Limits Continuity and Differentiability

257 questions · Mathematics · JEE Main

Limits Continuity and Differentiability

257 questions · Mathematics · JEE Main

For $α, β, γ \in R$ , if $lim_{x \to 0} \frac{x ^{2} s i n α x + ( γ - 1 ) e ^{x^{2}}}{s i n 2 x - β x} = 3$ , then $β + γ - α$ is equal to :2025 · 2 Apr · Shift 1 · Q43 · MCQ
$I f x \to 0 lim \frac{c o s ( 2 x ) + a c o s ( 4 x ) - b}{x ^{4}} i s f ini t e, t h e n (a + b) i s e q u a l t o :$ 2025 · 2 Apr · Shift 2 · Q32 · MCQ
Let $f (x) = {(1 + a x)^{1/ x}, x 0$ be continuous at $x = 0$ . Then $e^{a} b c$ is equal to:2025 · 3 Apr · Shift 1 · Q36 · MCQ
If $x \to 0 lim (\frac{t a n x}{x})^{\frac{1}{x ^{2}}} = p$ , then $96 lo g_{e} p$ is equal to 2025 · 3 Apr · Shift 2 · Q46 · Numerical
Let $f : R \to R$ be a continuous function satisfying $f (0) = 1$ and $f (2 x) - f (x) = x$ for all $x \in R$ . If $lim_{n \to \infty} {f (x) - f (\frac{x}{2 ^{n}})} = G (x)$ , then $\sum_{r = 1}^{10} G (r^{2})$ …2025 · 4 Apr · Shift 1 · Q26 · MCQ
If $lim_{x \to 1^{+}} \frac{( x - 1 ) ( 6 + λ c o s ( x - 1 )) + μ s i n ( 1 - x )}{( x - 1 ) ^{3}} = - 1$ , where $λ, μ \in R$ , then $λ + μ$ is equal to2025 · 4 Apr · Shift 1 · Q42 · MCQ
Let $m$ and $n$ be the number of points at which the function $f (x) = max {x, x^{3}, x^{5}, \dots x^{21}}, x \in R$ , is not differentiable and not continuous, respectively. Then $m + n$ is equal to…2025 · 4 Apr · Shift 1 · Q47 · Numerical
Let $f$ be a differentiable function on $R$ such that $f (2) = 1, f^{'} (2) = 4$ . Let $x \to 0 lim (f (2 + x))^{3/ x} = e^{α}$ . Then the number of times the curve $y = 4 x^{3} - 4 x^{2} - 4 (α - 7) x - α$ meets…2025 · 4 Apr · Shift 2 · Q45 · MCQ
$lim_{x \to 0^{+}} \frac{t a n ( 5 ( x ) ^{\frac{1}{3}} ) l o g _{e} ( 1 + 3 x ^{2} )}{( t a n ^{- 1} 3 x ) ^{2} ( e ^{5 (x)^{\frac{4}{3}}} - 1 )}$ is equal to2025 · 7 Apr · Shift 1 · Q31 · MCQ
The number of points of discontinuity of the function $f (x) = [\frac{x ^{2}}{2}] - [x], x \in [0, 4]$ , where $[\cdot]$ denotes the greatest integer function, is .2025 · 7 Apr · Shift 1 · Q48 · Numerical
If the function $f (x) = \frac{t a n ( t a n x ) - s i n ( s i n x )}{t a n x - s i n x}$ is continuous at $x = 0$ , then $f (0)$ is equal to .2025 · 7 Apr · Shift 2 · Q47 · Numerical
For $t > - 1$ , let $α_{t}$ and $β_{t}$ be the roots of the equation $((t + 2)^{1/7} - 1) x^{2} + ((t + 2)^{1/6} - 1) x + ((t + 2)^{1/21} - 1) = 0 . If t \to - 1^{+} lim α_{t} = a and t \to - 1^{+} lim β_{t} = b,$ …2025 · 7 Apr · Shift 2 · Q50 · Numerical
Given below are two statements: Statement I: $x \to 0 lim (\frac{t a n ^{- 1} x + l o g _{e} \frac{1 + x}{1 - x} - 2 x}{x ^{5}}) = \frac{2}{5}$ Statement II: $x \to 1 lim (x^{\frac{2}{1 - x}}) = \frac{1}{e ^{2}}$ …2025 · 8 Apr · Shift 2 · Q36 · MCQ
If $\sum_{r = 1}^{n} T_{r} = \frac{( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ( 2 n + 5 )}{64}$ , then $lim_{n \to \infty} \sum_{r = 1}^{n} (\frac{1}{T _{r}})$ is equal to :2025 · 22 Jan · Shift 1 · Q44 · MCQ
Let the function, $f (x) = {- 3 ax^{2} - 2, a^{2} + b x, x < 1 x ⩾ 1$ be differentiable for all $x \in R$ , where $a > 1, b \in R$ . If…2025 · 22 Jan · Shift 1 · Q45 · Numerical
If $lim_{x \to \infty} ((\frac{e}{1 - e}) (\frac{1}{e} - \frac{x}{1 + x}))^{x} = α$ , then the value of $\frac{l o g _{e} α}{1 + l o g _{e} α}$ equals :2025 · 22 Jan · Shift 2 · Q39 · MCQ
If the function $f (x) = {\frac{2}{x} {sin (k_{1} + 1) x + sin (k_{2} - 1) x}, x 0$ is continuous at $x = 0$ , then $k_{1}^{2} + k_{2}^{2}$ is equal to :2025 · 23 Jan · Shift 1 · Q34 · MCQ
$x \to \infty lim \frac{( 2 x ^{2} - 3 x + 5 ) ( 3 x - 1 ) ^{\frac{x}{2}}}{( 3 x ^{2} + 5 x + 4 ) ( 3 x + 2 ) ^{x}}$ is equal to :2025 · 23 Jan · Shift 2 · Q37 · MCQ
$lim_{x \to 0} cosec x (2 cos^{2} x + 3 cos x - cos^{2} x + sin x + 4)$ is:2025 · 24 Jan · Shift 1 · Q26 · MCQ
Let $f : R - {0} \to R$ be a function such that $f (x) - 6 f (\frac{1}{x}) = \frac{35}{3 x} - \frac{5}{2}$ . If the $x \to 0 lim (\frac{1}{α x} + f (x)) = β; α, β \in R$ …2025 · 24 Jan · Shift 1 · Q27 · MCQ
Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f (x) = [x] + ∣ x - 2∣, - 2 < x < 3$ , is not continuous and not differentiable. Then $m + n$ is equal to…2025 · 24 Jan · Shift 2 · Q42 · MCQ
Let $f (x) = {3 x, x 2$ where [.] denotes greatest integer function. If $α$ and $β$ are the number of points, where $f$ is not continuous and is not differentiable, respectively,…2025 · 28 Jan · Shift 1 · Q47 · Numerical
Let $f (x) = n \to \infty lim r = 0 \sum n (\frac{t a n ( x / 2 ^{r + 1} ) + t a n ^{3} ( x / 2 ^{r + 1} )}{1 - t a n ^{2} ( x / 2 ^{r + 1} )})$ Then $x \to 0 lim \frac{e ^{x} - e ^{f (x)}}{( x - f ( x ))}$ …2025 · 28 Jan · Shift 2 · Q46 · Numerical
