1
IIT-JEE 1997
Subjective
+2
-0
Calculate the equilibrium constant for the reaction
Fe2+ + Ce4+ $$\leftrightharpoons$$ Fe3+ + Ce3+
(given $$E_{C{e^{4 + }}/C{e^{3 + }}}^o$$ = 1.44 V; $$E_{F{e^{3 + }}/F{e^{2 + }}}^o$$ = 0.68 V)
2
IIT-JEE 1997
Fill in the Blanks
+2
-0
The real roots of the equation $$\,{\cos ^7}x + {\sin ^4}x = 1$$ in the interval $$\left( { - \pi ,\pi } \right)$$ are ...., ...., and ______.
3
IIT-JEE 1997
Subjective
+5
-0
Let $${z_1}$$ and $${z_2}$$ be roots of the equation $${z^2} + pz + q = 0\,$$ , where the coefficients p and q may be complex numbers. Let A and B represent $${z_1}$$ and $${z_2}$$ in the complex plane. If $$\angle AOB = \alpha \ne 0\,$$ and OA = OB, where O is the origin, prove that $${p^2} = 4q\,{\cos ^2}\left( {{\alpha \over 2}} \right)$$.
4
IIT-JEE 1997
Subjective
+5
-0
Prove that the values of the function $${{\sin x\cos 3x} \over {\sin 3x\cos x}}$$ do not lie between $${1 \over 3}$$ and 3 for any real $$x.$$
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