JEE Advanced 2006 Question Paper with Solutions

Let $\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k$ and $\overrightarrow c = \widehat i + \widehat j - \widehat k.$ A vector in the plane of $\overrightarrow a$ and $\overrightarrow b$ whose projection on $\overrightarrow c$ is ${1 \over {\sqrt 3 }},$ is

A student performs an experiment for determination of g ( = 4 π 2 l T 2 ) . The error in length l is △ l and in the time T is △ T and n is number of times the reading is taken.The reading of g is most accurate for

In a screw gauge, the zero of main scale coincides with the fifth division of circular scale in figure (i).The circular division of screw gauge is 50. It moves 0.5 mm on main scale in one rotation.The diameter of the ball in figure (ii) is

Match the folowing : (A) $\,\,\,$ Two rays $x + y = \left| a \right|$ and $ax - y=1$ intersects each other in the $\,\,\,\,\,\,\,\,\,\,$ first quadrant in interval $a \in \left( {{a_0},\,\,\infty } \right),$ the value of ${{a_0}}$ is (B) $\,\,\,$ Point $\left( {\alpha ,\beta ,\gamma } \right)$ lies on the plane $x+y+z=2.$ $\,\,\,\,\,\,\,\,\,\,\,$ Let $\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k,\widehat k \times \left( {\widehat k \times \overrightarrow a } \right) = 0,$ then $\gamma =$ (C) $\,\,\,$ $\left| {\int\limits_0^1 {\left( {1 - {y^2}} \right)dy} } \right| + \left| {\int\limits_1^0 {\left( {{y^2} - 1} \right)dy} } \right|$ (D) $\,\,\,$ If $\sin A\,\,\sin B\,\,\sin C + \cos A\,\,\cos B = 1,$ then the value of $\sin C =$ (p) $\,\,\,$ $2$ (q) $\,\,\,$ ${4 \over 3}$ (r) $\,\,\,$ $\left| {\int\limits_0^1 {\sqrt {1 - xdx} } } \right| + \left| {\int\limits_{ - 1}^0 {\sqrt {1 + xdx} } } \right|$ (s) $\,\,\,$ $1$

A solid sphere of radius $R$ has moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$ . If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to $I$ , then the value of $r$ is equal to

According to Bohr's theory E n = Total energy, K n = Kinetic Energy, V n = Potential Energy, r n = Radius of n th orbit Match the following Column I (A) V n /K n = ? (B) If radius of n th orbit $\propto E_n^x,x = ?$ (C) Angular momentum in lowest orbital (D) ${1 \over {{r_n}}} \propto {Z^y},y = ?$ Column II (p) 0 (q) -1 (r) -2 (s) 1

MgSO 4 on reaction with NH 4 OH and Na 2 HPO 4 forms a white crystalline precipitate. What is its formula?

Let $a,\,b,\,c$ be the sides of triangle where $a \ne b \ne c$ and $\lambda \in R$ . If the roots of the equation ${x^2} + 2\left( {a + b + c} \right)x + 3\lambda \left( {ab + bc + ca} \right) = 0$ are real, then

ABCD is a square of side length 2 units. C 1 is the circle touching all the sides of the square ABCD and C 2 is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point. If P is any point of C 1 and Q is another point on C 2 , then P A 2 + P B 2 + P C 2 + P D 2 Q A 2 + Q B 2 + Q C 2 + Q D 2 is equal to

ABCD is a square of side length 2 units. $C_1$ is the circle touching all the sides of the square ABCD and $C_2$ is the circumcircle of square ABCD. L is a fixed line in the same plane and R is a fixed point. If a circle is such that it touches the line L and the circle $C_1$ externally, such that both the circles are on the same side of the line, then the locus of centre of the circle is

One angle of an isosceles $\Delta$ is ${120^ \circ }$ and radius of its incircle $= \sqrt 3$ . Then the area of the triangle in sq. units is

The equations of the common tangents to the parabola $y = {x^2}$ and $y = - {\left( {x - 2} \right)^2}$ is/are

For a twice differentiable function $f(x),g(x)$ is defined as $4\sqrt {65} g\left( x \right) = \left( {f'{{\left( x \right)}^2} + f''\left( x \right)} \right)\,\,f\left( x \right)$ on $\,\,\,\left[ {a,\,\,\,e} \right].$ If for $a < b < c < d < e,\,f\left( a \right) = 0,f\left( b \right) = 2,f\left( c \right) = - 1,f\left( d \right) = 2,f\left( e \right) = 0$ then find the minimum number of zeros of $g(x)$ .

$f(x)$ is cubic polynomial with $f(2)=18$ and $f(1)=-1$ . Also $f(x)$ has local maxima at $x=-1$ and $f'(x)$ has local minima at $x=0$ , then

A plane which is perpendicular to two planes $2x - 2y + z = 0$ and $x - y + 2z = 4,$ passes through $(1, -2, 1).$ The distance of the plane from the point $(1, 2, 2)$ is

Let the definite integral be defined by the formula $\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 2}\left( {f\left( a \right) + f\left( b \right)} \right).}$ For more accurate result for $c \in \left( {a,b} \right),$ we can use $\int\limits_a^b {f\left( x \right)dx = \int\limits_a^c {f\left( x \right)dx + \int\limits_c^b {f\left( x \right)dx = F\left( c \right)} } }$ so that for $c = {{a + b} \over 2},$ we get $\int\limits_a^b {f\left( x \right)dx = {{b - a} \over 4}\left( {f\left( a \right) + f\left( b \right) + 2f\left( c \right)} \right).}$ $\int\limits_0^{\pi /2} {\sin x\,dx = }$

A curve $y=f(x)$ passes through $(1,1)$ and at $P(x,y),$ tangent cuts the $x$ -axis and $y$ -axis at $A$ and $B$ respectively such that $BP:AP=3:1,$ then

The species present in solution when CO 2 is dissolved in water are

We have taken a saturated solution of AgBr. K sp of AgBr is 12 $\times$ 10 -14 . If 10 -7 mole of AgNO 3 are added to 1 litre of this solution find conductivity (specific conductance) of this solution in terms of 10 -7 S m -1 units. Given, molar conductance of Ag + , Br - and $NO_3^-$ are 6 $\times$ 10 -3 Sm 2 mol -1 , 8 $\times$ 10 -3 Sm 2 mol -1 and 7 $\times$ 10 -3 Sm 2 mol -1

75.2 g of C 6 H 5 OH (phenol) is dissolved in a solvent of K f = 14. If the depression in freezing point is 7 K then find the % of phenol that dimerises.