-
question_answer1)
If \[f(x)={{x}^{\alpha }}log\text{ }x\] and \[f(0)=0\], then the value of a for which Rolle's theorem can be applied in [0, 1] is
A)
-2 done
clear
B)
-1 done
clear
C)
0 done
clear
D)
½ done
clear
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question_answer2)
Let \[f\] be a function which is continuous and differentiable for all real x. If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\text{ }\in [2,\text{ }4],\] then
A)
\[f(4)<8\] done
clear
B)
\[f(4)\ge 8\] done
clear
C)
\[f(4)\ge 12\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer3)
If \[f(0)=0,f'(0)=2\], then the derivative of \[y=f(f(f(f(x)))\] at \[x=0\] is
A)
2 done
clear
B)
8 done
clear
C)
16 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer4)
Let \[f(x)=\frac{{{({{e}^{x}}-1)}^{2}}}{\sin \left( \frac{x}{a} \right)\log \left( 1+\frac{x}{4} \right)}\] for \[x\ne 0,\] and \[f(0)=12\]. If f is continuous at \[x=0\], then the value of a is equal to
A)
1 done
clear
B)
-1 done
clear
C)
2 done
clear
D)
3 done
clear
View Solution play_arrow
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question_answer5)
If \[y=lo{{g}_{10}}x+lo{{g}_{x}}10+lo{{g}_{x}}x+lo{{g}_{10}}10\] then what is \[{{\left( \frac{dy}{dx} \right)}_{x=10}}\] equal to?
A)
10 done
clear
B)
2 done
clear
C)
1 done
clear
D)
0 done
clear
View Solution play_arrow
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question_answer6)
The function \[f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\] is not defined at \[x=\pi \]. The value of \[f(\pi )\] so that \[f(x)\] is continuous at \[x=\pi \] is
A)
\[-\frac{1}{2}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[-1\] done
clear
D)
\[1\] done
clear
View Solution play_arrow
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question_answer7)
Which one of the following functions is differentiable for all real values of x?
A)
\[\frac{x}{\left| x \right|}\] done
clear
B)
\[x\left| x \right|\] done
clear
C)
\[\frac{1}{\left| x \right|}\] done
clear
D)
\[\frac{1}{x}\] done
clear
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question_answer8)
Let \[0<x<\pi \] and y(x) be given by\[(1+\sin \,x){{y}^{3}}-(\cos \,x){{y}^{2}}+2(1+\sin \,x)\] \[y-2\,\cos \,x=0\]. The derivative of y with respect to \[\tan \frac{x}{2}\] at \[x=\frac{\pi }{2}\] is:
A)
\[\frac{1}{2}\] done
clear
B)
\[-\frac{1}{2}\] done
clear
C)
\[2\] done
clear
D)
\[-2\] done
clear
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question_answer9)
If\[y={{\log }^{n}}x\], where \[{{\log }^{n}}\] means log log log... (repeated n time), then \[x\,log\text{ }x\text{ }log\text{ }x\text{ }lo{{g}^{2}}x\text{ }lo{{g}^{3}}x\]\[....{{\log }^{n-1}}x{{\log }^{n}}x\frac{dy}{dx}\] is equal to
A)
\[\log x\] done
clear
B)
\[{{\log }^{n}}x\] done
clear
C)
\[\frac{1}{\log \,x}\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer10)
Let \[f(x)=g(x).\frac{{{e}^{1/x}}-{{e}^{-1/x}}}{{{e}^{1/x}}+{{e}^{-1/x}}}\], where g is a continuous function then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] does not exist if
A)
g(x) is any constant function done
clear
B)
g(x)=x done
clear
C)
\[g(x)={{x}^{2}}\] done
clear
D)
g(x) = x h (x), where h(x) is a polynomial. done
clear
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question_answer11)
If \[f(x)=\left\{ \begin{align} & \frac{x\log \cos x}{\log (1+{{x}^{2}})},x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\text{then}\,\,\text{f(x)is}\] is
A)
Continuous as well as differentiable at x = 0 done
clear
B)
Continuous but not differentiable at x = 0 done
clear
C)
Differentiable but not continuous at x = 0 done
clear
D)
Neither continuous nor differentiable at x = 0 done
clear
View Solution play_arrow
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question_answer12)
Let \[f(x)=\left\{ \begin{align} & {{5}^{1/x}},x<0 \\ & \lambda [x],x\ge 0 \\ \end{align} \right.\] and \[\lambda \in R\], then at x = 0
A)
f is discontinuous done
clear
B)
f is continuous only, if \[\lambda =0\] done
clear
C)
f is continuous only, whatever \[\lambda \] may be done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer13)
Consider the following statements:
1. The function f (x) = greatest integer \[\le x,\text{ }x\in R\]is a continuous function. |
2. All trigonometric functions are continuous on R. |
Which of the statements given above is/are correct? |
A)
1 only done
clear
B)
2 only done
clear
C)
Both 1 and 2 done
clear
D)
Neither 1 nor 2 done
clear
View Solution play_arrow
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question_answer14)
What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2(1-\cos x)}{{{x}^{2}}}\] equal to?
