\[\sum\limits_{m=1}^{n}{{{\tan }^{-1}}\left( \frac{2m}{{{m}^{4}}+{{m}^{2}}+2} \right)}\]is equal to
A)
\[{{\tan }^{-1}}\left( \frac{{{n}^{2}}+n}{{{n}^{2}}+n+2} \right)\]
done
clear
B)
\[{{\tan }^{-1}}\left( \frac{{{n}^{2}}-n}{{{n}^{2}}-n+2} \right)\]
done
clear
C)
\[{{\tan }^{-1}}\left( \frac{{{n}^{2}}+n+2}{{{n}^{2}}+n} \right)\]
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
The value of \[log\tan \left( \frac{\pi }{4}+\frac{ix}{2} \right)\]is
A)
\[i\,\,{{\tan }^{-1}}(\sinh x)\]
done
clear
B)
\[-i\,\,{{\tan }^{-1}}(\sinh x)\]
done
clear
C)
\[i\,\,{{\tan }^{-1}}(coshx)\]
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
Let R be a relation over the set \[N\times N\]and it is defined by \[(a,b)\,\,R(c,d)\Rightarrow a+d=b+c\].Then R is
A)
Reflexive only
done
clear
B)
Symmetric only
done
clear
C)
Transitive only
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[{{e}^{f(x)}}=\frac{10+x}{10-x},\] \[x\in (-10,10)\] and \[f(x)=kf\,\,\left( \frac{200x}{100+{{x}^{2}}} \right)\], then find the value of k.
A)
0.9
done
clear
B)
0.8
done
clear
C)
0.7
done
clear
D)
0.6
done
clear
E)
None of these
done
clear
View Answer play_arrow
The domain of the function \[f(x)=\frac{{{\sin }^{-1}}(3-x)}{\text{In}(|x|-2)}\]
A)
\[\left[ 2,4 \right]\]
done
clear
B)
\[(2,3)\bigcup (3,4)\]
done
clear
C)
\[[2,\infty )\]
done
clear
D)
\[(-\infty ,-3)\bigcup [2,\infty )\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
True statement for \[\underset{x\to 0}{\mathop{Lim}}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{2+3x}-\sqrt{2-3x}}\] is
A)
Lies between 0 and \[\frac{1}{2}\]
done
clear
B)
Lies between \[\frac{1}{2}\]and 1
done
clear
C)
Greater than 1
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
Find the values of a and b such that\[\underset{x\to 0}{\mathop{\lim }}\,\,\,\frac{x\,\,(1+a\,\,\cos x)-b\,\,\sin \,\,x}{{{x}^{3}}}=1\].
A)
\[\frac{5}{2},\]\[\frac{3}{2}\]
done
clear
B)
\[\frac{5}{2},\]\[\frac{-3}{2}\]
done
clear
C)
\[\frac{-5}{2},\]\[\frac{-3}{2}\]
done
clear
D)
\[\frac{-5}{2},\]\[\frac{3}{2}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The area of the triangle formed by the coordinate axis and a tangent to the curve \[xy={{a}^{2}}\]at the point \[({{x}_{1}},{{y}_{1}})\] on it, is
A)
\[\frac{{{a}^{2}}\,\,{{x}_{1}}}{{{y}_{1}}}\]
done
clear
B)
\[\frac{{{a}^{2}}\,\,{{y}_{1}}}{{{x}_{1}}}\]
done
clear
C)
\[2{{a}^{2}}\]
done
clear
D)
\[4{{a}^{2}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
\[\int{\frac{dx}{1-\cos x-\sin x}}\]is equal to
A)
\[\log \left| 1+\cot \frac{x}{2} \right|+c\]
done
clear
B)
\[\log \left| 1-\tan \frac{x}{2} \right|+c\]
done
clear
C)
\[\log \left| 1-\cot \frac{x}{2} \right|+c\]
done
clear
D)
\[\log \left| 1+\tan \frac{x}{2} \right|+c\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[\int{\frac{1}{(\sin \,\,x+4)(\sin \,\,x+1)}\,\,dx=\frac{A}{\tan \frac{x}{2}-1}+B\,\,{{\tan }^{-1}}[f\,\,(x)]+c,}\]then
A)
\[A=\frac{1}{5},\] \[B=\frac{-2}{5\sqrt{15}},\] \[f\,\,(x)=\frac{4\,\,\tan x-3}{\sqrt{15}}\]