The value of $n \to \infty lim (k = 1 \sum n \frac{k ^{3} + 6 k ^{2} + 11 k + 5}{( k + 3 )!})$ is :2025 · 29 Jan · Shift 1 · Q35 · MCQ
Let [t] be the greatest integer less than or equal to t. Then the least value of p ∈ N for which $x \to 0^{+} lim (x ([\frac{1}{x}] + [\frac{2}{x}] + \dots + [\frac{p}{x}]) - x^{2} ([\frac{1}{x ^{2}}] + [\frac{2 ^{2}}{x ^{2}}] + \dots + [\frac{9 ^{2}}{x ^{2}}]) \geq 1$ …2025 · 29 Jan · Shift 1 · Q46 · Numerical
Let the function $f (x) = (x^{2} - 1) x^{2} - a x + 2 + cos ∣ x ∣$ be not differentiable at the two points $x = α = 2$ and $x = β$ . Then the distance of the point $(α, β)$ from the line $12 x + 5 y + 10 = 0$ is equal to :2025 · 29 Jan · Shift 2 · Q45 · MCQ
Let $f : R \to R$ be defined as : $f (x) = {\frac{a - b c o s 2 x}{x ^{2}}; x 1$ If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential…2024 · 1 Feb · Shift 1 · Q45 · MCQ
Let ${x}$ denote the fractional part of $x$ and $f (x) = \frac{c o s ^{- 1} ( 1 - { x } ^{2} ) s i n ^{- 1} ( 1 - { x })}{{ x } - { x } ^{3}}, x e q 0$ . If $L$ and $R$ respectively denotes the left hand limit and the right hand limit…2024 · 1 Feb · Shift 1 · Q55 · Numerical
Let $f (x) = 2 x^{2} + 5 x ∣ - 3∣, x \in R$ . If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m + n$ is equal to :2024 · 1 Feb · Shift 2 · Q41 · MCQ
Let $f (x) = {x - 1, x is even, 2 x, x is odd, x \in N$ . If for some $a \in N, f (f (f (a))) = 21$ , then $x \to a^{-} lim {\frac{∣ x ∣ ^{3}}{a} - [\frac{x}{a}]}$ …2024 · 1 Feb · Shift 2 · Q50 · MCQ
Let $f : R \to R$ be a function given by $f (x) = {\frac{1 - c o s 2 x}{x ^{2}}, x 0$ where $α, β \in R$ . If $f$ is continuous at $x = 0$ , then $α^{2} + β^{2}$ is equal…2024 · 4 Apr · Shift 1 · Q50 · MCQ
If $lim_{x \to 1} \frac{( 5 x + 1 ) ^{1/3} - ( x + 5 ) ^{1/3}}{( 2 x + 3 ) ^{1/2} - ( x + 4 ) ^{1/2}} = \frac{m 5}{n ( 2 n ) ^{2/3}}$ , where $gcd (m, n) = 1$ , then $8 m + 12 n$ …2024 · 4 Apr · Shift 1 · Q54 · Numerical
If the function $f (x) = {\frac{7 2 ^{x} - 9 ^{x} - 8 ^{x} + 1}{2 - 1 + c o s x}, a lo g_{e} 2 lo g_{e} 3 x e q 0, x = 0$ is continuous at $x = 0$ , then the value of $a^{2}$ is equal to2024 · 4 Apr · Shift 2 · Q34 · MCQ
If the function $f (x) = \frac{s i n 3 x + α s i n x - β c o s 3 x}{x ^{3}}, x \in R$ , is continuous at $x = 0$ , then $f (0)$ is equal to :2024 · 5 Apr · Shift 1 · Q44 · MCQ
Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f (1) = 1$ and $lim_{t \to x} \frac{t ^{2} f ( x ) - x ^{2} f ( t )}{t - x} = 1$ for each $x > 0$ . Then $2 f (2) + 3 f (3)$ is equal to .2024 · 5 Apr · Shift 1 · Q55 · Numerical
Let , $f : [- 1, 2] \to R$ be given by $f (x) = 2 x^{2} + x + [x^{2}] - [x]$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . The number of points, where $f$ is not continuous, is :2024 · 5 Apr · Shift 2 · Q42 · MCQ
Let $a > 0$ be a root of the equation $2 x^{2} + x - 2 = 0$ . If $lim_{x \to \frac{1}{a}} \frac{16 ( 1 - c o s ( 2 + x - 2 x ^{2} ) )}{( 1 - a x ) ^{2}} = α + β 17$ , where $α, β \in Z$ , then $α + β$ …2024 · 5 Apr · Shift 2 · Q56 · Numerical
$lim_{n \to \infty} \frac{( 1 ^{2} - 1 ) ( n - 1 ) + ( 2 ^{2} - 2 ) ( n - 2 ) + \dots + ( ( n - 1 ) ^{2} - ( n - 1 ) ) \cdot 1}{( 1 ^{3} + 2 ^{3} + \dots\dots + n ^{3} ) - ( 1 ^{2} + 2 ^{2} + \dots\dots + n ^{2} )}$ is equal to :2024 · 6 Apr · Shift 2 · Q34 · MCQ
Let $[t]$ denote the greatest integer less than or equal to $t$ . Let $f : [0, \infty) \to R$ be a function defined by $f (x) = [\frac{x}{2} + 3] - [x]$ . Let $S$ be the set of all points in the…2024 · 6 Apr · Shift 2 · Q50 · Numerical
The value of $lim_{x \to 0} 2 (\frac{1 - c o s x c o s 2 x 3 c o s 3 x \dots\dots . 10 c o s 10 x}{x ^{2}})$ is .2024 · 8 Apr · Shift 1 · Q60 · Numerical
For $a, b > 0$ , let $f (x) = {\frac{t a n (( a + 1 ) x ) + b t a n x}{x}, x 0$ be a continuous function at $x = 0$ . Then $\frac{b}{a}$ is equal to :2024 · 8 Apr · Shift 2 · Q42 · MCQ
If $α = lim_{x \to 0^{+}} (\frac{e ^{t a n x} - e ^{x}}{t a n x - x})$ and $β = lim_{x \to 0} (1 + sin x)^{\frac{1}{2} c o t x}$ are the roots of the quadratic…2024 · 8 Apr · Shift 2 · Q58 · Numerical
Let $f : (0, π) \to R$ be a function given by $f (x) = ⎩ ⎨ ⎧ (\frac{8}{7})^{\frac{t a n 8 x}{t a n 7 x}}, a - 8, (1 + ∣ cot x ∣)^{\frac{b}{a} ∣ t a n x ∣}, 0 < x < \frac{π}{2} x = \frac{π}{2} \frac{π}{2} < x < π$ …2024 · 9 Apr · Shift 1 · Q53 · Numerical
$lim_{x \to 0} \frac{e - ( 1 + 2 x ) ^{\frac{1}{2 x}}}{x}$ is equal to2024 · 9 Apr · Shift 2 · Q45 · MCQ
Consider the function. $f (x) = ⎩ ⎨ ⎧ \frac{a ( 7 x - 12 - x ^{2} )}{b ∣ x ^{2} - 7 x + 12 ∣} b, x 3, x = 3,$ where $[x]$ denotes the greatest integer less…2024 · 27 Jan · Shift 1 · Q39 · MCQ