A)
0 done
clear
B)
1/2 done
clear
C)
¼ done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer15)
The number of points at which the function \[f(x)=\left| x-0.5 \right|+\left| x-1 \right|+\tan x\] does not have a derivative in the interval (0, 2) is
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
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question_answer16)
Let \[f(x+y)=f(x)+f(y)\] and \[f(x)={{x}^{2}}g(x)\] for all\[x,\text{ }y\in R\], where g(x) is continuous function. Then f?(x) is equal to
A)
g'(x) done
clear
B)
g(0) done
clear
C)
g(0)+g'(x) done
clear
D)
0 done
clear
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question_answer17)
Given \[f:[-2a,2a]\to R\] is an odd function such that the left hand derivative at x = a is zero and \[f(x)=f(2a-x)\forall x\in (a,2a),\] then its left had derivative at \[x=-a\] is
A)
0 done
clear
B)
a done
clear
C)
-a done
clear
D)
Does not exist done
clear
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question_answer18)
If \[f(xy)=f(x).f(y)\] for all x, y and f(x) is continuous at x = 2, then f(x) is not necessarily continuous in:
A)
\[(-\infty ,\infty )\] done
clear
B)
\[(0,\infty )\] done
clear
C)
\[(-\infty ,0)\] done
clear
D)
\[(2,\infty )\] done
clear
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question_answer19)
Let \[y={{t}^{10}}+1\] and \[x={{t}^{8}}+1\], then \[\frac{{{d}^{2}}y}{d{{x}^{2}}}\] is equal to:
A)
\[\frac{5}{2}t\] done
clear
B)
\[20{{t}^{8}}\] done
clear
C)
\[\frac{5}{16{{t}^{6}}}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer20)
Consider the function \[f(x)=\left\{ \begin{matrix} ax-2 & for & -2<x<-1 \\ -1 & for & -1\le x\le 1 \\ a+2{{(x-1)}^{2}} & for & 1<x<2 \\ \end{matrix} \right.\] What is the value of a for which f(x) is continuous at x =-1 and x=1?
A)
-1 done
clear
B)
1 done
clear
C)
0 done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer21)
Which one of the following is correct in respect of the function \[f(x)=\frac{{{x}^{2}}}{\left| x \right|}\] for \[x\ne 0\] and f(0) = 0?
A)
f (x) is discontinuous every where done
clear
B)
f (x) is continuous every where done
clear
C)
f(x) is continuous at x = 0 only done
clear
D)
f(x) is discontinuous at x = 0 only done
clear
View Solution play_arrow
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question_answer22)
If f(x) is differentiable everywhere, then which one of the following is correct?
A)
\[\left| f \right|\] is differentiable everywhere done
clear
B)
\[{{\left| f \right|}^{2}}\]is differentiable everywhere done
clear
C)
\[f\left| f \right|\]is not differentiable at some points done
clear
D)
None of the above done
clear
View Solution play_arrow
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question_answer23)
Suppose \[f(x)\] is differentiable at \[x=1\] and \[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f(1+h)=5\] then \[f'(1)\] equals
A)
3 done
clear
B)
4 done
clear
C)
5 done
clear
D)
6 done
clear
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question_answer24)
If \[f''(x)<0,\forall x\in (a,b),\] then \[f'(x)=0\] occurs
A)
Exactly once in (a, b) done
clear
B)
At most once in (a, b) done
clear
C)
At least once in (a, b) done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer25)
Let \[f'(x)=-1+\left| x-2 \right|,\] and \[g(x)=1-\left| x \right|;\] then the set of all points where fog is discontinuous is:
A)
{0, 2} done
clear
B)
{0, 1, 2} done
clear
C)
{0} done
clear
D)
An empty set done
clear
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question_answer26)
The value of p for which the function\[f(x)=\left\{ \begin{matrix} \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},x\ne 0 \\ 12{{(log\,4)}^{3}},x=0 \\ \end{matrix} \right.\]may be continuous at \[x=0\], is
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer27)
If\[f(x)=\left\{ \begin{matrix} mx+1x\le \frac{\pi }{2} \\ \sin x+nx>\frac{\pi }{2} \\ \end{matrix}\,\,\,\text{is}\,\,\text{continuous}\,\,\text{at} \right.\]\[x=\frac{\pi }{2}\], then which one of the following is correct?
