If \[A=\left\{ (x,y):{{x}^{2}}+{{y}^{2}}=25 \right\}\] and\[B=\left\{ (x,y):{{x}^{2}}+9{{y}^{2}}=144 \right\},\] then \[A\bigcap B\]contains
A)
One point
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B)
Three points
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C)
Two points
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D)
Four points
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E)
None of these
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On the set N of all natural numbers define the relation R by aRb if and only if the G.C.D. of a and b is 2, then R is
A)
reflexive, but not symmetric
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B)
symmetric only
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C)
reflexive and transitive
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D)
reflexive, symmetric and transitive
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E)
None of these
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The value of \[(A\bigcup B\bigcup C)\bigcap (A\bigcap {{B}^{C}}\bigcap {{C}^{C}})\bigcap {{C}^{C}},\] is
A)
\[B\bigcap {{C}^{C}}\]
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B)
\[{{B}^{c}}\bigcap {{C}^{C}}\]
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C)
\[B\bigcap C\]
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D)
\[(A\bigcap B\bigcap C)\]
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E)
None of these
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The rational number, which equals the number \[2.\overline{357}\]with recurring decimal is
A)
\[\frac{2355}{1001}\]
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B)
\[\frac{2355}{888}\]
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C)
\[\frac{2355}{999}\]
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D)
\[\frac{2350}{888}\]
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E)
None of these
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Sum of the first n terms of the series\[\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.......\] is equal to
A)
\[{{2}^{n}}-n-1\]
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B)
\[1-{{2}^{-n}}\]
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C)
\[n+{{2}^{-n}}-1\]
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D)
\[{{2}^{n+1}}\]
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E)
None of these
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Suppose a, b, c are in A. P and \[{{a}^{2}},\]\[{{b}^{2}},\]\[{{c}^{2}}\]are in G.P. If a< b< c and \[a+b+c=\frac{3}{2},\] then the value of a is
A)
\[\frac{1}{2\sqrt{2}}\]
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B)
\[\frac{1}{2\sqrt{3}}\]
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C)
\[\frac{1}{2}-\frac{1}{\sqrt{3}}\]
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D)
\[\frac{1}{2}-\frac{1}{\sqrt{2}}\]
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E)
None of these
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The equation of the circle passing through (1, 1) and the points of intersection of\[{{x}^{2}}+{{y}^{2}}+13x-3y=0\] and \[2{{x}^{2}}+2{{y}^{2}}+4x-7y-25=0\] is
A)
\[4{{x}^{2}}+4{{y}^{2}}-30x-10y-25=0\]
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B)
\[4{{x}^{2}}+4{{y}^{2}}+30x-13y-25=0\]
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C)
\[4{{x}^{2}}+4{{y}^{2}}-17x-10y+25=0\]
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D)
All of these
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E)
None of these
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If the circles \[{{x}^{2}}+{{y}^{2}}+2x+2ky+6=0,\]\[{{x}^{2}}+{{y}^{2}}+2ky+k=0\] intersect orthogonally, then k is.
A)
\[2\,\,or-\frac{3}{2}\]
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B)
\[-2\,\,or-\frac{3}{2}\]
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C)
\[2\,\,or\frac{3}{2}\]
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D)
\[-2\,\,or\frac{3}{2}\]
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E)
None of these
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If A and B are two events such that \[P(A)>0,\] and \[P(B)\ne 1,\] then \[P\left( \frac{\overline{A}}{\overline{B}} \right)\] is equal to
A)
\[1-P\,\,\left( \frac{A}{B} \right)\]
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B)
\[1-P\,\,\left( \frac{\overline{A}}{B} \right)\]
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C)
\[\frac{1-P\,\,(A\cup B)}{P\,\,(\overline{B})}\]
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D)
\[\frac{P\,\,(\overline{A})}{P\,\,(\overline{B})}\]
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E)
None of these
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Let A, B, C be three mutually independent events. Consider the two statements \[{{S}_{1}}\]and \[{{S}_{2}}\]
\[{{S}_{1}}:A\]and \[B\cup C\]are independent \[{{S}_{2}}:A\]and \[B\cap C\] are independent
Then,
A)
Both \[{{S}_{1}}\]and \[{{S}_{2}}\]are true
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B)
Only \[{{S}_{1}}\] is true
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C)
Only \[{{S}_{2}}\] is true
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D)
Neither \[{{S}_{1}}\] nor \[{{S}_{2}}\]is true
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E)
None of these
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If \[f\,\,(x)=\left| \begin{matrix} 1 & x & x+1 \\ 2\,\,x & x\,\,(x-1) & (x+1)\,\,x \\ 3x\,\,(x-1) & x(x-1)\,\,(x-2) & (x+1)\,\,\times \,\,(x-1) \\ \end{matrix} \right|\]then f(100) is equal to
A)
0
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B)
1
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C)
100
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D)
\[-100\]
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E)
None of these
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If A and B are square matrices of equal degree, then which one is correct among the following?
