Phrases Continuity Previous Year Question Paper and Solution with PDF

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Phrases Continuity PYQ

Qus : 1 Phrases PYQ

Test the continuity of the function at x = 2

$f(x)= \begin{cases} \frac{5}{2}-x & \text{ if } x<2 \\ 1 & \text{ if } x=2 \\ x-\frac{3}{2}& \text{ if } x>2 \end{cases}$

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Phrases Previous Year PYQ Phrases NIMCET 2020 PYQ

Solution

LHL ≠ f(2)

Qus : 2 Phrases PYQ

Let $f(x) = \lfloor x^2 - 3 \rfloor$ where $\lfloor \cdot \rfloor$ is the greatest integer function. Number of points in $(1,2)$ where $f$ is discontinuous:

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Phrases Previous Year PYQ Phrases NIMCET 2009 PYQ

Solution

Discontinuity occurs when $x^2 - 3$ is an integer

Let $x^2 - 3 = k \Rightarrow x = \sqrt{k+3}$

We need $1 < x < 2$ → square both sides: $1 < \sqrt{k+3} < 2$

$\Rightarrow 1 < k+3 < 4$

$\Rightarrow -2 < k < 1$

Possible integer values: $k = -1, 0$

So number of discontinuities = $2$

Qus : 3 Phrases PYQ

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Phrases Previous Year PYQ Phrases NIMCET 2012 PYQ

Solution

Qus : 4 Phrases PYQ

Let

,where [x]denotes the greatest integer

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Phrases Previous Year PYQ Phrases NIMCET 2019 PYQ

Solution

Qus : 5 Phrases PYQ

$f(x)=x+|x|$ is continuous for

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Phrases Previous Year PYQ Phrases NIMCET 2022 PYQ

Solution

Qus : 6 Phrases PYQ

If $f\colon R\rightarrow R$ is defined by $f(x)=\begin{cases}{\frac{x+2}{{x}^2+3x+2}} & {,\, if\, x\, \in R-\{-1,-2\}} \\ {-1} & {,if\, x=-2} \\ {0} & {,if\, x=-1}\end{cases}$ , then f(x) is continuous on the set

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Phrases Previous Year PYQ Phrases NIMCET 2021 PYQ

Solution

Qus : 7 Phrases PYQ

If $ f(x)= \begin{cases} x \sin\left(\frac{1}{x}\right), & x \ne 0 \\ 0, & x = 0 \end{cases} $ then

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Phrases Previous Year PYQ Phrases NIMCET 2010 PYQ

Solution

Solution: lim_{x→0} x·sin(1/x) = 0 ⇒ f is continuous at 0. f'(0) = lim_{x→0} [x sin(1/x)]/x = sin(1/x) But sin(1/x) has no limit as x→0⁺ or x→0⁻. ⇒ Both f'(0+) and f'(0-) do not exist.

Qus : 8 Phrases PYQ

$f(x) = x + |x|$ is continuous for

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Phrases Previous Year PYQ Phrases NIMCET 2018 PYQ

Solution

Qus : 9 Phrases PYQ

Let $g:\mathbb{R}\rightarrow \mathbb{R}$ and $h:\mathbb{R}\rightarrow \mathbb{R}$, be two functions such that $h(x) = sgn(g(x))$. Then select which of the following is not true?( $\mathbb{R}$ denotes the set of all real numbers, sgn stands for signum function)

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Phrases Previous Year PYQ Phrases NIMCET 2025 PYQ

Solution

Let $g:\mathbb{R}\to\mathbb{R}$ and $h:\mathbb{R}\to\mathbb{R}$ be such that $h(x) = \operatorname{sgn}(g(x))$.

Recall: $\operatorname{sgn}(t) = \begin{cases} 1, & t>0\\ 0, & t=0\\ -1, & t<0 \end{cases}$

Check each statement:

1) "The domain of $h(x)$ is the same as the domain of $g(x)$." $\Rightarrow$ True, because $\operatorname{sgn}(g(x))$ is defined for every $x$ where $g(x)$ is defined.

2) "The domain of continuity of $h(x)$ equals the domain of continuity of $g(x) - \{x\in\mathbb{R} : g(x)=0\}$."

At points where $g(x)\neq 0$, $h(x)$ is locally constant ($1$ or $-1$), hence continuous there (provided $g$ itself is continuous).

At points where $g(x)=0$, $h(x)$ jumps from $-1$ to $1$, so it is discontinuous. $\Rightarrow$

This statement is true.

3) "The domain of $h(x)$ is different from the domain of $g(x)$ at the same point."

Since for every $x$ in the domain of $g$, $h(x)=\operatorname{sgn}(g(x))$ is defined, the domains are exactly the same; they never differ.

$\Rightarrow$ This statement is false.

4) " $h(x)$ is discontinuous at $g(x)=0$."

At any $x_0$ where $g(x_0)=0$, the left and right limits of $h(x)$ are $-1$ and $1$, not equal to $h(x_0)=0$.

$\Rightarrow$ $h$ is discontinuous there, so this statement is true.

Therefore, the statement which is **not true** is: $\boxed{\text{Option 3}}$

Qus : 10 Phrases PYQ

Consider the function $f$ defined by $f(x)=\begin{cases} x^2-1, & x<3 \ 2ax, & x\geq 3 \end{cases}$ for all real number $x$. If $f$ is continuous at $x=3$, then value of $a$ is

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Phrases Previous Year PYQ Phrases NIMCET 2016 PYQ

Solution

Qus : 11 Phrases PYQ

is a continuous function at x = 0, then the value of k is

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Phrases Previous Year PYQ Phrases NIMCET 2020 PYQ

Solution

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