JEE Advance - Mathematics (2007)

1
Let $${H_1},{H_2},....,{H_n}$$ be mutually exclusive and exhaustive events with $$P\left( {{H_1}} \right) > 0,i = 1,2,.....,n.$$ Let $$E$$ be any other event with $$0 < P\left( E \right) < 1.$$
STATEMENT-1:
$$P\left( {{H_1}|E} \right) > P\left( {E|{H_1}} \right).P\left( {{H_1}} \right)$$ for $$i=1,2,....,n$$ because
STATEMENT-2: $$\sum\limits_{i = 1}^n {P\left( {{H_i}} \right)} = 1.$$
Respondre
(D)
Statement-1 is False, Statement-2 is True
2
Let $$ABCD$$ be a quadrilateral with area $$18$$, with side $$AB$$ parallel to the side $$CD$$ and $$2AB=CD$$. Let $$AD$$ be perpendicular to $$AB$$ and $$CD$$. If a circle is drawn inside the quadrilateral $$ABCD$$ touching all the sides, then its radius is
Respondre
(B)
$$2$$
3
Let $$(x, y)$$ be such that $${\sin ^{ - 1}}\left( {ax} \right) + {\cos ^{ - 1}}\left( y \right) + {\cos ^{ - 1}}\left( {bxy} \right) = {\pi \over 2}$$.
Column $$I$$
(A) If $$a=1$$ and $$b=0,$$ then $$(x, y)$$
(B) If $$a=1$$ and $$b=1,$$ then $$(x, y)$$
(C) If $$a=1$$ and $$b=2,$$ then $$(x, y)$$
(D) If $$a=2$$ and $$b=2,$$ then $$(x, y)$$

Column $$II$$
(p) lies on the circle $${x^2} + {y^2} = 1$$
(q) lies on $$\left( {{x^2} - 1} \right)\left( {{y^2} - 1} \right) = 0$$
(r) lies on $$y=x$$
(s) lies on $$\left( {4{x^2} - 1} \right)\left( {{y^2} - 1} \right) = 0$$
Respondre
A
B
C
D
4
The tangent to the curve $$y = {e^x}$$ drawn at the point $$\left( {c,{e^c}} \right)$$ intersects the line joining the points $$\left( {c - 1,{e^{c - 1}}} \right)$$ and $$\left( {c + 1,{e^{c + 1}}} \right)$$
Respondre
(A)
on the left of $$x=c$$
5
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
The line $$y=x$$ meets $$y = k{e^x}$$ for $$k \le 0$$ at
Respondre
(C)
two points
6
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
The positive value of $$k$$ for which $$k{e^x} - x = 0$$ has only one root is
Respondre
(A)
$${1 \over e}$$
7
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
For $$k>0$$, the set of all values of $$k$$ for which $$k{e^x} - x = 0$$ has two distinct roots is
Respondre
(A)
$$\left( {0,{1 \over e}} \right)$$
8
Let $$F(x)$$ be an indefinite integral of $$si{n^2}x.$$
STATEMENT-1: The function $$F(x)$$ satisfies $$F\left( {x + \pi } \right) = F\left( x \right)$$
for all real $$x$$. because

STATEMENT-2: $${\sin ^2}\left( {x + \pi } \right) = {\sin ^2}x$$ for all real $$x$$.
Respondre
(D)
Statement-1 is False, Statement-2 is True.
9
Match the integrals in Column $$I$$ with the values in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.
Column $$I$$
(A) $$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}} $$
(B) $$\int\limits_0^1 {{{dx} \over {\sqrt {1 - {x^2}} }}} $$
(C) $$\int\limits_2^3 {{{dx} \over {1 - {x^2}}}} $$
(D) $$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}} $$

Column $$II$$
(p) $${1 \over 2}\log \left( {{2 \over 3}} \right)$$
(q) $$2\log \left( {{2 \over 3}} \right)$$
(r) $${{\pi \over 3}}$$
(s) $${{\pi \over 2}}$$
Respondre
(C)
(A) - (s), (B) - (s), (C) - (p), (D) - (r)
10
One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is
Respondre
(C)
$${2 \over 5}$$
11
STATEMENT - 1: for eachreal $$t$$, there exists a point $$c$$ in $$\left[ {t,t + \pi } \right]$$ such that $$f'\left( c \right) = 0$$ because
STATEMENT - 2: $$f\left( t \right) = f\left( {t + 2\pi } \right)$$ for each real $$t$$.
Respondre
(B)
Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
12
The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$ - {\lambda ^2}\widehat i + \widehat j + \widehat k,$$ $$\widehat i - {\lambda ^2}\widehat j + \widehat k$$ and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$$ are coplanar, is
Respondre
(C)
two
13
Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ be unit vectors such that $${\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}$$ Which one of the following is correct ?
Respondre
(B)
$$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a \ne \overrightarrow 0 $$
14
Consider the following linear equations $$ax+by+cz=0;$$ $$\,\,\,$$ $$bx+cy+az=0;$$ $$\,\,\,$$ $$cx+ay+bz=0$$
Match the conditions/expressions in Column $$I$$ with statements in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS.$$

$$\,\,\,$$ Column $$I$$
(A)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$
(B)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(C)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(D)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$

$$\,\,\,$$ Column $$II$$
(p)$$\,\,\,$$ the equations represents planes meeting only at asingle point
(q)$$\,\,\,$$ the equations represents the line $$x=y=z.$$
(r)$$\,\,\,$$ the equations represent identical planes.
(s) $$\,\,\,$$ the equations represents the whole of the three dimensional space.
Respondre
(A)
A -> r, B -> q, C -> p, D -> s
15
Consider the planes $$3x-6y-2z=15$$ and $$2x+y-2z=5.$$
STATEMENT-1: The parametric equations of the line of intersection of the given planes are $$x=3+14t,y=1+2t,z=15t.$$ because

