List of practice Questions

\( (\vec{a}+2\vec{b}-\vec{c}) \cdot ((\vec{a}-\vec{b}) \times (\vec{a}-\vec{b}-\vec{c})) = \)

Variance of the following discrete frequency distribution is \begin{tabular}{|l|c|c|c|c|c|} \hline Class Interval & 0-2 & 2-4 & 4-6 & 6-8 & 8-10
\hline Frequency (\(f_i\)) & 2 & 3 & 5 & 3 & 2
\hline \end{tabular}

For a positive real number p, if the perpendicular distance from a point \( -\vec{i} + p\vec{j} - 3\vec{k} \) to the plane \( \vec{r} \cdot (2\vec{i} - 3\vec{j} + 6\vec{k}) = 7 \) is 6 units, then p =

In \( \triangle ABC \), if \( r = 3 \) and \( R = 5 \) then \( \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \)

If \( \vec{a} = 2\vec{i} - \vec{j} + 6\vec{k} \); \( \vec{b} = \vec{i} - \vec{j} + \vec{k} \) and \( \vec{c} = 3\vec{j} - \vec{k} \), then \( \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \)

If the vector \( \vec{v} = \vec{i} - 7\vec{j} + 2\vec{k} \) is along the internal bisector of the angle between the vectors \( \vec{a} \) and \( \vec{b} = -2\vec{i} - \vec{j} + 2\vec{k} \) and the unit vector along \( \vec{a} \) is \( \hat{a} = x\vec{i} + y\vec{j} + z\vec{k} \) then \( x = \)

Let \( \vec{a} = 2\vec{i} + \vec{j} - 2\vec{k} \) and \( \vec{b} = \vec{i} + \vec{j} \) be two vectors. If \( \vec{c} \) is a vector such that \( \vec{a} \cdot \vec{c} = |\vec{c}| \), \( |\vec{c} - \vec{a}| = 2\sqrt{2} \) and the angle between \( \vec{a} \times \vec{b} \) and \( \vec{c} \) is \( 30^\circ \), then \( |(\vec{a} \times \vec{b}) \times \vec{c| = \)}

An aeroplane is flying at a constant speed, parallel to the horizontal ground at a height of 5 kms. A person on the ground observed that the angle of elevation of the plane is changed from \(15^\circ\) to \(30^\circ\) in the duration of 50 seconds, then the speed of the plane (in kmph) is

If \( \text{sech}^{-1}x = \log 2 \) and \( \text{cosech}^{-1}y = -\log 3 \), then \( (x+y) = \)

If the sides a,b,c of the triangle ABC are in harmonic progression, then \( \text{cosec}^2 A/2, \text{cosec}^2 B/2, \text{cosec}^2 C/2 \) are in

If \( x \ne (2n+1)\frac{\pi{4} \), then the general solution of \( \cos x + \cos 3x = \sin x + \sin 3x \) is}

If \( \frac{1}{2} \sin^{-1} \left( \frac{3\sin 2\theta}{5+4\cos 2\theta} \right) = \tan^{-1 x \) then \( x = \)}

If \(2\sin x - \cos 2x = 1\), then \( (3 - 2\sin^2x) = \)

If \( \left(\frac{\sin 3\theta}{\sin \theta}\right)^2 - \left(\frac{\cos 3\theta}{\cos \theta}\right)^2 = a \cos b\theta \), then \( a : b = \)

If \( \alpha, \beta \) are the acute angles such that \( \frac{\sin \alpha}{\sin \beta} = \frac{6}{5} \) and \( \frac{\cos \alpha}{\cos \beta} = \frac{9}{5\sqrt{5}} \) then \( \sin \alpha = \)

If \( \frac{ax+5}{(x^2+b)(x+3) = \frac{x+21}{12(x^2+b)} + \frac{c}{12(x+3)} \), then \( b^2 = \)}

The coefficient of \( x^3 \) in the power series expansion of \( \frac{1+4x-3x^2}{(1+3x)^3 \) is}

Let 'a' be a non-zero real number. If the equation whose roots are the squares of the roots of the cubic equation \( x^3 - ax^2 + ax - 1 = 0 \) is identical with this cubic equation, then 'a' =

If the number of diagonals of a regular polygon is 35, then the number of sides of the polygon is

If four letters are chosen from the letters of the word ASSIGNMENT and are arranged in all possible ways to form 4 letter words (with or without meaning), then total number of such words that can be formed is

When the roots of \( x^3 + \alpha x^2 + \beta x + 6 = 0 \) are increased by 1, if one of the resultant values is the least root of \( x^4 - 6x^3 + 11x^2 - 6x = 0 \), then

If \( {}^{27}P_{r+7} = 7722 \cdot {}^{25}P_{r+4} \), then r =