IIT-JEE 1998

Number of divisor of the form 4\(n\)\(+ 2\left( {n \ge 0} \right)\) of the integer 240 is

If \(x > 1,y > 1,z > 1\) are in G.P., then \({1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}\) are in

Let \({T_r}\) be the \({r^{th}}\) term of an A.P., for \(r=1, 2, 3, ....\) If for some positive integers \(m\), \(n\) we have
\({T_m} = {1 \over n}\) and \({T_n} = {1 \over m},\) then \({T_n} = {1 \over m},\) equals

Let \(n\) be an odd integer. If \(\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,}\) for every value of \(\theta ,\) then

If \(\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)\) and \(S\left( {a,b} \right)\) are the vertices of a parrallelogram \(PQRS,\) then

Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

If the vertices \(P, Q, R\) of a triangle \(PQR\) are rational points, which of the following points of the triangle \(PQR\) is (are) always rational point(s)?

The diagonals of a parralleogram \(PQRS\) are along the lines \(x + 3y = 4\) and \(6x - 2y = 7\). Then \(PQRS\) must be a.

Which of the following number(s) is /are rational?

The number of values of \(x\,\,\) in the interval \(\left[ {0,\,5\pi } \right]\) satisfying the equation \(3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0\) is

Prove that \(\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha\)

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

Which of the following expressions are meaningful?

For any two vectors \(u\) and \(v,\) prove that
(a) \({\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}\) and
(b) $$\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v...

For three vectors \(u,v,w\) which of the following expression is not equal to any of the remaining three?

If \(a = i + j + k,\overrightarrow b = 4i + 3j + 4k\) and \(c = i + \alpha j + \beta k\) are linearly dependent vectors and \(\left| c \right| = \sqrt 3 ,\) then

Let [\({\mathrm\varepsilon}_\mathrm o\)] denote the dimentional formula of the permittivity of the vacuum, and [\(\mu_o\)] that of the permeability of the vacuum.If M = mass, L = length, T = time and I = electric current,

The SI unit of inductance, the henry can be written as