Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), \[\left( 8,-\,2 \right)\] and \[\left( 2,-\,2 \right)\].
A)
(5, 5)
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B)
(7, 2)
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C)
(5, 2)
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D)
(2, 5)
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E)
None of these
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Find the equation of the circle with centre \[\left( 4,-\,5 \right)\] and radius\[\sqrt{53}\].
A)
\[{{x}^{2}}+{{y}^{2}}-5x+10y=10\]
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B)
\[{{x}^{2}}+{{y}^{2}}-6x+12y=12\]
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C)
\[{{x}^{2}}+{{y}^{2}}-7x+14y=12\]
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D)
\[{{x}^{2}}+{{y}^{2}}-8x+10y=12\]
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E)
None of these
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Find the angle between the lines \[3y-\sqrt{3}x-12=0\] and \[y-\sqrt{3}x+9=0\]
A)
\[30{}^\circ \]
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B)
\[90{}^\circ \]
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C)
\[150{}^\circ \]
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D)
(A) and (C) Both
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E)
None of these
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If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13, 15} C = {11, 13, 15, 17}, D = {17, 19}, then find the value of \[(A\bigcup D)\,\,\bigcap \,\,(B\bigcup C)\].
A)
{7, 9, 11, 15}
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B)
{7, 9, 11, 17}
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C)
{5, 7, 11, 17, 19}
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D)
{3, 5, 11, 19}
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E)
None of these
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In a group of students 200 know Hindi, 125 know English and 75 know both. Each of the students knows either Hindi or English. How many students are there in the group?
A)
125
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B)
225
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C)
250
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D)
300
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E)
None of these
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If the set A and B are defind as \[A=\left\{ (x,y):y=\frac{1}{x},0\ne x\in R \right\}\] \[B=\{(x,y):-x,\,\,x\in R\}\] then
A)
\[A\bigcap B=A\]
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B)
\[A\bigcap B=B\]
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C)
\[A\bigcap B=\varnothing \]
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D)
All of these
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E)
None of these
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If A, B, C be three sets such that \[A\bigcup B=A\bigcup C\] and \[A\bigcap B=A\bigcap C,\] then
A)
A = B
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B)
B = C
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C)
A = C
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D)
A = B = C
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E)
None of these
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If \[\log {}_{7}2=m,\] then \[\log {}_{49}28\] is equal to
A)
\[2\,\,\left( 1+2m \right)\]
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B)
\[\frac{1+2m}{2}\]
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C)
\[\frac{2}{1+2m}\]
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D)
\[1+m\]
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E)
None of these
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If \[x={{2}^{1/3}}-{{2}^{-1/3}},\] then \[2{{x}^{3}}+6x=\]
A)
0
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B)
2
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C)
3
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D)
4
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E)
None of these
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If \[\sin \theta +\cos \theta =1,\] than the general value of \[\theta \] is.
A)
\[2n\,\pi \]
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B)
\[n\,\pi +{{(-\,1)}^{n}}\,\,\frac{\pi }{4}\,\,-\,\,\frac{\pi }{4}\]
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C)
\[2n\,\pi +\frac{\pi }{2}\]
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D)
\[2n\,\pi +\,\,\frac{2\pi }{2}\]
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E)
None of these
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The most general value of \[\theta \] satisfying the equations \[\sin \theta =\sin \alpha \] and \[\cos \theta =\cos \alpha \] is.
A)
\[2n\,\pi +\alpha \]
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B)
\[2n\,\pi +\pi /4\]
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C)
\[n\,\pi +\alpha \]
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D)
\[n\,\pi -\alpha \]
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E)
None of these
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General solution of \[ten5\,\theta =\cot 2\theta \] is.
A)
\[\theta =\frac{n\pi }{7}+\frac{\pi }{14}\]
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B)
\[\theta =\frac{n\pi }{7}+\frac{\pi }{5}\]
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C)
\[\theta =\frac{n\pi }{7}+\frac{\pi }{2}\]
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D)
\[\theta =n\,\pi +{{(-\,1)}^{n}}\,\,\frac{\pi }{3},n\in z\]
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E)
None of these
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The equation \[sin\,x+cos\,x=2\] has
A)
One solution
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B)
Two solution
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C)
Infinite number of solutions.
