Aspire's Library

A Place for Latest Exam wise Questions, Videos, Previous Year Papers,
Study Stuff for MCA Examinations
  • MCA NIMCET
    Previous Year Questions

MCA NIMCET Previous Year Questions (PYQs)

MCA NIMCET Differentibility PYQ

MCA NIMCET PYQ







Go to Discussion
MCA NIMCET Previous Year PYQMCA NIMCET NIMCET 2019 PYQ

Solution


MCA NIMCET PYQ
Let $f(x)=\begin{cases}{{x}^2\sin \frac{1}{x}} & {,\, x\ne0} \\ {0} & {,x=0}\end{cases}$
Then which of the follwoing is true





Go to Discussion
MCA NIMCET Previous Year PYQMCA NIMCET NIMCET 2024 PYQ

Solution

MCA NIMCET PYQ
Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f(0)=\frac{1}{\pi}$ and $f(x)=\frac{x}{e^{\pi x}-1}$ for $x\ne0$, then





Go to Discussion
MCA NIMCET Previous Year PYQMCA NIMCET NIMCET 2024 PYQ

Solution

Analysis of Continuity and Differentiability

Function:

\[ f(x) = \begin{cases} \dfrac{x}{e^{\pi x} - 1}, & x \neq 0 \\ \dfrac{1}{\pi}, & x = 0 \end{cases} \]

✅ Continuity at \( x = 0 \):

\[ \lim_{x \to 0} f(x) = \frac{1}{\pi} = f(0) \quad \Rightarrow \quad \text{Function is continuous at } x = 0 \]

✏️ Differentiability at \( x = 0 \):

\[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = -\frac{1}{2} \]

✅ Final Result:

  • Function is continuous at \( x = 0 \)
  • Function is differentiable at \( x = 0 \)
  • \( f'(0) = \boxed{-\frac{1}{2}} \)
MCA NIMCET PYQ
The set of points, where  is differential in 





Go to Discussion
MCA NIMCET Previous Year PYQMCA NIMCET NIMCET 2018 PYQ

Solution

MCA NIMCET PYQ
Let $\mathbb{R}\rightarrow\mathbb{R}$ be any function defined as $f(x)=\begin{cases}{{x}^{\alpha}\sin \frac{1}{{x}^{\beta}}} & {,x\ne0} \\ {0} & {,x=0}\end{cases}$, $\alpha , \beta \in \mathbb{R}$. Which of the following is true? ($\mathbb{R}$ denotes the set of all real numbers)





Go to Discussion
MCA NIMCET Previous Year PYQMCA NIMCET NIMCET 2025 PYQ

Solution

MCA NIMCET PYQ
The slope of the function \[ f(x) = \begin{cases} x^2 \sin\!\left(\dfrac{1}{x}\right), & \text{if } x \ne 0, \\[8pt] 0, & \text{if } x = 0 \end{cases} \]






Go to Discussion
MCA NIMCET Previous Year PYQMCA NIMCET NIMCET 2017 PYQ

Solution

$f'(0)=\lim_{x\to0}\dfrac{f(x)-f(0)}{x}$ 

$=\lim_{x\to0}\dfrac{x^2\sin(1/x)}{x}$ 

$=\lim_{x\to0}x\sin(\frac{1}{x})=0$

For $x\ne0$,

$f'(x)=2x\sin\left(\dfrac{1}{x}\right)-\cos\left(\dfrac{1}{x}\right)$

Answer: $\boxed{f'(0)=0}$ ✅


MCA NIMCET

Online Test Series,
Information About Examination,
Syllabus, Notification
and More.

Click Here to
View More

MCA NIMCET

Online Test Series,
Information About Examination,
Syllabus, Notification
and More.

Click Here to
View More
Ask Your Question or Put Your Review.

loading...