List of practice Questions

If the line joining points $ \vec{r}_1 = \hat{i} + 2\hat{j} $ and $ \vec{r}_2 = \hat{j} - 2\hat{k} $ intersects the plane through the points $ \vec{A} = 2\hat{i} - \hat{j},\ \vec{B} = -2\hat{j} + 3\hat{k},\ \vec{C} = \hat{k} - 2\hat{i} $ at $ T $, then find $ \vec{r}_T \cdot (\hat{i} + \hat{j} + \hat{k}) $

The vector equation of a plane passing through the line of intersection of the planes $$ \vec{r} \cdot (\hat{i} - 2\hat{k}) = 3,\quad \vec{r} \cdot (\hat{j} + \hat{k}) = 5 $$ and also passing through the point $ \vec{r} = \hat{i} + 2\hat{j} + 3\hat{k} $ is:

If $ \vec{a} = \hat{i} + \hat{j}, \vec{b} = 2\hat{j} - \hat{k} $ are two vectors such that $ \vec{r} \times \vec{a} = \vec{b} \times \vec{a}, \vec{r} \times \vec{b} = \vec{a} \times \vec{b} $, then the unit vector in the direction of $ \vec{r} $ is:

If $ \tan\left( \frac{\pi}{4} + \alpha \right) = \tan^3\left( \frac{\pi}{4} + \beta \right) $, then compute: $$ \tan(\alpha + \beta) \cot(\alpha - \beta) = ? $$

If $ A + B + C + D = 2\pi $, then $$ \sin A + \sin B + \sin C + \sin D = ? $$

If $ \theta = \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) + \tan^{-1} \left( \frac{1}{13} \right) + \tan^{-1} \left( \frac{1}{21} \right) + \tan^{-1} \left( \frac{1}{31} \right) $, then $ \tan \theta = ? $

In triangle $ ABC $, if $ C = 120^\circ $, $ c = \sqrt{19} $, and $ b = 3 $, then $ a = ? $

If a team of 4 persons is to be selected out of 4 married couples to play mixed doubles tennis game, then the number of ways of forming a team in which no married couple appears is

Evaluate: $$ \sin \frac{\pi}{12} \cdot \sin \frac{2\pi}{12} \cdot \sin \frac{3\pi}{12} \cdot \sin \frac{4\pi}{12} \cdot \sin \frac{5\pi}{12} \cdot \sin \frac{6\pi}{12} $$

Two values of $ (-8 - 8\sqrt{3}i)^{1/4} $ are

Let $ f(x) = x^2 + 2bx + 2c^2 $ and $ g(x) = -x^2 - 2cx + b^2 $, $ x \in \mathbb{R} $. If $ b $ and $ c $ are non-zero real numbers such that $ \min f(x)>\max g(x) $, then $$ \left| \frac{c}{b} \right| $$ lies in the interval

If $ x^2 - 4x + 5 + a>0 $ for all $ x \in \mathbb{R} $ whenever $ a \in (\alpha, \beta) $, then $ 4\beta + \alpha = $

The polynomial equation of degree 5 whose roots are the roots of the equation $$ x^5 - 3x^4 + 11x^2 - 12x + 4 = 0 $$ each increased by 2 is

If the system of equations $ 2x + 3y - 3z = 3,\ x + 2y + \alpha z = 1,\ 2x - y + z = \beta $ has infinitely many solutions, then $ \frac{\alpha}{\beta} = \frac{\beta}{\alpha} $

Identify the correct statements from the following:
I) Work is a path function.
II) Enthalpy is an extensive property.
III) Lattice enthalpy of ionic compounds can be obtained from Born–Haber cycle.