A)
\[A+B=B+A\]
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B)
\[A+B=A-B\]
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C)
\[AB=B-A\]
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D)
\[AB=BA\]
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E)
None of these
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If \[A=\left[ \begin{matrix} a & b \\ b & a \\ \end{matrix} \right]\]and \[{{A}^{2}}=\left[ \begin{matrix} \alpha & \beta \\ \beta & \alpha \\ \end{matrix} \right],\]then
A)
\[\alpha ={{a}^{2}}+{{b}^{2}},\]\[\beta =ab\]
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B)
\[\alpha =2ab,\]\[\beta ={{a}^{2}}+{{b}^{2}}\]
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C)
\[\alpha ={{a}^{2}}+{{b}^{2}},\]\[\beta ={{a}^{2}}-{{b}^{2}}\]
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D)
\[\alpha ={{a}^{2}}+{{b}^{2}},\]\[\beta =2ab\]
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E)
None of these
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Let \[f:R\to R\] be any function. Define \[g:R\to R\]by \[g(x)=\,|f(x)|\] for all x. Then g is
A)
onto if f is onto
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B)
one-one if f is one-one
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C)
continuous if f is continuous
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D)
differentiable if f is differentiable.
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E)
None of these
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Let \[E=\{1,\,2,\,3,\,4\}\] and \[F=\{\,1,\,2\}\]. Then the number of onto functions from E to F is
A)
14
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B)
16
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C)
12
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D)
8
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E)
None of these
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If \[y={{(\sin x)}^{\tan x}},\] then \[\frac{dy}{dx}\] is equal to
A)
\[{{(\sin \,\,x)}^{\tan x}}(1+{{\sec }^{2}}x\,\,\log \,\,\sin \,\,x)\]
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B)
\[\tan \,\,x\,\,{{(\sin \,\,x)}^{\tan x-1}}\cos \,\,x\]
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C)
\[{{(\sin x)}^{\tan x}}{{\sec }^{2}}x\,\,\log \,\,\sin \,\,x\]
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D)
\[\tan \,x\,{{(\sin \,x)}^{\tan x-1}}\]
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E)
None of these
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Let \[f:(0,\infty )\to R\] and \[F(x)=\int\limits_{0}^{x}{f(t)\,dt.}\] If \[F({{x}^{2}})={{x}^{2}}(1+x),\] then f(4) equals
A)
\[\frac{5}{4}\]
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B)
\[7\]
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C)
\[4\]
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D)
\[2\]
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E)
None of these
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If \[{{x}^{2}}+{{y}^{2}}=1,\] then
A)
\[yy''-2\,\,{{(y')}^{2}}+1=0\]
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B)
\[yy''+{{(y')}^{2}}+1=0\]
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C)
\[yy''+{{(y')}^{2}}-1=0\]
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D)
\[yy''+2\,\,{{(y')}^{2}}+1=0\]
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E)
None of these
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The value of the integral \[\int{\frac{{{\cos }^{3}}x+{{\cos }^{5}}x}{{{\sin }^{2}}x+{{\sin }^{4}}x}}\,\,dx\] is
A)
\[\sin x-6{{\tan }^{-1}}(\sin x)+c\]
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B)
\[\sin x-2{{(\sin x)}^{-1}}+c\]
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C)
\[\sin x-2{{(\sin x)}^{-1}}-{{6}^{\tan -1}}(\sin x)+c\]
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D)
\[\sin x-2{{(\sin x)}^{-1}}+{{5}^{\tan -1}}(\sin x)+c\]
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E)
None of these
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If \[\int\limits_{\sin x}^{1}{{{t}^{2}}f(t)\,dt=1-\sin x,}\] then \[f\left( \frac{1}{\sqrt{3}} \right)\] is
A)
\[\frac{1}{3}\]
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B)
\[\frac{1}{\sqrt{3}}\]
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C)
\[3\]
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D)
\[\sqrt{3}\]
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E)
None of these
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The value of the integral \[\int\limits_{0}^{\pi /2}{\frac{\sqrt{\cot x}}{\sqrt{\cot x}+\sqrt{\tan x}}\,\,}dx\] is
A)
\[\pi /4\]
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B)
\[\pi /2\]
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C)
\[\pi \]
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D)
All of these
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E)
None of these
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The area bounded by the curves \[y=\,|x|-1\] and \[y=-|x|+1\] is
A)
\[1\]
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B)
