List of practice Questions

If \( P(\vec{p}) \), \( Q(\vec{q}) \), \( R(\vec{r}) \), and \( S(\vec{s}) \) are four points such that \( 3\vec{p} + 8\vec{q} = 6\vec{r} + 5\vec{s} \), then the lines PQ and RS are

Let \( \vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{b} = \hat{i} + \hat{j} \), 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 \( |(\vec{a} \times \vec{b}) \times \vec{c}| \) is equal to

Consider a tetrahedron with faces \( F_1, F_2, F_3, F_4 \). Let \( \vec{v_1}, \vec{v_2}, \vec{v_3}, \vec{v_4} \) be area vectors perpendicular to these faces in the outward direction, then \( |\vec{v_1} + \vec{v_2} + \vec{v_3} + \vec{v_4}| \) equals

If \( V \) is the volume of the parallelepiped with edges \( \vec{a}, \vec{b}, \vec{c} \), then the volume of the parallelepiped with edges \( \vec{\alpha}, \vec{\beta}, \vec{\gamma} \) (defined by dot products) is

Define the length of \( a\hat{i} + b\hat{j} + c\hat{k} \) as \( |a| + |b| + |c| \). This definition coincides with the usual definition if and only if

If \( \vec{u} = \hat{i} + \hat{j}, \vec{v} = \hat{i} - \hat{j}, \vec{w} = \hat{i} + 2\hat{j} + 3\hat{k} \) and \( \hat{n} \) is a unit vector such that \( \vec{u} \cdot \hat{n} = 0 \) and \( \vec{v} \cdot \hat{n} = 0 \), then \( |\vec{w} \cdot \hat{n}| \) is

\( \lim_{x \to 0} \frac{\sqrt{1 - \cos(2x)}}{\sqrt{2} \cdot x} \) is

Radius of the circle \( \vec{r}^2 + \vec{r} \cdot (2\hat{i} - 2\hat{j} - 4\hat{k}) - 19 = 0, \vec{r} \cdot (\hat{i} - 2\hat{j} + 2\hat{k}) + 8 = 0 \) is

If \( x^y = e^{x - y} \), then \( \frac{dy}{dx} \) is equal to

The function \( f(x) = \frac{ax + b}{(x-1)(x-4)} \) has a local maxima at \( (2, -1) \), then

If \( z = \tan(y + ax) + \sqrt{y} - ax \), then \( z_{xx} - a^2 z_{yy} \) is equal to

\( \int \frac{x^2 + 4x^4 + 16}{dx} \) is equal to

If the solution of the differential equation \( \frac{dy}{dx} = ax + 32y + f \) represents a circle, then the value of \( a \) is

Solution of the differential equation \[ x = 1 + xy \frac{dy}{dx} + \frac{(xy)^2}{2!} \left(\frac{dy}{dx}\right)^2 + \frac{(xy)^3}{3!} \left(\frac{dy}{dx}\right)^3 + \cdots \] is

Two bodies A and B having temperatures \( 327^\circ C \) and \( 427^\circ C \) are radiating heat to the surrounding. The surrounding temperature is \( 27^\circ C \). The ratio of rates of heat radiation of A to that of B is

If \( \cos x = \frac{2\cos y - 1}{2 - \cos y} \), where \( x, y \in (0, \pi) \), then \( \tan \frac{x}{2} \cot \frac{y}{2} \) is equal to:

The value of \[ \sin^{-1}\left[\cos\left(\sin^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right) + \cos^{-1}\left(\frac{\sqrt{12}}{4}\right) + \sec^{-1}\left(\sqrt{2}\right)\right)\right] \] is

The headquarters of NATO is situated at

The first Indian woman to swim across the English channel is

Which of the following river is related with Bhakra-Nagal Project?

Namdhapha National Park is located in

Which of the following is known as 'evening star'?

Which of the following was the founder of Indian National Congress?