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 )
Απάντηση
(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
Απάντηση
(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
Απάντηση
(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
Απάντηση
(C)
6
11
A triangle with vertices $$\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)$$ is :
Απάντηση
(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
Απάντηση
(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
Απάντηση
(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 :
Απάντηση
(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
Απάντηση
(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 :
Απάντηση
(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 :
Απάντηση
(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
Απάντηση
(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
Απάντηση
(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 :
Απάντηση
(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 :
Απάντηση
(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 :
Απάντηση
(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 $$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$ :
Απάντηση
(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 $$\left| {\overrightarrow a } \right|:\left| {\overrightarrow b } \right|:\left| {\overrightarrow c } \right|$$ =
Απάντηση
(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 :
Απάντηση
(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 :
Απάντηση
(D)
720
34
The period of $${\sin ^2}\theta $$ is
Απάντηση
(B)
$$\pi $$
35
Which one is not periodic?
Απάντηση
(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
Απάντηση
(B)
$$ - \overline \omega $$
37
If $$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by :
Απάντηση
(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 :
Απάντηση
(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
Απάντηση
(A)
$$3{x^2} - 19x + 3 = 0$$
40
Product of real roots of equation $${t^2}{x^2} + \left| x \right| + 9 = 0$$
Απάντηση
(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
Απάντηση
(A)
$$a + b + 4 = 0$$
42
If $$p$$ and $$q$$ are the roots of the equation $${x^2} + px + q = 0,$$ then
Απάντηση
(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
Απάντηση
(A)
less than 1
44
The coefficients of $${x^p}$$ and $${x^q}$$ in the expansion of $${\left( {1 + x} \right)^{p + q}}$$ are
Απάντηση
(A)
equal
45
The positive integer just greater than $${\left( {1 + 0.0001} \right)^{10000}}$$ is
Απάντηση
(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 :
Απάντηση
(C)
374
47
The sum of integers from 1 to 100 that are divisible by 2 or 5 is :
Απάντηση
(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 :
Απάντηση
(D)
216
49
If 1, $${\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)$$ are in A.P. then x equals
Απάντηση
(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, $$then\,\left| {\matrix{
{\log \,l} & p & 1 \cr
{\log \,m} & q & 1 \cr
{\log \,n} & r & 1 \cr
} } \right|\,equals$$
Απάντηση
(D)
0
51
Fifth term of a GP is 2, then the product of its 9 terms is
Απάντηση
(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
Απάντηση
(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 :
