IIT-JEE 1998

The geometry and the type of hybrid orbital present about the central atom in BF3 is

Read the following Assertion and Reason and answer as per the options given below
Assetion: LiCl is predominantly a covalent compound
Reason : Electronegativity difference between Li and Cl is too small

Interpret the non-linear shape of H2S molecule and non-planar shape of PCl3 using valence shell electron pair repulsion (VSEPR) theory. (Atomic numbers : H = 1, P = 15, S = 16, Cl = 17)

The rate constant of a reaction is 1.5 \(\times\) 107 s-1 at 50oC and 4.5 \(\times\) 107 s-1 at 100oC. Evaluate the Arrhenius parameters A and Ea.

Calculate the equilibrium constant for the reaction:
2Fe3+ + 3I- \(\leftrightharpoons\) 2Fe2+ + \(I_3^-\). The standard reduction potentials in acidic conditions are 0.78 V and 0.54 V respectively for Fe3+ | Fe2+ and \(I_3^-\) | I- couples.

Find the solubility product of a saturated solution of Ag2CrO4 in water at 298 K if the emf of the cell Ag|Ag+ (satd. Ag2CrO4 soln.) || Ag+ (0.1 M) | Ag is 0.164 V at 298 K.

Hydrogen peroxide acts both as an oxidising and as a reducing agent in alkaline solution towards certain first row transition metal ions. Illustrate both these properties of H2O2 using chemical equations.

Highly pure dilute solution of sodium in liquid ammonia

Work out the following using chemical equation :
Chlorination of calcium hydroxide produces bleaching powder.

A solution of a nonvolatile solute in water freezes at -0.30oC. The vapour pressure of pure water at 298 K s 23.51 mm Hg and Kf for water is 1.86 K kg mol-1. Calculate the vapour pressure of this solution at 298 K.

An aqueous solution containing 0.10 g KIO3 (formula weight = 214.0) was treated with an excess of KI solution. The solution was acidified with HCl. The liberated I2 consumed 45.0 mL of thiosulphate solution to decolourise the blue strach-io...

The orbital diagram in which the Aufbau principle is violated is

Decrease in atomic number is observed during

Which of the following statement(s) is (are) correct?

ASSERTION:
Nuclide \({}_{13}^{30}Al\) is less stable than \({}_{20}^{40}Ca\)
REASON:
Nuclides having odd number of protons and neutrons are generally unstable.

The energy of an electron in the first Bohr orbit of H atom is -13.6 eV. The possible energy value(s) of the excited state(s) for electrons in Bohr orbits of hydrogen is (are)

Suppose \(f(x)\) is a function satisfying the following conditions
(a) \(f(0)=2,f(1)=1\),
(b) \(f\)has a minimum value at \(x=5/2\), and
(c) for all \(x\),
$$$f'\left( x \right) = \matrix{
{2ax} & {2ax - 1} & {2ax + b + 1} \cr
b ...

If \(f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},\) for every real number \(x\), then the minimum value of \(f\)

Let \(h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}\) for every real number \(x\). Then

A curve \(C\) has the property that if the tangent drawn at any point \(P\) on \(C\) meets the co-ordinate axes at \(A\) and \(B\), then \(P\) is the mid-point of \(AB\). The curve passes through the point \((1, 1)\). Determine the equation...

The number of values of \(x\) where the function
\(f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)\) attains its maximum is

If the circle \({x^2}\, + \,{y^2} = \,{a^2}\) intersects the hyperbola \(xy = {c^2}\) in four points \(P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R\,({x_3},\,{y_3}),\,S\,({x_4},\,{y_4}),\) then

The number of common tangents to the circles \({x^2}\, + \,{y^2} = 4\) and \({x^2}\, + \,{y^2}\, - 6x\, - 8y = 24\) is

\(C_1\) and \(C_2\) are two concentric circles, the radius of \(C_2\) being twice that of \(C_1\). From a point P on \(C_2\), tangents PA and PB are drawn to \(C_1\). Prove that the centroid of the triangle PAB lies on \(C_1\).