If $a = x \to 0 lim \frac{1 + 1 + x ^{4} - 2}{x ^{4}}$ and $b = x \to 0 lim \frac{s i n ^{2} x}{2 - 1 + c o s x}$ , then the value of $a b^{3}$ is :2024 · 27 Jan · Shift 1 · Q40 · MCQ
Consider the function $f : (0, 2) \to R$ defined by $f (x) = \frac{x}{2} + \frac{2}{x}$ and the function $g (x)$ defined by $g (x) = {min ⌊ f (t)}, \frac{3}{2} + x, 0 < t \leq x and 0 < x \leq 1 1 < x < 2 . Then,$ …2024 · 27 Jan · Shift 2 · Q33 · MCQ
$If lim_{x \to 0} \frac{3 + α s i n x + β c o s x + l o g _{e} ( 1 - x )}{3 t a n ^{2} x} = \frac{1}{3}, then 2 α - β is equal to :$ 2024 · 27 Jan · Shift 2 · Q41 · MCQ
Let $f (x) = lim_{r \to x} {\frac{2 r ^{2} [ ( f ( r ) ) ^{2} - f ( x ) f ( r ) ]}{r ^{2} - x ^{2}} - r^{3} e^{\frac{f ( r )}{r}}}$ be differentiable in $(- \infty, 0) \cup (0, \infty)$ and $f (1) = 1$ . Then the value of ea, such that…2024 · 29 Jan · Shift 2 · Q59 · Numerical
If the function $f (x) = {\frac{1}{∣ x ∣}, a x^{2} + 2 b, ∣ x ∣ ⩾ 2 ∣ x ∣ < 2$ is differentiable on $R$ , then $48 (a + b)$ is equal to .2024 · 30 Jan · Shift 1 · Q60 · Numerical
$lim_{x \to 0} \frac{e ^{2∣ s i n x ∣} - 2∣ s i n x ∣ - 1}{x ^{2}}$ 2024 · 31 Jan · Shift 1 · Q31 · MCQ
Let $g (x)$ be a linear function and $f (x) = {g (x) (\frac{1 + x}{2 + x})^{\frac{1}{x}}, x \leq 0, x > 0$ , is continuous at $x = 0$ . If $f^{'} (1) = f (- 1)$ , then the value $g (3)$ …2024 · 31 Jan · Shift 1 · Q44 · MCQ
Consider the function $f : (0, \infty) \to R$ defined by $f (x) = e^{- ∣ l o g_{e} x ∣}$ . If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m + n$ is2024 · 31 Jan · Shift 2 · Q47 · MCQ
If $lim_{x \to 0} \frac{a x ^{2} e ^{x} - b l o g _{e} ( 1 + x ) + c x e ^{- x}}{x ^{2} s i n x} = 1$ , then $16 (a^{2} + b^{2} + c^{2})$ is equal to .2024 · 31 Jan · Shift 2 · Q56 · Numerical
Let $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d > 0$ is its common difference, then $lim_{n \to \infty} \frac{d}{n} (\frac{1}{a _{1} + a _{2}} + \frac{1}{a _{2} + a _{3}} + \dots\dots\dots + \frac{1}{a _{n - 1} + a _{n}})$ …2023 · 6 Apr · Shift 1 · Q32 · MCQ
Let $a \in Z$ and $[t]$ be the greatest integer $\leq t$ . Then the number of points, where the function $f (x) = [a + 13 sin x], x \in (0, π)$ is not differentiable, is .2023 · 6 Apr · Shift 1 · Q39 · Numerical
$lim_{x \to 0} ((\frac{( 1 - c o s ^{2} ( 3 x )}{c o s ^{3} ( 4 x )}) (\frac{s i n ^{3} ( 4 x )}{( l o g _{e} ( 2 x + 1 ) ) ^{5}}))$ is equal to .2023 · 8 Apr · Shift 1 · Q25 · MCQ
If $α > β > 0$ are the roots of the equation $a x^{2} + b x + 1 = 0$ , and $lim_{x \to \frac{1}{α}} (\frac{1 - c o s ( x ^{2} + b x + a )}{2 ( 1 - α x ) ^{2}})^{\frac{1}{2}} = \frac{1}{k} (\frac{1}{β} - \frac{1}{α}), then k is equal to$ …2023 · 8 Apr · Shift 2 · Q32 · MCQ
Let $k$ and $m$ be positive real numbers such that the function $f (x) = {3 x^{2} + k x + 1, m x^{2} + k^{2}, 0 < x < 1 x \geq 1$ is differentiable for all $x > 0$ …2023 · 8 Apr · Shift 2 · Q39 · Numerical
Let $f : (- 2, 2) \to R$ be defined by $f (x) = {x [x], (x - 1) [x], - 2 < x < 0 0 \leq x \leq 2$ where $[x]$ denotes the greatest integer function. If m and n…2023 · 10 Apr · Shift 1 · Q39 · Numerical
Let $f (x) = [x^{2} - x] + ∣ - x + [x] ∣$ , where $x \in R$ and $[t]$ denotes the greatest integer less than or equal to $t$ . Then, $f$ is :2023 · 11 Apr · Shift 1 · Q34 · MCQ
Let $f$ and $g$ be two functions defined by $f (x) = {x + 1, ∣ x - 1∣, x < 0 x \geq 0$ and $g (x) = {x + 1, 1, x < 0 x \geq 0$ Then $(g \circ f) (x)$ …2023 · 11 Apr · Shift 2 · Q33 · MCQ
Let $[x]$ be the greatest integer $\leq x$ . Then the number of points in the interval $(- 2, 1)$ , where the function $f (x) = ∣ [x] ∣ + x - [x]$ is discontinuous, is .2023 · 12 Apr · Shift 1 · Q41 · Numerical
If $lim_{x \to 0} \frac{e ^{a x} - c o s ( b x ) - \frac{c x e ^{- c x}}{2}}{1 - c o s ( 2 x )} = 17$ , then $5 a^{2} + b^{2}$ is equal to2023 · 13 Apr · Shift 2 · Q25 · MCQ
Let $[x]$ denote the greatest integer function and $f (x) = max {1 + x + [x], 2 + x, x + 2 [x]}, 0 \leq x \leq 2$ . Let $m$ be the number of points in $[0, 2]$ , where $f$ is not continuous and $n$ be the number of points in $(0, 2)$ , where $f$ is not…2023 · 15 Apr · Shift 1 · Q29 · MCQ
$t \to 0 lim (1^{\frac{1}{s i n ^{2} t}} + 2^{\frac{1}{s i n ^{2} t}} + ... + n^{\frac{1}{s i n ^{2} t}})^{s i n^{2} t}$ is equal to2023 · 24 Jan · Shift 1 · Q27 · MCQ
Let $f (x) = {x^{2} sin (\frac{1}{x}) 0, x e 0, x = 0$ Then at $x = 0$ 2023 · 24 Jan · Shift 1 · Q36 · MCQ
The set of all values of $a$ for which $x \to a lim ([x - 5] - [2 x + 2]) = 0$ , where [ $α$ ] denotes the greatest integer less than or equal to $α$ is equal to2023 · 24 Jan · Shift 2 · Q28 · MCQ
The value of $n \to \infty lim \frac{1 + 2 - 3 + 4 + 5 - 6 + ..... + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n}{2 n ^{4} + 4 n + 3 - n ^{4} + 5 n + 4}$ is :2023 · 25 Jan · Shift 1 · Q28 · MCQ
