If \[X=\{{{4}^{n}}-3n-1:n\in N\}\] and \[Y=\{9\,\,(n-1):n\in N\}\], then \[X\bigcup Y\] is equal to
A)
X
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B)
Y
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C)
N
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D)
All of these
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E)
None of these
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If A, B and C are any three sets, then \[A\,\,\times \,\,(B\,\,\bigcup \,\,C)\]is equal to
A)
\[(A\,\,\times B)\,\,\bigcup \,\,(A\,\,\times C)\]
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B)
\[(A\,\,\bigcup \,\,B)\,\,\times \,\,(A\,\,\bigcap \,\,C)\]
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C)
\[(A\,\,\times B)\,\,\bigcap \,\,(A\,\,\times \,\,C)\]
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D)
All of these
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E)
None of these
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In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and chemistry, 7 in physics and chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is
A)
6
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B)
9
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C)
15
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D)
22
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E)
None of these
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If \[n\,\,\left( A \right)=3,\] \[n\,\,\left( B \right)=6\] and \[A\subseteq B,\] then the number of elements in \[A\bigcup B\]is equal to
A)
3
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B)
9
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C)
6
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D)
18
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E)
None of these
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The number of non-empty subsets of the set {1, 2, 3, 4} is
A)
15
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B)
14
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C)
16
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D)
17
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E)
None of these
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\[{{7}^{2{{\log }_{{{7}^{5}}}}}}\]is equal to
A)
\[{{\log }_{7}}35\]
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B)
5
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C)
25
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D)
\[{{\log }_{7}}25\]
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E)
None of these
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If \[\frac{3x+4}{{{x}^{2}}-3x+2}=\frac{A}{x-2}-\frac{B}{x-1},\] then(A,B)=
A)
(7, 10)
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B)
(10, 7)
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C)
\[\left( 10,-\,7 \right)\]
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D)
\[\left( -\,10,7 \right)\]
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E)
None of these
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If n is a positive integer, then \[{{\left( \frac{1+i}{1-i} \right)}^{4n+1}}=\]
A)
1
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B)
\[-\,1\]
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C)
i
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D)
\[-\,i\]
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E)
None of these
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If \[\frac{c+i}{c-i}=a+i\,b,\] where a, b, c are real, then \[{{a}^{2}}+{{b}^{2}}=\]
A)
1
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B)
\[-\,1\]
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C)
\[{{c}^{2}}\]
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D)
\[-\,{{c}^{2}}\]
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E)
None of these
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If x, y, z are in A.P. and \[{{\tan }^{-1}}x,\] \[{{\tan }^{-1}}y\] and \[{{\tan }^{-1}}z\] are also in A.P., then
A)
\[x=y=z\]
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B)
\[x=y=-\,z\]
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C)
\[x=1,\,\,y=2,\,\,z=3\]
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D)
\[x=2,\text{ }y=4,\text{ }z=6\]
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E)
None of these
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The sums of n terms of two arithmetic series are in the ratio \[2n+3:6n+5,\] then the ratio of their 13th terms is
A)
53 : 155
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B)
27 : 77
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C)
29 : 83
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D)
31 : 89
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E)
None of these
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If the product of three consecutive terms of G.P. is 216 and the sum of products of pair- wise terms is 156, then the numbers will be
A)
1, 3, 9
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B)
2, 6, 18
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C)
3, 9, 27
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D)
2, 4, 8
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E)
None of these
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\[\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+........+.......\frac{1}{n.\,\,(n+1)}\] equals
A)
\[\frac{1}{n\,(n+1)}\]
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B)
\[\frac{n}{n+1}\]
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C)
\[\frac{2\,n}{n+1}\]
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D)
\[\frac{2n}{n\,(n+1)}\]
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E)
None of these
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The equation \[\sqrt{{{x}^{2}}-4x+3}+\sqrt{{{x}^{2}}-9}\] \[=\sqrt{4{{x}^{2}}-14x+6}\]
A)
No solution
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B)
One solution
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C)
Two solutions
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D)
More than two solutions
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E)
None of these
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If \[\alpha ,\] \[\beta \] and \[\gamma \] are the roots of the equation\[{{x}^{3}}-8x+8=0,\] then \[\sum{{{\alpha }^{2}}}\] and \[\sum{\frac{1}{\alpha \beta }}\] are respectively
A)
0 and 6
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B)
16 and 8
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C)
\[-\,16\text{ }and\text{ }0\]
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D)
16 and 0
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E)
None of these
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In how many ways can mn letters be posted in n letter-boxes?