A)
m = 1, n = 0 done
clear
B)
\[m=\frac{n\pi }{2}+1\] done
clear
C)
\[n=m\left( \frac{\pi }{2} \right)\] done
clear
D)
\[m=n=\frac{\pi }{2}\] done
clear
View Solution play_arrow
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question_answer28)
Consider the function\[f(x)=\left\{ \begin{matrix} {{x}^{2}}, & x>2 \\ 3x-2, & x\le 2 \\ \end{matrix} \right.\]. Which one of the following statements is correct in respect of the above function?
A)
f(x) is derivable but not continuous at x = 2. done
clear
B)
f(x) is continuous but not derivable at x = 2. done
clear
C)
f(x) is neither continuous nor derivable at x = 2. done
clear
D)
f(x) is continuous as well as derivable at x = 2. done
clear
View Solution play_arrow
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question_answer29)
If the polynomial equation \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+.....+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}=0\],n positive integer, has two different real roots \[\alpha \]and \[\beta \], then between \[\alpha \text{ }and\text{ }\beta \], the equation \[n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{x}^{n-2}}+....+{{a}_{1}}=0\] has
A)
Exactly one root done
clear
B)
At most one root done
clear
C)
At least one root done
clear
D)
No root done
clear
View Solution play_arrow
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question_answer30)
If \[f(x)=\sqrt[3]{\frac{{{x}^{4}}}{\left| x \right|}},x\ne 0\] and f(0) = 0 is:
A)
Continuous for all x but not differentiable for any x done
clear
B)
Continuous and differentiable for all x done
clear
C)
Continuous for all x and differentiable for all \[x\ne 0\] done
clear
D)
Continuous and differentiable for all \[x\ne 0\] done
clear
View Solution play_arrow
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question_answer31)
Let \[f:[2,7]\to [0,\infty )\] be a continuous and differentiable function. Then, \[(f(7)-f(2))\frac{{{(f(7))}^{2}}+{{(f(2))}^{2}}+f(2)f(7)}{3}\] is, where \[c\in [2,7]\] [2, 7].
A)
\[5{{f}^{2}}(c)f'(c)\] done
clear
B)
\[5f'(c)\] done
clear
C)
\[f(c)f'(c)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer32)
If \[f(x)=\left\{ \begin{align} & \left( {{x}^{2}}/a \right)-a,\,\,when\,\,\,x<0 \\ & 0,\,\,when\,\,\,x=a,\,\,then \\ & a-\left( {{x}^{2}}/a \right),\,\,when\,\,x>a \\ \end{align} \right.\]
A)
\[\underset{x\to a}{\mathop{\lim }}\,f(x)=a\] done
clear
B)
\[f(x)\] is continuous at x = a done
clear
C)
\[f(x)\] is discontinuous at x = a done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer33)
The number of points in (1, 3), where \[f(x)={{a}^{[{{x}^{2}}]}},a>1\], is not differentiable, where [x] denotes the integral part of x.
A)
5 done
clear
B)
7 done
clear
C)
9 done
clear
D)
11 done
clear
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question_answer34)
A value of c for which conclusion of Mean Value Theorem holds for the function \[f(x)={{\log }_{e}}x\] on the interval [1, 3] is
A)
\[lo{{g}_{3}}e\] done
clear
B)
\[lo{{g}_{e}}3\] done
clear
C)
\[2lo{{g}_{3}}e\] done
clear
D)
\[\frac{1}{2}{{\log }_{3}}e\] done
clear
View Solution play_arrow
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question_answer35)
The derivative of ln \[(x+sin\text{ }x)\] with respect to \[(x+cos\text{ }x)\] is
A)
\[\frac{1+\cos x}{(x+\sin x)(1-\sin x)}\] done
clear
B)
\[\frac{1-\cos x}{(x+\sin x)(1+\sin x)}\] done
clear
C)
\[\frac{1-\cos x}{(x-\sin x)(1+\cos x)}\] done
clear
D)
\[\frac{1+\cos \,\,x}{(x-sin\,\,x)(1-cos\,\,x)}\] done
clear
View Solution play_arrow
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question_answer36)
What is the derivative of \[{{x}^{3}}\] with respect to\[{{x}^{2}}\]?
A)
\[3{{x}^{2}}\] done
clear
B)
\[\frac{3x}{2}\] done
clear
C)
x done
clear
D)
\[\frac{3}{2}\] done
clear
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question_answer37)
If the derivative of the function\[f(x)=\left\{ \begin{matrix} a{{x}^{2}}+b & x<-1 \\ b{{x}^{2}}+ax+4 & x\ge -1 \\ \end{matrix} \right.\]is everywhere continuous, then what are the values of a and b?
A)
a=2, b=3 done
clear
B)
a=3, b=2 done
clear
C)
a=-2, b=-3 done
clear
D)
a=-3, b=-2 done
clear
View Solution play_arrow
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question_answer38)
Consider the following: 1. \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\] exists. 2. \[\underset{x\to 0}{\mathop{\lim }}\,\,\,{{e}^{\frac{1}{x}}}\] does not exist. Which of the above is/are correct?