done
clear
B)
\[A=\frac{1}{5},\] \[B=\frac{1}{\sqrt{15}},\] \[f\,\,(x)=\frac{4\,\,\tan \left( \frac{x}{2} \right)+1}{\sqrt{15}}\]
done
clear
C)
\[A=\frac{2}{5},\] \[B=\frac{-2}{5},\] \[f\,\,(x)=\frac{4\,\,\tan x+1}{5}\]
done
clear
D)
\[A=\frac{2}{5},\] \[B=\frac{-2}{5\sqrt{15}},\]\[f\,\,(x)=\frac{4\,\,\tan \frac{x}{2}+1}{\sqrt{15}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The value of the integral \[\int_{1/4}^{4}{\left| {{\log }_{4}}x \right|}\,dx\]is
A)
\[\frac{1}{4}{{\log }_{4}}\,\,\left( \frac{{{e}^{9}}}{{{4}^{15}}} \right)\]
done
clear
B)
\[\frac{1}{4}{{\log }_{4}}\,\,\left( \frac{{{4}^{15}}}{{{e}^{9}}} \right)\]
done
clear
C)
\[\frac{1}{2}{{\log }_{4}}\,\,\left( \frac{{{e}^{9}}}{{{4}^{15}}} \right)\]
done
clear
D)
\[\frac{1}{2}{{\log }_{4}}\,\,\left( \frac{{{4}^{15}}}{{{e}^{9}}} \right)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If I is the greatest of the definite integrals
\[{{I}_{1}}=\int_{0}^{1}{{{e}^{-x}}{{\cos }^{2}}x\,dx,}\] \[{{I}_{2}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}\cos x\,dx.}\] \[{{I}_{3}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}dx,}\] \[{{I}_{4}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}/2}}dx,}\] then
A)
\[I={{I}_{1}}\]
done
clear
B)
\[I={{I}_{2}}\]
done
clear
C)
\[I={{I}_{3}}\]
done
clear
D)
\[I={{I}_{4}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If\[\int{\frac{{{x}^{2}}}{({{x}^{2}}+1)\,\,({{x}^{2}}+4)}dx=P\,\,{{\tan }^{-1}}x+Q\,\,{{\tan }^{-1}}\left( \frac{x}{2} \right)+c,}\]
A)
\[P=\frac{1}{3},\] \[Q=\frac{5}{3}\]
done
clear
B)
\[P=\frac{-1}{3},\]\[Q=\frac{-2}{3}\]
done
clear
C)
\[P=\frac{1}{3},\]\[Q=\frac{2}{3}\]
done
clear
D)
\[P=\frac{-1}{3},\]\[Q=\frac{2}{3}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The area bounded by the curves \[{{x}^{2}}+{{y}^{2}}=9\]and \[{{y}^{2}}=8x\]is
A)
\[\frac{2\sqrt{2}}{3}+\frac{9\pi }{2}+9{{\sin }^{-1}}\left( \frac{1}{3} \right)\]
done
clear
B)
\[\frac{2\sqrt{2}}{5}+\frac{9\pi }{2}-9{{\sin }^{-1}}\left( \frac{1}{3} \right)\]
done
clear
C)
0
done
clear
D)
\[16\pi \]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The degree of the differential equation \[3\frac{{{d}^{2}}y}{d{{x}^{2}}}={{\left\{ 1+{{\left( \frac{dy}{dx} \right)}^{2}} \right\}}^{3/2}}\]is
A)
1
done
clear
B)
2
done
clear
C)
3
done
clear
D)
6
done
clear
E)
None of these
done
clear
View Answer play_arrow
If integrating factor of \[x\,\,(1-{{x}^{2}})\,\,dy+(2{{x}^{2}}y-y-a{{x}^{3}})\,\,dx=0\] is \[{{e}^{\int{pdx}}},\] then P is equal to
A)
\[\frac{2{{x}^{2}}-a{{x}^{3}}}{x(1-{{x}^{2}})}\]
done
clear
B)
\[(2{{x}^{2}}-1)\]
done
clear
C)
\[\frac{2{{x}^{2}}-1}{a{{x}^{3}}}\]
done
clear
D)
\[\frac{2{{x}^{2}}-1}{x{{(1-x)}^{2}}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
One bag contains 5 white and 4 black balls. Another bag contains 7 white and 9 black balls. A ball is transferred from the first bag to the second and then a ball is drawn from second. Find the probability that the ball is white.
