AIEEE 2007

Let \(L\) be the line of intersection of the planes \(2x+3y+z=1\) and \(x+3y+2z=2.\) If \(L\) makes an angle \(\alpha\) with the positive \(x\)-axis, then cos \(\alpha\) equals

If \((2,3,5)\) is one end of a diameter of the sphere \({x^2} + {y^2} + {z^2} - 6x - 12y - 2z + 20 = 0,\) then the coordinates of the other end of the diameter are

If a line makes an angle of \(\pi /4\) with the positive directions of each of \(x\)-axis and \(y\)-axis, then the angle that the line makes with the positive direction of the \(z\)-axis is :

If \(p\) and \(q\) are positive real numbers such that \({p^2} + {q^2} = 1\), then the maximum value of \((p+q)\) is

The function \(f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)\) is an incresing function in

A value of \(c\) for which conclusion of Mean Value Theorem holds for the function \(f\left( x \right) = {\log _e}x\) on the interval \(\left[ {1,3} \right]\) is

The area enclosed between the curves \({y^2} = x\) and \(y = \left| x \right|\) is :

In the binomial expansion of \({\left( {a - b} \right)^n},\,\,\,n \ge 5,\) the sum of \({5^{th}}\) and \({6^{th}}\) terms is zero, then \(a/b\) equals

The sum of the series \({}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}}\) is

Consider a family of circles which are passing through the point \((-1, 1)\) and are tangent to \(x\)-axis. If \((h, k)\) are the coordinate of the centre of the circles, then the set of values of \(k\) is given by the interval :

If \(\,\left| {z + 4} \right|\,\, \le \,\,3\,\), then the maximum value of \(\left| {z + 1} \right|\) is :

The solution for \(x\) of the equation \(\int\limits_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }} = {\pi \over 2}}\) is

Let \(F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),\) where \(f\left( x \right) = \int\limits_l^x {{{\log t} \over {1 + t}}dt,}\) Then \(F(e)\) equals

Let \(I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx}\) and \(J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .\) Then which one of the following is true?

The differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is :

The largest interval lying in \(\left( { - {\pi \over 2},{\pi \over 2}} \right)\) for which the function
\(f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {{x \over 2} - 1} \right)\)\(+ \log \left( {\cos x} \right)\),
is define...

A tower stands at the centre of a circular park. \(A\) and \(B\) are two points on the boundary of the park such that \(AB(=a)\) subtends an angle of \({60^ \circ }\) at the foot of the tower, and the angle of elevation of the top of the to...

For the Hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\) , which of the following remains constant when \(\alpha\) varies\(=\)?

The normal to a curve at \(P(x,y)\) meets the \(x\)-axis at \(G\). If the distance of \(G\) from the origin is twice the abscissa of \(P\), then the curve is a :

\(\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}}\) equals

If sin-1\(\left( {{x \over 5}} \right)\) + cosec-1\(\left( {{5 \over 4}} \right)\) = \({\pi \over 2}\), then the value of x is :

Let \(f:R \to R\) be a function defined by
\(f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}\), then which of the following is true?

The function \(f:R/\left\{ 0 \right\} \to R\) given by
\(f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}\)
can be made continuous at \(x\) = 0 by defining \(f\)(0) as

Let \(A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|.\) If \(\,\,\left| {{A^2}} \right| = 25,\) then \(\,\left| \alpha \right|\) equals

If \(D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|\) for \(x \ne 0,y \ne 0,\) then \(D\) is :

The equation of a tangent to the parabola \({y^2} = 8x\) is \(y=x+2\). The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is :

The set S = {1, 2, 3, ........., 12} is to be partitioned into three sets A, B, C of equal size. Thus \(A \cup B \cup C = S,\,A \cap B = B \cap C = A \cap C = \phi\). The number of ways to partition S is

A pair of fair dice is thrown independently three times. The probability of getting a score of exactly \(9\) twice is :

Two aeroplanes \({\rm I}\) and \({\rm I}\)\({\rm I}\) bomb a target in succession. The probabilities of \({\rm I}\) and \({\rm I}\)\({\rm I}\) scoring a hit correctly are \(0.3\) and \(0.2,\) respectively. The second plane will bomb only if...

If the difference between the roots of the equation \({x^2} + ax + 1 = 0\) is less than \(\sqrt 5 ,\) then the set of possible values of \(a\) is

The sum of series \({1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......\) upto infinity is

In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals

The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys
and girls combined is 50. The percentage of boys in the class is

Let A \(\left( {h,k} \right)\), B\(\left( {1,1} \right)\) and C \((2, 1)\) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is \(1\) square unit, then the set of values which \('k'\) can take...

If one of the lines of \(m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0\) is a bisector of angle between the lines \(xy = 0,\) then \(m\) is :

Let \(P = \left( { - 1,0} \right),\,Q = \left( {0,0} \right)\) and \(R = \left( {3,3\sqrt 3 } \right)\) be three point. The equation of the bisector of the angle \(PQR\) is :

If \(\widehat u\) and \(\widehat v\) are unit vectors and \(\theta\) is the acute angle between them, then \(2\widehat u \times 3\widehat v\) is a unit vector for :

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k\) and \(\overrightarrow c = x\widehat i + \left( {x - 2} \right)\widehat j - \widehat k\,\,.\) If the vectors $$\ov...