A)
1 only done
clear
B)
2 only done
clear
C)
Both 1 and 2 done
clear
D)
Neither 1 nor 2 done
clear
View Solution play_arrow
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question_answer39)
If the function \[f(x)=\left[ \frac{{{(x-2)}^{3}}}{a} \right]\sin (x-2)+a\,\cos (x-2),\] [.] denotes the greatest integer function is continuous and differentiable in [4, 6], then
A)
\[a\in [8,\,64]\] done
clear
B)
\[a\in (0,\,8]\] done
clear
C)
\[a\in [64,\,\infty )\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer40)
Given \[f(x)=b({{[x]}^{2}}+[x])+1\] for \[x\ge -1\]\[=\sin (\pi (x+a))\] for \[x<-1\]where [x] denotes the integral part of x, then for what values of a, b, the function is continuous at x = -1?
A)
\[a=2n+(3/2);b\in R;n\in I\] done
clear
B)
\[a=4n+2;b\in R;n\in I\] done
clear
C)
\[a=4n+(3/2);b\in {{R}^{+1}};n\in I\] done
clear
D)
\[a=4n+1;b\in {{R}^{+}};n\in I\] done
clear
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question_answer41)
If \[y=\frac{1}{{{t}^{2}}+t-2}\] where \[t=\frac{1}{x-1}\], then find the number of points of discontinuities of \[y=f(x),\],
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer42)
If \[f(x)=\left| 1-x \right|,\] then the points where \[{{\sin }^{-1}}(f\left| x \right|)\] is non-differentiable are
A)
\[\left\{ 0,1 \right\}\] done
clear
B)
\[\left\{ 0,-1 \right\}\] done
clear
C)
\[\left\{ 0,1,-1 \right\}\] done
clear
D)
None of these done
clear
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question_answer43)
If \[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\], where \[0<x<\frac{\pi }{2}\], then \[\frac{dy}{dx}\] is equal to
A)
\[\frac{1}{2}\] done
clear
B)
2 done
clear
C)
\[\sin x+\cos x\] done
clear
D)
\[\sin x-\cos x\] done
clear
View Solution play_arrow
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question_answer44)
A function \[f:R\to R\] is defined as \[f(x)={{x}^{2}}\] for \[x\ge 0\] and \[f(x)=-x\] for \[x<0\]. Consider the following statements in respect of the above function:
1. The function is continuous at x = 0. |
2. The function is differentiable at x = 0. |
Which of the above statements is/are correct? |
A)
1 only done
clear
B)
2 only done
clear
C)
Both 1 and 2 done
clear
D)
Neither 1 nor 2 done
clear
View Solution play_arrow
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question_answer45)
A function f is defined as follows \[f(x)={{x}^{p}}\cos \left( \frac{1}{x} \right),x\ne 0f(0)=0\] What conditions should be imposed on p so that f may be continuous at x = 0?
A)
p = 0 done
clear
B)
p > 0 done
clear
C)
p < 0 done
clear
D)
No value of p done
clear
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question_answer46)
If \[f(x)=x+\frac{x}{1+x}+\frac{x}{{{(1+x)}^{2}}}+....to\,\,\infty \], then at \[x=0,f(x)\]
A)
Has no limit done
clear
B)
Is discontinuous done
clear
C)
Is continuous but not differentiable done
clear
D)
Is differentiable done
clear
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question_answer47)
Let f be a continuous function on R such that\[f(1/4n)=(\sin {{e}^{n}}){{e}^{-{{n}^{2}}}}+\frac{{{n}^{2}}}{{{n}^{2}}+1}\]. Then the value of f (0) is
A)
1 done
clear
B)
1/2 done
clear
C)
0 done
clear
D)
None of these done
clear
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question_answer48)
Suppose that \[f(0)=-3\] and \[f'(x)=\,\le 5\] for all values of x. Then, the largest value which f(2) can attain is????
