IIT-JEE 1987

The IUPAC name of the compound
CH2 = CH \(-\) CH(CH3)2

sp2 hybrid orbitals have equal s and p character.

In benzene, carbon uses all the three p-orbitals for hybridisation.

Hydrogen bond is maximum in

The brown ring complex compound is formulated as [Fe(H2O)5(NO)+]SO4. The oxidation state of iron is

(i) What is the weight of sodium bromate and molarity pf solution necessary to prepare 85.5 ml of 0.672 B solution when the half-cell reaction is
\(BrO_3^- + 6H^+ + 6e^- \to\) \(Br^- + 3H_2O\)
(ii) What would be the weight as well as mola...

The value of PV for 5.6 litres of an ideal gas is ________ RT, at NTP.

A spherical balloon of 21 cm diameter is to be filled up with hydrogen at N.T.P from a cylinder containing the gas at 20 atmospheres at 27oC. If the cylinder can hold 2.82 litres of water, calculate the number of balloons that can be filled...

Give reasons of the following:
Why is sodium chloride added during electrolysis of fused anhydrous magnesium chloride?

The electronegativity of the following elements increases in the order

In group IA, of alkali metals, the ionisation potential decreases on moving down the group. Therefore, lithium is a strongest reducing agent

Atomic radii of fluorine and neon in Angstrom units are respectively given by

The first ionisation potential in electron volts of nitrogen and oxygen atoms are respectively given by

Give reasons of the following :
Magnesium oxide is used for the lining of steel making furnace.

The metallic lustre exhibited by sodium is explained by

Sodium when burnt in excess of oxygen gives sodium oxide.

Find the point on the curve \(\,\,\,4{x^2} + {a^2}{y^2} = 4{a^2},\,\,\,4 < {a^2} < 8\)
that is farthest from the point \((0, -2)\).

Let \(f\) and \(g\) be increasing and decreasing functions, respectively from \(\left[ {0,\infty } \right)\) to \(\left[ {0,\infty } \right)\). Let \(h\left( x \right) = f\left( {g\left( x \right)} \right).\) If \(h\left( 0 \right) = 0,\) t...

The smallest positive root of the equation, \(\tan x - x = 0\) lies in

The set of all \(x\) for which \(in\left( {1 + x} \right) \le x\) is equal to ..........

Find the area bounded by the curves, \({x^2} + {y^2} = 25,\,4y = \left| {4 - {x^2}} \right|\) and \(x=0\) above the \(x\)-axis.

The area of the triangle formed by the tangents from the point (4, 3) to the circle \({x^2} + {y^2} = 9\) and the line joining their points of contact is...................

The circle \({x^2}\, + \,{y^2} - \,4x\, - 4y + \,4 = 0\) is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is $$x\, + \,y\, - xy\, + k\,{\left( {{x^2}\, + \,{y^2}...

Let a given line \(L_1\) intersects the x and y axes at P and Q, respectively. Let another line \(L_2\), perpendicular to \(L_1\), cut the x and y axes at R and S, respectively. Show that the locus of the point of intersection of the lines ...

If the expression
\(${{\left[ {\sin \left( {{x \over 2}} \right) + \cos {x \over 2} + i\,\tan \left( x \right)} \right]} \over {\left[ {1 + 2\,i\,\sin \left( {{x \over 2}} \right)} \right]}}\)$
is real, then the set of all possible values o...

If \({{{z_1}}}\) and \({{{z_2}}}\) are two nonzero complex numbers such that \(\left| {{z_1}\, + {z_2}} \right| = \left| {{z_1}} \right|\, + \left| {{z_2}} \right|\,\), then Arg \({z_1}\) - Arg \({z_2}\) is equal to

The value of \(\sum\limits_{k = 1}^6 {(\sin {{2\pi k} \over 7}} - i\,\cos \,{{2\pi k} \over 7})\) is

\(f\left( x \right) = \left| {\matrix{ {\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr {{{\cos }^2}x} & {{{\cos }^2}x} & {\cos e{c^2}x} \cr 1 & {{{\cos }^2}x} & {{{\cos }^2}x} \cr } } \right|.\)
Then $$\int\limi...

Evaluate :\(\,\,\int {\left[ {{{{{\left( {\cos 2x} \right)}^{1/2}}} \over {\sin x}}} \right]dx}\)

Prove by mathematical induction that \(- 5 - {{\left( {2n} \right)!} \over {{2^{2n}}{{\left( {n!} \right)}^2}}} \le {1 \over {{{\left( {3n + 1} \right)}^{1/2}}}}\) for all positive integers \(n\).

A man takes a step forward with probability \(0.4\) and backwards with probability \(0.6\) Find the probability that at the end of eleven steps he is one step away from the starting point.

A polygon of nine sides, each of length \(2\), is inscribed in a circle. The radius of the circle is .................

In a triangle, the lengths of the two larger sides are \(10\) and \(9\), respectively. If the angles are in \(AP\). Then the length of the third side can be

Find the set of all \(x\) for which \({{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}\)

If \(a,\,b,\,c,\,d\) and p are distinct real numbers such that
\($\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \left( {{b^2} + {c^2} + {d^2}} \right) \le 0\)$
then \(a,\,b,\,c,\,d\)

The sides of a triangle inscribed in a given circle subtend angles \(\alpha\), \(\beta\) and \(\gamma\) at the centre. The minimum value of the arithmetic mean of $$cos\left[ {\alpha + {\pi \over 2}} \right],\,\cos \left[ {\beta + {\p...

The solution set of the system of equations \(X + Y = {{2\pi } \over 3},\) \(cox\,x + cos\,y = {3 \over 2},\) where x and y are real, is _____.

The set of all \(x\) in the interval \(\left[ {0,\,\pi } \right]\) for which \(2\,{\sin ^2}x - 3\) \(\sin x + 1 \ge 0,\) is _____.

The number of all possible triplets \(\left( {{a_1},\,{a_2},\,{a_3}} \right)\) such that \({a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\left( x \right) = 0\,\) for all \(x\) is

Let \(b = 4\widehat i + 3\widehat j\) and \(\overrightarrow c\) be two vectors perpendicular to each other in the \(xy\)-plane. All vectors in the same plane having projecttions \(1\) and \(2\) along \(\overrightarrow b\) and $$\overright...

The number of vectors of unit length perpendicular to vectors \(\overrightarrow a = \left( {1,1,0} \right)\) and \(\overrightarrow b = \left( {0,1,1} \right)\) is

If \(A, B, C, D\) are any four points in space, prove that -
$$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = ...

If the vectors \(a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k\) and \(\widehat i + \widehat j + c\widehat k\)
\(\left( {a \ne b \ne c \ne 1} \right)\) are coplannar, then the value of $${1 \over {\left( ...

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that :