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D)
No solutions
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E)
None of these
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A person standing on the bank of a river observes that the angle subtended by a tree on the opposite bank is \[60{}^\circ \]. When he steps back 40 metres from the bank, he finds the angle to be\[30{}^\circ \]. The breadth of the river is.
A)
20 m
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B)
30 m
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C)
40 m
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D)
60 m
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E)
None of these
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If \[1=\sqrt{-1},\] then \[1+{{i}^{2}}+{{i}^{3}}-{{i}^{6}}+{{i}^{8}}\] is equal to
A)
\[2+i\]
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B)
\[2i\]
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C)
\[3+i\]
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D)
\[-\,1\]
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E)
None of these
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If \[x+\frac{1}{x}=2\cos \theta ,\] then x is equal to
A)
\[\cos \theta +i\,\,\sin \theta \]
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B)
\[\cos \theta -i\,\,\sin \theta \]
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C)
\[\cos \theta \pm i\,\,\sin \theta \]
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D)
\[\sin \theta \,\,\pm i\,\,\cos \theta \]
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E)
None of these
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If smallest positive integer n for which \[{{(1+i\,)}^{2n}}={{(1+i\,)}^{2n}}\] is
A)
1
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B)
2
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C)
3
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D)
4
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E)
None of these
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If \[\left( a+i\,b \right)\,\,\left( c+i\,d \right)\,\,\left( e+i\,f \right)\,\,\left( g+i\,h \right)=A+i\,B,\] then \[({{a}^{2}}+{{b}^{2}})\,\,({{c}^{2}}+{{d}^{2}})\,\,({{e}^{2}}+{{f}^{2}})\,\,({{g}^{2}}+{{h}^{2}})=\]
A)
\[{{A}^{2}}+{{B}^{2}}\]
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B)
\[{{A}^{2}}-{{B}^{2}}\]
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C)
\[{{A}^{2}}\]
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D)
\[{{B}^{2}}\]
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E)
None of these
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If \[Z=\frac{1-i\,\sqrt{3}}{1+i\,\sqrt{3}}\] then arg (z)=
A)
\[60{}^\circ \]
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B)
\[120{}^\circ \]
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C)
\[240{}^\circ \]
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D)
\[300{}^\circ \]
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E)
None of these
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If the points \[{{P}_{1}}\] and \[{{P}_{2}}\] represent two complex numbers \[{{Z}_{1}}\] and \[{{Z}_{2}},\] then the point \[{{P}_{3}}\] represents the number.
A)
\[{{Z}_{1}}+{{Z}_{2}}\]
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B)
\[{{Z}_{1}}-{{Z}_{2}}\]
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C)
\[{{Z}_{1}}\times {{Z}_{2}}\]
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D)
\[{{Z}_{1}}\div {{Z}_{2}}\]
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E)
None of these
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If \[x+\frac{1}{x}=2\cos \theta ,\] then \[{{x}^{n}}+\frac{1}{{{x}^{n}}}\] is equal to.
A)
\[2\cos \,n\,\theta \]
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B)
\[2\sin \,n\,\theta \]
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C)
\[\cos \,n\,\theta \]
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D)
\[\sin \,n\,\theta \]
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E)
None of these
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If \[{{p}^{th}}\] term of an A.P. be q and \[{{q}^{th}}\] term be p, than its \[{{r}^{th}}\]term will be.
A)
\[p+q+r\]
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B)
\[p+q-r\]
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C)
\[p+rq\]
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D)
\[pq-r\]
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E)
None of these
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The sum of the series \[\frac{1}{2}+\frac{1}{3}+\frac{1}{6}+......\] to 9 terms is
A)
1
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B)
\[-\frac{1}{2}\]
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C)
\[-\frac{3}{2}\]
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D)
\[-\frac{5}{6}\]
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E)
None of these
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If \[{{(p+q)}^{th}}\] term of a G.P. be m and \[{{(p-q)}^{th}}\] term be n, then the \[{{p}^{th}}\]term will be.