A)
7 done
clear
B)
10 done
clear
C)
2 done
clear
D)
9 done
clear
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question_answer49)
Let f'(x) be continuous at x = 0 and f"(0) = 4. Then value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}\] is
A)
12 done
clear
B)
10 done
clear
C)
6 done
clear
D)
4 done
clear
View Solution play_arrow
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question_answer50)
If function \[\text{f(x)=}\left\{ \begin{align} & \text{x,if}\,\,\text{x}\,\,\text{is}\,\,\text{rational} \\ & \text{1-x,if}\,\,\text{x}\,\,\text{is}\,\,\text{irrational} \\ \end{align} \right.\text{,then}\] the number of points at which f(x) is continuous, is-
A)
\[\infty \] done
clear
B)
1 done
clear
C)
0 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer51)
Consider the following in respect of the function \[f(x)=\left\{ \begin{matrix} 2+x, & x\ge 0 \\ 2-x, & x<0 \\ \end{matrix} \right.\]
1. \[\underset{x\to 1}{\mathop{\lim \,f}}\,(x)\] does not exist. |
2. f(x) is differentiable at x = 0 |
3. f(x) is continuous at x = 0 |
Which of the above statements is/are correct? |
A)
1 only done
clear
B)
3 only done
clear
C)
2 and 3 only done
clear
D)
1 and 3 only done
clear
View Solution play_arrow
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question_answer52)
If \[{{x}^{a}}{{y}^{b}}={{(x-y)}^{a+b}},\] then the value of \[\frac{dy}{dx}-\frac{y}{x}\] is equal to
A)
\[\frac{a}{b}\] done
clear
B)
\[\frac{b}{a}\] done
clear
C)
1 done
clear
D)
0 done
clear
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question_answer53)
If \[f(x)={{(x+1)}^{\cot \,x}}\] is continuous at\[x=0\], then what is f (0) equal to?
A)
1 done
clear
B)
e done
clear
C)
\[\frac{1}{e}\] done
clear
D)
\[{{e}^{2}}\] done
clear
View Solution play_arrow
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question_answer54)
If the function \[f(x)=\left\{ \begin{align} & \frac{k\,\,\cos \,\,x}{\pi -2x},when\,x\ne \frac{\pi }{2} \\ & 3,when\,x=\frac{\pi }{2} \\ \end{align} \right.\,\,be\]continuous at \[x=\frac{\pi }{2},\] then k =
A)
3 done
clear
B)
6 done
clear
C)
12 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer55)
Which of the following functions is not differentiable at\[x=1\]?
A)
\[f(x)=({{x}^{2}}-1)\left| (x-1)(x-2) \right|\] done
clear
B)
\[f(x)=\sin (\left| x-1 \right|)-\left| x-1 \right|\] done
clear
C)
\[f(x)=\tan (\left| x-1 \right|)+\left| x-1 \right|\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer56)
If \[y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+.....\infty }}}\], then \[\frac{dy}{dx}=\]
A)
\[\frac{x}{2y-1}\] done
clear
B)
\[\frac{x}{2y+1}\] done
clear
C)
\[\frac{1}{x(2y-1)}\] done
clear
D)
\[\frac{1}{x(1-2y)}\] done
clear
View Solution play_arrow
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question_answer57)
Which of the following function(s) has/have removable discontinuity at \[x=1\]?
A)
\[f(x)=\frac{1}{In\left| x \right|}\] done
clear
B)
\[f(x)=\frac{1}{{{x}^{3}}-1}\] done
clear
C)
\[f(x)={{2}^{{{2}^{\frac{1}{1-x}}}}}\] done
clear
D)
\[f(x)=\frac{\sqrt{x+1}-\sqrt{2x}}{{{x}^{2}}-x}\] done
clear
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question_answer58)
What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-1}{x}\] equal to?
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
1 done
clear
D)
\[-\frac{1}{2}\] done
clear
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question_answer59)
Let \[f(x)=\left\{ \begin{matrix} 3x-4, & 0\le x\le 2 \\ 2x+\ell , & 2<x\le 9 \\ \end{matrix} \right.\] If f is continuous at x = 2, then what is the value of\[\ell \]?
A)
0 done
clear
B)
2 done
clear
C)
-2 done
clear
D)
-1 done
clear
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question_answer60)
Let \[f(x)=[{{x}^{3}}-3],[x]=\]G.I.F. Then the no. of points in the interval (1, 2) where function is discontinuous is
A)
5 done
clear
B)
4 done
clear
C)
6 done
clear
D)
3 done
clear
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question_answer61)
If \[\theta \] are the points of discontinuity of \[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,{{\cos }^{2n}}x\] then the value of sin \[\theta \] is
A)
0 done
clear
B)
1 done
clear
C)
-1 done
clear
D)
½ done
clear
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question_answer62)
Let \[f:R\to R\] be a function defined by f(x) max\[\{x,\text{ }{{x}^{3}}\}\]. The set of all points where f(x) is NOT differentiable is
A)
{-1, 1} done
clear
B)
{-1, 0} done
clear
C)
{0, 1} done
clear
D)
{-1, 0, 1} done
clear
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question_answer63)
The derivative of \[{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] with respect to \[{{\cos }^{-1}}\left[ \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right]\] is equal to:
A)
1 done
clear
B)
-1 done
clear
C)
2 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer64)
If \[y={{(1+1/x)}^{x}}\] then \[\frac{2\sqrt{{{y}_{2}}(2)+1/8}}{(log\,3/2-1/3)}\] is equal to-
A)
3 done
clear
B)
4 done
clear
C)
1 done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer65)
If \[f(x)=\frac{1}{1-x}\], then the points of discontinuity of the function \[f[f\{f(x)\}]\] are
A)
{0, -1} done
clear
B)
{0, 1} done
clear
C)
{1, -1} done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer66)
What is the derivative of \[\left| x-1 \right|\] at\[x=2\]?
