\[\underset{x\to 0}{\mathop{\lim }}\,\,\,\left( \frac{{{a}^{x}}-{{b}^{x}}}{x} \right)=\]
A)
\[\log \left( \frac{b}{a} \right)\]
done
clear
B)
\[\log \left( \frac{a}{b} \right)\]
done
clear
C)
\[\frac{a}{b}\]
done
clear
D)
\[\log {{a}^{b}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
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
done
clear
B)
B = C
done
clear
C)
A = C
done
clear
D)
A = B = C
done
clear
E)
None of these
done
clear
View Answer play_arrow
If the base of a triangle and the ratio of the lengths of the other two unequal sides are given, then the vertex lies on a\an
A)
Straight line
done
clear
B)
circle
done
clear
C)
Ellipse
done
clear
D)
parabola
done
clear
E)
None of these
done
clear
View Answer play_arrow
If a vertex of a triangle is (1,1) and the midpoint of two sides through this vertex are \[\left( -\,1,2 \right)~\] and (3, 2), then the centroid of the triangle is
A)
\[\left( \frac{1}{3},\frac{7}{3} \right)\]
done
clear
B)
\[\left( 1,\,\,\frac{7}{3} \right)\]
done
clear
C)
\[\left( -\frac{1}{3},\frac{7}{3} \right)\]
done
clear
D)
\[\left( -1,\,\,\frac{7}{3} \right)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The expression \[{{\{x+{{({{x}^{3}}-1)}^{1/2}}\}}^{5}}+{{\{x-{{({{x}^{3}}-1)}^{1/2}}\}}^{5}}\] is a polynomial of degree
A)
5
done
clear
B)
6
done
clear
C)
7
done
clear
D)
8
done
clear
E)
None of these
done
clear
View Answer play_arrow
Directions: Based on the following paragraph. If \[\cos \,\,\frac{\pi }{7},\] \[cos\,\,\frac{3\pi }{7},\] \[\cos \,\,\frac{5\pi }{7}\] are the roots of the equation \[8{{x}^{3}}-4{{x}^{2}}-4x+1=0\]
Find the value of\[\sec \,\,\frac{\pi }{7}+\sec \,\,\frac{3\pi }{7}+\sec \,\,\frac{5\pi }{7}\].
A)
2
done
clear
B)
4
done
clear
C)
8
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
Directions: Based on the following paragraph. If \[\cos \,\,\frac{\pi }{7},\] \[cos\,\,\frac{3\pi }{7},\] \[\cos \,\,\frac{5\pi }{7}\] are the roots of the equation \[8{{x}^{3}}-4{{x}^{2}}-4x+1=0\]
Find the value of \[\sin \,\,\frac{\pi }{17}\,\,\sin \,\,\frac{3\pi }{14}\,\,\sin \,\,\frac{5\pi }{14}\]
A)
\[\frac{1}{4}\]
done
clear
B)
\[\frac{1}{8}\]
done
clear
C)
\[\frac{\sqrt{7}}{4}\]
done
clear
D)
\[\frac{\sqrt{7}}{8}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If in a triangle ABC, cosA cosB + sinA sinB sinC =1, then a:b:c is equal to
A)
\[1:1:\sqrt{2}\]
done
clear
B)
\[1:1:\sqrt{3}\]
done
clear
C)
\[1:\sqrt{2}:1\]
done
clear
D)
\[1:\sqrt{3}:1\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[{{z}_{1}}\] and \[{{z}_{2}}\] are two non-zero complex numbers such that \[|{{z}_{1}}+{{z}_{2}}|\,=\,|{{z}_{1}}|+|{{z}_{2}}|,\] then \[arg\,\,({{z}_{1}})-\,\arg \,\,({{z}_{2}})\] is equal to
A)
\[-\pi \]
done
clear
B)
\[-\frac{\pi }{2}\]
done
clear
C)
\[0\]
done
clear
D)
\[\frac{\pi }{2}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Let \[0<P\left( A \right)<1,\] \[0<P\left( B \right)<1\] and \[P(A\bigcup B)=P\left( A \right)+P\left( B \right)-P\left( A \right).P\left( B \right),\] then
A)
\[P\,\,\left( B/A \right)=P\left( B \right)-P\left( A \right)\]