A)
\[m/n\]
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B)
\[\sqrt{mn}\]
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C)
\[mn\]
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D)
\[O\]
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E)
None of these
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The value of \[{{4}^{1/3}}{{.4}^{1/9}}{{.4}^{1/27}}......\] unto \[\infty \] is
A)
2
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B)
3
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C)
4
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D)
9
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E)
None of these
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If the roots of the equation \[a{{x}^{2}}+bx+c=0\] be \[\alpha \]and \[\,\beta ,\] then the roots of the equation \[c{{x}^{2}}+bx+a=0\] are
A)
\[-\,a,-\,b\]
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B)
\[a,\frac{1}{\beta }\]
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C)
\[\frac{1}{\alpha },\frac{1}{\beta }\]
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D)
\[\frac{1}{\alpha },b\]
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E)
All of these
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If the roots of \[{{x}^{2}}-bx+c=0\] are two consecutive integers, then \[{{b}^{2}}-4c\] is
A)
1
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B)
2
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C)
3
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D)
4
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E)
All of these
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If how many ways corn 5 prizes be distributed among four students when every student can take one or more prizes
A)
1024
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B)
625
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C)
120
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D)
720
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E)
None of these
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If how many ways 7mean and 7 woman can be seated around a round table such that no two woman can sit together?
A)
\[{{(7!)}^{2}}\]
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B)
\[7!\,\times 6!\]
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C)
\[{{(6!)}^{2}}\]
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D)
\[7!\]
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E)
None of these
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If \[^{43}{{C}_{r-6}}={{\,}^{43}}{{C}_{3r+1}},\] then the value of r is
A)
6
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B)
8
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C)
10
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D)
12
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E)
None of these
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The value of \[{{\left( \sqrt{5}+1 \right)}^{5}}-{{\left( \sqrt{5}-1 \right)}^{5}}\] is
A)
252
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B)
352
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C)
452
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D)
532
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E)
None of these
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In the expansion of \[{{\left( x-\frac{1}{x} \right)}^{6}},\] the constant term is
A)
\[-\,20\]
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B)
20
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C)
30
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D)
\[-\,30\]
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E)
None of these
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The equation of the line whose slope is 3 and which cuts off an intercept 3 from the positive \[x-ax\] is
A)
\[y=3x9\]
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B)
\[y=3x+3\]
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C)
\[y=3x+9~\]
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D)
\[y=3x+12\]
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E)
None of these
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The acute angle between the lines y = 3 and \[y=\sqrt{3}x+9\] is
A)
\[30{}^\circ \]
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B)
\[45{}^\circ \]
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C)
\[60{}^\circ \]
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D)
\[90{}^\circ \]
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E)
None of these
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Radius of the circle \[{{x}^{2}}+{{y}^{2}}+2x\,\cos \theta +2y\,\sin \theta -8=0\] is.
A)
1
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B)
3
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C)
\[2\,\sqrt{3}\]
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D)
\[\sqrt{10}\]
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E)
None of these
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The line \[y=mx+1\] is a tangent to the parabola\[{{y}^{2}}=4x,\] if.
A)
m = 1
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B)
m = 2
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C)
m = 4
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D)
m = 3
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E)
None of these
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In the ellipse, minor axis is 8 and eccentricity is \[\frac{\sqrt{5}}{3}\]. Then major axis is
A)
6
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B)
10
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C)
12
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D)
16
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E)
None of these
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\[\underset{x\to o}{\mathop{Lim}}\,\,\,\left( \frac{{{a}^{x}}-{{b}^{x}}}{x} \right)\]
A)
\[\log \left( \frac{b}{a} \right)\]
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B)
\[\log \left( \frac{a}{b} \right)\]
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C)
\[\frac{a}{b}\]
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D)
\[\log \,{{a}^{b}}\]
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E)
None of these
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The chances of throwing a total of 3 or 5 or 11 with two dice is
A)
\[\frac{5}{36}\]
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B)
\[\frac{2}{9}\]
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C)
\[\frac{1}{9}\]
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D)
\[\frac{19}{39}\]
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E)
None of these
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If A and B are any two events, then the probability that exactly one of then occur is
A)
\[P\,(A)+P\,(B)-P\,(A\,\,\cap B)\]
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B)
\[P\,(A)+P\,(B)-2\,P\,(A\,\,\bigcap \,\,B)\]
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C)
\[P\,(A)+P\,(B)-P\,(A\,\,\bigcup \,\,B)\]
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D)
\[P\,(A)+P\,(B)-2\,P\,(A\,\,\bigcup \,\,B)\]
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E)
None of these
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