A)
-1 done
clear
B)
0 done
clear
C)
1 done
clear
D)
Derivative does not exist done
clear
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question_answer67)
Let \[f:R\to R\] be defined as \[f(x)=sin(\left| x \right|)\] Which one of the following is correct?
A)
f is not differentiable only at 0 done
clear
B)
f is differentiable at 9 only done
clear
C)
f is differentiable everywhere done
clear
D)
f is non-differentiable at many points done
clear
View Solution play_arrow
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question_answer68)
Let f be a function which is continuous and differentiable for all real x. If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\in [2,4],\] then
A)
\[f(4)<8\] done
clear
B)
\[f(4)\ge 8\] done
clear
C)
\[f(4)\ge 12\] done
clear
D)
None of these done
clear
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question_answer69)
The function \[f(x)=\sin ({{\log }_{e}}\left| x \right|),x\ne 0\], and 1 if \[x=0\]
A)
Is continuous at \[x=0\] done
clear
B)
Has removable discontinuity at \[x=0\] done
clear
C)
Has jump discontinuity at \[x=0\] done
clear
D)
Has oscillating discontinuity at \[x=0\] done
clear
View Solution play_arrow
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question_answer70)
\[\frac{{{d}^{n}}}{d{{x}^{n}}}(log\,x)=\]
A)
\[\frac{(n-1)!}{{{x}^{n}}}\] done
clear
B)
\[\frac{n!}{{{x}^{n}}}\] done
clear
C)
\[\frac{(n-2)!}{{{x}^{n}}}\] done
clear
D)
\[{{(-1)}^{n-1}}\frac{(n-1)!}{{{x}^{n}}}\] done
clear
View Solution play_arrow
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question_answer71)
If \[y={{(1+1/x)}^{x}}\] then \[\frac{2\sqrt{{{y}_{2}}(2)+1/8}}{(log\,3/2-1/3)}\] is equal to-
A)
3 done
clear
B)
4 done
clear
C)
1 done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer72)
The set of points of discontinuity of the function\[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{(2\,\sin \,x)}^{2}}^{n}}{{{3}^{n}}-{{(2\cos \,x)}^{2n}}}\] is given by
A)
R done
clear
B)
\[\left\{ n\pi \pm \frac{\pi }{3},n\in I \right\}\] done
clear
C)
\[\left\{ n\pi \pm \frac{\pi }{6},n\in I \right\}\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer73)
Which of the following function is continuous at for all value of x?
(i) \[f\left( x \right)\] =sgn\[({{x}^{3}}-x)\] |
(ii) \[f\left( x \right)\] =sgn \[(2\cos x-1)\] |
(iii) \[f\left( x \right)\] =sgn \[({{x}^{2}}-2x+3)\] |
A)
Only (i) done
clear
B)
Only (iii) done
clear
C)
Both (ii) and (iii) done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer74)
Which one of the following statements is correct in respect of the function\[f(x)={{x}^{3}}\sin x\]?
A)
f'(x) changes sign from positive to negative at x = 0 done
clear
B)
f '(x) changes sign from negative to positive at x = 0 done
clear
C)
Does not change sign at x = 0 done
clear
D)
\[f''(0)\ne 0\] done
clear
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question_answer75)
What is the set of all points, where the function \[f(x)=\frac{x}{1+\left| x \right|}\] is differentiable?
A)
\[(-\infty ,\infty )\] only done
clear
B)
\[(0,\infty )\] only done
clear
C)
\[(-\infty ,0)\cup (0,\infty )\]only done
clear
D)
\[(-\infty ,0)\] only done
clear
View Solution play_arrow
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question_answer76)
The number of points of non-differentiability for\[f(x)=max\,\{\left| \left| x \right|-1 \right|,1/2\}\] is
A)
4 done
clear
B)
3 done
clear
C)
2 done
clear
D)
5 done
clear
View Solution play_arrow
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question_answer77)
Which of the following functions have finite number of points of discontinuity? (where \[[\cdot ]\]represents greatest integer functions)
A)
\[tan\,x\] done
clear
B)
\[x\text{ }\!\![\!\!\text{ }x]\] done
clear
C)
\[\frac{\left| x \right|}{x}\] done
clear
D)
\[\sin \,[n\pi x]\] done
clear
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question_answer78)
If \[f''(x)=-f(x)\] and \[g(x)=f'(x)\] and \[F(x)={{\left( f\left( \frac{x}{2} \right) \right)}^{2}}+{{\left( g\left( \frac{x}{2} \right) \right)}^{2}}\] and given that \[F(5)=5\], then F (10) is equal to-
A)
5 done
clear
B)
10 done
clear
C)
0 done
clear
D)
15 done
clear
View Solution play_arrow
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question_answer79)
Let \[f:R\to R\] be a function defined by\[f(x)=min\{x+1,\left| x \right|+1\}\], Then which of the following is true?
