List of practice Questions

If $\begin{bmatrix} 4x & 5x-7 \\[0.3em] 4x & 2x+y\end{bmatrix}$=$\begin{bmatrix} x+6 & y \\[0.3em] 7y-13 & 7\end{bmatrix}$ then the value of 2x + y is:

If A = $\begin{bmatrix} 2 & 1 \\[0.3em] 1 & 4 \\[0.3em] -1 & 6 \\[0.3em] 0&7 \end{bmatrix}$ and B= $\begin{bmatrix} 1 & 2&3&0 \\[0.3em] 2 & -1 & 6&7\end{bmatrix}$ then

The equation $r^2=cos^2(\theta-\frac{\pi}{3})=2$ represents

The value of $\begin{vmatrix} \sin^2 14 \degree & \sin^2 66\degree & \tan 135\degree \\ \sin^2 66\degree & \tan 135\degree & \sin^2 14 \degree \\ \tan 135\degree & \sin^2 14 \degree & \sin^2 66\degree \end{vmatrix}$

Consider the following statements (A) and (B): \[ \text{(A):} \quad \int_a^b \frac{d}{dx} \left( f(x) \right) dx = \frac{d}{dx} \int_a^b f(x) dx \] \[ \text{(B):} \quad \frac{d}{dx} \left( \int f(x) dx \right) = f(x) + C \] Which one of the following is True?

Let $ [t] $ represent the greatest integer not exceeding $ t $. The number of discontinuous points of $ [10^t] $ in $ (0, 10) $ is:

Let $ f(x) = \frac{x}{\sqrt{1- x}}$ + $\frac{\sqrt{1- x}}{x}$. If $ \lim_{x _ 1^-} f(x) = l $ and $ \lim_{x \to m} f(x) = \frac{5}{2} $, then the set of all possible finite values of $ l $ and $ m $ is

Let $ x + y + 1 = 0 $ and $ x - y + 4 = 0 $ be the asymptotes of a hyperbola $ H $. If $ (1, 1) $ is a point on $ H $, then the length of the latus rectum of $ H $ is

If $ P(2, 3) $ and $ Q(-1, 2) $ are conjugate points with respect to the circle $ x^2 + y^2 + 2gx + 3y - 2 = 0 $ then the radius of the circle is

If a straight line is equally inclined at an angle $ \theta $ with all the three coordinate axes, then $ \tan \theta =$

The number of straight lines that can be drawn through the point $ (-3, 4) $, which are at a distance of 5 units from the point $ (2, -8) $, is:

If $ O $ is the origin and $ P, Q $ are points on the line $ 3x + 4y + 15 = 0 $ such that $ OP = OQ = 9 $, then the area of $ \triangle OPQ $ is:

A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 $ \mu C \, m^{-2} $. The charge on the sphere is nearly:

In a Young’s double slit experiment, a laser light of wavelength 560 nm produces an interference pattern with consecutive bright fringes’ separation of 7.2 mm. Now another light is used to produce an interference pattern with consecutive bright fringes’ separation of 8.1 mm. The wavelength of the second light is:

Which of the following statements regarding the damping force of a damped oscillator is NOT correct? Options:

The vector equation of any plane passing through the line of intersection of the planes $ \bar{r} \cdot \bar{n}_1 = q_1 $ and $ \bar{r} \cdot \bar{n}_2 = q_2 $ is given by $ \bar{r} \cdot (\bar{n}_1 + \lambda \bar{n}_2) = q_1 + \lambda q_2 $ for $ \lambda \in \bar{R} $. The vector equation of a plane passing through the point $ 2\bar{i} - 3\bar{j} + \bar{k} $ and the line of intersection of the planes $ \bar{r} \cdot (\bar{i} - 2\bar{j} + 3\bar{k}) = 5 $, $ \bar{r} \cdot (3\bar{i} + \bar{j} - 2\bar{k}) = 7 $ is

Let $ \bar{a}, \bar{b}, \bar{c} $ be three non-coplanar vectors. Then the point of intersection of the line joining the points $ \mathbf{a} + \bar{b} + \bar{c} $, $ \bar{a} - \bar{b} + 3\bar{c} $ and the line joining the points $ 2\bar{a} - \bar{b} + \bar{c} $, $ \bar{a} - 2\bar{b} + 4\bar{c} $ is

In $ \triangle ABC $, the expression $ \frac{a}{s-a} + \frac{b}{s-b} + \frac{c}{s-c} $ is equivalent to:

In $ \triangle ABC $, if $ r = 1 $, $ R = 4 $, and $ \Delta = 8 $, then \[ \frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = \]

If $ \cos A + \cos B + \cos C = 0 $ and $ \sin A + \sin B + \sin C = 0 $, then $ \cos(A - B) = $