AIEEE 2008

The line passing through the points \((5,1,a)\) and \((3, b, 1)\) crosses the \(yz\)-plane at the point \(\left( {0,{{17} \over 2}, - {{ - 13} \over 2}} \right)\) . Then

If the straight lines \(\,\,\,\,\,\) \(\,\,\,\,\,\) \({{x - 1} \over k} = {{y - 2} \over 2} = {{z - 3} \over 3}\) \(\,\,\,\,\,\) and\(\,\,\,\,\,\) \({{x - 2} \over 3} = {{y - 3} \over k} = {{z - 1} \over 2}\) intersects at a point, then th...

Suppose the cubic \({x^3} - px + q\) has three distinct real roots
where \(p>0\) and \(q>0\). Then which one of the following holds?

How many real solutions does the equation
\({x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0\) have?

The area of the plane region bounded by the curves \(x + 2{y^2} = 0\) and \(\,x + 3{y^2} = 1\) is equal to :

Statement - 1 : \(\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.}\)
Statement - 2 : $$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( ...

The differential equation of the family of circles with fixed radius \(5\) units and centre on the line \(y = 2\) is :

The point diametrically opposite to the point \(P(1, 0)\) on the circle \({x^2} + {y^2} + 2x + 4y - 3 = 0\) is :

The conjugate of a complex number is \({1 \over {i - 1}}\) then that complex number is :

The solution of the differential equation
\({{dy} \over {dx}} = {{x + y} \over x}\) satisfying the condition \(y(1)=1\) is :

A focus of an ellipse is at the origin. The directrix is the line \(x=4\) and the eccentricity is \({{1 \over 2}}\). Then the length of the semi-major axis is :

Let \(f:N \to Y\) be a function defined as f(x) = 4x + 3 where
Y = { y \(\in\) N, y = 4x + 3 for some x \(\in\) N }.
Show that f is invertible and its inverse is

\(AB\) is a vertical pole with \(B\) at the ground level and \(A\) at the top. \(A\) man finds that the angle of elevation of the point \(A\) from a certain point \(C\) on the ground is \({60^ \circ }\). He moves away from the pole along th...

The value of \(\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}}\) is

The value of \(cot\left( {\cos e{c^{ - 1}}{5 \over 3} + {{\tan }^{ - 1}}{2 \over 3}} \right)\) is :

Let \(f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr } } \right.\)
Then which one of the following is true?

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”,
and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p.
State...

The statement \(p \to \left( {q \to p} \right)\) is equivalent to

Let \(a, b, c\) be any real numbers. Suppose that there are real numbers \(x, y, z\) not all zero such that \(x=cy+bz,\) \(y=az+cx,\) and \(z=bx+ay.\) Then \({a^2} + {b^2} + {c^2} + 2abc\) is equal to :

Let \(A\) be a square matrix all of whose entries are integers.
Then which one of the following is true?

Let \(A\) be \(a\,2 \times 2\) matrix with real entries. Let \(I\) be the \(2 \times 2\) identity matrix. Denote by tr\((A)\), the sum of diagonal entries of \(a\). Assume that \({a^2} = I.\)
Statement-1 : If \(A \ne I\) and \(A \ne - I\)...

A parabola has the origin as its focus and the line \(x=2\) as the directrix. Then the vertex of the parabola is at :

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?

In a shop there are five types of ice-cream available. A child buys six ice-cream.
Statement - 1: The number of different ways the child can buy the six ice-cream is \({}^{10}{C_5}\).
Statement - 2: The number of different ways the ch...

A die is thrown. Let \(A\) be the event that the number obtained is greater than \(3.\) Let \(B\) be the event that the number obtained is less than \(5.\) Then \(P\left( {A \cup B} \right)\) is :

It is given that the events \(A\) and \(B\) are such that
\(P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2}\) and \(P\left( {B|A} \right) = {2 \over 3}.\) Then \(P(B)\) is :

STATEMENT - 1 : For every natural number \(n \ge 2,\)
\(${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .\)$
STATEMENT - 2 : For every natural number \(n \ge 2,\),
$$$\sqrt {n\left( {n + 1} \right...

The quadratic equations \({x^2} - 6x + a = 0\) and \({x^2} - cx + 6 = 0\) have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

Let R be the real line. Consider the following subsets of the plane \(R \times R\) :
\(S = \left\{ {(x,y):y = x + 1\,\,and\,\,0 < x < 2} \right\}\)
\(T = \left\{ {(x,y): x - y\,\,\,is\,\,an\,\,{\mathop{\rm int}} eger\,} \right\}\),
Which...

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following
gives possible values of a and b?

The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :

The non-zero vectors are \({\overrightarrow a ,\overrightarrow b }\) and \({\overrightarrow c }\) are related by \({\overrightarrow a = 8\overrightarrow b }\) and \({\overrightarrow c = - 7\overrightarrow b \,\,.}\) Then the angle betwee...

The vector \(\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\) lies in the plane of the vectors
\(\overrightarrow b = \widehat i + \widehat j\) and \(\overrightarrow c = \widehat j + \widehat k\) and bisects the ...