\[2\]
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C)
\[2\sqrt{2}\]
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D)
4
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E)
None of these
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The area bounded by the parabolas \[y=(x+{{1}^{2}})\] and \[y=(x-{{1}^{2}})\] and the line \[y=1/4\] is
A)
\[4\,sq.\,\,units\]
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B)
\[\frac{1}{6}sq.\,\,units\]
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C)
\[\frac{4}{3}sq.\,\,units\]
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D)
\[\frac{1}{3}sq.\,\,units\]
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E)
None of these
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If for a real number y, [y] is the greatest integer less than or equal to y, then the value of the integral \[\int\limits_{\pi /2}^{3\pi /2}{[2\sin x]\,\,dx}\] is
A)
\[-\pi \]
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B)
\[0\]
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C)
\[-\frac{\pi }{2}\]
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D)
\[\frac{\pi }{2}\]
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E)
None of these
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A solution of the differential equation \[{{\left( \frac{dy}{dx} \right)}^{2}}-x\frac{dy}{dx}+y=0\] is
A)
\[y=2\]
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B)
\[y=2x\]
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C)
\[y=2x-4\]
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D)
\[y=2{{x}^{2}}-4\]
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E)
None of these
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Statement I: The number of ways distributing 10 identical balls in 4 distinct boxes such that no box is empty is\[{}^{9}{{C}_{3}}\] Statement II: The number of ways of choosing any 3 places from 9 different places is\[{}^{9}{{C}_{3}}\].
A)
Statement I is true, Statement II is true Statement II is not a correct explanation for Statement I.
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B)
Statement I is true, Statement II is false
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C)
Statement I is false, Statement II is true
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D)
Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
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E)
None of these
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If \[y=y(x)\] and it follows the relation\[x\,\cos \,y+y\,\cos \,x=\pi ,\] then\[y''(0)\]
A)
\[1\]
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B)
\[-1\]
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C)
\[\pi \]
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D)
\[-\pi \]
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E)
None of these
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The scalar \[\overrightarrow{A}.(\overrightarrow{B}+\overrightarrow{C})\times (\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C})\] equals:
A)
\[0\]
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B)
\[[\overrightarrow{A}\,\,\overrightarrow{B}\,\,\overrightarrow{C}]+[\overrightarrow{B}\,\,\overrightarrow{C}\,\,\overrightarrow{A}]\]
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C)
\[[\overrightarrow{A}\,\,\overrightarrow{B}\,\,\overrightarrow{C}]\]
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D)
All of these
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E)
None of these
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Let a, b, c be distinct non-negative numbers. If the vectors and \[a\hat{i}+a\hat{j}+c\hat{k},\] \[\hat{i}+\hat{k}\] and \[c\hat{i}+c\hat{j}+b\hat{k}\] lie in a plane, then c is
A)
the Arithmetic Mean of a and b
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B)
the Geometric Mean of a and b
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C)
the Harmonic Mean of a and b
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D)
equal to zero
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E)
None of these
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If \[\overrightarrow{a},\] \[\overrightarrow{b}\]and \[\overrightarrow{c}\]are unit coplanar vectors, then the scalar triple product \[[2\overrightarrow{a}-\overrightarrow{b},2\overrightarrow{b}-\overrightarrow{c},2\overrightarrow{c}-\overrightarrow{a}]=\]
A)
\[0\]
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B)
\[1\]
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C)
\[-\sqrt{3}\]
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D)
\[\sqrt{3}\]
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E)
None of these
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The equation of tangent to the curve \[y=2\cos x\,\,at\,\,x=\frac{\pi }{4}\] is
A)
\[y-\sqrt{2}=2\sqrt{2}\,\,\left( x-\frac{\pi }{4} \right)\]
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B)
\[y+\sqrt{2}=\sqrt{2}\,\,\left( x-\frac{\pi }{4} \right)\]
