In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?
$$\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}$$, $$n \in N$$, ( [x] denotes the greatest integer less than or equal to x )
คำตอบ
(D)
does not exist
8
If $$f\left( 1 \right) = 1,{f'}\left( 1 \right) = 2,$$ then
$$\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}}$$ is
คำตอบ
(A)
$$2$$
9
$$f$$ is defined in $$\left[ { - 5,5} \right]$$ as
$$f\left( x \right) = x$$ if $$x$$ is rational
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = - x$$ if $$x$$ is irrational. Then
คำตอบ
(B)
$$f(x)$$ is discontinuous at every $$x,$$ except $$x = 0$$
10
If f(x + y) = f(x).f(y) $$\forall $$ x, y and f(5) = 2, f'(0) = 3, then
f'(5) is
คำตอบ
(C)
6
11
A triangle with vertices $$\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)$$ is :
คำตอบ
(A)
isosceles and right angled
12
Locus of mid point of the portion between the axes of
$$x$$ $$cos$$ $$\alpha + y\,\sin \alpha = p$$ where $$p$$ is constant is :
If $$y=f(x)$$ makes +$$ve$$ intercept of $$2$$ and $$0$$ unit on $$x$$ and $$y$$ axes and encloses an area of $$3/4$$ square unit with the axes then $$\int\limits_0^2 {xf'\left( x \right)dx} $$ is
คำตอบ
(D)
$$-3/4$$
14
If $$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},$$ then $$\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$$ is
คำตอบ
(A)
$${n^2}y$$
15
$${\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x,$$ then sin x is equal to :
คำตอบ
(A)
$${\tan ^2}\left( {{\alpha \over 2}} \right)$$
16
The maximum distance from origin of a point on the curve
$$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$
$$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$$ both $$a,b > 0$$ is
คำตอบ
(B)
$$a+b$$
17
If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then
$$\left| {\matrix{
a & b & {ax + b} \cr
b & c & {bx + c} \cr
{ax + b} & {bx + c} & 0 \cr
} } \right|$$ is equal to
The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $$3$$$$a$$ is :
คำตอบ
(C)
$${x^2}\, + \,{y^2} = 4{a^2}$$
23
The area bounded by the curves $$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$ and $$y = \left| {\ln \left| x \right|} \right|$$ is :
คำตอบ
(A)
$$4$$sq. units
24
The order and degree of the differential equation
$$\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$$ are
คำตอบ
(C)
$$(3,3)$$
25
The order and degree of the differential equation
$$\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$$ are
คำตอบ
(C)
$$(3,3)$$
26
A problem in mathematics is given to three students $$A,B,C$$ and their respective probability of solving the problem is $${1 \over 2},{1 \over 3}$$ and $${1 \over 4}.$$ Probability that the problem is solved is :
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(A)
$${3 \over 4}$$
27
$$A$$ and $$B$$ are events such that $$P\left( {A \cup B} \right) = 3/4$$,$$P\left( {A \cap B} \right) = 1/4,$$
$$P\left( {\overline A } \right) = 2/3$$ then $$P\left( {\overline A \cap B} \right)$$ is :
คำตอบ
(A)
$$5/12$$
28
If $$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$ and the angle between $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$\pi /6$$ then $${\left( {\overrightarrow a \times \overrightarrow b } \right)^2}$$ is equal to :
คำตอบ
(B)
$$16$$
29
If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then :
If $$\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3$$ thus what will be the value of $$\left| {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a } \right|,$$ given that $$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$ :
คำตอบ
(A)
$$25$$
31
$$\overrightarrow a = 3\widehat i - 5\widehat j$$ and $$\overrightarrow b = 6\widehat i + 3\widehat j$$ are two vectors and $$\overrightarrow c $$ is a vector such that $$\overrightarrow c = \overrightarrow a \times \overrightarrow b $$ then $$\left| {\overrightarrow a } \right|:\left| {\overrightarrow b } \right|:\left| {\overrightarrow c } \right|$$ =
คำตอบ
(B)
$$\sqrt {34} :\sqrt {45} :39$$
32
If the vectors $$\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$ and $$\widehat b = \widehat j$$ are such that $$\overrightarrow a ,\overrightarrow c $$ and $$\overrightarrow b $$ form a right handed system then $${\overrightarrow c }$$ is :
คำตอบ
(A)
$$z\widehat i - x\widehat k$$
33
Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :
คำตอบ
(D)
720
34
The period of $${\sin ^2}\theta $$ is
คำตอบ
(B)
$$\pi $$
35
Which one is not periodic?
คำตอบ
(B)
$$\cos \sqrt x + {\cos ^2}x$$
36
z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ then z equals
คำตอบ
(B)
$$ - \overline \omega $$
37
If $$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by :
คำตอบ
(C)
$${\mathop{\rm Re}\nolimits} (z) > 3$$
38
The locus of the centre of a circle which touches the circle $$\left| {z - {z_1}} \right| = a$$ and$$\left| {z - {z_2}} \right| = b\,$$ externally
($$z,\,{z_1}\,\& \,{z_2}\,$$ are complex numbers) will be :
คำตอบ
(B)
a hyperbola
39
If $$\alpha \ne \beta $$ but $${\alpha ^2} = 5\alpha - 3$$ and $${\beta ^2} = 5\beta - 3$$ then the equation having $$\alpha /\beta $$ and $$\beta /\alpha \,\,$$ as its roots is
คำตอบ
(A)
$$3{x^2} - 19x + 3 = 0$$
40
Product of real roots of equation $${t^2}{x^2} + \left| x \right| + 9 = 0$$
คำตอบ
(A)
is always positive
41
Difference between the corresponding roots of $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0$$ is same and $$a \ne b,$$ then
คำตอบ
(A)
$$a + b + 4 = 0$$
42
If $$p$$ and $$q$$ are the roots of the equation $${x^2} + px + q = 0,$$ then
คำตอบ
(A)
$$p = 1,\,\,q = - 2$$
43
If $$a,\,b,\,c$$ are distinct $$ + ve$$ real numbers and $${a^2} + {b^2} + {c^2} = 1$$ then $$ab + bc + ca$$ is
คำตอบ
(A)
less than 1
44
The coefficients of $${x^p}$$ and $${x^q}$$ in the expansion of $${\left( {1 + x} \right)^{p + q}}$$ are
คำตอบ
(A)
equal
45
The positive integer just greater than $${\left( {1 + 0.0001} \right)^{10000}}$$ is
คำตอบ
(D)
3
46
Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is :
คำตอบ
(C)
374
47
The sum of integers from 1 to 100 that are divisible by 2 or 5 is :
คำตอบ
(B)
3050
48
Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are :
คำตอบ
(D)
216
49
If 1, $${\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)$$ are in A.P. then x equals
คำตอบ
(B)
$$1 - \,{\log _3}\,4\,$$
50
l, m, n are the $${p^{th}}$$, $${q^{th}}$$ and $${r^{th}}$$ term of a G.P all positive, $$then\,\left| {\matrix{
{\log \,l} & p & 1 \cr
{\log \,m} & q & 1 \cr
{\log \,n} & r & 1 \cr
} } \right|\,equals$$
คำตอบ
(D)
0
51
Fifth term of a GP is 2, then the product of its 9 terms is
คำตอบ
(B)
512
52
Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is
คำตอบ
(B)
3/5
53
If the chord y = mx + 1 of the circle $${x^2}\, + \,{y^2} = 1$$ subtends an angle of measure $${45^ \circ }$$ at the major segment of the circle then value of m is :
