to

View answer and explanation

Correct option: A

Option A (7) is correct because the given expression can be simplified into a quadratic equation, and 7 is the positive root of that equation.

170 Medium

170 of 914

Application +4 | -1

The solution of the inequation

{4^{ - x + 0.5}} - {7.2^{ - x}} < 4,x \in R

View answer and explanation

Correct option: A

\left( { - 2,\infty } \right)

The correct answer is A (

\left( { - 2,\infty } \right)

) because by substituting

2^{-x} = t

, the inequality can be solved for t, and then back-substituted to find the range of x.

171 Medium

171 of 914

Application +4 | -1

x^2-x-2

is a factor of

x^4-p x^2-q=0

, then

(p+q)

equals

View answer and explanation

Correct option: B

The correct answer is B (1) because substituting the roots of

x^2-x-2

into

x^4-px^2-q=0

allows us to solve for

p+q

172 Easy

172 of 914

Understanding +4 | -1

a

and

b

are two different positive real numbers then which of the following statement is true?

View answer and explanation

Correct option: B

2\sqrt {ab} < a + b

Option B (

2\sqrt {ab} < a + b

) is correct because the Arithmetic Mean is always greater than or equal to the Geometric Mean for distinct positive numbers.

173 Medium

173 of 914

Application +4 | -1

x

be real, then the minimum value of

x^2-8 x+17

is:

View answer and explanation

Correct option: C

Option C (1) is correct because the minimum value of the quadratic expression

x^2-8 x+17

can be found by completing the square or using the formula

\frac{{4ac-b^2}}{{4a}}

174 Easy

174 of 914

Understanding +4 | -1

4 \le x \le 9,

then

View answer and explanation

Correct option: A

\left( {x - 4} \right)\left( {x - 9} \right) \le 0

Option B (

(x - 4)(x - 9) \ge 0

) is incorrect. Option A (

(x - 4)(x - 9) \le 0

) is correct because for

x

between 4 and 9 inclusive,

(x-4)

is non-negative and

(x-9)

is non-positive, making their product non-positive.

175 Easy

175 of 914

Application +4 | -1

p,q,r

and

s

are integers. If the A.M. of the roots of

{x^2} - px + {q^2} = 0

and GM of the roots of

{x^2} - rx + {s^2} = 0

are equal, then

View answer and explanation

Correct option: C

p

is an even integer

Option C (

p

is an even integer) is correct because the given condition leads to

p = 2|s|

, which implies that

p

is an even integer.

176 Medium

176 of 914

Application +4 | -1

The greatest negative integer satisfying

{x^2} - 4x - 77 < 0

and

{x^2} > 4,

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the greatest negative integer satisfying both inequalities is -3, which is not among the options.

177 Easy

177 of 914

Analyzing +4 | -1

x^2+a x+b

is an integer for every integer

x

, then

View answer and explanation

Correct option: C

a+b

is always an integer

Option C (

a+b

is always an integer) is correct because if

x^2+ax+b

is an integer for every integer

x

, then evaluating at

x=0

and

x=1

shows that

b

is an integer and

a+b

is an integer.

178 Medium

178 of 914

Application +4 | -1

If the roots of the equation

x^2-5 x+16=0

are

\alpha, \beta

and the roots of equation

x^2+p x+q=0

are

\alpha^2+\beta^2, \alpha \beta / 2

, then

View answer and explanation

Correct option: B

p=-1, q=-56

Option D (

p=-1, q=56

) is correct because by using Vieta's formulas and the relationships between the roots of the two equations, we can find the values of

p

and

q

179 Easy

179 of 914

Application +4 | -1

The number of real solutions of

x^2-4|x|-2=0

View answer and explanation

Correct option: B

Option B (2) is correct because substituting |x| = y results in a quadratic equation with two positive solutions for |x|, leading to two real solutions for x.

180 Easy

180 of 914

Application +4 | -1

p

and

q

are the roots of the equation

x^{2}+p x+q=0

, then

View answer and explanation

Correct option: A

p=1, q=-2

Option A (

p=1, q=-2

) is correct because these values satisfy both the sum and product of roots conditions for the given quadratic equation.

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