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Paper was held on Tue, Apr 11, 2000 9:00 AM
Chemistry
1
The number of nodal planes in a px orbital is
2
Amongst the following identify the species with an atom in +6 oxidation state
3
The electronic configuration of an element is 1s2, 2s2 2p6, 3s2 3p6 3d5, 4s1 This represents its
4
The correct order of radii is
5
The correct order of acidic strength is
6
Amongst H2O, H2S, H2Se and H2Te, the one with the highest boiling point is
7
Read the following statement and explanation and answer as per the options given below
ASSERTION : The first ionization energy of Be is greater than that of B.
REASON : 2p orbital is lower in energy than 2s.
ASSERTION : The first ionization energy of Be is greater than that of B.
REASON : 2p orbital is lower in energy than 2s.
8
Molecular shape of SF4, CF4 and XeF4 are
9
The hybridisation of atomic orbitals of nitrogen in $$NO_2^+$$, $$NO_3^-$$ and $$NH_4^+$$ are
Mathematics
1
For all $$x \in \left( {0,1} \right)$$
2
Let $$f\left( x \right) = \left\{ {\matrix{
{\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr
{1,} & {for} & {x = 0} \cr
} } \right.$$ then at $$x=0$$, $$f$$ has
3
If $$f\left( x \right) = \left\{ {\matrix{
{{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr
{2,} & {otherwise,} \cr
} } \right.$$ then $$\int\limits_{ - 2}^3 {f\left( x \right)dx = } $$
4
Let $$g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,} $$ where f is such that
$${1 \over 2} \le f\left( t \right) \le 1,$$ for $$t \in \left[ {0,1} \right]$$ and $$\,0 \le f\left( t \right) \le {1 \over 2},$$ for $$t \in \left[ {1,2} \right]$$.
Then $$g(2)$$ satisfies the inequality
$${1 \over 2} \le f\left( t \right) \le 1,$$ for $$t \in \left[ {0,1} \right]$$ and $$\,0 \le f\left( t \right) \le {1 \over 2},$$ for $$t \in \left[ {1,2} \right]$$.
Then $$g(2)$$ satisfies the inequality
5
The value of the integral $$\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx} $$ is :
6
If $${x^2} + {y^2} = 1,$$ then
7
If the vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ form the sides $$BC,$$ $$CA$$ and $$AB$$ respectively of a triangle $$ABC,$$ then
8
Let the vectors $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ be such that
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = 0.$$ Let $${P_1}$$ and $${P_2}$$ be planes determined
by the pairs of vectors $$\overrightarrow a .\overrightarrow b $$ and $$\overrightarrow c .\overrightarrow d $$ respectively. Then the angle between $${P_1}$$ and $${P_2}$$ is
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = 0.$$ Let $${P_1}$$ and $${P_2}$$ be planes determined
by the pairs of vectors $$\overrightarrow a .\overrightarrow b $$ and $$\overrightarrow c .\overrightarrow d $$ respectively. Then the angle between $${P_1}$$ and $${P_2}$$ is
9
If $$\overrightarrow a \,,\,\overrightarrow b $$ and $$\overrightarrow c $$ are unit coplanar vectors, then the scalar triple product $$\left[ {2\overrightarrow a - \overrightarrow b ,2\overrightarrow b - \overrightarrow c ,2\overrightarrow c - \overrightarrow a } \right] = $$
10
If the normal to the curve $$y = f\left( x \right)$$ and the point $$(3, 4)$$ makes an angle $${{{3\pi } \over 4}}$$ with the positive $$x$$-axis, then $$f'\left( 3 \right) = $$
11
The incentre of the triangle with vertices $$\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)$$ and $$\left( {2,\,0} \right)$$ is
12
If $${z_1},\,{z_2}$$ and $${z_3}$$ are complex numbers such that $$\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right| = 1,$$ then $$\left| {{z_1} + {z_2} + {z_3}} \right|$$ is
13
If $$\arg \left( z \right) < 0,$$ then $$\arg \left( { - z} \right) - \arg \left( z \right) = $$
14
If b > a, then the equation (x - a) (x - b) - 1 = 0 has
15
If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation
16
For the equation $$3{x^2} + px + 3 = 0$$. p > 0, if one of the root is square of the other, then p is equal to
17
If $$\alpha \,\text{and}\,\beta $$ $$(\alpha \, < \,\beta )$$ are the roots of the equation $${x^2} + bx + c = 0\,$$, where $$c < 0 < b$$, then
18
For $$2 \le r \le n,\,\,\,\,\left( {\matrix{
n \cr
r \cr
} } \right) + 2\left( {\matrix{
n \cr
{r - 1} \cr
} } \right) + \left( {\matrix{
n \cr
{r - 2} \cr
} } \right) = $$
19
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
20
Consider an infinite geometric series with first term a and common ratio $$r$$. If its sum is 4 and the second term is 3/4, then
21
Let $$PS$$ be the median of the triangle with vertices $$P(2, 2),$$ $$Q(6, -1)$$ and $$R(7, 3).$$ The equation of the line passing through $$(1, -1)$$ and parallel to $$PS$$ is
22
Let $$f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)$$. Then $$f\left( \theta \right)$$ is
23
The triangle PQR is inscribed in the circle $${x^2}\, + \,\,{y^2} = \,25$$. If Q and R have co-ordinates (3, 4) and ( - 4, 3) respectively, then $$\angle \,Q\,P\,R$$ is equal to
24
If the circles $${x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + \,k\, = \,0$$ intersect orthogonally, then k is
25
If $$x + y = k$$ is normal to $${y^2} = 12x,$$ then $$k$$ is
26
If the line $$x - 1 = 0$$ is the directrix of the parabola $${y^2} - kx + 8 = 0,$$ then one of the values of $$k$$ is
27
In a triangle $$ABC$$, $$2ac\,\sin {1 \over 2}\left( {A - B + C} \right) = $$
28
In a triangle $$ABC$$, let $$\angle C = {\pi \over 2}$$. If $$r$$ is the inradius and $$R$$ is the circumradius of the triangle, then $$2(r+R)$$ is equal to
29
A pole stands vertically inside a triangular park $$\Delta ABC$$. If the angle of elevation of the top of the pole from each corner of the park is same, then in $$\Delta ABC$$ the foot of the pole is at the
30
Consider the following statements in $$S$$ and $$R$$
$$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?
$$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?
31
Let $$f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.} $$ Then $$f$$ decreases in the interval
Physics
1
The dimension of $$\left( {{1 \over 2}} \right){\varepsilon _0}{E^2}$$
( $${\varepsilon _0}$$ : permittivity of free space, E electric field )
( $${\varepsilon _0}$$ : permittivity of free space, E electric field )
2
A wind-powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed v, the electrical power output will be proportional to
3
The period of oscillation of a simple pendulum of length $$L$$ suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination $$\alpha$$, is given by