The set \[(A\,\,\cup \,\,B\,\,\cup \,\,C)\,\,\cap \,\,(A\,\,\cap \,\,B'\,\,\cap \,\,C')\,\,\cap \,\,C\] is equal to ______.
A)
\[B\,\,\cap \,\,C'\]
done
clear
B)
\[A\,\,\cap \,\,C\]
done
clear
C)
\[B'\,\,\cap \,\,C'\]
done
clear
D)
\[B\,\,\cap \,\,C\]
done
clear
E)
None of these
done
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View Answer play_arrow
Let \[P=\{\theta :\sin \theta -\cos \theta \sqrt{2}\cos \theta \}\] and \[Q=\{\theta :\sin \theta +\cos \theta =\sqrt{2}\sin \theta \}\] be two sets. Then,
A)
\[P\subset Q\,\,and\,\,\,Q-P\ne \phi \]
done
clear
B)
\[Q\not\subset P\]
done
clear
C)
\[P\not\subset Q\]
done
clear
D)
\[P=Q\]
done
clear
E)
None of these
done
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View Answer play_arrow
The domain of the function \[f(x)=\frac{1}{\sqrt{|x|-\,x}}\] is_______.
A)
\[(-\,\infty ,\infty )\]
done
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B)
\[(-\,\infty ,\infty )-\{0\}\]
done
clear
C)
\[(-\,\infty ,0)\]
done
clear
D)
\[(\,0,\infty )\]
done
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E)
None of these
done
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View Answer play_arrow
If \[f(x)=\left( \frac{1+x}{1+x} \right)\] and \[g(x)=\frac{3x+{{x}^{3}}}{1+3{{x}^{2}}},\] then \[f\,\,\left( g\left( x \right) \right)\] is equal to ________.
A)
\[3f\,\left( x \right)\]
done
clear
B)
\[{{[f(x)]}^{2}}\]
done
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C)
\[f\left( 3x \right)\]
done
clear
D)
\[-\,f\left( x \right)\]
done
clear
E)
None of these
done
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View Answer play_arrow
The value of \[\frac{Sec\,8\theta -1}{Sec\,4\theta -1}\] is equal to _______.
A)
\[\frac{\tan 8\theta }{\tan 2\theta }\]
done
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B)
\[\frac{\tan 2\theta }{\tan 8\theta }\]
done
clear
C)
\[\frac{\tan 8\theta }{\tan 4\theta }\]
done
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D)
\[\frac{\tan 4\theta }{\tan 8\theta }\]
done
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E)
None of these
done
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View Answer play_arrow
If \[cos\,x=tan\,y,\] \[cot\,y=tan\,z\] and \[cot\,z=tan\,x,\]then sin x is equal to ____.
A)
\[\frac{\sqrt{5}+1}{4}\]
done
clear
B)
\[\frac{\sqrt{5}-1}{4}\]
done
clear
C)
\[\frac{\sqrt{5}-1}{2}\]
done
clear
D)
\[\frac{\sqrt{5}+1}{2}\]
done
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E)
None of these
done
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View Answer play_arrow
In a triangle, the lengths of two larger sides are 10 cm and 9 cm. If the angles of the triangle are in AP, then the length of the third side can be _____.
A)
\[3\pm \sqrt{2}\]
done
clear
B)
\[5\pm \sqrt{6}\]
done
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C)
\[3\pm \sqrt{6}\]
done
clear
D)
\[5\pm \sqrt{2}\]
done
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E)
None of these
done
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View Answer play_arrow
If \[\tan A=\left( \frac{1-\cos B}{\sin B} \right),\] then tan 2A is equal to_____.
A)
\[\tan \frac{B}{2}\]
done
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B)
\[2\,\tan \,B\]
done
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C)
tan B
done
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D)
cot B
done
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E)
None of these
done
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View Answer play_arrow
If 7 points out of 12 are in the same straight line, then the number of triangles formed is ______.
A)
185
done
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B)
176
done
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C)
191
done
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D)
181
done
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E)
None of these
done
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The sum of the series: \[1+\frac{3}{2!}+\frac{5}{4!}+\frac{7}{6!}+.........\infty \] is equal to ______.