A)
\[f(x)\] is differentiable everywhere done
clear
B)
\[f(x)\] is not differentiable at x = 0 done
clear
C)
\[f(x)\ge 1\] for all \[x\in R\] done
clear
D)
\[f(x)\] is not differentiable at \[x=1\] done
clear
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question_answer80)
The function \[f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\] is not defined at \[x=\pi \]. The value of \[f(\pi )\], so that \[f(x)\] is continuous at \[x=\pi \], is
A)
\[-\frac{1}{2}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[-1\] done
clear
D)
1 done
clear
View Solution play_arrow
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question_answer81)
If \[{{(x-a)}^{2}}+{{(y-b)}^{2}}={{c}^{2}}\], for some c > 0, then\[\frac{{{\left[ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right]}^{\frac{3}{2}}}}{\frac{{{d}^{2}}y}{d{{x}^{2}}}}\] is
A)
Is a constant dependent on a done
clear
B)
Is a constant dependent on b done
clear
C)
Is a constant independent of a and b done
clear
D)
0 done
clear
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question_answer82)
Which one of the following is correct in respect of the function \[f(x)=\left| x \right|+{{x}^{2}}\]
A)
\[f(x)\] is not continuous at x = 0 done
clear
B)
\[f(x)\] is differentiable at x = 0 done
clear
C)
\[f(x)\] is continuous but not differentiable at x = 0 done
clear
D)
None of the above done
clear
View Solution play_arrow
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question_answer83)
If \[f(x)=\cos \left[ \frac{\pi }{x} \right]\cos \left( \frac{\pi }{2}(x-1) \right);\] where [x] is the greatest integer function of x, then f(x) is continuous at
A)
x = 0 done
clear
B)
x = 1, 2 done
clear
C)
x = 0, 2, 4 done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer84)
If the mean value theorem is\[f(b)-f(a)=(b-a)f'(c)\]. Then, for the function \[{{x}^{2}}-2x+3\] in \[\left[ 1,\frac{3}{2} \right]\] the value of c is
A)
6/5 done
clear
B)
5/4 done
clear
C)
4/3 done
clear
D)
7/6 done
clear
View Solution play_arrow
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question_answer85)
Which of the following is correct for \[f(x)=\left\{ \begin{matrix} (x-e){{2}^{-{{2}^{\left( \frac{1}{(e-x)} \right)}},}} & x\ne e\,\,at\,\,x=e \\ 0, & x=e \\ \end{matrix} \right.\]
A)
f(x) is discontinuous at x = e done
clear
B)
f(x) is differentiable at x = e done
clear
C)
f(x) is non-differentiable at x = e done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer86)
The number of points at which the function \[f(x)=\frac{1}{{{\log }_{e}}\left| x \right|}\] is discontinuous, is
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer87)
If \[y={{\tan }^{-1}}\left( \frac{{{2}^{x}}}{1+{{2}^{2x+1}}} \right),\]then \[\frac{dy}{dx}at\,x=0\] is
A)
\[\frac{3}{5}\log \,2\] done
clear
B)
\[\frac{2}{5}\log \,2\] done
clear
C)
\[-\frac{3}{2}\log \,2\] done
clear
D)
\[\log \,2\left( \frac{-1}{10} \right)\] done
clear
View Solution play_arrow
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question_answer88)
If \[f(x)={{\log }_{x}}(In\,\,x)\], then at \[x=e,f'(x)\]equals-
A)
0 done
clear
B)
1 done
clear
C)
e done
clear
D)
1/e done
clear
View Solution play_arrow
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question_answer89)
If \[s=\sqrt{{{t}^{2}}+1}\], then \[\frac{{{d}^{2}}s}{d{{t}^{2}}}\] is equal to
A)
\[\frac{1}{s}\] done
clear
B)
\[\frac{1}{{{s}^{2}}}\] done
clear
C)
\[\frac{1}{{{s}^{3}}}\] done
clear
D)
\[\frac{1}{{{s}^{4}}}\] done
clear
View Solution play_arrow
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question_answer90)
What is the derivative of\[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\] with respect to \[{{\tan }^{-1}}x\]?
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
1 done
clear
D)
x done
clear
View Solution play_arrow
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question_answer91)
If the function\[f(x)=\frac{x(x-2)}{{{x}^{2}}-4},x\ne \pm 2\]is continuous at\[x=2\], then what is \[f(2)\] equal to?
