Algebra
Sets, Relations and FunctionsLogarithmsQuadratic Equations and InequalitiesSequence And SeriesBinomial TheoremMatricesDeterminantsPermutations and CombinationsProbabilityComplex NumbersVector AlgebraThree Dimensional GeometryStatisticsTrigonometry
Trigonometric Angles and EquationsInverse Trigonometric FunctionHeight and DistanceProperties of TrianglesCoordinate Geometry
Coordinate System and Straight LineCircleParabolaEllipseHyperbolaConic SectionCalculus
FunctionsLimit, Continuity and DifferentiabilityDifferentiationApplication of DerivativesIndefinite IntegrationDefinite IntegrationArea Under The CurvesDifferential EquationsLimit, Continuity and Differentiability
Practice QuestionsMCQ (Single Correct Answer)
1
If $$f(x) = \sqrt {25 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{f(x) - f(1)} \over {x - 1}}$$ equal to ?
NDA 2015 Paper 2
2
Consider the function
$$f(x) = \left\{ {\matrix{ {ax - 2,} & {for} & { - 2 < x < - 1} \cr { - 1,} & {for} & { - 1 \le x \le 1} \cr {a + 2{{(x - 1)}^2},} & {for} & {1 < x < 2} \cr } } \right.$$
What is the value of a for which f(x) is continuous at x = $$-$$1 and x = 1?
$$f(x) = \left\{ {\matrix{ {ax - 2,} & {for} & { - 2 < x < - 1} \cr { - 1,} & {for} & { - 1 \le x \le 1} \cr {a + 2{{(x - 1)}^2},} & {for} & {1 < x < 2} \cr } } \right.$$
What is the value of a for which f(x) is continuous at x = $$-$$1 and x = 1?
NDA 2015 Paper 2
3
The function $$f(x) = {{1 - \sin x + \cos x} \over {1 + \sin x + \cos x}}$$ is not defined at x = $$\pi$$. The value of f($$\pi$$), so that f(x) is continuous at x = $$\pi$$, is
NDA 2015 Paper 2
4
Consider the following functions
1. $$f(x) = \left\{ \matrix{ {1 \over x},\,if\,x \ne 0 \hfill \cr 0,\,if\,x = 0 \hfill \cr} \right.$$
$$f(x) = \left\{ {\matrix{ {2x + 5,} & {if\,x > 0} \cr {{x^2} + 2x + 5,} & {if\,x \le 0} \cr } } \right.$$
Which of the above functions is/are derivable at x = 0 ?
1. $$f(x) = \left\{ \matrix{ {1 \over x},\,if\,x \ne 0 \hfill \cr 0,\,if\,x = 0 \hfill \cr} \right.$$
$$f(x) = \left\{ {\matrix{ {2x + 5,} & {if\,x > 0} \cr {{x^2} + 2x + 5,} & {if\,x \le 0} \cr } } \right.$$
Which of the above functions is/are derivable at x = 0 ?
NDA 2015 Paper 2
5
If $$f(x) = {{\sin ({e^{x - 2}} - 1)} \over {\ln (x - 1)}}$$, then $$\mathop {\lim }\limits_{x \to 2} $$ f(x) is equal to
NDA 2015 Paper 2
6
The value of A is
NDA 2015 Paper 2
7
The value of B is
NDA 2015 Paper 2
8
Consider the following in respect of the function
$$f(x) = \left\{ \matrix{ 2 + x,x \ge 0 \hfill \cr 2 - x,x < 0 \hfill \cr} \right.$$.
I. $$\mathop {\lim }\limits_{x \to 1} f(x)$$ does not exist.
II. f(x) is differentiable at x = 0.
III. f(x) is continuous at x = 0.
Which of the above statements is/are correct?
$$f(x) = \left\{ \matrix{ 2 + x,x \ge 0 \hfill \cr 2 - x,x < 0 \hfill \cr} \right.$$.
I. $$\mathop {\lim }\limits_{x \to 1} f(x)$$ does not exist.
II. f(x) is differentiable at x = 0.
III. f(x) is continuous at x = 0.
Which of the above statements is/are correct?
NDA 2016 Paper 2
9
Let f : A $$\to$$ R, where A = R \ {0} is such that $$f(x) = {{x + |x|} \over x}$$. On which one of the following sets is f(x) continuous?
NDA 2016 Paper 2
10
Which of the following statements is correct?
NDA 2016 Paper 2
11
Which one of the following statements is correct?
NDA 2016 Paper 2
12
Which of the following statements are correct?