If the function $f (x) = ⎩ ⎨ ⎧ (1 + ∣ cos x ∣)^{\frac{λ}{∣ c o s x ∣}} μ e^{\frac{c o t 6 x}{c o t 4 x}},,, 0 < x < \frac{π}{2} x = \frac{π}{2} \frac{π}{2} < x < π$ …2023 · 25 Jan · Shift 2 · Q26 · MCQ
Let $x = 2$ be a root of the equation $x^{2} + p x + q = 0$ and $f (x) = {\frac{1 - c o s ( x ^{2} - 4 p x + q ^{2} + 8 q + 16 )}{( x - 2 p ) ^{4}}, 0, x e 2 p x = 2 p$ Then $x \to 2 p^{+} lim [f (x)]$ …2023 · 29 Jan · Shift 1 · Q40 · MCQ
Suppose $f : R \to (0, \infty)$ be a differentiable function such that $5 f (x + y) = f (x) \cdot f (y), \forall x, y \in R$ . If $f (3) = 320$ , then $\sum_{n = 0}^{5} f (n)$ is equal to :2023 · 30 Jan · Shift 1 · Q28 · MCQ
Let $f, g$ and $h$ be the real valued functions defined on $R$ as $f (x) = {\frac{x}{∣ x ∣}, 1, x e q 0 x = 0 g (x) = {\frac{s i n ( x + 1 )}{( x + 1 )}, 1, x e q - 1 x = - 1$ …2023 · 30 Jan · Shift 2 · Q28 · MCQ
$x \to \infty lim \frac{( 3 x + 1 + 3 x - 1 ) ^{6} + ( 3 x + 1 - 3 x - 1 ) ^{6}}{( x + x ^{2} - 1 ) ^{6} + ( x - x ^{2} - 1 ) ^{6}} x^{3}$ 2023 · 31 Jan · Shift 2 · Q35 · MCQ
The number of points where the function $f (x) = ⎩ ⎨ ⎧ ∣2 x^{2} - 3 x - 7∣ [4 x^{2} - 1] ∣ x + 1∣ + ∣ x - 2∣ if if if x \leq - 1 - 1 < x < 1 x \geq 1$ …2022 · 24 Jun · Shift 1 · Q39 · Numerical
Let $f (x) = ⎩ ⎨ ⎧ \frac{sin ( x - [ x ])}{x - [ x ]} max {2 x, 3 [∣ x ∣]} 1, x \in (- 2, - 1), ∣ x ∣ < 1, otherwise$ where $[t]$ denotes greatest integer $\leq t$ . If $m$ …2022 · 24 Jun · Shift 2 · Q26 · MCQ
If $n \to \infty lim (n^{2} - n - 1 + n α + β) = 0$ , then $8 (α + β)$ is equal to :2022 · 25 Jul · Shift 1 · Q25 · MCQ
Let $f (x) = {4 x^{2} - 8 x + 5, if 8 x^{2} - 6 x + 1 ⩾ 0 [4 x^{2} - 8 x + 5], if 8 x^{2} - 6 x + 1 < 0,$ where $[α]$ denotes the greatest integer less…2022 · 25 Jul · Shift 1 · Q40 · Numerical
$x \to \frac{π}{4} lim \frac{8 2 - ( c o s x + s i n x ) ^{7}}{2 - 2 s i n 2 x}$ is equal to2022 · 25 Jul · Shift 2 · Q27 · MCQ
Let f(x) be a polynomial function such that $f (x) + f^{'} (x) + f^{''} (x) = x^{5} + 64$ . Then, the value of $x \to 1 lim \frac{f ( x )}{x - 1}$ is equal to:2022 · 25 Jun · Shift 1 · Q29 · MCQ
$x \to \frac{π}{2} lim (tan^{2} x ((2 sin^{2} x + 3 sin x + 4)^{\frac{1}{2}} - (sin^{2} x + 6 sin x + 2)^{\frac{1}{2}}))$ is equal to2022 · 25 Jun · Shift 2 · Q27 · MCQ
Let $f (x) = [2 x^{2} + 1]$ and $g (x) = {2 x - 3, 2 x + 3, x < 0 x \geq 0$ , where [t] is the greatest integer $\leq$ t. Then, in the open interval ( $-$ …2022 · 25 Jun · Shift 2 · Q38 · Numerical
Let f : R $\to$ R be a continuous function such that $f (3 x) - f (x) = x$ . If $f (8) = 7$ , then $f (14)$ is equal to :2022 · 26 Jul · Shift 1 · Q25 · MCQ
If the function $f (x) = {\frac{l o g _{e} ( 1 - x + x ^{2} ) + l o g _{e} ( 1 + x + x ^{2} )}{s e c x - c o s x} k,, x \in (\frac{- π}{2}, \frac{π}{2}) - {0} x = 0$ …2022 · 26 Jul · Shift 1 · Q30 · MCQ
If $f (x) = {x + a ∣ x - 4∣,, x \leq 0 x > 0$ and $g (x) = {x + 1 (x - 4)^{2} + b,, x < 0 x \geq 0$ …2022 · 26 Jul · Shift 1 · Q31 · MCQ
Let $f (x) = {x^{3} - x^{2} + 10 x - 7, - 2 x + lo g_{2} (b^{2} - 4), x \leq 1 x > 1$ . Then the set of all values of b, for which f(x) has maximum value at x = 1,…2022 · 26 Jul · Shift 1 · Q32 · MCQ
Let $β = x \to 0 lim \frac{α x - ( e ^{3 x} - 1 )}{α x ( e ^{3 x} - 1 )}$ for some $α \in R$ . Then the value of $α + β$ is :2022 · 26 Jul · Shift 2 · Q25 · MCQ
$x \to \frac{1}{2} lim \frac{s i n ( c o s ^{- 1} x ) - x}{1 - t a n ( c o s ^{- 1} x )}$ is equal to :2022 · 26 Jun · Shift 1 · Q25 · MCQ
Let f, g : R $\to$ R be two real valued functions defined as $f (x) = {- ∣ x + 3∣ e^{x},, x < 0 x \geq 0$ and $g (x) = {x^{2} + k_{1} x 4 x + k_{2},, x < 0 x \geq 0$ …2022 · 26 Jun · Shift 1 · Q26 · MCQ
$x \to 0 lim \frac{c o s ( s i n x ) - c o s x}{x ^{4}}$ is equal to :2022 · 26 Jun · Shift 2 · Q27 · MCQ
Let f(x) = min {1, 1 + x sin x}, 0 $\leq$ x $\leq$ 2 $π$ . If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to2022 · 26 Jun · Shift 2 · Q28 · MCQ
If for $p e q q e q 0$ , the function $f (x) = \frac{7 p ( 729 + x ) - 3}{3 729 + q x - 9}$ is continuous at $x = 0$ , then :2022 · 27 Jul · Shift 2 · Q25 · MCQ
Let a be an integer such that $x \to 7 lim \frac{18 - [ 1 - x ]}{[ x - 3 a ]}$ exists, where [t] is greatest integer $\leq$ t. Then a is equal to :2022 · 27 Jun · Shift 1 · Q23 · MCQ
Let [t] denote the greatest integer $\leq$ t and {t} denote the fractional part of t. The integral value of $α$ for which the left hand limit of the function $f (x) = [1 + x] + \frac{α ^{2 [x] + {x}} + [ x ] - 1}{2 [ x ] + { x }}$ …2022 · 27 Jun · Shift 2 · Q40 · Numerical
