Let $y=f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y=-\frac{1}{2}$ Then $S=\left\{x \in R : \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$ :

Bag I contains 3 red, 4 black, and 3 white balls and Bag II contains 2 red, 5 black, and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball drawn is found to be black in colour. Then the probability that the transferred ball is red, is:

Let $ a, b, c, d $ be non-zero numbers. If the point of intersection of the lines \[ 4ax + 2ay + c = 0 \text{and} 5bx + 2by + d = 0 \] lies in the fourth quadrant and is equidistant from the two, then the relation between $ a, b, c, d $ is

For a group of 100 candidates, the mean and standard deviation of scores were found to be 40 and 15 respectively. Later on, it was found that the scores 25 and 35 were misread as 52 and 53 respectively. Then the corrected mean and standard deviation corresponding to the corrected figures are

The locus of the mid-point of all chords of the parabola $ y^2 = 4x $ which are drawn through its vertex is:

Let $ \mathbf{A} = 2\hat{i} + \hat{j} - 2\hat{k} $ and $ \mathbf{B} = \hat{i} + \hat{j} $. If $ \mathbf{C} $ is a vector such that $ |\mathbf{C} - \mathbf{A}| = 3 $ and the angle between $ \mathbf{A} \times \mathbf{B} $ and $ \mathbf{C} $ is $ 30^\circ $, then $ [(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}] = 3 $, the value of $ \mathbf{A} \cdot \mathbf{C} $ is equal to:

Between any two real roots of the equation $ e^x \sin x = 1 $, the equation $ e^x \cos x = -1 $ has:

The range of values of $ \theta $ in the interval $ \left( 0, \pi \right) $ such that the points $ (3, 2) $ and $ (\cos \theta, \sin \theta) $ lie on the same sides of the line $ x + y - 1 = 0 $, is: