Relation between Continuity and Differentiability

Q. The relation $R$ on the set of natural numbers $N$ is defined as $xRy⟺x^2-4xy+3y^2=0,x,y\in N$ then $R$ is

1. reflexive but neither symmetric nor transitive relation.
2. an equivalence relation
3. symmetric but neither reflexive nor transitive relation
4. transitive but neither reflexive nor symmetric relation

Q. Let a relation $R$ on the set $N$ of natural numbers be defined as $(x,y)\iff x^2-4xy+3y^2=0\forall x,y\in N.$ Then the relation $R$ is

1. reflexive
2. symmetric
3. transitive
4. an equivalence relation

Q. If $x=1+a+a^2+....\infty$ and $y=1+b+b^2+....\infty ,$ then the value of $1+ab++a^2{b}^2+....\infty$ is
$(0<a<1,0<b<1)$

1. $\frac{xy}{1-x-y}$
2. $\frac{xy}{x+y+1}$
3. $\frac{xy}{2xy+x+y-1}$
   
4. $\frac{xy}{x+y-1}$

Q. The function
f(x) = $\{\begin{array}{cc}\frac{1-\cos 3x}{x^2},& ifx\ne 0\\ K,& ifx=0\end{array}$
is continuous at x = 0 for

कौनसे विकल्प के लिए फलन पर सतत है?

1. K = 9
   
   K = 9 के लिए
2. K = $\frac{9}{2}$
   
   K = $\frac{9}{2}$ के लिए
3. K = 18
   
   K = 18 के लिए
4. No real value of K
   
   K के किसी भी वास्तविक मान के लिए नहीं

Q.

$\cos (i^{5}x)=?$

1. $i\cos hx$

2. $–i\cos hx$

3. $\cos hx$

4. $–\cos hx$

Q. The value of k for which f(x) is continuous at x = 0 where $f\left(x\right)=\{\begin{array}{cc}\frac{\sin 2x+\log (1-3x)}{2x};& 0<x<\frac{1}{3}\\ k;& x=0\end{array}$

k के किस मान के लिए x = 0 पर f(x) सतत है, जहाँ $f\left(x\right)=\{\begin{array}{cc}\frac{\sin 2x-\log (1-3x)}{2x};& 0<x<\frac{1}{3}\\ k;& x=0\end{array}$ है

1. $\frac{1}{4}$
2. 2
3. 1
4. $\frac{-1}{2}$

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Given f(x) = [·] then which of the following is/are incorrect? ([×] represent, greatest integer function)

दिया है f(x) = [·] तब निम्नलिखित में से कौनसा/कौनसे विकल्प गलत है/हैं? ([×] महत्तम पूर्णांक फलन को दर्शाता है)

1. f(x) is continuous in R
   
   f(x), R में सतत है
2. Domain of f(x) is equal to R
   
   f(x) का प्रांत R के बराबर है
3. f(x) is periodic
   
   f(x) आवर्ती फलन है
4. Range of f(x) is [0, 1)
   
   f(x) का परिसर [0, 1) है

Q. If x = θcosθ – sinθ and y = θsinθ + cosθ, where 'θ' is a parameter, then $\frac{dy}{dx}$ equals

यदि x = θcosθ – sinθ तथा y = θsinθ + cosθ, जहाँ 'θ' एक प्राचल है, तब $\frac{dy}{dx}$ का मान है

1. –θtanθ
2. θsinθ
3. tanθ
4. –cotθ

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Let f(x) = sinx and g(x) = tanx, which of the following is/are correct?

माना f(x) = sinx तथा g(x) = tanx, निम्नलिखित में से कौनसा/कौनसे विकल्प सही है/हैं?

1. f(x) is continuous in R
   
   f(x), R में सतत है
2. g(x) is continuous in R
   
   g(x), R में सतत है
3. f(x) – g(x) is continuous in R
   
   f(x) – g(x), R में सतत है
4. $g\left(\frac{\pi }{4}\right)\cdot f\left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$

Q. Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

Consider the function f(x) = |sinx| + |cosx|, x ∈ R.