Let $f : [0, 1] \to R$ be a twice differentiable function in $(0, 1)$ such that $f (0) = 3$ and $f (1) = 5$ . If the line $y = 2 x + 3$ intersects the graph of $f$ at only two distinct points in $(0, 1)$ , then the least number of points…2022 · 28 Jul · Shift 1 · Q40 · Numerical
$x \to 0 lim (\frac{( x + 2 c o s x ) ^{3} + 2 ( x + 2 c o s x ) ^{2} + 3 s i n ( x + 2 c o s x )}{( x + 2 ) ^{3} + 2 ( x + 2 ) ^{2} + 3 s i n ( x + 2 )})^{\frac{100}{x}}$ is equal to .2022 · 28 Jul · Shift 1 · Q45 · Numerical
The function $f : R \to R$ defined by $f (x) = n \to \infty lim \frac{c o s ( 2 π x ) - x ^{2 n} s i n ( x - 1 )}{1 + x ^{2 n + 1} - x ^{2 n}}$ is continuous for all x in :2022 · 28 Jul · Shift 2 · Q25 · MCQ
Let f : R $\to$ R be defined as $f (x) = ⎩ ⎨ ⎧ [e^{x}], a e^{x} + [x - 1], b + [sin (π x)], [e^{- x}] - c, x < 0 0 \leq x < 1 1 \leq x < 2 x \geq 2$ where a, b, c $\in$ …2022 · 28 Jun · Shift 1 · Q29 · MCQ
Let f, g : R $\to$ R be functions defined by $f (x) = {[x] ∣1 - x ∣,, x < 0 x \geq 0$ and $g (x) = {e^{x} - x (x - 1)^{2} - 1,, x < 0 x \geq 0$ …2022 · 28 Jun · Shift 2 · Q27 · MCQ
The value of $n \to \infty lim 6 tan {r = 1 \sum n tan^{- 1} (\frac{1}{r ^{2} + 3 r + 3})}$ is equal to :2022 · 28 Jun · Shift 2 · Q38 · MCQ
If $x \to 1 lim \frac{s i n ( 3 x ^{2} - 4 x + 1 ) - x ^{2} + 1}{2 x ^{3} - 7 x ^{2} + a x + b} = - 2$ , then the value of (a $-$ b) is equal to .2022 · 28 Jun · Shift 2 · Q44 · Numerical
If $x \to 0 lim \frac{α e ^{x} + β e ^{- x} + γ s i n x}{x s i n ^{2} x} = \frac{2}{3}$ , where $α, β, γ \in R$ , then which of the following is NOT correct?2022 · 29 Jul · Shift 1 · Q29 · MCQ
The number of points, where the function $f : R \to R$ , $f (x) = ∣ x - 1∣ cos ∣ x - 2∣ sin ∣ x - 1∣ + (x - 3) x^{2} - 5 x + 4$ , is NOT differentiable, is :2022 · 29 Jul · Shift 1 · Q38 · MCQ
$Let the function f (x) = {\frac{l o g _{e} ( 1 + 5 x ) - l o g _{e} ( 1 + α x )}{x} 10; if x e q 0; if x = 0 be continuous at x = 0.$ Then $α$ is…2022 · 29 Jul · Shift 2 · Q26 · MCQ
If $[t]$ denotes the greatest integer $\leq t$ , then the number of points, at which the function $f (x) = 4∣2 x + 3∣ + 9 [x + \frac{1}{2}] - 12 [x + 20]$ is not differentiable in the open interval $(- 20, 20)$ , is .2022 · 29 Jul · Shift 2 · Q40 · Numerical
The value of $x \to 1 lim \frac{( x ^{2} - 1 ) s i n ^{2} ( π x )}{x ^{4} - 2 x ^{3} + 2 x - 1}$ is equal to:2022 · 29 Jun · Shift 2 · Q26 · MCQ
Suppose $x \to 0 lim \frac{F ( x )}{x ^{3}}$ exists and is equal to L, where $F (x) = a + sin \frac{x}{2} - b cos x 0 - b cos x 0 a + sin \frac{x}{2} 0 a + sin \frac{x}{2} - b cos x$ …2022 · 30 Jun · Shift 1 · Q35 · Numerical
Let $f (x) = x^{6} + 2 x^{4} + x^{3} + 2 x + 3$ , x $\in$ R. Then the natural number n for which $x \to 1 lim \frac{x ^{n} f ( 1 ) - f ( x )}{x - 1} = 44$ is .2021 · 1 Sep · Shift 2 · Q39 · Numerical
Let [t] denote the greatest integer $\leq$ t. The number of points where the function $f (x) = [x] x^{2} - 1 + sin (\frac{π}{[ x ] + 3}) - [x + 1], x \in (- 2, 2)$ is not continuous is …2021 · 1 Sep · Shift 2 · Q45 · Numerical
Let $S_{k} = r = 1 \sum k tan^{- 1} (\frac{6 ^{r}}{2 ^{2 r + 1} + 3 ^{2 r + 1}})$ . Then $k \to \infty lim S_{k}$ is equal to :2021 · 16 Mar · Shift 1 · Q24 · MCQ
Let the functions f : R $\to$ R and g : R $\to$ R be defined as : $f (x) = {x + 2, x^{2}, x < 0 x \geq 0$ and $g (x) = {x^{3}, 3 x - 2, x < 1 x \geq 1$ …2021 · 16 Mar · Shift 1 · Q31 · MCQ
If $x \to 0 lim \frac{a e ^{x} - b c o s x + c e ^{- x}}{x s i n x} = 2$ , then a + b + c is equal to .2021 · 16 Mar · Shift 1 · Q41 · Numerical
Let f : S $\to$ S where S = (0, $\infty$ ) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $\to$ R be defined as g(x) = loge f(x), then the value of |g''(5) $-$ g''(1)| is equal to :2021 · 16 Mar · Shift 2 · Q29 · MCQ
Let $α \in$ R be such that the function $f (x) = {\frac{c o s ^{- 1} ( 1 - { x } ^{2} ) s i n ^{- 1} ( 1 - { x })}{{ x } - { x } ^{3}}, α, x e 0 x = 0$ …2021 · 16 Mar · Shift 2 · Q35 · MCQ
Let f : R $\to$ R and g : R $\to$ R be defined as $f (x) = {x + a, ∣ x - 1∣, x < 0 x \geq 0$ and $g (x) = {x + 1, (x - 1)^{2} + b, x < 0 x \geq 0$ …2021 · 16 Mar · Shift 2 · Q43 · Numerical
The value of $x \to 0^{+} lim \frac{c o s ^{- 1} ( x - [ x ] ^{2} ) . s i n ^{- 1} ( x - [ x ] ^{2} )}{x - x ^{3}}$ , where [ x ] denotes the greatest integer $\leq$ x is :2021 · 17 Mar · Shift 1 · Q24 · MCQ
If the function $f (x) = \frac{c o s ( s i n x ) - c o s x}{x ^{4}}$ is continuous at each point in its domain and $f (0) = \frac{1}{k}$ , then k is .2021 · 17 Mar · Shift 1 · Q42 · Numerical
The value of the limit $θ \to 0 lim \frac{t a n ( π c o s ^{2} θ )}{s i n ( 2 π s i n ^{2} θ )}$ is equal to :2021 · 17 Mar · Shift 2 · Q29 · MCQ