I. (fof) (x) = f(x).
II. (gog) (x) = g(x) only when x = 0.
III. (go(fog)) (x) can taken only three values.
Select the correct answer using the code given below.
I. (fof) (x) = f(x).
II. (gog) (x) = g(x) only when x = 0.
III. (go(fog)) (x) can taken only three values.
Select the correct answer using the code given below.
NDA 2016 Paper 2
13
Which of the following statements is/are correct?
I. g(x) is differentiable at x = 0
II. g(x) is differentiable at x = 2.
Select the correct answer using the code given below.
I. g(x) is differentiable at x = 0
II. g(x) is differentiable at x = 2.
Select the correct answer using the code given below.
NDA 2016 Paper 2
14
What is the value of the differential coefficient of g(x) at x = $$-$$2 ?
NDA 2016 Paper 2
15
Which of the following statements are correct?
I. g(x) is continuous at x = 0.
II. g(x) is continuous at x = 2.
III. g(x) is continuous at x = $$-$$1.
Select the correct answer using the code given below.
I. g(x) is continuous at x = 0.
II. g(x) is continuous at x = 2.
III. g(x) is continuous at x = $$-$$1.
Select the correct answer using the code given below.
NDA 2016 Paper 2
16
Let $$f(x) = \left\{ {\matrix{
{{{{e^x} - 1} \over x},} & {x > 0} \cr
{0,} & {x = 0} \cr
} } \right.$$, be a real valued function, then which of the following statements is/are correct?
I. f(x) is right continuous at x = 0
II. f(x) is discontinuous at x = 1
Select the correct answer using the code given below.
I. f(x) is right continuous at x = 0
II. f(x) is discontinuous at x = 1
Select the correct answer using the code given below.
NDA 2016 Paper 2
17
Which of the following statements are not correct?
1. y as a function of x is not defined for all real x.
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.
Select the correct answer using the codes given below.
1. y as a function of x is not defined for all real x.
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.
Select the correct answer using the codes given below.
NDA 2016 Paper 1
18
What is the derivative of y as a function of x with respect to x for x < 0 ?
NDA 2016 Paper 1
19
What is $$\mathop {\lim }\limits_{x \to {0^ + }} f(x)$$ equal to
NDA 2016 Paper 1
20
What is $$\mathop {\lim }\limits_{x \to {0^ - }} f(x)$$ equal to?
NDA 2016 Paper 1
21
Consider the following statements
1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at $$x = {\pi \over 2}$$
Which of the above statements is/are correct?
1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at $$x = {\pi \over 2}$$
Which of the above statements is/are correct?
NDA 2016 Paper 1
22
Consider the following statements
1. The function f(x) is differentiable at x = 0.
2. The function f(x) is differentiable at x = $${\pi \over 2}$$.
Which of the above statements is/are correct?
1. The function f(x) is differentiable at x = 0.
2. The function f(x) is differentiable at x = $${\pi \over 2}$$.
Which of the above statements is/are correct?
NDA 2016 Paper 1
23
If $$\mathop {\lim }\limits_{x \to 0} \phi (x) = {a^2}$$, where a $$\ne$$ 0, then what is $$\mathop {\lim }\limits_{x \to 0} \phi \left( {{x \over a}} \right)$$ equal to ?
NDA 2016 Paper 1
24
What is $$\mathop {\lim }\limits_{x \to 0} {e^{ - {1 \over {{x^2}}}}}$$ equal to ?
NDA 2016 Paper 1
25
Consider the function $$f(x) = \left| {x - 1} \right| + {x^2}$$ where x $$\in$$ R, then which one of the following statements is correct?
NDA 2016 Paper 1
26
A function is defined as follows
$$f(x) = \left\{ {\matrix{ { - {x \over {\sqrt {{x^2}} }},} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$
Which one of the following is correct in respect of the above function?
$$f(x) = \left\{ {\matrix{ { - {x \over {\sqrt {{x^2}} }},} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$
Which one of the following is correct in respect of the above function?
NDA 2017 Paper 2
27
Consider the following statements
1. $$x + {x^2}$$ is continuous at x = 0
2. $$x + \cos {1 \over x}$$ is discontinuous at x = 0
3. $${x^2} + \cos {1 \over x}$$ is continuous at x = 0
Which of the above are correct?
1. $$x + {x^2}$$ is continuous at x = 0
2. $$x + \cos {1 \over x}$$ is discontinuous at x = 0
3. $${x^2} + \cos {1 \over x}$$ is continuous at x = 0
Which of the above are correct?
