JEE MAIN - Mathematics (2004)

1
The domain of the function
$$f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}$$
回答
(B)
[2, 3)
2
The range of the function f(x) = $${}^{7 - x}{P_{x - 3}}$$ is
回答
(D)
{1, 2, 3}
3
If $$f:R \to S$$, defined by
$$f\left( x \right) = \sin x - \sqrt 3 \cos x + 1$$,
is onto, then the interval of $$S$$ is
回答
(A)
[-1, 3]
4
The graph of the function y = f(x) is symmetrical about the line x = 2, then
回答
(B)
$$f\left( {2 + x} \right) = f\left( {2 - x} \right)$$
5
Let $$f(x) = {{1 - \tan x} \over {4x - \pi }}$$, $$x \ne {\pi \over 4}$$, $$x \in \left[ {0,{\pi \over 2}} \right]$$.

If $$f(x)$$ is continuous in $$\left[ {0,{\pi \over 2}} \right]$$, then $$f\left( {{\pi \over 4}} \right)$$ is
回答
(C)
$$-{1 \over 2}$$
6
If $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}$$, then the value of $$a$$ and $$b$$, are
回答
(B)
$$a$$ = 1 and $$b$$ $$ \in R$$
7
Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.
Which of these is/are correct?
回答
(C)
only (a) and (b)
8
In a series of 2n observations, half of them equal $$a$$ and remaining half equal $$–a$$. If the standard deviation of the observations is 2, then $$|a|$$ equals
回答
(A)
2
9
Let $$\alpha ,\,\beta $$ be such that $$\pi < \alpha - \beta < 3\pi $$.
If $$sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$$ and $$\cos \alpha + \cos \beta = - {{27} \over {65}}$$ then the value of $$\cos {{\alpha - \beta } \over 2}$$ :
回答
(D)
$$ - {3 \over {\sqrt {130} }}$$
10
The value of $$\int\limits_{ - 2}^3 {\left| {1 - {x^2}} \right|dx} $$ is
回答
(D)
$${28 \over 3}$$
11
If $$x = {e^{y + {e^y} + {e^{y + .....\infty }}}}$$ , $$x > 0,$$ then $${{{dy} \over {dx}}}$$ is
回答
(C)
$${{1 - x} \over x}$$
12
A point on the parabola $${y^2} = 18x$$ at which the ordinate increases at twice the rate of the abscissa is
回答
(A)
$$\left( {{9 \over 8},{9 \over 2}} \right)$$
13
statement about the matrix $$A$$ is
回答
(A)
$${A^2} = 1$$
14
the inverse of matrix $$A$$, then $$\alpha $$ is
回答
(A)
$$5$$
15
$$\left| {\matrix{ {\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr {\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr {\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr } } \right|,$$ is
回答
(D)
$$0$$
16
If $$\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,} $$ then value of
$$(A, B)$$ is
回答
(B)
$$\left( { \cos \alpha ,\sin \alpha } \right)$$
17
$$\int {{{dx} \over {\cos x - \sin x}}} $$ is equal to
回答
(A)
$${1 \over {\sqrt 2 }}\log \left| {\tan \left( {{x \over 2} + {{3\pi } \over 8}} \right)} \right| + C$$
18
The eccentricity of an ellipse, with its centre at the origin, is $${1 \over 2}$$. If one of the directrices is $$x=4$$, then the equation of the ellipse is :
回答
(B)
$$3{x^2} + 4{y^2} = 12$$
19
The value of $$I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx} $$ is
回答
(C)
$$2$$
20
If $$\int\limits_0^\pi {xf\left( {\sin x} \right)dx = A\int\limits_0^{\pi /2} {f\left( {\sin x} \right)dx,} } $$ then $$A$$ is
回答
(B)
$$\pi $$
21
If $$f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {xg\left\{ {x\left( {1 - x} \right)} \right\}dx} $$
and $${I_2} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {g\left\{ {x\left( {1 - x} \right)} \right\}dx} ,$$ then the value of $${{{I_2}} \over {{I_1}}}$$ is
回答
(D)
$$2$$
22
The area of the region bounded by the curves
$$y = \left| {x - 2} \right|,x = 1,x = 3$$ and the $$x$$-axis is :
回答
(D)
$$1$$
23
Solution of the differential equation $$ydx + \left( {x + {x^2}y} \right)dy = 0$$ is
回答
(B)
$$ - {1 \over {xy}} + \log y = C$$
24
The probability that $$A$$ speaks truth is $${4 \over 5},$$ while the probability for $$B$$ is $${3 \over 4}.$$ The probability that they contradict each other when asked to speak on a fact is :
回答
(C)
$${7 \over 20}$$
25
A particle acted on by constant forces $$4\widehat i + \widehat j - 3\widehat k$$ and $$3\widehat i + \widehat j - \widehat k$$ is displaced from the point $$\widehat i + 2\widehat j + 3\widehat k$$ to the point $$\,5\widehat i + 4\widehat j + \widehat k.$$ The total work done by the forces is :
回答
(D)
$$40$$ units
26
Let $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w $$ be such that $$\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow v } \right|2,\,\,\,\left| {\overrightarrow w } \right|3.$$ If the projection $${\overrightarrow v }$$ along $${\overrightarrow u }$$ is equal to that of $${\overrightarrow w }$$ along $${\overrightarrow u }$$ and $${\overrightarrow v },$$ $${\overrightarrow w }$$ are perpendicular to each other then $$\left| {\overrightarrow u - \overrightarrow v + \overrightarrow w } \right|$$ equals :
回答
(C)
$${\sqrt {14} }$$
27
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-zero vectors such that no two of these are collinear. If the vector $$\overrightarrow a + 2\overrightarrow b $$ is collinear with $$\overrightarrow c $$ and $$\overrightarrow b + 3\overrightarrow c $$ is collinear with $$\overrightarrow a $$ ($$\lambda $$ being some non-zero scalar) then $$\overrightarrow a + 2\overrightarrow b + 6\overrightarrow c $$ equals to :
回答
(A)
$\overrightarrow{0}$
28
Let $${{T_r}}$$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, $$m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\,$$ then a - d equals
回答
(D)
0
29
If $$u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta } $$