A)
0 done
clear
B)
\[\frac{1}{2}\] done
clear
C)
1 done
clear
D)
2 done
clear
View Solution play_arrow
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question_answer92)
If the functions \[f(x)\] and \[g(x)\] are continuous in [a, b] and differentiable in (a, b), then the equation \[\left| \begin{matrix} f(a) & f(b) \\ g(a) & g(b) \\ \end{matrix} \right|=(b-a)\left| \begin{matrix} f(a) & f'(x) \\ g(a) & g'(x) \\ \end{matrix} \right|\] has in the interval [a, b]
A)
At least one root done
clear
B)
Exactly one root done
clear
C)
At most one root done
clear
D)
No root done
clear
View Solution play_arrow
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question_answer93)
If \[u=f({{x}^{3}}),v=g({{x}^{2}}),f'(x)=\cos x\] and \[g'(x)=\sin x,\] then \[\frac{du}{dv}=\]
A)
\[\frac{1}{2}x\cos {{x}^{3}}\cos ec\,{{x}^{2}}\] done
clear
B)
\[\frac{3}{2}x\cos {{x}^{3}}\cos ec\,{{x}^{2}}\] done
clear
C)
\[\frac{1}{2}x\sec {{x}^{3}}\sin \,{{x}^{2}}\] done
clear
D)
\[\frac{3}{2}x\sec {{x}^{3}}\cos ec\,{{x}^{2}}\] done
clear
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question_answer94)
What is the value of k for which the following function f(x) is continuous for all x? \[f(x)=\left\{ \begin{align} & \frac{{{x}^{3}}-3x+2}{{{(x-1)}^{2}}},for\,\,x\ne 1 \\ & k\,\,\,\,\,\,\,\,,for\,\,x=1 \\ \end{align} \right.\]
A)
3 done
clear
B)
2 done
clear
C)
1 done
clear
D)
-1 done
clear
View Solution play_arrow
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question_answer95)
Suppose\[f(x)={{e}^{ax}}+{{e}^{bx}}\], where\[a\ne b\], and that \[f{{\,}^{n}}(x)-2f'(x)-15f(x)=0\] for all x. Then the product ab is
A)
25 done
clear
B)
9 done
clear
C)
-15 done
clear
D)
-9 done
clear
View Solution play_arrow
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question_answer96)
Let \[3f(x)-2f(1/x)=x,\] then \[f'(2)\] is equal to
A)
\[\frac{2}{7}\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[2\] done
clear
D)
\[\frac{7}{2}\] done
clear
View Solution play_arrow
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question_answer97)
If \[{{I}_{n}}=\frac{{{d}^{n}}}{d{{x}^{n}}}({{x}^{n}}\log \,x)\], then \[{{I}_{n}}-n{{I}_{n-1}}=\]
A)
n done
clear
B)
\[n-1\] done
clear
C)
\[n!\] done
clear
D)
\[\left( n-1 \right)!\] done
clear
View Solution play_arrow
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question_answer98)
If \[\left. \begin{align} & f(x)=sin\text{ }x,when\,\,x\,\,is\,\,rational \\ & \,\,\,\,\,\,\,\,\,\,=\,\cos x,when\,\,x\,\,is\,\,irrational \\ \end{align} \right\}\]
A)
Discontinuous at \[x=n\pi +\pi /4\] done
clear
B)
Continuous at \[x=n\pi +\pi /4\] done
clear
C)
Discontinuous at all x done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer99)
Let \[f(x)=\left\{ \begin{align} & \frac{1-{{\sin }^{3}}x}{3{{\cos }^{2}}x},x<\frac{\pi }{2} \\ & p,x=\frac{\pi }{2} \\ & \frac{q(1-\sin x)}{{{(\pi -2x)}^{2}}},x>\frac{\pi }{2} \\ \end{align} \right.\] If f(x) is continuous at \[x=\frac{\pi }{2},(p,q)=\]
A)
\[(1,4)\] done
clear
B)
\[\left( \frac{1}{2},2 \right)\] done
clear
C)
\[\left( \frac{1}{2},4 \right)\] done
clear
D)
None of these done
clear
View Solution play_arrow
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question_answer100)
If \[y=\frac{(a-x)\sqrt{a-x}-(b-x)\sqrt{x-b}}{\sqrt{a-x}+\sqrt{x-b}}\], then \[\frac{dy}{dx}\] wherever it is defined is
A)
\[\frac{x+(a+b)}{\sqrt{(a-x)(x-b)}}\] done
clear
B)
\[\frac{2x-a-b}{2\sqrt{a-x}\sqrt{x-b}}\] done
clear
C)
\[-\frac{(a+b)}{2\sqrt{(a-x)(x-b)}}\] done
clear
D)
\[\frac{2x+(a+b)}{2\sqrt{(a-x)(x-b)}}\] done
clear
View Solution play_arrow