NDA 2017 Paper 2
28
A function is defined in (0, $$\infty$$) by
$$f(x) = \left( {\matrix{ {1 - {x^2}} & {for} & {0 < x \le 1} \cr {\ln x} & {for} & {1 < x \le 2} \cr {\ln 2 - 1 + 0.5x} & {for} & {2 < x < \infty } \cr } } \right.$$
Which one of the following is correct in respect of the derivative of the function, i.e., f'(x) ?
$$f(x) = \left( {\matrix{ {1 - {x^2}} & {for} & {0 < x \le 1} \cr {\ln x} & {for} & {1 < x \le 2} \cr {\ln 2 - 1 + 0.5x} & {for} & {2 < x < \infty } \cr } } \right.$$
Which one of the following is correct in respect of the derivative of the function, i.e., f'(x) ?
NDA 2017 Paper 2
29
If $$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\sin x} \over x} = l$$ and $$\mathop {\lim }\limits_{x \to \infty } {{\cos x} \over x} = m$$, then which one of the following is correct?
NDA 2017 Paper 2
30
The left-hand derivative of f(x) = [x] sin ($$\pi$$x) at x = k, where k is an integer and [x] is the greatest integer function, is
NDA 2017 Paper 2
31
The set of all points, where the function $$f(x) = \sqrt {1 - {e^{ - {x^2}}}} $$ is differentiable, is
NDA 2017 Paper 2
32
If $$f(x) = x(\sqrt x - \sqrt {x + 1} )$$, then f(x) is
NDA 2017 Paper 2
33
Let g be the greatest integer function. Then, the function f(x) = (g(x))2 $$-$$ g(x) is discontinuous at
NDA 2017 Paper 2
34
What is $$\mathop {\lim }\limits_{x \to 0} {{{e^x} - (1 + x)} \over {{x^2}}}$$ equal to ?
NDA 2017 Paper 1
35
If $$F(x) = \sqrt {9 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{F(x) - F(1)} \over {x - 1}}$$ equal to ?
NDA 2017 Paper 1
36
Let f(x + y) = f(x) f(y) for all x and y. Then, what is f'(5) equal to [where f' (x) is the derivative of f(x)]?
NDA 2017 Paper 1
37
Let f(x) be defined as follows
$$f(x) = \left\{ {\matrix{ {2x + 1,} & { - 3 < x < - 2} \cr {x - 1,} & { - 2 \le x < 0} \cr {x + 2,} & {0 \le x < 1} \cr } } \right.$$
Which one of the following statements is correct in respect of the above function?
$$f(x) = \left\{ {\matrix{ {2x + 1,} & { - 3 < x < - 2} \cr {x - 1,} & { - 2 \le x < 0} \cr {x + 2,} & {0 \le x < 1} \cr } } \right.$$
Which one of the following statements is correct in respect of the above function?
NDA 2017 Paper 1
38
Consider the following statements
1. If $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ both exist, then $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists.
2. If $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists, then both $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ must exist.
Which of the above statement is/are correct?
1. If $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ both exist, then $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists.
2. If $$\mathop {\lim }\limits_{x \to a} \{ f(x)g(x)\} $$ exists, then both $$\mathop {\lim }\limits_{x \to a} f(x)$$ and $$\mathop {\lim }\limits_{x \to a} g(x)$$ must exist.
Which of the above statement is/are correct?
NDA 2017 Paper 1
39
Consider the function $$f(x) = \left\{ {\matrix{
{{{\sin 2x} \over {5x}},} & {if\,x \ne 0} \cr
{{2 \over {15}},} & {if\,x = 0} \cr
} } \right.$$
Which one of the following is correct in respect of the function?
Which one of the following is correct in respect of the function?
NDA 2018 Paper 2
40
For the function f(x) = | x $$-$$ 3 |, which one of the following is not correct?
NDA 2018 Paper 2
41
If the function $$f(x) = {{2x - {{\sin }^{ - 1}}x} \over {2x + {{\tan }^{ - 1}}x}}$$ is continuous at each point in its domain then what is the value of f(0) ?
NDA 2018 Paper 2
42
If $$f(x) = \sqrt {25 - {x^2}} $$, then what is $$\mathop {\lim }\limits_{x \to 1} {{f(x) - f(1)} \over {x - 1}}$$ equal to ?
NDA 2018 Paper 2
43
What is $$\mathop {\lim }\limits_{\theta \to 0} {{\sqrt {1 - \cos \theta } } \over \theta }$$ equal to ?
