Limits, Continuity & Differentiability · JEE Main previous year questions

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2024 33 questions

Q222024 · 9 Apr 2024 · Shift 1· Numerical answer
Let $lim_{n \to \infty} (\frac{n}{n ^{4} + 1} - \frac{2 n}{( n ^{2} + 1 ) n ^{4} + 1} + \frac{n}{n ^{4} + 16} - \frac{8 n}{( n ^{2} + 4 ) n ^{4} + 16} + \dots + \frac{n}{n ^{4} + n ^{4}} - \frac{2 n \cdot n ^{2}}{( n ^{2} + n ^{2} ) n ^{4} + n ^{4}})$ be $\frac{π}{k}$ , using only the principal values of the inverse trigonometric functions. Then $k^{2}$ is equal to ______.
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Answer: 32
Q232024 · 9 Apr 2024 · Shift 1· Numerical answer
Let $f : (0, π) \to R$ be a function given by $f (x) = ⎩ ⎨ ⎧ (\frac{8}{7})^{\frac{t a n 8 x}{t a n 7 x}}, a - 8, (1 + ∣ cot x ∣)^{\frac{b}{a} ∣ t a n x ∣}, 0 < x < \frac{π}{2} x = \frac{π}{2} \frac{π}{2} < x < π$ where $a, b \in Z$ . If $f$ is continuous at $x = \frac{π}{2}$ , then $a^{2} + b^{2}$ is equal to ______.
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Answer: 81
Q102024 · 9 Apr 2024 · Shift 2
$x \to \frac{π}{2} lim (\frac{\int _{x^{3}}^{(π /2)^{3}} ( s i n ( 2 t ^{1/3} ) + c o s ( t ^{1/3} )) d t}{( x - \frac{π}{2} ) ^{2}})$ is equal to
1. A. $\frac{3 π ^{2}}{2}$
2. B. $\frac{9 π ^{2}}{8}$
3. C. $\frac{5 π ^{2}}{9}$
4. D. $\frac{11 π ^{2}}{10}$
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Answer: B
Q152024 · 9 Apr 2024 · Shift 2
$x \to 0 lim \frac{e - ( 1 + 2 x ) ^{\frac{1}{2 x}}}{x}$ is equal to
1. A. $\frac{- 2}{e}$
2. B. $e - e^{2}$
3. C.0
4. D. $e$
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Answer: D
Q242024 · 9 Apr 2024 · Shift 2· Numerical answer
For a differentiable function $f : R \to R$ , suppose $f^{'} (x) = 3 f (x) + α$ , where $α \in R$ , $f (0) = 1$ and $x \to - \infty lim f (x) = 7$ . Then $9 f (- lo g_{e} 3)$ is equal to ____.
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Answer: 61
Q302024 · 8 Apr 2024 · Shift 1· Numerical answer
The value of $lim_{x \to 0} 2 (\frac{1 - c o s x c o s 2 x 3 c o s 3 x \dots 10 c o s 10 x}{x ^{2}})$ is ________.
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Answer: 55
Q122024 · 8 Apr 2024 · Shift 2
For $a, b > 0$ , let $f (x) = ⎩ ⎨ ⎧ \frac{t a n (( a + 1 ) x ) + b t a n x}{x}, 3, \frac{a x + b ^{2} x ^{2} - a x}{b a x x}, x < 0 x = 0 x > 0$ be a continuous function at $x = 0$ . Then $\frac{b}{a}$ is equal to :
1. A.4
2. B.5
3. C.8
4. D.6
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Answer: D
Q282024 · 8 Apr 2024 · Shift 2· Numerical answer
If $α = x \to 0^{+} lim (\frac{e ^{t a n x} - e ^{x}}{t a n x - x})$ and $β = x \to 0 lim (1 + sin x)^{\frac{1}{2} c o t x}$ are the roots of the quadratic equation $a x^{2} + b x - e = 0$ , then $12 lo g_{e} (a + b)$ is equal to ____.
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Answer: 6
Q52024 · 6 Apr 2024 · Shift 2
$lim_{n \to \infty} \frac{( 1 ^{2} - 1 ) ( n - 1 ) + ( 2 ^{2} - 2 ) ( n - 2 ) + \dots + (( n - 1 ) ^{2} - ( n - 1 )) \cdot 1}{( 1 ^{3} + 2 ^{3} + \dots + n ^{3} ) - ( 1 ^{2} + 2 ^{2} + \dots + n ^{2} )}$ is equal to :
1. A. $\frac{2}{3}$
2. B. $\frac{1}{2}$
3. C. $\frac{3}{4}$
4. D. $\frac{1}{3}$
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Answer: D
Q122024 · 5 Apr 2024 · Shift 2
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 :
1. A.5
2. B.6
3. C.4
4. D.3
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Answer: C
Q262024 · 5 Apr 2024 · Shift 2· Numerical answer
Let $a > 0$ be a root of the equation $2 x^{2} + x - 2 = 0$ . If $x \to \frac{1}{a} lim \frac{16 ( 1 - c o s ( 2 + x - 2 x ^{2} ) )}{( 1 - a x ) ^{2}} = α + β 17$ , where $α, β \in Z$ , then $α + β$ is equal to _________.