फलन f(x) = |sinx| + |cosx|, x ∈ R पर विचार कीजिए।

Q. Value of $f^{\prime }\left(\frac{3\pi }{4}\right)$ is

प्रश्न - $f^{\prime }\left(\frac{3\pi }{4}\right)$ का मान है

1. $-\sqrt{2}$
2. 0
3. $2\sqrt{2}$
4. $\sqrt{2}$

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न


Value of k if $f\left(x\right)=\{\begin{array}{cc}{(1+x)}^{\frac{1}{\sin x}},& x\ne 0\\ k,& x=0\end{array}$ is continuous at x = 0

यदि x = 0 पर $f\left(x\right)=\{\begin{array}{cc}{(1+x)}^{\frac{1}{\sin x}},& x\ne 0\\ k,& x=0\end{array}$ सतत है, तब k का मान है

1. ln(e^e)
2. e
3. e^lne
4. (lne)^e

Q. If f(x) = $\left(e^x-e^{-2x}\right)\sin 2x-3x$, then f(0) + f′(0) is equal to

यदि f(x) = $\left(e^x-e^{-2x}\right)\sin 2x-3x$, तब f(0) + f′(0) का मान है

1. –1
2. 2
3. Zero
   
   शून्य
4. –3

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Which of the following statement(s) is/are true?

निम्नलिखित में से कौनसा/कौनसे कथन सही है/हैं?

1. The function f(x) = |x| + |x + 1| is continuous ∀ x ∈ R
   
   फलन f(x) = |x| + |x + 1| सतत है ∀ x ∈ R
2. The function f(x) = |x| + |x + 1| is differentiable ∀ x ∈ R
   
   फलन f(x) = |x| + |x + 1| अवकलनीय है ∀ x ∈ R
3. The function f(x) = |x| + |x + 1| is discontinuous at x = 0, –1
   
   x = 0, –1 पर फलन f(x) = |x| + |x + 1| असतत है
4. The function f(x) = |x| + |x + 1| is non-differentiable at x = 0, –1
   
   x = 0, –1 पर फलन f(x) = |x| + |x + 1| अन-अवकलनीय है

Q.

Is the function, f(x) = min {x, x²} ᵿx ϵ R continuous?

Q.

If $f\left(x\right)=|lo{g}_{10}^x|\forall xϵ(0,\infty )$ then

1. f(x) is continuous in $(0,\infty )$
2. f(x) is differentiable in $(0,\infty )$
3. f(x) is continuous but not differentiable in $(0,\infty )$
4. None of these

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{1-\sqrt{2}\sin x}{\pi -4x},& x\ne \frac{\pi }{4}\\ a,& x=\frac{\pi }{4}\end{array}$ is continuous at $x=\frac{\pi }{4},$ then value of $a$ is

1. $\frac{1}{2\sqrt{2}}$
2. $\frac{1}{\sqrt{2}}$
3. $\sqrt{2}$
4. $\frac{1}{4}$

Q.

If h(x) = min {x, x²} for x ϵ R. Find LHD and RHD at x = 1.

Q. If $x+\frac{1}{x}<0$, then least integral value of $x^2+2$ is
(where $x\in R$)

Q.

if $f\left(x\right)=x^2-16,g\left(x\right)=16-x^2$how do you find $(f-g)\left(x\right)$

Q. If $y={\left(\cos x\right)}^{x^2}$, then $\frac{dy}{dx}$ is equal to

यदि $y={\left(\cos x\right)}^{x^2}$, तब $\frac{dy}{dx}$ का मान है

1. ${\left(\cos x\right)}^{x^2}\left(\sin x-2x\right)$
2. ${\left(\cos x\right)}^{x^2}\left(2x\ln \cos x-x^2\tan x\right)$
3. ${\left(\cos x\right)}^{x^2}\left(-x^2\sin x+2x\cos x\right)$
4. ${\left(\cos x\right)}^{x^2}\left(x^2\ln \cos x-2x\cos x\right)$

Q.

How do you find $g\left(3\right)$ given $g\left(x\right)=x^2-x$

Q. Let $f\left(x\right)=\frac{\tan \left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}$, where $[.]$ denotes the greatest integer function. Then

1. $f\left(x\right)$ is continuous and differentiable at all $x\in R$
2. $f\left(x\right)$ is continuous at all $x\in R$ but not differentiable at infinitely many points
   
3. $f\left(x\right)$ is neither continuous nor differentiable at a finite number of points
   
4. $f\left(x\right)$ is neither continuous nor differentiable at infinitely many points