JEE Advanced 2007 Question Paper with Solutions

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

Let the vectors $\overrightarrow {PQ} ,\,\,\overrightarrow {QR} ,\,\,\overrightarrow {RS} ,\,\,\overrightarrow {ST} ,\,\,\overrightarrow {TU} ,$ and $\overrightarrow {UP} ,$ represent the sides of a regular hexagon. 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 $\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 \,\,.$

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 ?

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 e M 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

STATEMENT 1 The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times the product of its pressure and its volume. Because STATEMENT 2 The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

The question contains STATEMENT - 1 (Assertion) and STATEMENT - 2 (Reason) and has 4 choices (a), (b), (c) and (d) out of which ONLY ONE is correct. STATEMENT - 1 Alkali metals dissolve in liquid ammonia to give blue solutions. because STATEMENT - 2 Alkali metals is liquid ammonia give solvated species of the type [M(NH 3 ) n ] + (M = alkali metals)

Among the following the paramagnetic compound is

The species having bond order different from that in CO is

Consider a titration of potassium dichromate solution with acidified Mohr’s salt solution using diphenylamine as indicator. The number of moles of Mohr’s salt required per mole of dichromate is

The lines ${L_1}:y - x = 0$ and ${L_2}:2x + y = 0$ intersect the line ${L_3}:y + 2 = 0$ at $P$ and $Q$ respectively. The bisector of the acute angle between ${L_1}$ and ${L_2}$ intersects ${L_3}$ at $R$ . 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.

A man walks a distance of 3 units from the origin towards the north-east ( N 45 ∘ E ) direction. From there, he walks a distance of 4 units towards the north-west ( N 45 ∘ W ) direction to reach a point P. Then the position of P in the Argand plane is

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

Let V r denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let T r = V r + 1 − V r − 2 a n d Q r = T r + 1 − T r f o r r = 1 , 2 , . . . The sum V 1 + V 2 +...+ V n is

Let O ( 0 , 0 ) , P ( 3 , 4 ) , Q ( 6 , 0 ) be the vertices of the triangles O P Q . The point R inside the triangle O P Q is such that the triangles O P R , P Q R , O Q R are of equal area. The coordinates of R are

The phases of the light wave at c , d , e and f are ϕ c , ϕ d , ϕ e and ϕ f respectively. It is given that ϕ c ≠ ϕ f .

Two wires each carrying a steady current I are shown in four configurations in Column I. Some of the resulting effects are described in Column II. Match the statements in Column I with the statements in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS. .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; overflow:hidden;padding:10px 5px;word-break:normal;} .tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px; font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;} .tg .tg-c3ow{border-color:inherit;text-align:center;vertical-align:top} .tg .tg-7btt{border-color:inherit;font-weight:bold;text-align:center;vertical-align:top} .tg .tg-0pky{border-color:inherit;text-align:left;vertical-align:top} Column I Column II (A) Point P is situated midway between the wires. (P) The magnetic fields (B) at P due to the currents in the wire are in same direction. (B) Point P is situated at the mid-point of the line joining the centers of the circular wires, which have same radii. (Q) The magnetic fields (B) at P due to the currents in the wires are in opposite directions. (C) Point P is situated at the mid-point of the line joining the centers of the circular wires, which have same radii. (R) There is no magnetic field at P. (D) Point P is situated at the common center of the wires. (S) The wires repel each other.

A magnetic field B → = B 0 j ^ exists in the region a < x < 2 a and B → = − B 0 j ^ , in the region 2 a < x < 3 a , where B 0 is a positive constant. A positive point charge moving with a velocity v → = v 0 i ^ , where v 0 is a positive constant, enters the magnetic field at x = a . The trajectory of the charge in this region can be like,

STATEMENT 1 A vertical iron rod has a coil of wire wound over it at the bottom end. An alternating current flows in the coil. The rod goes through a conducting ring as shown in the figure. The ring can float at a certain height above the coil. Because STATEMENT 2 In the above situation, a current is induced in the ring which interacts with the horizontal component of the magnetic field to produce an average force in the upward direction.

Speed of the light is

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

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

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.

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 ( x ) = 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

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

This question contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has 4 choices (a), (b), (c), (d) out of which ONLY ONE is correct. STATEMENT-1 : Band gap in germanium is small. because STATEMENT-2 : The energy gap of each germanium atomic energy level is infinitesimally small.

Consider a reaction aG + bH $\to$ Products. When concentration of both the reactants G and H is doubled, the rate increases by eight times. However, when concentration of G is doubled keeping the concentration of H fixed, the rate is doubled. The overall order of the reaction is