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C)
\[y-\sqrt{2}=-\sqrt{2}\,\,\left( x-\frac{\pi }{4} \right)\]
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D)
\[y-\sqrt{2}=\sqrt{2}\,\,\left( x-\frac{\pi }{4} \right)\]
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E)
None of these
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The order and degree of the differential equation, \[y=x\frac{dy}{dx}+\sqrt{{{a}^{2}}{{\left( \frac{dy}{dx} \right)}^{2}}+{{b}^{2}}}\] are
A)
(1, 2)
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B)
(2, 1)
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C)
(1, 1)
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D)
(2, 2)
done
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E)
None of these
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The shortest distance between lines \[\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\] and \[\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\]is
A)
\[\sqrt{30}\]
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B)
\[2\sqrt{30}\]
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C)
\[5\sqrt{30}\]
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D)
\[3\sqrt{30}\]
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E)
None of these
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Local maximum value of the function \[\frac{\log x}{x}\] is
A)
\[e\]
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B)
\[1\]
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C)
\[\frac{1}{e}\]
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D)
\[2e\]
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E)
None of these
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For a moderately skewed distribution, quartile deviation and the standard deviation are related by
A)
\[S.D=\frac{2}{3}Q.D\]
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B)
\[S.D=\frac{3}{2}Q.D\]
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C)
\[S.D=\frac{3}{4}Q.D\]
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D)
\[S.D=\frac{4}{3}Q.D\]
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E)
None of these
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Two lines \[\frac{x-{{x}_{1}}}{{{I}_{i}}}=\frac{y-{{y}_{1}}}{{{m}_{i}}}=\frac{z-{{z}_{1}}}{{{n}_{i}}}(i=1,\,\,2)\] are perpendicular to each other if their direction ratios satisfy
A)
\[{{I}_{i}}={{m}_{i}}={{n}_{i}}\]
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B)
\[{{I}_{1}}{{I}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}=0\]
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C)
\[\frac{{{I}_{1}}}{{{I}_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{{{n}_{1}}}{{{n}_{2}}}=0\]
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D)
All of these
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E)
None of these
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The equation of the plane through (1, 2, 3) and parallel to the plane \[2x+3y-4z=0\] is
A)
\[2x+3y+4z+4=4\]
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B)
\[2x+3y+4z+4=0\]
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C)
\[2x-3y+4z+4=0\]
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D)
\[2x+3y-4z+4=0\]
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E)
None of these
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Which of the following Venn diagram corresponds to the statement "All mothers are women"? (M is the set of all mothers, W is the set of all women)
A)
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B)
done
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C)
done
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D)
done
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E)
None of these
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If for two functions g and f, gof of both injective and surjective, then which of the following is true?
A)
g and f should be injective and surjective
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B)
g should be injective and surjective
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C)
f should be injective and surjective
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D)
None of them may be surjective and injective
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E)
None of these
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The function \[f(x)=\left\{ \begin{matrix} x,if\,\,\,\,\,\,0\le x\le 1 \\ 1,if\,\,\,\,\,\,1<x\le 2 \\ \end{matrix} \right.\] is
A)
Continuous at all \[x,\] \[0\le x\le 2\] and differentiate at all x, except x = 1 in the interval [0, 2]
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B)
Continuous and differentiable at all x in [0, 2]
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C)
Not continuous at any point in [0, 2]
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D)
Not differentiable at any point [0, 2]
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E)
None of these
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