then the difference between the maximum and minimum values of $${u^2}$$ is given by :
回答
(A)
$${\left( {a - b} \right)^2}$$
30
The value of $$I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx} $$ is
回答
(C)
$$2$$
31
If $$x = {e^{y + {e^y} + {e^{y + .....\infty }}}}$$ , $$x > 0,$$ then $${{{dy} \over {dx}}}$$ is
回答
(C)
$${{1 - x} \over x}$$
32
A point on the parabola $${y^2} = 18x$$ at which the ordinate increases at twice the rate of the abscissa is
回答
(A)
$$\left( {{9 \over 8},{9 \over 2}} \right)$$
33
If $$\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$$, then z lies on :
回答
(B)
the imaginary axis
34
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
回答
(B)
$${x^2} - 18x + 16 = 0$$
35
If $$\left( {1 - p} \right)$$ is a root of quadratic equation $${x^2} + px + \left( {1 - p} \right) = 0$$ then its root are
回答
(C)
$$ 0,-1$$
36
If one root of the equation $${x^2} + px + 12 = 0$$ is 4, while the equation $${x^2} + px + q = 0$$ has equal roots,
then the value of $$'q'$$ is
回答
(D)
$${{49} \over 4}$$
37
How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order
回答
(C)
360
38
The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is
回答
(B)
21
39
The coefficient of the middle term in the binomial expansion in powers of $$x$$ of $${\left( {1 + \alpha x} \right)^4}$$ and $${\left( {1 - \alpha x} \right)^6}$$ is the same if $$\alpha $$ equals
回答
(C)
$${{ - 3} \over {10}}$$
40
The coefficient of $${x^n}$$ in expansion of $$\left( {1 + x} \right){\left( {1 - x} \right)^n}$$ is
回答
(B)
$${\left( { - 1} \right)^n}\left( {1 - n} \right)$$
41
The equation of the straight line passing through the point $$(4, 3)$$ and making intercepts on the co-ordinate axes whose sum is $$-1$$ is :
回答
(A)
$${x \over 2} - {y \over 3} = 1$$ and $${x \over -2} +{y \over 1} = 1$$
42
Let $$A\left( {2, - 3} \right)$$ and $$B\left( {-2, 1} \right)$$ be vertices of a triangle $$ABC$$. If the centroid of this triangle moves on the line $$2x + 3y = 1$$, then the locus of the vertex $$C$$ is the line :
回答
(D)
$$2x + 3y = 9$$
43
If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference $$10\,\pi $$, then the equation of the circle is :
回答
(D)
$${x^2}\, + \,{y^2} - \,2x\, + \,2y - \,23\,\, = 0$$
44
If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2} = 4$$ orthogonally, then the locus of its centre is :
回答
(B)
$$2ax\, + 2by\, - ({a^2}\, + \,{b^2} + 4) = 0$$
45
Intercept on the line y = x by the circle $${x^2}\, + \,{y^2} - 2x = 0$$ is AB. Equation of the circle on AB as a diameter is :
回答
(D)
$$\,{x^2}\, + \,{y^2} - \,x\, - \,y\,\, = 0$$
46
If $$a \ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$, then :
回答
(D)
$${d^2} + {\left( {2b + 3c} \right)^2} = 0$$
47
Let $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ be non-zero vectors such that $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right|\left| {\overrightarrow c } \right|\overrightarrow a \,\,.$$ If $$\theta $$ is the acute angle between the vectors $${\overrightarrow b }$$ and $${\overrightarrow c },$$ then $$sin\theta $$ equals :
回答
(A)
$${{2\sqrt 2 } \over 3}$$
48
A line with direction cosines proportional to $$2,1,2$$ meets each of the lines $$x=y+a=z$$ and $$x+a=2y=2z$$ . The co-ordinates of each of the points of intersection are given by :
回答
(B)
$$\left( {3a,2a,3a} \right),\left( {a,a,a} \right)$$
49
If the straight lines
$$x=1+s,y=-3$$$$ - \lambda s,$$ $$z = 1 + \lambda s$$ and $$x = {t \over 2},y = 1 + t,z = 2 - t,$$ with parameters $$s$$ and $$t$$ respectively, are co-planar, then $$\lambda $$ equals :
回答
(D)
$$-2$$
50
Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :
回答
(C)
not symmetric