A)
\[\frac{8}{17}\]
done
clear
B)
\[\frac{5}{9}\]
done
clear
C)
\[\frac{4}{9}\]
done
clear
D)
\[\frac{40}{153}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
A dice is thrown \[(2n+1)\] times. The probability of getting 1, 3 or 4 at most n times is
A)
\[\frac{1}{2}\]
done
clear
B)
\[\frac{1}{3}\]
done
clear
C)
\[\frac{1}{4}\]
done
clear
D)
\[\frac{2}{3}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Three squares of a chess board are chosen at random, the probability that two are of one colour and one of another is
A)
\[\frac{8}{21}\]
done
clear
B)
\[\frac{32}{12}\]
done
clear
C)
\[\frac{16}{21}\]
done
clear
D)
\[\frac{21}{16}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
A pack of playing cards was found to contain only 51 cards. If the first 13 cards which are examined are all red, then the probability that the missing card is black, is
A)
\[\frac{1}{3}\]
done
clear
B)
\[\frac{2}{3}\]
done
clear
C)
\[\frac{1}{2}\]
done
clear
D)
\[\frac{{{25}_{{{c}_{13}}}}}{{{51}_{{{c}_{13}}}}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Forces of magnitudes 3 and 2 units acting in the directions \[5i+3j+4k\] and \[3i+4j-5k\]respectively act on a particle which is displaced from the points \[(1,-1,-1)\] to \[(3,\,\,3,\,\,1)\].The work done by the forces is equal to
A)
\[50\sqrt{2}\]unit
done
clear
B)
\[40\sqrt{2}\]unit
done
clear
C)
\[\frac{57}{5}\sqrt{2}\]unit
done
clear
D)
\[8\sqrt{2}\]unit
done
clear
E)
None of these
done
clear
View Answer play_arrow
The points D, E, F, divide BC, CA and AB of the triangle ABC in the ratio \[1:4,\]\[3:2\] and \[3:7\] respectively and the point K divides AB in the ratio \[1:3,\] then \[\left( \overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF} \right):\overrightarrow{CK}\]is equal to
A)
\[1:1\]
done
clear
B)
\[2:5\]
done
clear
C)
\[5:2\]
done
clear
D)
\[7:5\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
A vector n of magnitude 8 units is inclined to x-axis at \[45{}^\circ ,\] y-axis at \[60{}^\circ \]and an acute angle with z-axis. If a plane passes through a point \[(\sqrt{2},\,\,-1,\,\,1)\]and is normal to n, then its equation in vector form is
A)
\[r.(\sqrt{2}\,\,i+j+k)=4\]
done
clear
B)
\[r.\,\,(\sqrt{2}i+j+k)=2\]
done
clear
C)
\[r.\,\,(i+j+k)\]
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
If a line makes angles \[\alpha ,\] \[\beta ,\] \[\gamma ,\] \[\delta \] with the four diagonals of a cube, then the value of \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma +{{\cos }^{2}}\delta =\]
A)
1
done
clear
B)
\[\frac{4}{3}\]
done
clear
C)
2
done
clear
D)
Zero
done
clear
E)
None of these
done
clear
View Answer play_arrow
The point of intersection of the lines\[\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}\] and \[\frac{x-2}{1}=\frac{y-4}{3}=\frac{z-6}{5}\] is
A)
\[\left( \frac{1}{2},\,\,\frac{1}{2},\,\,\frac{-3}{2} \right)\]
done
clear
B)
\[\left( \frac{-1}{2},\,\,\frac{-1}{2},\,\,\frac{3}{2} \right)\]
done
clear
C)
\[\left( \frac{1}{2},\,\,\frac{-1}{2},\,\,\frac{-3}{2} \right)\]
done
clear
D)
\[\left( \frac{-1}{2},\,\,\frac{1}{2},\,\,\frac{3}{2} \right)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The equation of line of intersection of the planes \[4x+4y-5z=12,\]\[8x+12y-13z=32\] can be written as
A)
\[\frac{x}{2}=\frac{y-1}{3}=\frac{z-2}{4}\]
done
clear
B)
\[\frac{x}{2}=\frac{y}{3}=\frac{z-2}{4}\]
done
clear
C)
\[\frac{x-1}{2}=\frac{y-2}{3}=\frac{z}{4}\]
done
clear
D)
