Limits Continuity And Differentiability - Calculus - Mathematics Previous Year Questions

1

2016 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2016

\(\mathop {\lim }\limits_{x \to 1} {\left( {{{1 + x} \over {2 + x}}} \right)^{{{(1 - \sqrt x )} \over {(1 - x)}}}}\) is equal to

A

1

B

does not exist

C

\(\sqrt {{2 \over 3}}\)

D

2

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2

2016 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2016

If \(y = (1 + x)(1 + {x^2})(1 + {x^4})...(1 + {x^{2n}})\), then the value of \(\left( {{{dy} \over {dx}}} \right)\) at x = 0 is

A

0

B

\(-\)1

C

1

D

2

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3

2016 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2016

The value of

\(\mathop {\lim }\limits_{n \to \infty } \left\{ {{{\sqrt {n + 1} + \sqrt {n + 2} + ... + \sqrt {2n - 1} } \over {{n^{3/2}}}}} \right\}\) is

A

\({2 \over 3}(2\sqrt 2 - 1)\)

B

\({2 \over 3}(\sqrt 2 - 1)\)

C

\({2 \over 3}(\sqrt 2 + 1)\)

D

\({2 \over 3}(2\sqrt 2 + 1)\)

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4

2017 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2017

Let \(f(x) = \left\{ {\matrix{ \[{{{{x^p}} \over {{{(\sin x)}^q}}},} & {if\,0 < x \le {\pi \over 2}} \cr\] {0,} & {if\,x = 0} \cr } } \right.\), \((p,q \in R)\). Then, Lagrange's mean value theorem is applicable to f(x) in closed interval [0, x]

A

for all p, q

B

only when p > q

C

only when p < q

D

for no value of p, q

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5

2017 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2017

\(\mathop {\lim }\limits_{x \to 0} {(\sin x)^{2\tan x}}\) is equal to

A

2

B

1

C

0

D

does not exist

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6

2017 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2017

If f'' (0) = k, k \(\ne\) 0, then the value of

\(\mathop {\lim }\limits_{x \to 0} {{2f(x) - 3f(2x) + f(4x)} \over {{x^2}}}\) is

A

k

B

2k

C

3k

D

4k

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7

2017 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2017

Let f : R \(\to\) R be twice continuously differentiable. Let f(0) = f(1) = f'(0) = 0. Then,

A

f''(x) \(\ne\) 0 for all x

B

f''(c) = 0 for some c \(\in\) R

C

f''(x) \(\ne\) 0 if x \(\ne\) 0

D

f'(x) > 0 for all x

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8

2017 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2017

Let for all x > 0, \(f(x) = \mathop {\lim }\limits_{n \to \infty } n({x^{1/n}} - 1)\), then

A

\(f(x) + f\left( {{1 \over x}} \right) = 1\)

B

f(xy) = f(x) + f(y)

C

f(xy) = xf(y) + yf(x)

D

f(xy) = xf(x) + yf(y)

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9

2017 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2017

Consider the non-constant differentiable function f one one variable which obeys the relation \({{f(x)} \over {f(y)}} = f(x - y)\). If f' (0) = p and f' (5) = q, then f' (\(-\)5) is

A

\({{{p^2}} \over q}\)

B

\({q \over p}\)

C

\({p \over q}\)

D

q

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10

2018 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2018

Let f : [a, b] \(\to\) R be such that f is differentiable in (a, b), f is continuous at x = a and x = b and moreover f(a) = 0 = f(b). Then

A

there exists at least one point c in (a, b) such that f'(c) = f(c)

B

f'(x) = f(x) does not hold at any point in (a, b)

C

at every point of (a, b), f'(x) > f(x)

D

at every point of (a, b), f'(x) < f(x)

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11

2018 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2018

Let \(f(x) = 3{x^{10}} - 7{x^8} + 5{x^6} - 21{x^3} + 3{x^2} - 7\).