STATEMENT-2: The vector $${14\widehat i + 2\widehat j + 15\widehat k}$$ is parallel to the line of intersection of given planes.
Respondre
(D)
Statement-1 is False, Statement-2 is True.
16
STATEMENT-1: $$\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 .$$ because
STATEMENT-2: $$\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $$ and $$\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 \,\,.$$
Respondre
(C)
Statement-1 is True, Statement-2 is False
17
Which one of the following statements is correct ?
Respondre
(C)
$${G_1} = {G_2}\, = {G_3} = ...$$
18
If $$\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,$$ then all the values of $${z \over {1 - {z^2}}}$$ lie on
Respondre
(D)
the y-axis
19
in the interval $$\left[ {0,2\pi } \right]$$
Respondre
(C)
two
20
Let $$\alpha ,\,\beta $$ be the roots of the equation $${x^2} - px + r = 0$$ and $${\alpha \over 2},\,2\beta $$ be the roots of the equation $${x^2} - qx + r = 0$$. Then the value of $$r$$
Respondre
(D)
$${2 \over 9}\left( {2p - q} \right)\left( {2q - p} \right)$$
21
The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is
Respondre
(C)
96
22
The sum $${V_1}$$+$${V_2}$$ +...+$${V_n}$$ is
Respondre
(B)
$${1 \over {12}}n(n + 1)\,(3{n^2} + n + 2)$$
23
$${T_r}$$ is always
Respondre
(D)
a composite number
24
Which one of the following statements is correct ?
Respondre
(B)
$${H_1} < {H_2}\, < {H_3} < ...$$
25
Which one of the following statements is correct ?
Respondre
(A)
$${A_1} > {A_2}\, > {A_3} > ...$$
26
Which one of the following is a correct statement?
Respondre
(B)
$${Q_1},\,\,{Q_2},\,\,{Q_3},...$$ are A.P. with common difference 6
27
A man walks a distance of 3 units from the origin towards the north-east ($$N\,{45^ \circ E }$$) direction. From there, he walks a distance of 4 units towards the north-west $$\left( {N\,{{45}^ \circ }\,W} \right)$$ direction to reach a point P. Then the position of P in the Argand plane is
Respondre
(D)
$$\left( {3 + 4i} \right){e^{i\pi /4}}$$
28
Let $$O\left( {0,0} \right),P\left( {3,4} \right),Q\left( {6,0} \right)$$ be the vertices of the triangles $$OPQ$$. The point $$R$$ inside the triangle $$OPQ$$ is such that the triangles $$OPR$$, $$PQR$$, $$OQR$$ are of equal area. The coordinates of $$R$$ are
Respondre
(C)
$$\left( {3,{4 \over 3}} \right)$$
29
Statement-1: The ratio $$PR$$ : $$RQ$$ equals $$2\sqrt 2 :\sqrt 5 $$. because
Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.
Respondre
(C)
Statement-1 is True, Statement-2 is False.
30
A hyperbola, having the transverse axis of length $$2\sin \theta ,$$ is confocal with the ellipse $$3{x^2} + 4{y^2} = 12.$$ Then its equation is
Respondre
(A)
$${x^2}\cos e{c^2}\theta - {y^2}{\sec ^2}\theta = 1$$
31
Match the statements in Column $$I$$ with the properties in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.
Column $$I$$
(A) Two intersecting circles
(B) Two mutually external circles
(C) Two circles, one strictly inside the other
(D) Two branches vof a hyperbola

Column $$II$$
(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normal
Respondre
(A)
A - p, q; B - p, q; C - q, r; D - q, r
32
The ratio of the areas of the triangles $$PQS$$ and $$PQR$$ is
Respondre
(C)
$$1:4$$
33
The radius of the incircle of the triangle $$PQR$$ is
Respondre
(D)
$$2$$
34
STATEMENT-1: The curve $$y = {{ - {x^2}} \over 2} + x + 1$$ is symmetric with respect to the line $$x=1$$. because
STATEMENT-2: A parabola is symmetric about its axis.
Respondre
(A)
Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
35
$${{{d^2}x} \over {d{y^2}}}$$ equals
Respondre
(D)
$$ - \left( {{{{d^2}y} \over {d{x^2}}}} \right){\left( {{{dy} \over {dx}}} \right)^{ - 3}}$$
36
Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II.
Column I

(A) GM_eM_s ,
G $$ \to $$ universal gravitational constant, M_e $$ \to $$ mass of the earth, M_s $$ \to $$ mass of the Sun

(B) $${{3RT} \over M}$$,
R $$ \to $$ universal gas constant, T $$ \to $$ absolute temperature, M $$ \to $$ molar mass

(C) $${{{F^2}} \over {{q^2}{B^2}}}$$ ,
F $$ \to $$ force, q $$ \to $$ charge, B $$ \to $$ magnetic field

(D) $${{G{M_e}} \over {{R_e}}}$$,
G $$ \to $$ universal gravitational constant, M_e $$ \to $$ mass of the earth, R_e $$ \to $$ radius of the earth

Column II

(p) (volt) (coulomb) (metre)

(q) (kilogram) (metre)³ (second)⁻²

(r) (meter)²(second)⁻²

(s) (farad) (volt)² (kg)⁻¹
Respondre
(A)
A $$ \to $$ (p) & (q), B $$ \to $$ (r) & (s), C $$ \to $$ (r) & (s), D $$ \to $$ (r) & (s)