\[\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z}{4}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[\overrightarrow{p,}\]\[\overrightarrow{q,}\]\[\overrightarrow{r}\]are three mutually perpendicular vectors of the same magnitude. If a vector\[\overrightarrow{x}\] satisfies the equation, \[\overrightarrow{p}\times \left[ (\overrightarrow{x}-\overrightarrow{q})\times \overrightarrow{p} \right]+\overrightarrow{q}\times \left[ (\overrightarrow{x}-\overrightarrow{r)}\times \overrightarrow{q} \right]+\overrightarrow{r}\times \left[ (\overrightarrow{x}-\overrightarrow{p})\times \overrightarrow{r} \right]=0\]then \[\overrightarrow{x}\]is given by
A)
\[\frac{1}{2}\,\,(\overrightarrow{p}+\overrightarrow{q}-2\overrightarrow{r})\]
done
clear
B)
\[\frac{1}{2}\,\,(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})\]
done
clear
C)
\[\frac{1}{3}\,\,(\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})\]
done
clear
D)
\[\frac{1}{3}\,\,(2\overrightarrow{p}+\overrightarrow{q}-\overrightarrow{r})\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Let \[\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\] \[\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\] and \[\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}\]. A vector in the plane of \[\overrightarrow{a}\] and \[\overrightarrow{b}\] whose projection on \[\overrightarrow{c}\] is \[\frac{1}{\sqrt{3}},\]is
A)
\[4\hat{i}-\hat{j}+4\hat{k}\]
done
clear
B)
\[3\hat{i}+\hat{j}-3\hat{k}\]
done
clear
C)
\[2\hat{i}+\hat{j}-2\hat{k}\]
done
clear
D)
\[4\hat{i}+\hat{j}-4\hat{k}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The points \[\left( 0,\,\,\frac{8}{3} \right),\]\[(1,\,\,3)\]and (82, 30) are vertices of
A)
an obtuse angled triangle
done
clear
B)
an acute angled triangle
done
clear
C)
a right angled triangle
done
clear
D)
Both (a) and (b)
done
clear
E)
None of these
done
clear
View Answer play_arrow
A random variable X has the probability distribution
X 1 2 3 4 5 6 7 8 P(X) 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05
For the events E = {X is a prime number} and \[F=\{X<4\},\]then probability of \[P(E\cup F)\] is
A)
0.87
done
clear
B)
0.77
done
clear
C)
0.35
done
clear
D)
0.50
done
clear
E)
None of these
done
clear
View Answer play_arrow
In a certain code language, '617' means 'sweet and hot', '735' means 'coffee is sweet' and '263' means "tea is hot'. Which of the following would mean 'coffee is hot'?
A)
731
done
clear
B)
536
done
clear
C)
367
done
clear
D)
753
done
clear
E)
None of these
done
clear
View Answer play_arrow
Directions: Read the following information carefully and answer the questions given below: Ravi is son of Aman's father's sister. Sahil is son of Divya who is mother of Gaurav and grandmother of Aman. Ashok is father of Tanya and grandfather of Ravi. Divya is wife of Ashok.
How is Ravi related to Divya?
A)
Nephew
done
clear
B)
Grandson
done
clear
C)
Son
done
clear
D)
Data inadequate
done
clear
E)
None of these
done
clear
View Answer play_arrow
Directions: Read the following information carefully and answer the questions given below: Ravi is son of Aman's father's sister. Sahil is son of Divya who is mother of Gaurav and grandmother of Aman. Ashok is father of Tanya and grandfather of Ravi. Divya is wife of Ashok.
How is Gaurav's wife related to Tanya?
A)
Niece
done
clear
B)
Sister
done
clear
C)
Sister-in-law
done
clear
D)
Data inadequate
done
clear
E)
None of these
done
clear
View Answer play_arrow
Directions: Study the following information and answer the questions given below it: (i) Kailash, Govind and Harinder are intelligent. (ii) Kailash, Rajesh and Jitendra are hard- working. (iii) Rajesh, Harinder and Jitendra are honest. (iv) Kailash, Govind and Jitendra are ambitious.
Which of the following persons is neither hard-working nor ambitious?
