JEE Advance - Mathematics (2007)
- 1Let $${H_1},{H_2},....,{H_n}$$ 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 $$i=1,2,....,n$$ becauseSTATEMENT-2: $$\sum\limits_{i = 1}^n {P\left( {{H_i}} \right)} = 1.$$
Trả lời(D)Statement-1 is False, Statement-2 is True - 3Let $$(x, y)$$ 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 $$\left( {4{x^2} - 1} \right)\left( {{y^2} - 1} \right) = 0$$Trả lờiABCD - 5If 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
Trả lời(C)two points - 6If 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
Trả lời(A)$${1 \over e}$$ - 7If 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
Trả lời(A)$$\left( {0,{1 \over e}} \right)$$ - 8Let $$F(x)$$ 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 $$x$$. becauseSTATEMENT-2: $${\sin ^2}\left( {x + \pi } \right) = {\sin ^2}x$$ for all real $$x$$.
Trả lời(D)Statement-1 is False, Statement-2 is True. - 9Match 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) $${{\pi \over 2}}$$Trả lời(C)(A) - (s), (B) - (s), (C) - (p), (D) - (r) - 11STATEMENT - 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 $$t$$.Trả lời(B)Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 - 13Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ be unit vectors such that $${\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}$$ Which one of the following is correct ?Trả lời(B)$$\overrightarrow a \times \overrightarrow b = b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a \ne \overrightarrow 0 $$
- 14Consider 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.Trả lời(A)A -> r, B -> q, C -> p, D -> s - 15Consider 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.
Trả lời(D)Statement-1 is False, Statement-2 is True. - 16STATEMENT-1: $$\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 .$$ because
STATEMENT-2: $$\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $$ and $$\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 \,\,.$$Trả lời(C)Statement-1 is True, Statement-2 is False - 27A 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 isTrả lời(D)$$\left( {3 + 4i} \right){e^{i\pi /4}}$$
- 28Let $$O\left( {0,0} \right),P\left( {3,4} \right),Q\left( {6,0} \right)$$ 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 $$R$$ areTrả lời(C)$$\left( {3,{4 \over 3}} \right)$$
- 31Match 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 hyperbolaColumn $$II$$
(p) have a common tangent
(q) have a common normal
(r) do not have a common tangent
(s) do not have a common normalTrả lời(A)A - p, q; B - p, q; C - q, r; D - q, r - 36Some 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) GMeMs ,
G $$ \to $$ universal gravitational constant, Me $$ \to $$ mass of the earth, Ms $$ \to $$ 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, Me $$ \to $$ mass of the earth, Re $$ \to $$ radius of the earthColumn II
(p) (volt) (coulomb) (metre)
(q) (kilogram) (metre)3 (second)−2
(r) (meter)2(second)−2
(s) (farad) (volt)2 (kg)−1
Trả lời(A)A $$ \to $$ (p) & (q), B $$ \to $$ (r) & (s), C $$ \to $$ (r) & (s), D $$ \to $$ (r) & (s)