IIT-JEE 1983

The coefficient of \({x^4}\) in \({\left( {{x \over 2} - {3 \over {{x^2}}}} \right)^{10}}\) is

If \({\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + .....\) then \(a=..........\) and \(n =............\)

If \({\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ..... + {C_n}{x^n}\) then show that the sum of the products of the \({C_i}s\) taken two at a time, represented \(\sum\limits_{0 \le i < j \le n} {\sum {{C_i}{C_j}} }\) is equ...

Use mathematical Induction to prove : If \(n\) is any odd positive integer, then \(n\left( {{n^2} - 1} \right)\) is divisible by 24.

Given positive integers \(r > 1,\,n > 2\) and that the coefficient of \(\left( {3r} \right)\)th and \(\left( {r + 2} \right)\)th terms in the binomial expansion of \({\left( {1 + x} \right)^{2n}}\) are equal. Then

m men and n women are to be seated in a row so that no two women sit together. If \(m > n\), then show that the number of ways in which they can be seated is \(\,{{m!(m + 1)!} \over {(m - n + 1)!}}\)

Fifteen coupons are numbered \(1, 2 ........15,\) respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is \(9,\) is

Cards are drawn one by one at random from a well - shuffled full pack of \(52\) playing cards until \(2\) aces are obtained for the first time. If \(N\) is the number of cards required to be drawn, then show that $${P_r}\left\{ {N = n} \rig...

\(A, B, C\) are events such that
\(P\left( A \right) = 0.3,P\left( B \right) = 0.4,P\left( C \right) = 0.8\)
\(P\left( {AB} \right) = 0.08,P\left( {AC} \right) = 0.28;\,\,P\left( {ABC} \right) = 0.09\)
If $$P\left( {A \cup B \cup C} \right)...

If the letters of the word "Assassin" are written down at random in a row, the probability that no two S's occur together is \(1/35\)

The ex-radii \({r_1},{r_2},{r_3}\) of \(\Delta\)\(ABC\) are H.P. Show that its sides \(a, b, c\) are in A.P.

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is \({15^ \circ }\). The distance of the boat from the foot of the light house is

If one root of the quadratic equation \(a{x^2} + bx + c = 0\) is equal to the \(n\)-th power of the other, then show that
\(${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0\)$

Find all real values of \(x\) which satisfy \({x^2} - 3x + 2 > 0\) and \({x^2} - 2x - 4 \le 0\)

The equation \(2{x^2} + 3x + 1 = 0\) has an irrational root.

The rational number, which equals the number \(2\overline {357}\) with recurring decimal is

Find three numbers \(a,b,c\) between \(2\) and \(18\) such that
(i) their sum is \(25\)
(ii) the numbers \(2,\) \(a, b\) are consecutive terms of an A.P. and
(iii) the numbers \(b,c,18\) are consecutive terms of a G.P.

The coordinates of \(A, B, C\) are \((6, 3), (-3, 5), (4, -2)\) respectively, and \(P\) is any point \((x, y)\). Show that the ratio of the area of the triangles \(\Delta\) \(PBC\) and \(\Delta\)\(ABC\) is $$\left| {{{x + y - 2} \over 7...

The vertices of a triangle are \(\left[ {a{t_1}{t_2},\,\,a\left( {{t_1} + {t_2}} \right)} \right],\,\,\left[ {a{t_2}{t_3},a\left( {{t_2} + {t_3}} \right)} \right],\,\,\left[ {a{t_3}{t_1},\,a\left( {{t_3} + {t_1}} \right)} \right]\). Find ...

Given the points \(A\left( {0,4} \right)\) and \(B\left( {0, - 4} \right)\), the equation of the locus of the point \(P\left( {x,y} \right)\) such that \(\left| {AP - BP} \right| = 6\) is .............

The straight lines \(x + y = 0,\,3x + y - 4 = 0,\,x + 3y - 4 = 0\) form a triangle which is

The end \(A, B\) of a straight line segment of constant length \(c\) slide upon the fixed rectangular axes \(OX, OY\) respectively. If the rectangle \(OAPB\) be completed, then show that the locus of the foot of the perpendicular drawn from...

The straight line \(5x + 4y = 0\) passes through the point of intersection of the straight lines \(x + 2y - 10 = 0\) and \(2x + y + 5 = 0.\)

Show that \($16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi } \over {15}}} \right)\cos \left( {{{16\pi } \over {15}}} \right) = 1\)$

Find all solutions of \(4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x\)

If \(\tan \,A = \left( {1 - \cos B} \right)/\sin B,\) then \(tan2A = tan\,B\).

If \(X.A=0, X.B=0, X.C=0\) for some non-zero vector \(X,\) then \(\left[ {A\,B\,C} \right] = 0\)

A river is flowing from west to east at a speed of 5 meters per minute. A man on the south bank of the river, capable of swimming at 10 meters per minute in still water, wants to swim across the river in the shortest time. He should swim in...

A particle moves in a circle of radius R. In half the period of revolution its displacement is ________ and distance covered is _________.

Two balls of different masses are thrown vertically upwards with the same speed. They pass through the point of projection in their downward motion with the same speed. (Neglect air resistance)

Match the physical quantities given in column I with dimension expressed in terms of mass (M), length (L), time (T), and charge (Q) given in column II and write the correct answer against the matched quantity in a tabular form in your answe...