IIT-JEE 1983

A vector \(\overrightarrow A\) has components \({A_1},{A_2},{A_3}\) in a right -handed rectangular Cartesian coordinate system \(oxyz.\) The coordinate system is rotated about the \(x\)-axis through an angle \({\pi \over 2}.\) Find the co...

The points with position vectors \(60i+3j,\) \(40i-8j,\) \(ai-52j\) are collinear if

The unit vector perpendicular to the plane determined by \(P\left( {1, - 1,2} \right),\,Q\left( {2,0, - 1} \right)\) and \(R\left( {0,2,1} \right)\) is ...........

The volume of the parallelopiped whose sides are given by
\(\overrightarrow {OA} = 2i - 2j,\,\overrightarrow {OB} = i + j - k,\,\overrightarrow {OC} = 3i - k,\) is

The area of the triangle whose vertices are \(A(1, -1, 2), B(2, 1, -1), C(3, -1, 2)\) is ..........

Show that \(1+x\) \(In\left( {x + \sqrt {{x^2} + 1} } \right) \ge \sqrt {1 + {x^2}}\) for all \(x \ge 0\)

Find the coordinates of the point on the curve \(y = {x \over {1 + {x^2}}}\)
where the tangent to the curve has the greatest slope.

If \(y = a\,\,In\,x + b{x^2} + x\) has its extreamum values at \(x=-1\) and \(x=2\), then

The function \(y = 2{x^2} - In\,\left| x \right|\) is monotonically increasing for values of \(x\left( {x \ne 0} \right)\) satisfying the inequalities ......... and monotonically decreasing for values of \(x\) satisfying the inequalities .....

If \(x-r\) is a factor of the polynomial \(f\left( x \right) = {a_n}{x^4} + ..... + {a_0},\) repeated \(m\) times \(\left( {1 < m \le n} \right)\), then \(r\) is a root of \(\left( x \right) = 0\) repeated \(m\) times.

If \(a+b+c=0\), then the quadratic equation \(3a{x^2} + 2bx + c = 0\) has

The normal to the curve \(\,x = a\left( {\cos \theta + \theta \sin \theta } \right)\), \(y = a\left( {\sin \theta - \theta \cos \theta } \right)\) at any point \('\theta '\) is such that

The larger of \(\cos \left( {In\,\,\theta } \right)\) and \(In\) \(\left( {\cos \,\,\theta } \right)\) If \({e^{ - \pi /2}} < \theta < {\pi \over 2}\) is ..................

\(AB\) is a diameter of a circle and \(C\) is any point on the circumference of the circle. Then

Find the area bounded by the \(x\)-axis, part of the curve \(y = \left( {1 + {8 \over {{x^2}}}} \right)\) and
the ordinates at \(x=2\) and \(x=4\). If the ordinate at \(x=a\) divides the area into two equal parts, find \(a\).

The point of intersection of the line 4x - 3y - 10 = 0 and the circle \({x^2} + {y^2} - 2x + 4y - 20 = 0\) are ........................and ...................

The equation of the circle passing through (1, 1) and the points of intersection of \({x^2} + {y^2} + 13x - 3y = 0\) and \(2{x^2} + 2{y^2} + 4x - 7y - 25 = 0\) is

The centre of the circle passing through the point (0, 1) and touching the curve \(\,y = {x^2}\) at (2, 4) is

Through a fixed point (h, k) secants are drawn to the circle \(\,{x^2}\, + \,{y^2} = \,{r^2}\). Show that the locus of the mid-points of the secants intercepted by the circle is \(\,{x^2}\, + \,{y^2}\) = \(hx + ky\).

Prove that the complex numbers \({{z_1}}\), \({{z_2}}\) and the origin form an equilateral triangle only if \(z_1^2 + z_2^2 - {z_1}\,{z_2} = 0\).

If \(z = x + iy\) and \(\omega = \left( {1 - iz} \right)/\left( {z - i} \right),\) then \(\,\left| \omega \right| = 1\) implies that, in the complex plane,

The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order if and only if

Evaluate : \(\int\limits_0^{\pi /4} {{{\sin x + \cos x} \over {9 + 16\sin 2x}}dx}\)

The value of the integral \(\int\limits_0^{\pi /2} {{{\sqrt {\cot x} } \over {\sqrt {\cot x} + \sqrt {\tan x} }}dx}\) is

If \(\left( {a + bx} \right){e^{y/x}} = x,\) then prove that \({x^3}{{{d^2}y} \over {d{x^2}}} = {\left( {x{{dy} \over {dx}} - y} \right)^2}\)

The derivative of an even function is always an odd function.

Evaluate : \(\int {{{\left( {x - 1} \right){e^x}} \over {{{\left( {x + 1} \right)}^3}}}dx}\)

The value of \(\tan \left[ {{{\cos }^{ - 1}}\left( {{4 \over 5}} \right) + {{\tan }^{ - 1}}\left( {{2 \over 3}} \right)} \right]\) is

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

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