NDA 2018 Paper 2
44
Let f(x + y) = f(x) f(y) and f(x) = 1 + xg(x) $$\phi$$ (x), where $$\mathop {\lim }\limits_{x \to 0} $$ g(x) = a and $$\mathop {\lim }\limits_{x \to 0} $$ $$\phi$$ (x) = b. Then, what is f'(x) equal to ?
NDA 2018 Paper 2
45
What is $$\mathop {\lim }\limits_{x \to {\pi \over 6}} {{2{{\sin }^2}x + \sin x - 1} \over {2{{\sin }^2}x - 3\sin x + 1}}$$ equal to ?
NDA 2018 Paper 2
46
Consider the function $$f(x) = \left\{ {\matrix{
{{x^2}In|x|} & {x \ne 0} \cr
0 & {x = 0} \cr
} } \right.$$ What is f' (0) equal to ?
NDA 2018 Paper 1
47
If $$f(x) = {{{x^2} - 9} \over {{x^2} - 2x - 3}},x \ne 3$$ is continuous at x = 3, then which one of the following is correct?
NDA 2018 Paper 1
48
What is $$\mathop {\lim }\limits_{x \to 0} {{\tan x} \over {\sin 2x}}$$ equal to
NDA 2018 Paper 1
49
What is $$\mathop {\lim }\limits_{h \to 0} {{\sqrt {2x + 3h} - \sqrt {2x} } \over {2h}}$$ equal to ?
NDA 2018 Paper 1
50
For what value of k is the function $$f(x) = \left\{ \matrix{
2x + {1 \over 4},x < 0 \hfill \cr
k,x = 0 \hfill \cr
{\left( {x + {1 \over 2}} \right)^2},x > 0 \hfill \cr} \right.$$ continuous?
NDA 2019 Paper 2
51
What is the value of $$\mathop {\lim }\limits_{x \to 0} {{\sin x^\circ } \over {\tan 3x^\circ }}$$ ?
NDA 2019 Paper 2
52
Consider the following statements in respect of the function $$f(x) = \sin \left( {{1 \over x}} \right)$$ for x $$\ne$$ 0 and f(0) = 0 :
1. $$\mathop {\lim }\limits_{x \to 0} f(x)$$ exits
2. f(x) is continuous at x = 0
hich of the above statement is/are correct?
1. $$\mathop {\lim }\limits_{x \to 0} f(x)$$ exits
2. f(x) is continuous at x = 0
hich of the above statement is/are correct?
NDA 2019 Paper 2
53
$$\mathop {\lim }\limits_{x \to 0} {{1 - {{\cos }^3}4x} \over {{x^2}}}$$ is equal to
NDA 2019 Paper 1
54
The value of k which makes $$f(x) = \left\{ {\matrix{
{\sin x,} & {x \ne 0} \cr
{k,} & {x = 0} \cr
} } \right.$$ continuous at x = 0, is
NDA 2019 Paper 1
55
Which one of the following is correct regarding $\rm \lim\limits_{x\rightarrow 3}\frac{|x-3|}{x-3}$?
NDA Mathematics 1st September 2024
56
What is $\rm \lim\limits_{x\rightarrow \frac{\pi}{2}}(\sec \theta-\tan \theta)$ equal to?
NDA Mathematics 1st September 2024
57
Consider the following statements :
I. f(x) is continuous at x = 0.
Il. f(x) is continuous at x = 1
Which of the statements given above is/are correct?
NDA Mathematics 1st September 2024
58
Consider the following statements :
1. $f(x)$ is increasing in the interval $(e, \infty)$
2. $f(x)$ is decreasing in the interval $(1, e)$
3. $9 \ln 7 > 7 \ln 9$
Which of the statements given above are correct ?
NDA Mathematics 21 April 2024
59
What is $\lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x)$ equal to?
NDA Mathematics 21 April 2024
60
What is f(0.5) equal to ?
NDA Mathematics 3 September 2023
61
What is $\displaystyle \lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$ equal to?
NDA Mathematics 3 September 2023
62
What is $\displaystyle \lim _{x \rightarrow 1} f(x)^{\frac{1}{g(x)}}$ equal to?
NDA Mathematics 3 September 2023
63
What is $\displaystyle \lim _{x \rightarrow 0+}$ h(x) equal to ?
NDA Mathematics 3 September 2023
64
What is $\displaystyle \lim _{x \rightarrow 0-}$ h(x) equal to ?