done
clear
B)
\[P\,\,\left( A'-B' \right)=P\,\,\left( A' \right)-P\,\,\left( B' \right)\]
done
clear
C)
\[P\,(A\bigcup B)'=P\,\,(A')\,\,P\,\,(B')\]
done
clear
D)
\[P\,\,\left( A/B \right)=P\,\,\left( A \right)-P\,\,\left( B \right)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If a circle passes through the point (a, b)and cuts the circle \[{{x}^{2}}+{{y}^{2}}={{k}^{2}}\] orthogonally, then the equation of the locus of its centre is
A)
\[2ax+2by-({{a}^{2}}+{{b}^{2}}+{{k}^{2}})=0\]
done
clear
B)
\[2ax+2by-({{a}^{2}}-{{b}^{2}}+{{k}^{2}})=0\]
done
clear
C)
\[{{x}^{2}}+{{y}^{2}}-3ax-4by+{{a}^{2}}+{{b}^{2}}-{{k}^{2}})=0\]
done
clear
D)
\[{{x}^{2}}+{{y}^{2}}-2ax-3by+({{a}^{2}}-{{b}^{2}}-{{k}^{2}})=0\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If the equation \[16{{x}^{2}}-3{{y}^{2}}-32x+12y-44=0\]represents a hyperbola, then
A)
The length of whose transverse axis is \[2\sqrt{3}\]
done
clear
B)
The length of whose conjugate axis is 8
done
clear
C)
Whose centre is (1, 2)
done
clear
D)
Whose eccentricity is\[\sqrt{\frac{19}{3}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The curve given by \[x+y={{e}^{xy}}\] has a tangent parallel to the y-axis at the point
A)
(0, 1)
done
clear
B)
(1, 0)
done
clear
C)
(1, 1)
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
The equation of common tangent to the parabola \[{{y}^{2}}=4ax\] and \[{{x}^{2}}=4ay\] is
A)
\[x+y=0\]
done
clear
B)
\[xy+a=0\]
done
clear
C)
\[x+y+a=0\]
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
\[\frac{\tan x}{\tan \,\,3x}\] never lies between
A)
\[-\frac{1}{3}\,\,and\,\,0\]
done
clear
B)
\[\frac{1}{3}\,\,and\,\,3\]
done
clear
C)
\[-\frac{1}{3}\,\,and\,\,\frac{1}{3}\]
done
clear
D)
\[-\,3\,\,and\,\,3\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The middle term in the expansion of \[{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}}+2 \right)}^{n}}\]is
A)
\[\frac{n}{{{[(n/2)!]}^{2}}}\]
done
clear
B)
\[\frac{(2n)!}{{{[(n/2)!]}^{2}}}\]
done
clear
C)
\[\frac{1.3.5...(2n+1)!}{n!}{{2}^{n}}\]
done
clear
D)
\[\frac{(2n)!}{{{(n!)}^{2}}}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If function f(x) is differentiable at \[x=a,\] then \[\underset{x\to a}{\mathop{\lim }}\,\,\,\frac{{{x}^{2}}\,f\,(a)-{{a}^{2}}\,f\,(x)}{x-a}\]
A)
\[2a\,f(a)+{{a}^{2}}f'(a)\]
done
clear
B)
\[-{{a}^{2}}f'(a)\]
done
clear
C)
\[af(a)-{{a}^{2}}f'(a)\]
done
clear
D)
\[2af(a)-{{a}^{2}}f'(a)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If a, b, c are in AP, then \[a+\frac{1}{bc},\] \[b+\frac{1}{ca},\] \[c+\frac{1}{ab}\] are in
A)
AP
done
clear
B)
GP
done
clear
C)
AGP
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently, is
A)
40
done
clear
B)
60
done
clear
C)
80
done
clear
D)
100
done
clear
E)
None of these
done
clear
View Answer play_arrow
Negation of "Ram is in class X or Rashmi is in class XII" is
A)
Ram is not in class X and Rashmi is not in class XII
done
clear
B)
Ram is not in class X but Ram is class in XII
done
clear
C)
Either Ram is not in class X or Rashmi is not in class XII