A)
\[e\]
done
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B)
\[\frac{1}{e}\]
done
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C)
\[{{e}^{2}}\]
done
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D)
\[\frac{1}{{{e}^{2}}}\]
done
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E)
None of these
done
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If the sum of the roots of the equation \[a{{x}^{2}}+bx+c=0\] is equal to the sum of the reciprocals of their square, then \[b{{c}^{2}},\] \[c{{a}^{2}}\] and \[a{{b}^{2}},\]are in _______.
A)
G.P.
done
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B)
H.P.
done
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C)
A.P.
done
clear
D)
Arithmetic-geometric progression
done
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E)
None of these
done
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View Answer play_arrow
For all \[n\in N\], the sums \[{{s}_{n}}={{n}^{3}}+3{{n}^{2}}+5n+3\] is divisible by ______.
A)
2
done
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B)
6
done
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C)
8
done
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D)
3
done
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E)
None of these
done
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View Answer play_arrow
If \[x=-\,5+2\,\sqrt{-\,4},\] then the value of \[{{x}^{4}}+9{{x}^{3}}+35{{x}^{2}}-x+4\] is_______.
A)
\[-\,152\]
done
clear
B)
160
done
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C)
\[-\,160\]
done
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D)
\[-\,142\]
done
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E)
None of these
done
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View Answer play_arrow
The solution set of the given in equation \[\frac{-1}{|x|-2}\ge 1\] (where\[x\in R,\,x\ne \pm \,2\]) is______.
A)
\[[-\,2,-\,1]\,\,\cup \,\,[1,2]\]
done
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B)
\[(-\,2,-\,1]\,\,\cup \,\,(1,2)\]
done
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C)
\[[-\,2,-\,1)\,\,\cup \,\,(1,2]\]
done
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D)
\[(-\,2,-\,1]\,\,\cup \,\,[1,2)\]
done
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E)
None of these
done
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View Answer play_arrow
The number of integral terms in the expansion of \[{{\left( {{5}^{1/2}}+{{7}^{1/8}} \right)}^{1024}}\] is _______.
A)
125
done
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B)
127
done
clear
C)
129
done
clear
D)
131
done
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E)
None of these
done
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View Answer play_arrow
The number of integers greater than 6000 that can be formed, using the digits 3, 5, 6, 7 and 8 without repetition, is ______.
A)
216
done
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B)
512
done
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C)
192
done
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D)
228
done
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E)
None of these
done
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View Answer play_arrow
Find the locus of point, so that the join of \[\left( -\,5,1 \right)\] and (3, 2) subtends a right angle at the moving point.
A)
\[{{x}^{2}}+{{y}^{2}}+3x-2y-13=0\]
done
clear
B)
\[{{x}^{2}}+{{y}^{2}}+2x-3y-13=0\]
done
clear
C)
\[{{x}^{2}}+{{y}^{2}}-2x+3y+13=0\]
done
clear
D)
\[{{x}^{2}}+{{y}^{2}}-3x+2y+13=0\]
done
clear
E)
None of these
done
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If the straight line through the point P (3, 4) makes an angle \[\frac{\pi }{6}\] with x-axis and meets the line \[~12x+5y+10=0\] at Q, find the length of PQ.
A)
\[\frac{132}{12\,\sqrt{3}+5}\]
done
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B)
\[\frac{123}{8\,\sqrt{3}+7}\]
done
clear
C)
\[\frac{115}{6\,\sqrt{2}+5}\]
done
clear
D)
\[\frac{110}{4\,\sqrt{2}+7}\]
done
clear
E)
None of these
done
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View Answer play_arrow
Two vertices of a triangle are \[\left( 3,-\,1 \right)\] and \[\left( -\,2,3 \right)\]and its orthocenter is at the origin. Find the co-ordinates of the third vertex.
A)
\[\left( \frac{-36}{7},\frac{-43}{7} \right)\]
done
clear
B)
\[\left( \frac{-36}{7},\frac{-45}{7} \right)\]
done
clear
C)
\[\left( \frac{36}{5},\frac{44}{5} \right)\]
done
clear
D)
\[\left( \frac{39}{5},\frac{49}{5} \right)\]
done
clear
E)
None of these
done
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View Answer play_arrow
Length of the chord cutoff by \[y=2x+1\] from the circle \[{{x}^{2}}+{{y}^{2}}=4\] is equal to _______.