A)
Kailash
done
clear
B)
Govind
done
clear
C)
Harinder
done
clear
D)
Rajesh
done
clear
E)
None of these
done
clear
View Answer play_arrow
Directions: Study the following information and answer the questions given below it: (i) Kailash, Govind and Harinder are intelligent. (ii) Kailash, Rajesh and Jitendra are hard- working. (iii) Rajesh, Harinder and Jitendra are honest. (iv) Kailash, Govind and Jitendra are ambitious.
Which of the following persons is neither honest nor hard-working but is ambitious?
A)
Kailash
done
clear
B)
Govind
done
clear
C)
Rajesh
done
clear
D)
Harinder
done
clear
E)
None of these
done
clear
View Answer play_arrow
Which of the set of the letters when sequentially placed at the gaps in the given letter series shall complete it? __ __N__L__ __O___M___KP__N__L___.
A)
L M O N P K M K O P
done
clear
B)
M O L N P K M O K P
done
clear
C)
P O M K P N L O M K
done
clear
D)
OMPPKNMOLK
done
clear
E)
None of these
done
clear
View Answer play_arrow
Which of the following symbols should be exchanged to make the given equation correct? \[472~\,\times \,326442+38=240\]
A)
\[\div \,\,and\,\,+\]
done
clear
B)
\[\div \,\,and\,\,\times \]
done
clear
C)
\[+\,\,and\,\,\times \]
done
clear
D)
\[+\,\,and\,\,-\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Saroj performing yoga with her head down and leg up. Her face is towards the east. In which direction will her left hand be?
A)
East
done
clear
B)
West
done
clear
C)
North
done
clear
D)
South
done
clear
E)
None of these
done
clear
View Answer play_arrow
Which one of the alternative figures will complete the figure pattern?
A)
done
clear
B)
done
clear
C)
done
clear
D)
done
clear
E)
None of these
done
clear
View Answer play_arrow
How many parallelograms are there in this figure?
A)
49
done
clear
B)
46
done
clear
C)
44
done
clear
D)
39
done
clear
E)
None of these
done
clear
View Answer play_arrow
The system of homogeneous equations
\[tx+(t+1)y+(t-1)z=0\] \[(t+1)x+ty+(t+2)z=0\] \[(t-1)x+(t+2)y+tz=0\] has non-trivial solutions for
A)
exactly three real values of t
done
clear
B)
exactly two real values of t
done
clear
C)
exactly one real value of t
done
clear
D)
infinite number of value of t.
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[x=-\,5\] is a root of \[\left| \begin{matrix} 2x+1 & 4 & 8 \\ 2 & 2x & 2 \\ 7 & 6 & 2x \\ \end{matrix} \right|=0,\] the other two roots are
A)
3, 3.5
done
clear
B)
1, 3.5
done
clear
C)
3, 6
done
clear
D)
2, 6
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[f(x)=\left| \begin{matrix} {{\sec }^{2}}x & 1 & 1 \\ {{\cos }^{2}}x & {{\cos }^{2}}x & \cos e{{c}^{2}}x \\ 1 & {{\cos }^{2}}x & {{\cot }^{2}}x \\ \end{matrix} \right|,\] then
A)
\[\int\limits_{-\pi /4}^{\pi /4}{f\,\,(x)\,\,dx=\frac{1}{16}\,\,(3\pi +8)}\]
done
clear
B)
\[f'\,\,(\pi /2)=0\]
done
clear
C)
Maximum value of f(x) is 1.
done
clear
D)
Minimum value of f(x) is 0.