A)
\[{{(mn)}^{n}}\]
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B)
\[{{m}^{mn}}\]
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C)
\[{{n}^{mn}}\]
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D)
All of these
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E)
None of these
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\[{}^{47}{{C}_{4}}+\sum\limits_{r=1}^{5}{^{52-r}}{{C}_{3}}=\]
A)
\[{}^{47}{{C}_{6}}\]
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B)
\[{}^{52}{{C}_{5}}\]
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C)
\[{}^{52}{{C}_{4}}\]
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D)
All of these
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E)
None of these
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If the coefficient of \[{{r}^{th}}\]term and \[{{(r+4)}^{th}}\]term are equal in the expansion of \[{{(1+x)}^{20}},\] then the value of r will be
A)
7
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B)
8
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C)
9
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D)
10
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E)
None of these
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If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+......+{{C}_{n}}{{x}^{n}},\] then the value of \[{{C}_{0}}+2{{C}_{1}}x+3{{C}_{2}}+......+(n+1){{C}_{n}}\]will be
A)
\[(n+2){{2}^{n-1}}\]
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B)
\[(n+1){{2}^{n}}\]
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C)
\[(n+1){{2}^{n-1}}\]
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D)
\[(n+2){{2}^{n}}\]
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E)
None of these
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Which of the following relations is possible?
A)
\[\sin \theta =\frac{5}{3}\]
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B)
\[\tan 1125{}^\circ =1\]
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C)
\[\cos \theta =\frac{1+{{p}^{2}}}{1-{{p}^{2}}},\,\,(P\ne \pm \,1)\]
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D)
\[\sec \theta =\frac{1}{2}\]
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E)
None of these
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If \[\sec \theta =m\] and \[\tan \theta =n\], then \[\frac{1}{m}\left[ (m+n)+\frac{1}{m+n} \right]=\]
A)
2
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B)
2m
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C)
2n
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D)
mn
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E)
None of these
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If \[\sin \theta +\cos ec\theta =2\], then \[{{\sin }^{2}}\theta +\cos e{{c}^{2}}\theta =\]
A)
1
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B)
3
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C)
2
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D)
4
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E)
None of these
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The value of \[\cos 12{}^\circ +\cos 84{}^\circ +\cos 156{}^\circ +\cos 132{}^\circ \] is
A)
1
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B)
\[\frac{1}{2}\]
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C)
\[-\frac{1}{2}\]
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D)
\[\frac{1}{8}\]
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E)
None of these
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Which of the following number(s) is/are rational?
A)
\[\sin 15{}^\circ \]
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B)
\[\cos 15{}^\circ \]
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C)
\[sin15{}^\circ \cos 15{}^\circ \]
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D)
\[sin15{}^\circ \cos 75{}^\circ \]
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E)
None of these
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The mid-point of the line joining the points \[\left( -\,10,8 \right)\] and \[\left( -\,6,12 \right)\] divides the line joining the points \[\left( 4,-\,2 \right)\] and \[\left( -\,2,4 \right)\] in the ratio of:
A)
1 : 2 internally
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B)
1 : 2 externally
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C)
2 : 1 internally
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D)
2 : 1 externally
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E)
None of these
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The equation of the straight line passing through the point (3, 2) and perpendicular to the line y = x is
A)
\[x-y=5\]
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B)
\[x+y=5\]
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C)
\[x+y=\text{1}\]
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D)
\[xy=\text{1}\]
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E)
None of these
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The line which is parallel to x-axis and crosses the curve \[y=\sqrt{x}\]y at an angle of \[45{}^\circ \] is equal to
A)
\[x=\frac{1}{4}\]
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B)
\[y=\frac{1}{4}\]
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C)
\[y=\frac{1}{2}\]
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D)
y = 1
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E)
None of these
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Angle between the line \[\frac{x}{a}+\frac{y}{b}=1\]and \[\frac{x}{a}-\frac{y}{b}=1\]
A)
\[2{{\tan }^{-1}}\frac{b}{a}\]
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B)
\[{{\tan }^{-1}}\frac{2ab}{{{a}^{2}}+{{b}^{2}}}\]
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C)
\[{{\tan }^{-1}}\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}\]
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D)
\[2{{\tan }^{-1}}\frac{a}{b}\]
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E)
None of these
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The equation \[{{y}^{2}}-{{x}^{2}}-2x-1=0\] represents
A)
A hyperbola
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B)
An ellipse
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C)
A pair of straight lines
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D)
A rectangular hyperbola
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E)
None of these
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Which of the following points lie on the parabola\[{{x}^{2}}=4ay\]?
A)
\[x=a{{t}^{2}},\,y=2\,at\]
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B)
\[x=2\,at,\,y=at\]
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C)
\[x=2\,a{{t}^{2}},\,y=at\]
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D)
\[x=2\,at,\,y=a{{t}^{2}}\]
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E)
None of these
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The eccentricity of the hyperbola \[5{{x}^{2}}-4{{y}^{2}}+20x+8y=4\] is
A)
\[\sqrt{2}\]
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B)
\[\frac{3}{2}\]
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C)
\[8\,\sqrt{2}\]
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D)
\[4\,\sqrt{2}\]
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E)
None of these
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It is given that f'(a) exists, then \[\underset{x\to a}{\mathop{\lim }}\,\,\,\frac{x.f(a)-a.f(x)}{x-a}\] is equal to
A)
\[f\,\left( a \right)+a\,f'\left( a \right)\]
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B)
\[f\left( a \right)a\text{ }f\left( a \right)\]
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C)
\[f\left( a \right)\]
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D)
\[-\text{ }f'\left( a \right)\]
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E)
None of these
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If \[f\left( a \right)=2,\] \[f\left( a \right)=1,\] \[g\left( a \right)=-1;\] \[g'\left( a \right)=2,\] then \[\underset{x\to a}{\mathop{\lim }}\,\,\,\frac{g(x)\,f(a)-g(a)\,f(x)}{x-a}\]
A)
3
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B)
5
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C)
0
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D)
\[-\,3\]
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E)
None of these
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Two cards are drawn one by one at random from a pack of 52 cards. The probability that both of them are king, is
A)
\[\frac{2}{13}\]
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B)
\[\frac{1}{169}\]
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C)
\[\frac{1}{221}\]
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D)
\[\frac{30}{221}\]
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E)
None of these
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If A and Bare two mutually exclusive events, then P (A + B) =
A)
\[P\,\,\left( A \right)+P\,\,\left( B \right)-P\,\,\left( AB \right)\]
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B)
\[P\,\,\left( A \right)-P\,\,\left( B \right)\]
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C)
\[P\,\,\left( A \right)+P\,\,\left( B \right)\]
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D)
\[P\,\,\left( A \right)+P\,\,\left( B \right)+P\,\,\left( AB \right)\]
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E)
None of these
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In a certain code \['Z-P'\] means Z is the father of 'P', 'P + Q' means 'P' is the mother of 'Q', \['J\times Q'\]means 'J' is the brother of 'Q' and \['Q-Z'\] means 'Q' is the grand-daughter of 'Z'. What is the relation of 'Z' to 'J'?
A)
Father
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B)
Sister
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C)
Grandfather
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D)
Grandmother
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E)
None of these
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Shailesh and Mohan start from a fixed point. Shailesh moves 3 km northward, turns right and then covers 4 km. Mohan moves 5 km westwards, turns right and walks 3 km. The distance between Shailesh and Mohan now is
A)
10 km
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B)
9 km
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C)
8 km
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D)
6 km
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E)
None of these
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Find the missing character from among the given alternatives.
A)
860
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B)
1140
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C)
2880
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D)
3240
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E)
None of these
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Five children were administered psychological tests to know their intellectual levels. In the report psychologists pointed out that the child A is less intelligent than the child B. The child C is less intelligent than the child D. The child B is less intelligent than the child C and Child A is more intelligent than the child E. Which child is the most intelligent?
A)
A
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B)
B
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C)
D
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D)
E
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E)
None of these
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Which of the following sets is best represented in the given diagram?
A)
Animals, Insects, Cockroaches
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B)
Animals, Males, Females and Hermaphrodites
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C)
States, Districts, Union Territories
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D)
Country, States, Districts
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E)
None of these
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