A)
\[2\sqrt{\frac{13}{7}}\]
done
clear
B)
\[2\sqrt{\frac{19}{5}}\]
done
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C)
\[3\sqrt{\frac{13}{7}}\]
done
clear
D)
\[3\sqrt{\frac{19}{5}}\]
done
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E)
None of these
done
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The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then \[P\,\,\left( A' \right)+P\,\,\left( B' \right)\] is equal to ____.
A)
\[2+2p-q\]
done
clear
B)
\[2\,-\,3p\,+\,q\]
done
clear
C)
\[2-2p+q\]
done
clear
D)
\[2+3p-q\]
done
clear
E)
None of these
done
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View Answer play_arrow
The locus of chords of contact of perpendicular tangents to the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] touch another fixed ellipse is _______.
A)
\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{1}{{{a}^{2}}+{{b}^{2}}}\]
done
clear
B)
\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{2}{{{a}^{2}}-{{b}^{2}}}\]
done
clear
C)
\[\frac{{{x}^{2}}}{{{a}^{4}}}+\frac{{{y}^{2}}}{{{b}^{4}}}=\frac{1}{{{a}^{2}}+{{b}^{2}}}\]
done
clear
D)
\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{2}{{{a}^{2}}+{{b}^{2}}}\]
done
clear
E)
None of these
done
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View Answer play_arrow
How many real tangent(s) can be drawn from the point (4, 3) to the hyperbola \[\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1\]
A)
0
done
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B)
1
done
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C)
2
done
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D)
3
done
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E)
None of these
done
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View Answer play_arrow
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{2}-\sqrt{1+\cos x}}\] is _______.
A)
\[4\,\sqrt{2}\,{{(\log \,3)}^{2}}\]
done
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B)
\[8\sqrt{2}\,{{(\log \,3)}^{2}}\]
done
clear
C)
\[2\,\sqrt{2}\,{{(\log \,3)}^{2}}\]
done
clear
D)
\[3\,\sqrt{2}\,{{(\log \,3)}^{2}}\]
done
clear
E)
None of these
done
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View Answer play_arrow
\[\underset{x\to 2}{\mathop{\lim }}\,\frac{\sqrt{1+\sqrt{2+x}-\sqrt{3}}}{x-2}\] is equal to_______.
A)
\[\frac{1}{4\,\sqrt{3}}\]
done
clear
B)
\[\frac{1}{8\,\sqrt{3}}\]
done
clear
C)
\[\frac{2}{3\,\sqrt{3}}\]
done
clear
D)
\[\frac{3}{4}\]
done
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E)
None of these
done
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View Answer play_arrow
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (1+3{{x}^{2}})}{x({{e}^{5x}}-1)}\] is equal to ????.
A)
\[\frac{3}{5}\]
done
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B)
\[\frac{-3}{5}\]
done
clear
C)
\[\frac{5}{3}\]
done
clear
D)
\[\frac{-5}{3}\]
done
clear
E)
None of these
done
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View Answer play_arrow
If p and q are statements with truth values false and true, respectively, then \[(\sim p\vee \sim q)\wedge r\] the truth value of is ________.
A)
true
done
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B)
false
done
clear
C)
false, if r is true
done
clear
D)
false, if r is false
done
clear
E)
None of these
done
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View Answer play_arrow
A five digit number is formed by the digits 1, 2, 3, 4, 5 without repetition. Find the probability that the number is divisible by 4.
A)
\[\frac{1}{3}\]
done
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B)
\[\frac{1}{5}\]
done
clear
C)
\[\frac{1}{7}\]
done
clear
D)
\[\frac{2}{5}\]
done
clear
E)
None of these
done
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View Answer play_arrow
Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27, respectively, find the correct standard deviation.
A)
11.34
done
clear
B)
10.24
done
clear
C)
16.44
done
clear
D)
9.34
done
clear
E)
None of these
done
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View Answer play_arrow
The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is _______.