The value of the sum \(\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})\) , where i = \(\sqrt { - 1}\), equals

If \(\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy\) , then

If \({\omega}\) is an imaginary cube root of unity, then \({(1\, + \omega \, - {\omega ^2})^7}\) equals

If \(\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} }\) then the value of \(f(1)\) is

Prove that \(\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.\)
Hence or otherwise, evaluate the integral
\(\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.}\)

Let \(f\left( x \right) = x - \left[ x \right],\) for every real number \(x\), where \(\left[ x \right]\) is the integral part of \(x\). Then \(\int_{ - 1}^1 {f\left( x \right)\,dx}\) is

The order of the differential equation whose general solution is given by
\(y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},\) where
\({C_1},{C_2},{C_3},{C_4},{C_5},\) are arbitrary constants, is

If\(\,\,\,\) \(y = {{a{x^2}} \over {\left( {x - a} \right)\left( {x - b} \right)\left( {x - c} \right)}} + {{bx} \over {\left( {x - b} \right)\left( {x - c} \right)}} + {c \over {x - c}} + 1\),
prove that $${{y'} \over y} = {1 \over x}\lef...

The number of values of \(c\) such that the straight line \(y=4x + c\) touches the curve \(\left( {{x^2}/4} \right) + {y^2} = 1\) is

If \(P=(x, y)\), \({F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)\) and \(16{x^2} + 25{y^2} = 400,\) then \(P{F_1} + P{F_2}\) equals

The angle between a pair of tangents drawn from a point \(P\) to the parabola \({y^2} = 4ax\) is \({45^ \circ }\). Show that the locus of the point \(P\) is a hyperbola.

If \({a_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}},\,\,\,then\,\,\,\sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}}} }\) equals

Let \(p\) be a prime and \(m\) a positive integer. By mathematical induction on \(m\), or otherwise, prove that whenever \(r\) is an integer such that \(p\) does not divide \(r\), \(p\) divides \({}^{np}{C_r},\)
[Hint: You may use the fact ...

An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

If from each of the three boxes containing \(3\) white and \(1\) black, \(2\) white and \(2\) black, \(1\) white and \(3\) black balls, one ball is drawn at random, then the probability that \(2\) white and \(1\) black ball will be drawn is

If \(\overline E\) and \(\overline F\) are the complementary events of events \(E\) and \(F\) respectively and if \(0 < P\left( F \right) < 1,\) then

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

Three players, \(A,B\) and \(C,\) toss a coin cyclically in that order (that is \(A, B, C, A, B, C, A, B,...\)) till a head shows. Let \(p\) be the probability that the coin shows a head. Let \(\alpha ,\,\,\,\beta\) and \(\gamma\) be, res...

If \(E\) and \(F\) are events with \(P\left( E \right) \le P\left( F \right)\) and \(P\left( {E \cap F} \right) > 0,\) then

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

Let \({C_1}\) and \({C_2}\) be the graphs of the functions \(y = {x^2}\) and \(y = 2x,\) \(0 \le x \le 1\) respectively. Let \({C_3}\) be the graph of a function \(y=f(x),\) \(0 \le x \le 1,\) \(f(0)=0.\) For a point \(P\) on \({C_1},\) let...

A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals

Let \({A_0}{A_1}{A_2}{A_3}{A_4}{A_5}\) be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \({A_0}{A_1},{A_0}{A_2}\) and \({A_0}{A_4}\) is

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose \({60^ \circ }\) and \({30^ \circ }\) are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is ...

Prove that a triangle \(ABC\) is equilateral if and only if \(\tan A + \tan B + \tan C = 3\sqrt 3\).

If in a triangle \(PQR\), \(\sin P,\sin Q,\sin R\) are in \(A.P.,\) then