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
General solution of the equation \[(\sqrt{3}-1)\,\,\sin \theta +(\sqrt{3}+1)\,\,\cos \theta =2\] is
A)
\[2n\pi \,\,\pm \,\,\frac{\pi }{4}\,\,+\,\,\frac{\pi }{12}\]
done
clear
B)
\[n\pi +{{(-1)}^{n}}\,\,\frac{\pi }{4}\,\,+\,\,\frac{\pi }{12}\]
done
clear
C)
\[2n\pi \,\,\pm \,\,\frac{\pi }{4}\,\,-\,\,\frac{\pi }{12}\]
done
clear
D)
\[n\pi +\,\,{{(-1)}^{n}}\,\,\frac{\pi }{4}\,\,-\,\,\frac{\pi }{12}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The two curves \[{{x}^{3}}-3x{{y}^{2}}+2=0\] and \[3{{x}^{2}}y-{{y}^{3}}-2=0\]
A)
Cut at right angles
done
clear
B)
touch each other
done
clear
C)
Cut at an angle \[\frac{\pi }{3}\]
done
clear
D)
Cut at an angle \[\frac{\pi }{4}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[\frac{3x+4}{{{x}^{2}}-3x+2}=\frac{A}{x-2}-\frac{B}{x-1},\] then (A, B) =
A)
(7, 10)
done
clear
B)
(10, 7)
done
clear
C)
\[\left( 10,-\,7 \right)\]
done
clear
D)
\[\left( -\,10,7 \right)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
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
done
clear
B)
Y
done
clear
C)
N
done
clear
D)
All of these
done
clear
E)
None of these
done
clear
View Answer play_arrow
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
done
clear
B)
B
done
clear
C)
D
done
clear
D)
E
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[A={{\sin }^{2}}x+{{\cos }^{4}}x,\] then for all real x
A)
\[\frac{13}{16}\,\,\le \,\,A\,\,\le \,\,1\]
done
clear
B)
\[1\le A\le 2\]
done
clear
C)
\[\frac{3}{4}\le A\le \frac{13}{16}\]
done
clear
D)
\[\frac{3}{4}\le A\le 1\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
\[\underset{x\to 2}{\mathop{\lim }}\,\,\,\left( \frac{\sqrt{1-\{\cos 2\,\,(x-2)\}}}{x-2} \right)\]
A)
equals \[\sqrt{2}\]
done
clear
B)
equals \[-\sqrt{2}\]
done
clear
C)
equals \[\frac{1}{\sqrt{2}}\]
done
clear
D)
does not exist
done
clear
E)
None of these
done
clear
View Answer play_arrow
If C and D are two events such that \[C\subset D\] and \[P(D)\ne 0,\] then the correct statement among the following is
A)
\[P\,(C|D)\ge P\,(C)\]
done
clear
B)
\[P\,(C|D)<P\,(C)\]
done
clear
C)
\[P\,(C|D)=\frac{P\,(D)}{P\,(C)}\]
done
clear
D)
\[P\,(C|D)=P\,(C)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The lines \[x+y=\left| a \right|\] and \[ax-y=1\] intersect each other in the first quadrant. Then, the set of all possible values of a is the interval
A)
\[\left( -\,1,1 \right]\]
done
clear
B)
(0,\[\infty \])
done
clear
C)
[1,\[\infty \])
done
clear
D)
\[(-1,\infty )\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
The equation of the tangent to the curve \[y=x+\frac{4}{{{x}^{2}}},\] that is parallel to the x-axis, is
A)
y = 0
done
clear
B)
y = 1
done
clear
C)
y = 2
done
clear
D)
y = 3
done
clear
E)
None of these
done
clear
View Answer play_arrow
For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is
A)
there is a regular polygon with \[\frac{r}{R}=\frac{1}{2}\]
done
clear
B)
there is a regular polygon with \[\frac{r}{R}=\frac{1}{\sqrt{2}}\]
done
clear
C)
there is a regular polygon with \[\frac{r}{R}=\frac{2}{3}\]
done
clear
D)
there is a regular polygon with \[\frac{r}{R}=\frac{\sqrt{3}}{2}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
A person is to count 4500 currency notes. Let \[{{a}_{n}}\]denotes the number of notes he counts in the nth minute. If \[{{a}_{1}}={{a}_{2}}=....={{a}_{10}}=150\] and \[{{a}_{10}},{{a}_{11}},......\] are in AP with common difference \[-\,2,\] then the time taken by him to count all notes, is
A)
24 min
done
clear
B)
34 min
done
clear
C)
125 min
done
clear
D)
135 min
done
clear
E)
None of these
done
clear
View Answer play_arrow
The distance of the point \[\left( 1,-\,5,9 \right)\] from the plane \[~x-y+z=5\] measured along a straight line \[x=y=z\] is
A)
\[3\,\sqrt{5}\]
done
clear
B)
\[10\,\sqrt{3}\]
done
clear
C)
\[5\,\sqrt{3}\]
done
clear
D)
\[3\,\sqrt{10}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[\omega \] is an imaginary cube root of unity, then value of the expression \[2\,\,(1+\omega )\,\,(1+{{\omega }^{2}})+3\,\,(2+\omega )\] \[(2+{{\omega }^{2}})+.....+(n+1)\,\,(n+\omega )\,\,(n+{{\omega }^{2}})\] is
A)
\[\frac{1}{4}\,\,{{n}^{2}}\,\,{{(n+1)}^{2}}+n\]
done
clear
B)
\[\frac{1}{4}\,\,{{n}^{2}}\,\,{{(n+1)}^{2}}-n\]
done
clear
C)
\[\frac{1}{4}\,\,n\,\,{{(n+1)}^{2}}-n\,\]
done
clear
D)
\[\frac{1}{4}\,\,{{n}^{2}}\,\,{{(n-1)}^{2}}-n\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[x={{2}^{1/3}}-{{2}^{1/3}},\] then \[2{{x}^{3}}+6x=\]
A)
0
done
clear
B)
2
done
clear
C)
3
done
clear
D)
4
done
clear
E)
None of these
done
clear
View Answer play_arrow
If \[{}^{43}{{C}_{r-6}}={}^{43}{{C}_{3r+1}}\], then the value of r is
A)
6
done
clear
B)
8
done
clear
C)
10
done
clear
D)
12
done
clear
E)
None of these
done
clear
View Answer play_arrow
If A = {a, b, c}, B = {b, c, d} and C = {a, d, c} then \[(A-B)\times (B\cap C)=\]
A)
{(a, c), (a, d)}
done
clear
B)
{(a, b), (c, d)}
done
clear
C)
{(c, a), (a, d)}
done
clear
D)
{(a, c), (b, d)}
done
clear
E)
None of these
done
clear
View Answer play_arrow
A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is \[45{}^\circ \]. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to \[30{}^\circ \]. Then the speed (in m/sec) of the bird is
A)
\[20\sqrt{2}\]
done
clear
B)
\[40\,\,\left( \sqrt{2}-1 \right)\]
done
clear
C)
\[40\,\,\left( \sqrt{3}-\sqrt{2} \right)\]
done
clear
D)
\[20\,\,\left( \sqrt{3}-1 \right)\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centered at (0, y), passing through origin and touching the circle C externally, then the radius of T is equal to
A)
\[\frac{1}{2}\]
done
clear
B)
\[\frac{\sqrt{3}}{\sqrt{2}}\]
done
clear
C)
\[\frac{\sqrt{3}}{2}\]
done
clear
D)
\[\frac{1}{4}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow
\[{{\log }_{e}}3-\frac{{{\log }_{e}}9}{{{2}^{2}}}+\frac{{{\log }_{e}}27}{{{3}^{2}}}-\frac{{{\log }_{e}}81}{{{4}^{2}}}+..........\] is
A)
\[{{\log }_{e}}3\]
done
clear
B)
\[{{\log }_{e}}2\]
done
clear
C)
\[({{\log }_{e}}3)\,\,({{\log }_{e}}2)\]
done
clear
D)
\[\frac{{{\log }_{e}}5}{{{\log }_{e}}3}\]
done
clear
E)
None of these
done
clear
View Answer play_arrow