NDA Mathematics 3 September 2023
65
What is the value of a ?
NDA Mathematics 3 September 2023
66
What is the value of b?
NDA Mathematics 3 September 2023
67
If the function f(x) is differentiable at x = 1, then what is the value of (a + b)?
NDA Mathematics 3 September 2023
68
What is $\displaystyle \lim _{x \rightarrow 0} $ f(x) equal to ?
NDA Mathematics 3 September 2023
69
What is $\lim _{x \rightarrow 0} \frac{f(x)}{x}$ equal to ?
NDA Mathematics 16 April 2023
70
What is $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}$ equal to ?
NDA Mathematics 16 April 2023
71
What is the value of p ?
NDA Mathematics 16 April 2023
72
What is the value of q ?
NDA Mathematics 16 April 2023
73
What is $\lim\limits_{x \rightarrow 5} \frac{5-x}{|x-5|}$ equal to ?
NDA Mathematics 16 April 2023
74
What is $\lim\limits_{x \rightarrow 1} \frac{x^9-1}{x^3-1}$ equal to ?
NDA Mathematics 16 April 2023
75
What is $\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}$ equal to ?
NDA Mathematics 4 September 2022
76
What is $\displaystyle\lim_{x\rightarrow \frac{\pi}{2}} \frac{4x−2\pi}{\cos x}$ equal to ?
NDA Mathematics 4 September 2022
77
If f(x) = $\frac{x^2+x+|x|}{x}$, then what is $\displaystyle\lim_{x \rightarrow 0}$ f(x) equal to ?
NDA Mathematics 4 September 2022
78
What is $\displaystyle\lim_{h \rightarrow 0} \frac{\sin^2(x+h)−\sin^2x}{h}$ equal to ?
NDA Mathematics 4 September 2022
79
Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?
NDA Mathematics 10 April 2022
80
What is $\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2$ equal to ?
NDA Mathematics 10 April 2022
81
What is $\lim\limits_{x \to 1}\frac{x^3 - 1}{\sqrt x - 1}$ equal to?
NDA Mathematics 10 April 2022
82
What is $\rm \displaystyle\lim_{n \rightarrow \infty} \frac{a^n+b^n}{a^n-b^n}$ where a > b > 1, equal to?
NDA Mathematics 14 November 2021
83
Let $\rm f(x) = \left\{\begin{matrix} 1+\frac{x}{2k}, & 0 < x < 2\\\ kx, & 2 \le x < 4 \end {matrix}\right.$
If $\displaystyle\lim_{x\rightarrow 2}$ f(x) exists, then what is the value of k?
NDA Mathematics 14 November 2021
84
Consider the following statements in respect of f(x) = |x| - 1
1. f(x) is continuous at x = 1.
2. f(x) is differentiable at x = 0.
Which of the above statements is/are correct?
NDA Mathematics 14 November 2021
85
If $f(x) = \frac{[x]}{|x|}$, x ≠ 0, where [⋅] denotes the greatest integer function, then what is the right-hand limit of f(x) at x = 1?
NDA Mathematics 14 November 2021
86
Consider the following statements in respect of the function $\rm f(x) = sin \left(\frac{1}{x^2}\right)$, x ≠ 0:
1. It is continuous at x = 0, if f(0) = 0.
2. It is continuous at $x = \frac{2}{\sqrt{\pi}}$.
Which of the above statements is/are correct?
NDA Mathematics 14 November 2021
87
If a differentiable function f(x) satisfies $\mathop {\lim }\limits_{x \to - 1} \dfrac{f(x)+1}{x^2-1}=-\dfrac{3}{2}$ then what is $\mathop {\lim }\limits_{x \to - 1} f(x)$ equal to?
NDA Mathematics 18 April 2021
88
If the function $\rm f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a + bx,\;\;}&{x < 1}\\ {5,}&{x = 1}\\ {b - ax,}&{x > 1} \end{array}} \right.$ is continuous, then what is the value of (a + b)?
NDA Mathematics 18 April 2021
89
If $\rm\mathop {\lim }\limits_{x \to a} \frac{a^x -x^a}{x^x -a^a}= - 1$, then what is the value of a?
NDA Mathematics 18 April 2021
90
What is $\rm\mathop {\lim }\limits_{x \to 2}\frac{x^3 + x^2}{x^2 + 3x + 2}$ equal to?
NDA Mathematics 18 April 2021
91
If f(x) = e|x|, then which one of the following is correct?
NDA Mathematics 18 April 2021