Maximum and Minimum Values of Quadratic Expressions

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JEE Main 2014

If $a \in R$ and the equation $- 3 (x - [x])^{2} + 2 (x - [x]) + a^{2} = 0$ (where $[x]$ denotes the greatest integer $\leq x$ ) has no integral solution, then all possible values of $a$ lie in the interval:

(- 2, - 1)

(B)

(- \infty, - 2) \cup (2, \infty)

(C)

(- 1, 0) \cup (0, 1)

(D)

(1, 2)

JEE Main 2009

LEVELJEE Main

If the roots of the equation $b x^{2} + c x + a = 0$ be imaginary, then for all real values of $x$ , the expression $3 b^{2} x^{2} + 6 b c x + 2 c^{2}$ is

(A)

less than

4 ab

(B)

greater than

- 4 ab

(C)

less than

- 4 ab

(D)

greater than

4 ab

JEE Main 2006

LEVELJEE Main

If $x$ is real, the maximum value of $\frac{3 x ^{2} + 9 x + 17}{3 x ^{2} + 9 x + 7}$ is

For all 'x', $x^{2} + 2 a x + 10 - 3 a > 0$ , then the interval in which 'a' lies is

(A)

a < - 5

(B)

- 5 < a < 2

(C)

a > 5

(D)

2 < a < 5

JEE Advanced 2003

LEVELJEE Main

If $f (x) = x^{2} + 2 b x + 2 c^{2}$ and $g (x) = - x^{2} - 2 c x + b^{2}$ such that $min f (x) > max g (x)$ , then the relation between $b$ and $c$ , is

(A)

no real value of

b

c

(B)

0 < c < b 2

(C)

∣ c ∣ < ∣ b ∣ 2

(D)

∣ c ∣ > ∣ b ∣ 2

JEE Advanced 2001

LEVELJEE Main

Let $f (x) = (1 + b^{2}) x^{2} + 2 b x + 1$ and let $m (b)$ be the minimum value of $f (x)$ . As $b$ varies, the range of $m (b)$ is

(A)

[0, 1]

(B)

(0, 1/2]

(C)

[1/2, 1]

(D)

(0, 1]

JEE Advanced 1994

LEVELJEE Main

The number of points of intersection of two curves $y = 2 sin x$ and $y = 5 x^{2} + 2 x + 3$ is

\infty

JEE Advanced 1990

LEVELJEE Main

Let $f (x)$ be a quadratic expression which is positive for all the real values of $x$ . If $g (x) = f (x) + f^{'} (x) + f^{''} (x)$ , then for any real $x$ ,

(A)

g (x) < 0

(B)

g (x) > 0

(C)

g (x) = 0

(D)

g (x) \geq 0

JEE Advanced 1984

LEVELBoard

If $a^{2} + b^{2} + c^{2} = 1$ , then $ab + b c + c a$ lies in the interval

(A)

[1/2, 2]

(B)

[- 1, 2]

(C)

[- 1/2, 1]

(D)

[- 1, 1/2]

JEE Advanced 1984

LEVELJEE Main

For real $x$ , the function $\frac{( x - a ) ( x - b )}{x - c}$ will assume all real values provided

* Multiple Correct Options

(A)

a > b > c

(B)

a < b < c

(C)

a > c > b

(D)

a < c < b

JEE Advanced 1979

LEVELBoard

If $x, y$ and $z$ are real and different and $u = x^{2} + 4 y^{2} + 9 z^{2} - 6 y z - 3 z x - 2 x y$ , then $u$ is always

The entire graphs of the equation $y = x^{2} + k x - x + 9$ is strictly above the x-axis if and only if

(A)

k < 7

(B)

- 5 < k < 7

(C)

k > - 5

Find the range of values of $t$ for which $2 sin t = \frac{1 - 2 x + 5 x ^{2}}{3 x ^{2} - 2 x - 1}$ , $t \in [- π /2, π /2]$ .

JEE Advanced 1998

LEVELJEE Main

Let $f (x) = A x^{2} + B x + C$ where $A, B, C$ are real numbers. Prove that if $f (x)$ is an integer whenever $x$ is an integer, then the numbers $2 A, A + B$ and $C$ are all integers. Conversely, prove that if the numbers $2 A, A + B$ and $C$ are all integers then $f (x)$ is an integer whenever $x$ is an integer.

