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