done
clear
E)
All of these
done
clear
View Answer play_arrow
If \[{{a}_{1}},\]\[{{a}_{2}},\]\[{{a}_{3}},\]____., \[{{a}_{n}}\]is an A.P. with common difference d, then \[\tan \left[ {{\tan }^{-1}}\left( \frac{d}{1+{{a}_{1}}{{a}_{2}}} \right)+{{\tan }^{-1}}\left( \frac{d}{1+{{a}_{2}}{{a}_{3}}} \right)+.....+{{\tan }^{-1}}\left( \frac{d}{1+{{a}_{n-1}}{{a}_{n}}} \right) \right]=\]
A)
\[\frac{(n-1)\,\,d}{{{a}_{1}}+{{a}_{2}}}\]
done
clear
B)
\[\frac{(n-1)\,\,d}{1+{{a}_{1}}{{a}_{n}}}\]
done
clear
C)
\[\frac{nd}{1+{{a}_{1}}{{a}_{n}}}\]
done
clear
D)
\[\frac{{{a}_{n}}-{{a}_{1}}}{{{a}_{n}}+{{a}_{1}}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The rank of the matrix \[\left| \begin{matrix} -1 & 2 & 5 \\ 2 & -4 & a-4 \\ 1 & -2 & a+1 \\ \end{matrix} \right|\] is
A)
2 if \[a=1\]
done
clear
B)
1 if \[a=-6\]
done
clear
C)
1 if \[a=6\]
done
clear
D)
Both (A) and (B)
done
clear
E)
None of these
done
clear
View Answer play_arrow
The equation of a curve is y = f(x). The tangents at (1, f(1)), (2, f(2)) and (3, f(3)) make angles \[\frac{\pi }{6},\]\[\frac{\pi }{3}\] and \[\frac{\pi }{4}\] respectively with the positive direction of the x-axis. Then the value of \[\int_{2}^{3}{f'\,\,(x)\,\,f''\,\,(x)\,\,dx+\int_{2}^{3}{f''\,\,(x)\,dx}}\] is equal to
A)
\[-\frac{1}{\sqrt{3}}\]
done
clear
B)
\[\frac{1}{\sqrt{3}}\]
done
clear
C)
1
done
clear
D)
\[\frac{2}{\sqrt{3}}\]
done
clear
E)
None of these
done
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View Answer play_arrow
Two straight roads \[{{R}_{1}}\] and \[{{R}_{2}}\] diverge from a point A at an angle of\[120{}^\circ \]. Nitin starts walking from point A along \[{{R}_{1}}\]at a uniform speed of 3 km/hr. Prakash starts walking at the same time from A along \[{{R}_{2}}\]at a uniform speed of 2 km/h. They continue walking for 4 hours along their respective roads and reach point B and C on \[{{R}_{1}}\]and \[{{R}_{2}},\] respectively. There is a straight line path connecting B and C. Then Nitin returns to point A after walking along the line segments BC and CA. Prakash also returns to A after walking along line segments CB and BA. Their speeds remain unchanged. The time interval (in hours) between Nitin's and Prakash's return to the point A is:
A)
\[\frac{10\sqrt{19+26}}{3}\]
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B)
\[\frac{2\sqrt{19}+10}{3}\]
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C)
\[\frac{\sqrt{19}+26}{3}\]
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D)
\[\frac{\sqrt{19}+10}{3}\]
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E)
None of these
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Let a, b, c be non-zero real number such that\[\int_{0}^{1}{(1+{{\cos }^{8}}x)\,\,(a{{x}^{2}}+bx+c)\,\,dx}\]\[=\int_{0}^{2}{(1+{{\cos }^{8}}x)\,\,(a{{x}^{2}}+bx+c)\,\,dx}\]. Then the quadratic equation \[a{{x}^{2}}+bx+c=0\]has
A)
no root in (0, 2)
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B)
at least one root in (1, 2)
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C)
a double root (0, 2)
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D)
All of these
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E)
None of these
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If \[f\,\,(x)=\left\{ \begin{matrix} x\frac{{{e}^{(1/x)}}-{{e}^{(-1/x)}}}{{{e}^{(1/x)}}+{{e}^{(-1/x)}}},\,\,\,\,\,\,\,x\ne 0 \\ 0,\,\,\,\,\,\,x=0 \\ \end{matrix} \right.,\] then which of the following in true
A)
f is continuous and differentiable at every point
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B)
f is continuous at every point but is not differentiable
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C)
f is differentiable at every point
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D)
f is differentiable only at the origin
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E)
None of these
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If \[f\,\,(x)=\left\{ \begin{matrix} -{{x}^{2}},when\,x\underline{<}0 \\ 5x-4,when\,0<x\underline{<}1 \\ 4{{x}^{2}}-3x,when\,1<x<2 \\ 3x+4,when\,x\underline{>}2 \\ \end{matrix} \right.\,\,,\text{then}\]
A)
f (x) is continuous at x = 0
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B)
f (x) is continuous at x = 2
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C)
f (x) is discontinuous at x = 1
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D)
f (x) is discontinuous at x = 2
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E)
None of these
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