The value of $n \to \infty lim \frac{[ r ] + [ 2 r ] + ... + [ n r ]}{n ^{2}}$ , where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :2021 · 17 Mar · Shift 2 · Q31 · MCQ
If $x \to 0 lim \frac{s i n ^{- 1} x - t a n ^{- 1} x}{3 x ^{3}}$ is equal to L, then the value of (6L + 1) is2021 · 18 Mar · Shift 1 · Q30 · MCQ
If $f (x) = {\frac{1}{∣ x ∣} a x^{2} + b; ∣ x ∣ \geq 1; ∣ x ∣ < 1$ is differentiable at every point of the domain, then the values of a and b are respectively :2021 · 18 Mar · Shift 1 · Q36 · MCQ
Let $f : R \to R$ be a function defined as $f (x) = ⎩ ⎨ ⎧ \frac{sin ( a + 1 ) x + sin 2 x}{2 x} b \frac{x + b x ^{3} - x}{b x ^{5/2}} if x < 0 if x = 0 if x > 0$ …2021 · 18 Mar · Shift 2 · Q29 · MCQ
Let f : R $\to$ R satisfy the equation f(x + y) = f(x) . f(y) for all x, y $\in$ R and f(x) $e$ 0 for any x $\in$ R. If the function f is differentiable at x = 0 and f'(0) = 3, then $$\mathop {\lim }\limits_{h \to 0} {1 \over h}(f(h) -…2021 · 18 Mar · Shift 2 · Q42 · Numerical
Let a function f : R $\to$ R be defined as $f (x) = ⎩ ⎨ ⎧ sin x - e^{x} a + [- x] 2 x - b i f i f i f x \leq 0 0 < x < 1 x \geq 1$ …2021 · 20 Jul · Shift 1 · Q34 · MCQ
If the value of $x \to 0 lim (2 - cos x cos 2 x)^{(\frac{x + 2}{x ^{2}})}$ is equal to ea, then a is equal to .2021 · 20 Jul · Shift 1 · Q44 · Numerical
If $f : R \to R$ is given by $f (x) = x + 1$ , then the value of $n \to \infty lim \frac{1}{n} [f (0) + f (\frac{5}{n}) + f (\frac{10}{n}) + ...... + f (\frac{5 ( n - 1 )}{n})]$ …2021 · 20 Jul · Shift 2 · Q30 · MCQ
Let a function $g : [0, 4] \to R$ be defined as $g (x) = {0 \leq t \leq x max {t^{3} - 6 t^{2} + 9 t - 3} 4 - x 0 \leq x \leq 3 3 < x \leq 4$ then the number of…2021 · 20 Jul · Shift 2 · Q44 · Numerical
If $x \to 0 lim \frac{α x e ^{x} - β l o g _{e} ( 1 + x ) + γ x ^{2} e ^{- x}}{x s i n ^{2} x} = 10, α, β, γ \in R$ , then the value of $α$ + $β$ + $γ$ is .2021 · 20 Jul · Shift 2 · Q45 · Numerical
Let f : R $\to$ R be defined as $f (x) = {\frac{x ^{3}}{( 1 - c o s 2 x ) ^{2}} lo g_{e} (\frac{1 + 2 x e ^{- 2 x}}{( 1 - x e ^{- x} ) ^{2}}), α, x e 0 x = 0$ …2021 · 22 Jul · Shift 2 · Q36 · MCQ
Let f : R $\to$ R be a function defined as $f (x) = {3 (1 - \frac{∣ x ∣}{2}) 0 i f i f ∣ x ∣ \leq 2 ∣ x ∣ > 2$ Let g : R $\to$ R be given…2021 · 22 Jul · Shift 2 · Q44 · Numerical
If f : R $\to$ R is a function defined by f(x)= [x - 1] $cos (\frac{2 x - 1}{2}) π$ , where [.] denotes the greatest integer function, then f is :2021 · 24 Feb · Shift 1 · Q24 · MCQ
$n \to \infty lim tan {r = 1 \sum n tan^{- 1} (\frac{1}{1 + r + r ^{2}})}$ is equal to .2021 · 24 Feb · Shift 1 · Q39 · Numerical
$n \to \infty lim (1 + \frac{1 + \frac{1}{2} + ........ + \frac{1}{n}}{n ^{2}})^{n}$ is equal to :2021 · 25 Feb · Shift 1 · Q32 · MCQ
The number of points, at which the function f(x) = | 2x + 1 | $-$ 3| x + 2 | + | x2 + x $-$ 2 |, x $\in$ R is not differentiable, is .2021 · 25 Feb · Shift 1 · Q43 · Numerical
A function f is defined on [ $-$ 3, 3] as $f (x) = {min {∣ x ∣, 2 - x^{2}}, [∣ x ∣], - 2 \leq x \leq 2 2 < ∣ x ∣ \leq 3$ where [x] denotes the greatest integer $\leq$ x.…2021 · 25 Feb · Shift 2 · Q45 · Numerical
If $x \to 0 lim \frac{a x - ( e ^{4 x} - 1 )}{a x ( e ^{4 x} - 1 )}$ exists and is equal to b, then the value of a $-$ 2b is .2021 · 25 Feb · Shift 2 · Q47 · Numerical
Let f : R $\to$ R be defined as $f (x) = ⎩ ⎨ ⎧ \frac{λ ∣ x ^{2} - 5 x + 6 ∣}{μ ( 5 x - x ^{2} - 6 )}, e^{\frac{t a n ( x - 2 )}{x - [ x ]}}, μ, x < 2 x > 2 x = 2$ …2021 · 25 Jul · Shift 1 · Q27 · MCQ
Consider the function where P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to . Includes diagram2021 · 25 Jul · Shift 2 · Q41 · Numerical
Let a, b $\in$ R, b $\in$ 0, Define a function $f (x) = {a sin \frac{π}{2} (x - 1), \frac{t a n 2 x - s i n 2 x}{b x ^{3}}, f or x \leq 0 f or x > 0$ . If f is…2021 · 26 Aug · Shift 1 · Q42 · Numerical
Let [t] denote the greatest integer less than or equal to t. Let f(x) = x $-$ [x], g(x) = 1 $-$ x + [x], and h(x) = min{f(x), g(x)}, x $\in$ [ $-$ 2, 2]. Then h is :2021 · 26 Aug · Shift 2 · Q23 · MCQ
$x \to 2 lim (n = 1 \sum 9 \frac{x}{n ( n + 1 ) x ^{2} + 2 ( 2 n + 1 ) x + 4})$ is equal to :2021 · 26 Aug · Shift 2 · Q38 · MCQ
The value of $h \to 0 lim 2 {\frac{3 s i n ( \frac{π}{6} + h ) - c o s ( \frac{π}{6} + h )}{3 h ( 3 c o s h - s i n h )}}$ is :2021 · 26 Feb · Shift 1 · Q24 · MCQ
Let f(x) be a differentiable function at x = a with f'(a) = 2 and f(a) = 4. Then $x \to a lim \frac{x f ( a ) - a f ( x )}{x - a}$ equals :2021 · 26 Feb · Shift 2 · Q26 · MCQ
Let $f (x) = sin^{- 1} x$ and $g (x) = \frac{x ^{2} - x - 2}{2 x ^{2} - x - 6}$ . If $g (2) = x \to 2 lim g (x)$ , then the domain of the function fog is :2021 · 26 Feb · Shift 2 · Q35 · MCQ