JEE Main 2019 (12 January)

LEVELJEE Main

The number of integral values of $m$ for which the quadratic expression $(1 + 2 m) x^{2} - 2 (1 + 3 m) x + 4 (1 + m), x \in R$ , is always positive, is :

(A)

8

(B)

7

(C)

6

(D)

3

JEE Main 2019 (10 January)

LEVELJEE Main

The values of $λ$ such that sum of the squares of the roots of the quadratic equation $x^{2} + (3 - λ) x + 2 = λ$ has the least value is :

(A)

(B)

\frac{4}{9}

(C)

\frac{15}{8}

(D)

JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

The probability of selecting integers $a \in [- 5, 30]$ such that $x^{2} + 2 (a + 4) x - 5 a + 64 > 0$ , for all $x \in R$ , is:

(A)

7/36

(B)

2/9

(C)

1/6

(D)

1/4

JEE Main 2021 (25 February Shift 1)

LEVELJEE Main

The integer ' $k$ ', for which the inequality $x^{2} - 2 (3 k - 1) x + 8 k^{2} - 7 > 0$ is valid for every $x$ in $R$ is :

JEE Main 2022 (27 July Shift 1)

LEVELJEE Main

Let $f (x) = 2 x^{2} - x - 1$ and $S = {n \in Z : ∣ f (n) ∣ \leq 800}$ . Then, the value of $\sum_{n \in S} f (n)$ is equal to ________.

JEE Main 2022 (28 July Shift 1)

LEVELJEE Main

For $p, q \in R$ , consider the real valued function $f (x) = (x - p)^{2} - q, x \in R$ and $q > 0$ . Let $a_{1}, a_{2}, a_{3}$ and $a_{4}$ be in an arithmetic progression with mean $p$ and positive common difference. If $∣ f (a_{i}) ∣ = 500$ for all $i = 1, 2, 3, 4$ , then the absolute difference between the roots of $f (x) = 0$ is ________.

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} - 8 x + 12}$ . Then range of $f$ is

(A)

(- \infty, - 21/4] \cup [0, \infty)

(B)

(- \infty, - 21/4) \cup (0, \infty)

(C)

(- \infty, - 21/4] \cup [21/4, \infty)

(D)

(- \infty, - 21/4] \cup [1, \infty)

JEE Main 2024 (04 Apr Shift 1)

LEVELJEE Main

Let the sum of the maximum and the minimum values of the function $f (x) = \frac{2 x ^{2} - 3 x + 8}{2 x ^{2} + 3 x + 8}$ be $\frac{m}{n}$ , where $gcd (m, n) = 1$ . Then $m + n$ is equal to :

JEE Main 2025 (January)

LEVELJEE Main

Let $α$ and $β$ be the distinct roots of $2 x^{2} + (cos θ) x - 1 = 0,$ $θ \in (0, 2 π)$ . If m and M are the minimum and the maximum values of $α_{θ}^{4} + β_{θ}^{4}$ , then $16 (M + m)$ equals:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the roots of the equation bx2+cx+a=0 be imaginary, then for all real values of x, the expression 3b2x2+6bcx+2c2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x is real, the maximum value of 3x2+9x+73x2+9x+17​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For all 'x', x2+2ax+10−3a>0, then the interval in which 'a' lies is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=x2+2bx+2c2 and g(x)=−x2−2cx+b2 such that minf(x)>maxg(x), then the relation between b and c, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=(1+b2)x2+2bx+1 and let m(b) be the minimum value of f(x). As b varies, the range of m(b) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of points of intersection of two curves y=2sinx and y=5x2+2x+3 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x) be a quadratic expression which is positive for all the real values of x. If g(x)=f(x)+f′(x)+f′′(x), then for any real x,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a2+b2+c2=1, then ab+bc+ca lies in the interval

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For real x, the function x−c(x−a)(x−b)​ will assume all real values provided

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x,y and z are real and different and u=x2+4y2+9z2−6yz−3zx−2xy, then u is always