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Answer: 170
Q202024 · 4 Apr 2024 · Shift 1
Let $f : R \to R$ be a function given by $f (x) = ⎩ ⎨ ⎧ \frac{1 - c o s 2 x}{x ^{2}}, α, \frac{β 1 - c o s x}{x}, x < 0 x = 0, x > 0$ where $α, β \in R$ . If $f$ is continuous at $x = 0$ , then $α^{2} + β^{2}$ is equal to :
1. A.48
2. B.6
3. C.3
4. D.12
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Answer: D
Q242024 · 4 Apr 2024 · Shift 1· Numerical answer
If $x \to 1 lim \frac{( 5 x + 1 ) ^{1/3} - ( x + 5 ) ^{1/3}}{( 2 x + 3 ) ^{1/2} - ( x + 4 ) ^{1/2}} = \frac{m 5}{n ( 2 n ) ^{2/3}}$ , where $g cd (m, n) = 1$ , then $8 m + 12 n$ is equal to ____.
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Answer: 100
Q42024 · 4 Apr 2024 · Shift 2
If the function $f (x) = {\frac{7 2 ^{x} - 9 ^{x} - 8 ^{x} + 1}{2 - 1 + c o s x}, a lo g_{e} 2 lo g_{e} 3, x \neq = 0 x = 0$ is continuous at $x = 0$ , then the value of $a^{2}$ is equal to
1. A.968
2. B.1250
3. C.1152
4. D.746
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Answer: C
Q72024 · 4 Apr 2024 · Shift 2
Let $f (x) = \int_{0}^{x} (t + sin (1 - e^{t})) d t, x \in R$ . Then, $lim_{x \to 0} \frac{f ( x )}{x ^{3}}$ is equal to
1. A. $\frac{1}{6}$
2. B. $- \frac{1}{6}$
3. C. $\frac{2}{3}$
4. D. $- \frac{2}{3}$
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Answer: B
Q152024 · 1 Feb 2024 · Shift 1
Let $f : R \to R$ be defined as : $f (x) = ⎩ ⎨ ⎧ \frac{a - b c o s 2 x}{x ^{2}}; x^{2} + c x + 2; 2 x + 1; x < 0 0 \leq x \leq 1 x > 1$ If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differentiable then $m + a + b + c$ equals :
1. A.1
2. B.4
3. C.3
4. D.2
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Answer: D
Q252024 · 1 Feb 2024 · Shift 1· Numerical answer
Let ${x}$ denote the fractional part of $x$ and $f (x) = \frac{c o s ^{- 1} ( 1 - { x } ^{2} ) s i n ^{- 1} ( 1 - { x })}{{ x } - { x } ^{3}}, x \neq = 0$ . If $L$ and $R$ respectively denotes the left hand limit and the right hand limit of $f (x)$ at $x = 0$ , then $\frac{32}{π ^{2}} (L^{2} + R^{2})$ is equal to ____.
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Answer: 18
Q112024 · 1 Feb 2024 · Shift 2
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 :
1. A.5
2. B.3
3. C.2
4. D.0
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Answer: B
Q142024 · 1 Feb 2024 · Shift 2
Let $α$ be a non-zero real number. Suppose $f : R \to R$ is a differentiable function such that $f (0) = 2$ and $lim_{x \to - \infty} f (x) = 1$ . If $f^{'} (x) = α f (x) + 3$ , for all $x \in R$ , then $f (- lo g_{e} 2)$ is equal to :
1. A.7
2. B.9
3. C.3
4. D.5
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Answer: B
Q12024 · 31 Jan 2024 · Shift 1
$lim_{x \to 0} \frac{e ^{2∣ s i n x ∣} - 2∣ s i n x ∣ - 1}{x ^{2}}$
1. A.is equal to 1
2. B.does not exist
3. C.is equal to $- 1$
4. D.is equal to 2
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Answer: D
Q142024 · 31 Jan 2024 · Shift 1
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 $g (3)$ is
1. A. $lo g_{e} (\frac{4}{9}) - 1$
2. B. $\frac{1}{3} lo g_{e} (\frac{4}{9 e ^{1/3}})$
3. C. $lo g_{e} (\frac{4}{9 e ^{1/3}})$
4. D. $\frac{1}{3} lo g_{e} (\frac{4}{9}) + 1$
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Answer: C
Q172024 · 31 Jan 2024 · Shift 1
Let $a$ be the sum of all coefficients in the expansion of $(1 - 2 x + 2 x^{2})^{2023} (3 - 4 x^{2} + 2 x^{3})^{2024}$ and $b = x \to 0 lim \frac{0 \int x \frac{l o g ( 1 + t )}{t ^{2024} + 1} d t}{x ^{2}}$ . If the equation $c x^{2} + d x + e = 0$ and $2 b x^{2} + a x + 4 = 0$ have a common root, where $c, d, e \in R$ , then $d : c : e$ equals
1. A.2 : 1 : 4
2. B.1 : 1 : 4
3. C.1 : 2 : 4
4. D.4 : 1 : 4
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Answer: B
Q112024 · 31 Jan 2024 · Shift 2
Let $f : R \to (0, \infty)$ be a strictly increasing function such that $lim_{x \to \infty} \frac{f ( 7 x )}{f ( x )} = 1$ . Then, the value of $lim_{x \to \infty} [\frac{f ( 5 x )}{f ( x )} - 1]$ is equal to
1. A.0
2. B.4
3. C.1
4. D.7/5
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Answer: A
Q122024 · 31 Jan 2024 · Shift 2
The temperature $T (t)$ of a body at time $t = 0$ is $16 0^{\circ} F$ and it decreases continuously as per the differential equation $\frac{d T}{d t} = - K (T - 80)$ , where $K$ is a positive constant. If $T (15) = 12 0^{\circ} F$ , then $T (45)$ is equal to
1. A. $9 0^{\circ}$ F
2. B. $8 5^{\circ}$ F
3. C. $8 0^{\circ}$ F
4. D. $9 5^{\circ}$ F
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Answer: A
Q172024 · 31 Jan 2024 · Shift 2
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 $m + n$ is
1. A.0
2. B.1
3. C.2
4. D.3
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Answer: B