Let $f : R \to R$ be defined as $f (x) = ⎩ ⎨ ⎧ 2 sin (- \frac{π x}{2}) ∣ a x^{2} + x + b ∣ sin (π x) if x < - 1 if - 1 \leq x \leq 1 if x > 1$ …2021 · 26 Feb · Shift 2 · Q36 · MCQ
If $α$ , $β$ are the distinct roots of x2 + bx + c = 0, then $x \to β lim \frac{e ^{2 (x^{2} + b x + c)} - 1 - 2 ( x ^{2} + b x + c )}{( x - β ) ^{2}}$ is equal to :2021 · 27 Aug · Shift 1 · Q29 · MCQ
If $x \to \infty lim (x^{2} - x + 1 - a x) = b$ , then the ordered pair (a, b) is :2021 · 27 Aug · Shift 2 · Q35 · MCQ
Let $f : (- \frac{π}{4}, \frac{π}{4}) \to R$ be defined as $f (x) = ⎩ ⎨ ⎧ (1 + ∣ sin x ∣)^{\frac{3 a}{∣ s i n x ∣}} b e^{c o t 4 x / c o t 2 x},,, - \frac{π}{4} < x < 0 x = 0 0 < x < \frac{π}{4}$ …2021 · 27 Jul · Shift 1 · Q32 · MCQ
Let f : R $\to$ R be a function such that f(2) = 4 and f'(2) = 1. Then, the value of $x \to 2 lim \frac{x ^{2} f ( 2 ) - 4 f ( x )}{x - 2}$ is equal to :2021 · 27 Jul · Shift 1 · Q34 · MCQ
Let $f : [0, 3] \to R$ be defined by $f (x) = min {x - [x], 1 + [x] - x}$ where [x] is the greatest integer less than or equal to x. Let P denote the set containing all x $\in$ [0, 3] where f i discontinuous, and Q denote the set containing…2021 · 27 Jul · Shift 1 · Q47 · Numerical
The value of $x \to 0 lim (\frac{x}{8 1 - s i n x - 8 1 + s i n x})$ is equal to :2021 · 27 Jul · Shift 2 · Q31 · MCQ
Let $f : [0, \infty) \to [0, 3]$ be a function defined by $f (x) = {max {sin t : 0 \leq t \leq x}, 2 + cos x, 0 \leq x \leq π x > π$ Then which of the following…2021 · 27 Jul · Shift 2 · Q34 · MCQ
The function $f (x) = x^{2} - 2 x - 3 . e^{∣ 9 x^{2} - 12 x + 4 ∣}$ is not differentiable at exactly :2021 · 31 Aug · Shift 1 · Q25 · MCQ
If the function $f (x) = ⎩ ⎨ ⎧ \frac{1}{x} lo g_{e} (\frac{1 + \frac{x}{a}}{1 - \frac{x}{b}}) k \frac{c o s ^{2} x - s i n ^{2} x - 1}{x ^{2} + 1 - 1},,, x < 0 x = 0 x > 0$ …2021 · 31 Aug · Shift 1 · Q33 · MCQ
$x \to 0 lim \frac{s i n ^{2} ( π c o s ^{4} x )}{x ^{4}}$ is equal to :2021 · 31 Aug · Shift 1 · Q35 · MCQ
If $α = x \to \frac{π}{4} lim \frac{t a n ^{3} x - t a n x}{c o s ( x + \frac{π}{4} )}$ and $β = x \to 0 lim (cos x)^{c o t x}$ are the roots of the equation, ax2…2021 · 31 Aug · Shift 2 · Q26 · MCQ
Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then2021 · 31 Aug · Shift 2 · Q34 · MCQ
If a function f(x) defined by $f (x) = ⎩ ⎨ ⎧ a e^{x} + b e^{- x}, c x^{2}, a x^{2} + 2 c x, - 1 \leq x < 1 1 \leq x \leq 3 3 < x \leq 4$ be…2020 · 2 Sep · Shift 1 · Q23 · MCQ
If $x \to 1 lim \frac{x + x ^{2} + x ^{3} + ... + x ^{n} - n}{x - 1}$ = 820, (n $\in$ N) then the value of n is equal to .2020 · 2 Sep · Shift 1 · Q28 · Numerical
$x \to 0 lim (tan (\frac{π}{4} + x))^{\frac{1}{x}}$ is equal to :2020 · 2 Sep · Shift 2 · Q40 · MCQ
Let [t] denote the greatest integer $\leq$ t. If for some $λ \in$ R - {1, 0}, $x \to 0 lim \frac{1 - x + ∣ x ∣}{λ - x + [ x ]}$ = L, then L is equal to :2020 · 3 Sep · Shift 1 · Q21 · MCQ
If $x \to 0 lim {\frac{1}{x ^{8}} (1 - cos \frac{x ^{2}}{2} - cos \frac{x ^{2}}{4} + cos \frac{x ^{2}}{2} cos \frac{x ^{2}}{4})}$ = 2-k then the value of k is …2020 · 3 Sep · Shift 1 · Q39 · Numerical
$x \to a lim \frac{( a + 2 x ) ^{\frac{1}{3}} - ( 3 x ) ^{\frac{1}{3}}}{( 3 a + x ) ^{\frac{1}{3}} - ( 4 x ) ^{\frac{1}{3}}}$ ( $a e$ 0) is equal to :2020 · 3 Sep · Shift 2 · Q33 · MCQ
Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y. $x \to 0 lim \frac{f ( x )}{x} = 1$ , then f'(3) is equal to .2020 · 4 Sep · Shift 1 · Q26 · Numerical
Let $f : (0, \infty) \to (0, \infty)$ be a differentiable function such that f(1) = e and $t \to x lim \frac{t ^{2} f ^{2} ( x ) - x ^{2} f ^{2} ( t )}{t - x} = 0$ . If f(x) = 1, then x is equal…2020 · 4 Sep · Shift 2 · Q29 · MCQ
The function $f (x) = {\frac{π}{4} + tan^{- 1} x, \frac{1}{2} (∣ x ∣ - 1), ∣ x ∣ \leq 1 ∣ x ∣ > 1$ is :2020 · 4 Sep · Shift 2 · Q40 · MCQ
If $α$ is positive root of the equation, p(x) = x2 - x - 2 = 0, then $x \to α^{+} lim \frac{1 - c o s ( p ( x ) )}{x + α - 4}$ is equal to :2020 · 5 Sep · Shift 1 · Q23 · MCQ
Let $f (x) = x . [\frac{x}{2}]$ , for -10< x < 10, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to .2020 · 5 Sep · Shift 1 · Q27 · Numerical
If the function $f (x) = {k_{1} (x - π)^{2} - 1, k_{2} cos x, x \leq π x > π$ is twice differentiable, then the ordered pair…2020 · 5 Sep · Shift 1 · Q35 · MCQ
$x \to 0 lim \frac{x ( e ^{(1 + x^{2} + x^{4} - 1) / x} - 1 )}{1 + x ^{2} + x ^{4} - 1}$ 2020 · 5 Sep · Shift 2 · Q24 · MCQ
Let f : R $\to$ R be defined as $f (x) = ⎩ ⎨ ⎧ x^{5} sin (\frac{1}{x}) + 5 x^{2}, 0, x^{5} cos (\frac{1}{x}) + λ x^{2}, x < 0 x = 0 x > 0$ …2020 · 6 Sep · Shift 1 · Q26 · Numerical
For all twice differentiable functions f : R $\to$ R, with f(0) = f(1) = f'(0) = 02020 · 6 Sep · Shift 2 · Q23 · MCQ
Let f : R $\to$ R be a function defined by f(x) = max {x, x2}. Let S denote the set of all points in R, where f is not differentiable. Then :2020 · 6 Sep · Shift 2 · Q24 · MCQ
$x \to 2 lim \frac{3 ^{x} + 3 ^{3 - x} - 12}{3 ^{- x /2} - 3 ^{1 - x}}$ is equal to .2020 · 7 Jan · Shift 1 · Q30 · Numerical
Let S be the set of points where the function, ƒ(x) = |2-|x-3||, x $\in$ R is not differentiable. Then $x \in S \sum f (f (x))$ is equal to .2020 · 7 Jan · Shift 1 · Q33 · Numerical
If the function ƒ defined on $(- \frac{1}{3}, \frac{1}{3})$ by f(x) = ${\frac{1}{x} lo g_{e} (\frac{1 + 3 x}{1 - 2 x}), k, w h e n x e 0 w h e n x = 0$ …2020 · 7 Jan · Shift 2 · Q26 · Numerical
$x \to 0 lim (\frac{3 x ^{2} + 2}{7 x ^{2} + 2})^{\frac{1}{x ^{2}}}$ is equal to2020 · 8 Jan · Shift 1 · Q37 · MCQ
Let S be the set of all functions ƒ : [0,1] $\to$ R, which are continuous on [0,1] and differentiable on (0,1). Then for every ƒ in S, there exists a c $\in$ (0,1), depending on ƒ, such that2020 · 8 Jan · Shift 2 · Q34 · MCQ
Let ƒ be any function continuous on [a, b] and twice differentiable on (a, b). If for all x $\in$ (a, b), ƒ'(x) > 0 and ƒ''(x) < 0, then for any c $\in$ (a, b), $\frac{f ( c ) - f ( a )}{f ( b ) - f ( c )}$ is greater than :2020 · 9 Jan · Shift 1 · Q29 · MCQ
If $f (x) = ⎩ ⎨ ⎧ \frac{s i n ( a + 2 ) x + s i n x}{x}; b; \frac{( x + 3 x ^{2} ) ^{\frac{1}{3}} - x ^{\frac{1}{3}}}{x ^{\frac{4}{3}}}; x < 0 x = 0 x > 0$ …2020 · 9 Jan · Shift 1 · Q36 · MCQ
Let [t] denote the greatest integer $\leq$ t and $x \to 0 lim x [\frac{4}{x}] = A$ . Then the function, f(x) = [x2]sin( $π$ x) is discontinuous, when x is equal to :2020 · 9 Jan · Shift 2 · Q37 · MCQ
$x \to 0 lim \frac{s i n ^{2} x}{2 - 1 + c o s x}$ equals:2019 · 8 Apr · Shift 1 · Q34 · MCQ
Let ƒ : R $\to$ R be a differentiable function satisfying ƒ'(3) + ƒ'(2) = 0. Then $x \to 0 lim (\frac{1 + f ( 3 + x ) - f ( 3 )}{1 + f ( 2 - x ) - f ( 2 )})^{\frac{1}{x}}$ is equal to2019 · 8 Apr · Shift 2 · Q33 · MCQ
Let ƒ : [–1,3] $\to$ R be defined as $f (x) = ⎩ ⎨ ⎧ ∣ x ∣ + [x] x + ∣ x ∣ x + [x],,, - 1 \leq x < 1 1 \leq x < 2 2 \leq x \leq 3$ …2019 · 8 Apr · Shift 2 · Q43 · MCQ
Let ƒ(x) = 15 – |x – 10|; x $\in$ R. Then the set of all values of x, at which the function, g(x) = ƒ(ƒ(x)) is not differentiable, is :2019 · 9 Apr · Shift 1 · Q25 · MCQ
If the function ƒ defined on , $(\frac{π}{6}, \frac{π}{3})$ by $f (x) = {\frac{2 cosx o l imi t s - 1}{c o t x - 1}, k, x e \frac{π}{4} x = \frac{π}{4}$ …2019 · 9 Apr · Shift 1 · Q33 · MCQ
If the function $f (x) = {a ∣ π - x ∣ + 1, x \leq 5 b ∣ x - π ∣ + 3, x > 5$ is continuous at x = 5, then the value of a – b is :-2019 · 9 Apr · Shift 2 · Q27 · MCQ
If $f (x) = [x] - [\frac{x}{4}]$ ,x $\in$ 4 , where [x] denotes the greatest integer function, then2019 · 9 Apr · Shift 2 · Q37 · MCQ
$y \to 0 lim \frac{1 + 1 + y ^{4} - 2}{y ^{4}}$ 2019 · 9 Jan · Shift 1 · Q34 · MCQ
Let f : R $\to$ R be a function defined as $f (x) = ⎩ ⎨ ⎧ 5 a + b x b + 5 x 30;;;; x \leq 1 1 < x < 3 3 \leq x < 5 x \geq 5$ …2019 · 9 Jan · Shift 1 · Q38 · MCQ
For each x $\in$ R, let [x] be the greatest integer less than or equal to x. Then $x \to 0^{-} lim \frac{x ( [ x ] + ∣ x ∣ ) s i n [ x ]}{∣ x ∣}$ is…2019 · 9 Jan · Shift 2 · Q43 · MCQ
If $f (x) = ⎩ ⎨ ⎧ \frac{s i n ( p + 1 ) x + s i n x}{x} q \frac{x + x ^{2} - x}{x ^{\frac{3}{2}}}, x < 0, x = 0, x > 0$ is continuous at x =…2019 · 10 Apr · Shift 1 · Q31 · MCQ
Let f : R $\to$ R be differentiable at c $\in$ R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is :2019 · 10 Apr · Shift 1 · Q39 · MCQ
If $x \to 1 lim \frac{x ^{4} - 1}{x - 1} = x \to k lim \frac{x ^{3} - k ^{3}}{x ^{2} - k ^{2}}$ , then k is :2019 · 10 Apr · Shift 1 · Q42 · MCQ
If $x \to 1 lim \frac{x ^{2} - a x + b}{x - 1} = 5$ , then a + b is equal to :2019 · 10 Apr · Shift 2 · Q26 · MCQ
For each t $\in$ R , let [t] be the greatest integer less than or equal to t Then $x \to 1^{+} lim \frac{( 1 - ∣ x ∣ + s i n ∣ 1 - x ∣ ) s i n ( \frac{π}{2} [ 1 - x ] )}{∣ 1 - x ∣ . [ 1 - x ]}$ …2019 · 10 Jan · Shift 1 · Q30 · MCQ
Let $f (x) = {max {∣ x ∣, x^{2}} 8 - 2 ∣ x ∣ ∣ x ∣ \leq 2 2 < ∣ x ∣ \leq 4$ Let S be the set…2019 · 10 Jan · Shift 1 · Q31 · MCQ
Let f : ( $-$ 1, 1) $\to$ R be a function defined by f(x) = max ${- ∣ x ∣, - 1 - x^{2}} .$ If K be the set of all points at which f is not differentiable, then K has exactly -2019 · 10 Jan · Shift 2 · Q25 · MCQ
Let [x] denote the greatest integer less than or equal to x. Then $x \to 0 lim \frac{t a n ( π s i n ^{2} x ) + ( ∣ x ∣ - s i n ( x [ x ] ) ) ^{2}}{x ^{2}}$ …2019 · 11 Jan · Shift 1 · Q22 · MCQ
Let $f (x) = {- 1 x^{2} - 1, - 2 \leq x < 0 0 \leq x \leq 2$ and $g (x) = ∣ f (x) ∣ + f (∣ x ∣) .$ …2019 · 11 Jan · Shift 1 · Q36 · MCQ