Then \(\mathop {\lim }\limits_{h \to 0} {{f(1 - h) - f(1)} \over {{h^3} + 3h}}\)

A

does not exist

B

is \({{50} \over 3}\)

C

is \({{53} \over 3}\)

D

is \({{22} \over 3}\)

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12

2018 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2018

Let f : [a, b] \(\to\) R be differentiable on [a, b] and k \(\in\) R. Let f(a) = 0 = f(b). Also let J(x) = f'(x) + kf(x). Then

A

J(x) > 0 for all x \(\in\) [a, b]

B

J(x) < 0 for all x \(\in\) [a, b]

C

J(x) = 0 has at least one root in (a, b)

D

J(x) = 0 through (a, b)

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13

2018 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2018

Let f : R \(\to\) R be a twice continuously differentiable function such that f(0) = f(1) = f'(0) = 0. Then

A

f''(0) = 0

B

f''(c) = 0 for some c\(\in\)R

C

if c \(\ne\) 0, then f''(c) \(\ne\) 0

D

f'(x) > 0 for all x \(\ne\) 0

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14

2018 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2018

Let \(f(x) = \left\{ {\matrix{ \[{ - 2\sin x,} & {if\,x \le - {\pi \over 2}} \cr\] \[{A\sin x + B,} & {if\, - {\pi \over 2} < x < {\pi \over 2}} \cr\] \[{\cos x} & {if\,x \ge {\pi \over 2}} \cr\] } } \right.\). Then,

A

f is discontinuous for all A and B

B

f is continuous for all A = \(-\) 1 and B = 1

C

f is continuous for all A = 1 and B = \(-\) 1

D

f is continuous for all real values of A, B

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15

2019 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2019

$$\mathop {\lim }\limits_{x \to {0^ + }} ({x^n}\ln x),\,n > 0$$

A

does not exist

B

exists and is zero

C

exists and is 1

D

exists and is e^\(-\)1

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16

2019 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2019

The limit of the interior angle of a regular polygon of n sides as n \(\to\) \(\infty\) is

A

\(\pi\)

B

\({\pi \over 3}\)

C

\({3\pi \over 2}\)

D

\({2\pi \over 3}\)

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17

2019 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2019

A particle starts at the origin and moves 1 unit horizontally to the right and reaches P₁, then it moves \({1 \over 2}\) unit vertically up and reaches P₂, then it moves \({1 \over 4}\) unit horizontally to right and reaches P₃, then it moves \({1 \over 8}\) unit vertically down and reaches P₄, then it moves \({1 \over 16}\) unit horizontally to right and reaches P₅ and so on. Let P_n = (x_n, y_n) and \(\mathop {\lim }\limits_{n \to \infty } {x_n} = \alpha\) and \(\mathop {\lim }\limits_{n \to \infty } {y_n} = \beta\). Then, (\(\alpha\), \(\beta\)) is

A

(2, 3)

B

\(\left( {{4 \over 3},{2 \over 5}} \right)\)

C

\(\left( {{2 \over 5},1} \right)\)

D

\(\left( {{4 \over 3},3} \right)\)

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18

2019 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2019

$$\mathop {\lim }\limits_{x \to {0^ + }} {({e^x} + x)^{1/x}}$$

A

Does not exist finitely

B

is 1

C

is e²

D

is 2

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19

2019 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2019

Let \(f:[1,3] \to R\) be a continuous function that is differentiable in (1, 3) an

f'(x) = | f(x) |² + 4 for all x\(\in\) (1, 3). Then,

A

f(3) \(-\) f(1) = 5 is true

B

f(3) \(-\) f(1) = 5 is false

C

f(3) \(-\) f(1) = 7 is false

D

f(3) \(-\) f(1) > 0 only at one point of (1, 3)

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20

2019 · Mathematics · Calculus · Limits Continuity And Differentiability

WB JEE 2019

The value of \(\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]\) is

A

\({{[q]} \over p}\)

B

0

C

1

D

\(\infty\)

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