A)
\[\sqrt{7}\]
done
clear
B)
\[\sqrt{\frac{52}{7}}\]
done
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C)
\[\frac{52}{7}\]
done
clear
D)
\[7\]
done
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E)
None of these
done
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View Answer play_arrow
Five students A, B, C, D and E are sitting in a row, D is on the right of E. B is on the left of E but is on the right of A. D is on the left of C. Who is sitting in the middle?
A)
A
done
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B)
B
done
clear
C)
C
done
clear
D)
E
done
clear
E)
None of these
done
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View Answer play_arrow
Robert starts walking straight towards East. After walking 75 m he turns to the left and walks 80 m straight. Again he turns to the left and walks a distance of 30 m straight, again he turns to the left and walks a distance of 20 m. How far is he from his starting point?
A)
85 m
done
clear
B)
65 m
done
clear
C)
55 m
done
clear
D)
75 m
done
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E)
None of these
done
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View Answer play_arrow
Direction: In each of the following number series, a wrong number is given. Find out that number. 258, 130, 66, 34, 18, 8, 6
A)
130
done
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B)
66
done
clear
C)
34
done
clear
D)
8
done
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E)
None of these
done
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View Answer play_arrow
Direction: In each of the following number series, a wrong number is given. Find out that number.s 2, 3, 6, 15, 37.5, 157.5, 630
A)
3
done
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B)
6
done
clear
C)
15
done
clear
D)
37.5
done
clear
E)
None of these
done
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View Answer play_arrow
Direction: Study the figure given below carefully and answer the questions that follow:
What is the sum of the numbers which belong to two figures only?
A)
6
done
clear
B)
15
done
clear
C)
20
done
clear
D)
All of these
done
clear
E)
None of these
done
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View Answer play_arrow
Direction: Study the figure given below carefully and answer the questions that follow:
What is the product of the number which belong to three figures only?
A)
27
done
clear
B)
162
done
clear
C)
648
done
clear
D)
All of these
done
clear
E)
None of these
done
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View Answer play_arrow
Direction: Study the following information and answer the questions given below it. Eight friends P, Q, R, S, T, U, V and W are sitting around a circle facing the centre. T is third to the left of V who is to the immediate right of Q who is third to the left of P. W is second to the right of U who is not an immediate neighbour of T. S is not an immediate neighbour of Q.
Which of the following is the correct position of Q with respect to S?
A)
Second to the right
done
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B)
Second to the left
done
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C)
Third to the right
done
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D)
Third to the left
done
clear
E)
None of these
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View Answer play_arrow
What is T's position with respect to R?
A)
To the immediate right
done
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B)
To the immediate left
done
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C)
Second to the right
done
clear
D)
Cannot be determined
done
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E)
None of these
done
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View Answer play_arrow
Which of the following pairs has the first person to the immediate left of second person?
A)
VQ
done
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B)
PU
done
clear
C)
RT
done
clear
D)
WS
done
clear
E)
None of these
done
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View Answer play_arrow
Who is second to the right of Q?
A)
U
done
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B)
P
done
clear
C)
W
done
clear
D)
S
done
clear
E)
None of these
done
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View Answer play_arrow
\[\cos \frac{\pi }{7}+\cos \frac{3\pi }{7}+\cos \frac{5\pi }{7}\] equals to _______.
A)
1
done
clear
B)
\[\frac{1}{\sqrt{2}}\]
done
clear
C)
\[\frac{1}{2}\]
done
clear
D)
\[\frac{1}{4}\]
done
clear
E)
None of these
done
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View Answer play_arrow
The solution of the equation \[\left| x\,-\,1 \right|\,+\,\left| x \right|\,+\,\left| x\,+\,1 \right|\,=\,x\,+\,2\] is_______.
A)
\[x\in [-1,\,1]\]
done
clear
B)
\[x\in [-1,\,2]\]
done
clear
C)
\[x\in [0,\,1]\]
done
clear
D)
\[x\in [0,\,2]\]
done
clear
E)
None of these
done
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View Answer play_arrow
If the point of intersection of the \[\frac{{{x}^{2}}}{{{a}^{2}}}\,+\,\frac{{{y}^{2}}}{{{b}^{2}}}\,=\,1\] and \[\frac{{{x}^{2}}}{{{\alpha }^{2}}}+\frac{{{y}^{2}}}{{{\beta }^{2}}}=1\] are at the extremities of the conjugate diameters of the former, then ______.