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The entire graphs of the equation y=x2+kx−x+9 is strictly above the x-axis if and only if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the range of values of t for which 2sint=3x2−2x−11−2x+5x2​, t∈[−π/2,π/2].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral values of m for which the quadratic expression (1+2m)x2−2(1+3m)x+4(1+m),x∈R, is always positive, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The values of λ such that sum of the squares of the roots of the quadratic equation x2+(3−λ)x+2=λ has the least value is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The probability of selecting integers a∈[−5,30] such that x2+2(a+4)x−5a+64>0, for all x∈R, is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integer 'k', for which the inequality x2−2(3k−1)x+8k2−7>0 is valid for every x in R is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=2x2−x−1 and S={n∈Z:∣f(n)∣≤800}. Then, the value of ∑n∈S​f(n) is equal to ________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R−{2,6}→R be real valued function defined as f(x)=x2−8x+12x2+2x+1​. Then range of f is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the sum of the maximum and the minimum values of the function f(x)=2x2+3x+82x2−3x+8​ be nm​, where gcd(m,n)=1. Then m+n is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be the distinct roots of 2x2+(cos θ)x−1=0, θ∈(0,2π). If m and M are the minimum and the maximum values of αθ4​+βθ4​, then 16(M+m) equals:

If the roots of the equation $b x^{2} + c x + a = 0$ be imaginary, then for all real values of $x$ , the expression $3 b^{2} x^{2} + 6 b c x + 2 c^{2}$ is

If $x$ is real, the maximum value of $\frac{3 x ^{2} + 9 x + 17}{3 x ^{2} + 9 x + 7}$ is

For all 'x', $x^{2} + 2 a x + 10 - 3 a > 0$ , then the interval in which 'a' lies is

If $f (x) = x^{2} + 2 b x + 2 c^{2}$ and $g (x) = - x^{2} - 2 c x + b^{2}$ such that $min f (x) > max g (x)$ , then the relation between $b$ and $c$ , is

Let $f (x) = (1 + b^{2}) x^{2} + 2 b x + 1$ and let $m (b)$ be the minimum value of $f (x)$ . As $b$ varies, the range of $m (b)$ is

The number of points of intersection of two curves $y = 2 sin x$ and $y = 5 x^{2} + 2 x + 3$ is

Let $f (x)$ be a quadratic expression which is positive for all the real values of $x$ . If $g (x) = f (x) + f^{'} (x) + f^{''} (x)$ , then for any real $x$ ,

If $a^{2} + b^{2} + c^{2} = 1$ , then $ab + b c + c a$ lies in the interval

For real $x$ , the function $\frac{( x - a ) ( x - b )}{x - c}$ will assume all real values provided

If $x, y$ and $z$ are real and different and $u = x^{2} + 4 y^{2} + 9 z^{2} - 6 y z - 3 z x - 2 x y$ , then $u$ is always

The entire graphs of the equation $y = x^{2} + k x - x + 9$ is strictly above the x-axis if and only if

Find the range of values of $t$ for which $2 sin t = \frac{1 - 2 x + 5 x ^{2}}{3 x ^{2} - 2 x - 1}$ , $t \in [- π /2, π /2]$ .

The number of integral values of $m$ for which the quadratic expression $(1 + 2 m) x^{2} - 2 (1 + 3 m) x + 4 (1 + m), x \in R$ , is always positive, is :

The values of $λ$ such that sum of the squares of the roots of the quadratic equation $x^{2} + (3 - λ) x + 2 = λ$ has the least value is :

The probability of selecting integers $a \in [- 5, 30]$ such that $x^{2} + 2 (a + 4) x - 5 a + 64 > 0$ , for all $x \in R$ , is:

The integer ' $k$ ', for which the inequality $x^{2} - 2 (3 k - 1) x + 8 k^{2} - 7 > 0$ is valid for every $x$ in $R$ is :

Let $f (x) = 2 x^{2} - x - 1$ and $S = {n \in Z : ∣ f (n) ∣ \leq 800}$ . Then, the value of $\sum_{n \in S} f (n)$ is equal to ________.

Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} - 8 x + 12}$ . Then range of $f$ is

Let the sum of the maximum and the minimum values of the function $f (x) = \frac{2 x ^{2} - 3 x + 8}{2 x ^{2} + 3 x + 8}$ be $\frac{m}{n}$ , where $gcd (m, n) = 1$ . Then $m + n$ is equal to :

Let $α$ and $β$ be the distinct roots of $2 x^{2} + (cos θ) x - 1 = 0,$ $θ \in (0, 2 π)$ . If m and M are the minimum and the maximum values of $α_{θ}^{4} + β_{θ}^{4}$ , then $16 (M + m)$ equals: