The value of $$\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}$$ is
คำตอบ
(D)
1
2
The function $$f\left( x \right)$$ $$ = \log \left( {x + \sqrt {{x^2} + 1} } \right)$$, is
คำตอบ
(C)
an odd function
3
A function $$f$$ from the set of natural numbers to integers defined by
$$$f\left( n \right) = \left\{ {\matrix{
{{{n - 1} \over 2},\,when\,n\,is\,odd} \cr
{ - {n \over 2},\,when\,n\,is\,even} \cr
} } \right.$$$
is
คำตอบ
(D)
one-one and onto both
4
If $$f:R \to R$$ satisfies $$f$$(x + y) = $$f$$(x) + $$f$$(y), for all x, y $$ \in $$ R and $$f$$(1) = 7, then $$\sum\limits_{r = 1}^n {f\left( r \right)} $$ is
คำตอบ
(A)
$${{7n\left( {n + 1} \right)} \over 2}$$
5
Domain of definition of the function f(x) = $${3 \over {4 - {x^2}}}$$ + $${\log _{10}}\left( {{x^3} - x} \right)$$, is
คำตอบ
(A)
(-1, 0)$$ \cup $$(1, 2)$$ \cup $$(2, $$\infty $$)
6
If $$\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}$$ = k, the value of k is
คำตอบ
(D)
$${2 \over 3}$$
7
Let $$f(a) = g(a) = k$$ and their nth derivatives
$${f^n}(a)$$, $${g^n}(a)$$ exist and are not equal for some n. Further if
continuous for all x but not differentiable at x = 0
10
In an experiment with 15 observations on $$x$$, then following results were available:
$$\sum {{x^2}} = 2830$$, $$\sum x = 170$$
One observation that was 20 was found to be wrong and was replaced by the correct value
30. Then the corrected variance is :
คำตอบ
(D)
78.00
11
The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of
the set is increased by 2, then the median of the new set :
คำตอบ
(D)
remains the same as that of the original set
12
Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is :
คำตอบ
(A)
$${{2 \over 5}}$$
13
The trigonometric equation $${\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a$$ has a solution for :
คำตอบ
(D)
$$\left| a \right| \le {1 \over {\sqrt 2 }}$$
14
If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $$a>0,$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively such that $${p^2} = q$$ , then $$a$$ equals
คำตอบ
(D)
$$2$$
15
If $$A = \left[ {\matrix{
a & b \cr
b & a \cr
} } \right]$$ and $${A^2} = \left[ {\matrix{
\alpha & \beta \cr
\beta & \alpha \cr
} } \right]$$, then
If the system of linear equations
$$x + 2ay + az = 0;$$ $$x + 3by + bz = 0;\,\,x + 4cy + cz = 0;$$
has a non - zero solution, then $$a, b, c$$.
คำตอบ
(D)
are in H.P.
18
The area of the region bounded by the curves $$y = \left| {x - 1} \right|$$ and $$y = 3 - \left| x \right|$$ is :
คำตอบ
(D)
$$4$$ sq. units
19
Let $$f(x)$$ be a function satisfying $$f'(x)=f(x)$$ with $$f(0)=1$$ and $$g(x)$$ be a function that satisfies $$f\left( x \right) + g\left( x \right) = {x^2}$$. Then the value of the integral $$\int\limits_0^1 {f\left( x \right)g\left( x \right)dx,} $$ is
คำตอบ
(D)
$$e - {{{e^2}} \over 2} - {3 \over 2}$$
20
If $$f\left( {a + b - x} \right) = f\left( x \right)$$ then $$\int\limits_a^b {xf\left( x \right)dx} $$ is equal to
คำตอบ
(C)
$${{a + b} \over 2}\int\limits_a^b {f\left( x \right)dx} $$
21
The value of the integral $$I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx} $$ is
The degree and order of the differential equation of the family of all parabolas whose axis is $$x$$-axis, are respectively.
คำตอบ
(C)
$$1,2$$
24
Events $$A, B, C$$ are mutually exclusive events such that $$P\left( A \right) = {{3x + 1} \over 3},$$ $$P\left( B \right) = {{1 - x} \over 4}$$ and $$P\left( C \right) = {{1 - 2x} \over 2}$$ The set of possible values of $$x$$ are in the interval.
คำตอบ
(B)
$$\left[ {{1 \over 3},{1 \over 2}} \right]$$
25
If $$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$ then $$\overrightarrow a + \overrightarrow b + \overrightarrow c = $$
คำตอบ
(C)
$$0$$
26
Let $$\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j$$ and $$\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.$$ If $$\widehat n$$ is a unit vector such that $$\overrightarrow u .\widehat n = 0$$ and $$\overrightarrow v .\widehat n = 0\,\,,$$ then $$\left| {\overrightarrow w .\widehat n} \right|$$ is equal to :
คำตอบ
(A)
$$3$$
27
The vectors $$\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k$$ are the sides of triangle $$ABC.$$ The length of the median through $$A$$ is :
คำตอบ
(D)
$$\sqrt {33} $$
28
The two lines $$x=ay+b,z=cy+d$$ and $$x = a'y + b',z = c'y + d'$$ will be perpendicular, if and only if :
$$\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c $$ are $$3$$ vectors, such that
$$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$ , $$\left| {\overrightarrow a } \right| = 1\,\,\,\left| {\overrightarrow b } \right| = 2,\,\,\,\left| {\overrightarrow c } \right| = 3,$$,
then $${\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a }$$ is equal to :
คำตอบ
(C)
$$-7$$
31
A tetrahedron has vertices at $$O(0,0,0), A(1,2,1) B(2,1,3)$$ and $$C(-1,1,2).$$ Then the angle between the faces $$OAB$$ and $$ABC$$ will be :
If $$\left| {\matrix{
a & {{a^2}} & {1 + {a^3}} \cr
b & {{b^2}} & {1 + {b^3}} \cr
c & {{c^2}} & {1 + {c^3}} \cr
} } \right| = 0$$ and vectors $$\left( {1,a,{a^2}} \right),\,\,$$
$$\left( {1,b,{b^2}} \right)$$ and $$\left( {1,c,{c^2}} \right)\,$$ are non-coplanar, then the product $$abc$$ equals :
คำตอบ
(C)
$$-1$$
33
Consider points $$A, B, C$$ and $$D$$ with position
vectors $$7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k$$ and $$5\widehat i - \widehat j + 5\widehat k$$ respectively. Then $$ABCD$$ is a :
คำตอบ
(D)
None
34
Let $${Z_1}$$ and $${Z_2}$$ be two roots of the equation $${Z^2} + aZ + b = 0$$, Z being complex. Further , assume that the origin, $${Z_1}$$ and $${Z_2}$$ form an equilateral triangle. Then :
คำตอบ
(D)
$${a^2} = 3b$$
35
If $${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$ then :
คำตอบ
(B)
x = 4n , where n is any positive integer
36
If the sum of the roots of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the sum of the squares of their reciprocals, then $${a \over c},\,{b \over a}$$ and $${c \over b}$$ are in
คำตอบ
(D)
Harmonic Progression
37
The value of '$$a$$' for which one root of the quadratic equation
$$$\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0$$$
is twice as large as the other is
คำตอบ
(B)
$$ {2 \over 3}$$
38
The number of real solutions of the equation $${x^2} - 3\left| x \right| + 2 = 0$$ is
คำตอบ
(C)
4
39
The real number $$x$$ when added to its inverse gives the minimum sum at $$x$$ equal :
คำตอบ
(C)
1
40
If $$x$$ is positive, the first negative term in the expansion of $${\left( {1 + x} \right)^{27/5}}$$ is
คำตอบ
(D)
8th term.
41
The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by
คำตอบ
(B)
$$6!\, \times 5!$$
42
The number of integral terms in the expansion of $${\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}$$ is
คำตอบ
(C)
33
43
A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
คำตอบ
(C)
196
44
If $${}^n{C_r}$$ denotes the number of combination of n things taken r at a time, then the expression $$\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$$ equals
คำตอบ
(C)
$${}^{n + 2}{C_{r + 1}}$$
45
A square of side a lies above the $$x$$-axis and has one vertex at the origin. The side passing through the origin makes an angle $$\alpha \left( {0 < \alpha < {\pi \over 4}} \right)$$ with the positive direction of x-axis. The equation of its diagonal not passing through the origin is :
If $${x_1},{x_2},{x_3}$$ and $${y_1},{y_2},{y_3}$$ are both in G.P. with the same common ratio, then the points $$\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$$ and $$\left( {{x_3},{y_3}} \right)$$ :
คำตอบ
(B)
lie on a straight line
47
Locus of centroid of the triangle whose vertices are $$\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)$$ and $$\left( {1,0} \right),$$ where $$t$$ is a parameter, is :
If the equation of the locus of a point equidistant from the point $$\left( {{a_{1,}}{b_1}} \right)$$ and $$\left( {{a_{2,}}{b_2}} \right)$$ is
$$\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0$$ , then the value of $$'c'$$ is :
The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :
คำตอบ
(D)
$${x^2}\, + \,{y^2} - \,2x\, + \,2y\,\, = \,47$$
50
The foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the hyperbola $${{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}}$$ coincide. Then the value of $${b^2}$$ is :
คำตอบ
(D)
$$7$$
51
If $$f\left( y \right) = {e^y},$$ $$g\left( y \right) = y;y > 0$$ and
$$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,} $$ then :
Let $$f\left( x \right)$$ be a polynomial function of second degree. If $$f\left( 1 \right) = f\left( { - 1} \right)$$ and $$a,b,c$$ are in $$A.P, $$ then $$f'\left( a \right),f'\left( b \right),f'\left( c \right)$$ are in
คำตอบ
(B)
$$A.P$$
54
If $$z$$ and $$\omega $$ are two non-zero complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$Arg(z) - Arg(\omega ) = {\pi \over 2},$$ then $$\,\overline {z\,} \omega $$ is equal to