A)
\[\frac{{{a}^{2}}}{{{\alpha }^{2}}}+\frac{{{b}^{2}}}{{{\beta }^{2}}}=2\]
done
clear
B)
\[\frac{{{\alpha }^{2}}}{{{a}^{2}}}-\frac{{{\beta }^{2}}}{{{b}^{2}}}=2\]
done
clear
C)
\[\frac{{{a}^{2}}}{{{\alpha }^{2}}}-\frac{{{b}^{2}}}{{{\beta }^{2}}}=2\]
done
clear
D)
All the above
done
clear
E)
None of these
done
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View Answer play_arrow
If \[\sum\limits_{r=1}^{n}{\,{{T}_{r}}=\frac{n}{8}\,\,(n+1)\,\,(n+2)\,(n+3)},\] then find \[\sum\limits_{r=1}^{n}{\frac{1}{{{T}_{r}}}}\].
A)
\[\frac{n\,(n+3)}{2\,(n+1)\,\,(n+2)}\]
done
clear
B)
\[\frac{n\,(n+1)\,\,(n+2)}{4\,(n+3)}\]
done
clear
C)
\[\frac{8}{n\,(n+1)\,\,(n+2)\,\,(n+3)}\]
done
clear
D)
\[\frac{2\,(n+1)\,\,(n+2)}{n\,(n+3)}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Let\[{{a}_{n}}=\underbrace{111........1}_{n\,times},\] then which one among the following is not true?
A)
\[{{a}_{912}}\]is not prime
done
clear
B)
\[{{a}_{951}}\] is not prime
done
clear
C)
\[{{a}_{480}}\] is not prime
done
clear
D)
\[{{a}_{91}}\] is prime
done
clear
E)
None of these
done
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View Answer play_arrow
The point represented by the complex number \[\left( 2-i \right)\] is rotated about origin through an angle \[\frac{\pi }{2}\] in the clockwise direction, the new position of point is
A)
1 + 2i
done
clear
B)
\[\,1+2\,i\]
done
clear
C)
\[-\,1-\,\,2i\]
done
clear
D)
2 + i
done
clear
E)
None of these
done
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View Answer play_arrow
Let \[{{n}_{1}}<{{n}_{2}}<{{n}_{3}}<{{n}_{4}}<{{n}_{5}}\] be positive integers such that \[{{n}_{1}}+{{n}_{2}}+{{n}_{3}}+{{n}_{4}}+{{n}_{5}}=20\]. Then, the number of such distinct arrangements \[({{n}_{1}},\,\,{{n}_{2}},\,\,{{n}_{3}},\,\,{{n}_{4}},\,\,{{n}_{5}})\] is________.
A)
3
done
clear
B)
5
done
clear
C)
7
done
clear
D)
9
done
clear
E)
None of these
done
clear
View Answer play_arrow
The values of x between 0 and \[2\pi \] which satisfy the equation \[\sin x\sqrt{8{{\cos }^{2}}x}=1\] are in AP with common difference ________.
A)
\[\frac{\pi }{8}\]
done
clear
B)
\[\frac{\pi }{4}\]
done
clear
C)
\[\frac{\pi }{2}\]
done
clear
D)
\[\frac{3\pi }{8}\]
done
clear
E)
None of these
done
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View Answer play_arrow
If e is the eccentricity of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] and \[\theta \] is the angle between the asymptotes, then \[\cos \left( \frac{\theta }{2} \right)\] is equal to______.
A)
e
done
clear
B)
\[\frac{1}{e}\]
done
clear
C)
\[\frac{-\,1}{e}\]
done
clear
D)
\[-\,e\]
done
clear
E)
None of these
done
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View Answer play_arrow
The number of ways to 16 different things to three persons A, B, C so that B gets one more than A and C gets two more than B is ______.
A)
\[\frac{16!}{4!5!7!}\]
done
clear
B)
\[\frac{16!}{9!2!5!}\]
done
clear
C)
\[\frac{16!}{3!8!5!}\]
done
clear
D)
\[\frac{16!}{4!6!6!}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow