List of practice Questions

If \( x = \sqrt{2 \cosec^{-1} t} \) and \( y = \sqrt{2 \sec^{-1} t} \), \( |t| \geq 1 \), then \( \dfrac{dy}{dx} = \)
If \( \int \frac{5 \tan x}{\tan x - 2} \, dx = a x + b \log |\sin x - 2 \cos x| + c \), then \( a + b = \)
The circumradius of the triangle formed by the points \( (2, -1, 1) \), \( (1, -3, -5) \), and \( (3, -4, -4) \) is
Let \( A(2, 3, 5), B(-1, 3, 2), C(\lambda, 5, \mu) \) be the vertices of \( \triangle ABC \). If the median through the vertex \( A \) is equally inclined to the coordinate axes, then
Evaluate the limit: \[ \lim_{x \to \frac{\pi}{4}} \frac{2\sqrt{2} - \left(\cos x + \sin x\right)^3}{1 - \sin 2x} \]
If the smallest circle through the points of intersection of \( x^2 + y^2 = a^2 \) and \( x \cos \alpha + y \sin \alpha = p \), \( 0<p<a \), is \[ x^2 + y^2 - a^2 + \lambda(x \cos \alpha + y \sin \alpha - p) = 0 \] then \( \lambda = \)
Circles are drawn through the point \( (2, 0) \) to cut intercepts of length 5 units on the X-axis. If their centre lies in the first quadrant, then their equation is
If the locus of a point that divides a chord of slope 2 of the parabola \( y^2 = 4x \) internally in the ratio 1 : 2 is a parabola, then its vertex is
Assertion (A): The length of the latus rectum of an ellipse is 4. The focus and its corresponding directrix are respectively \( (1, -2) \) and the line \( 3x + 4y - 15 = 0 \). Then its eccentricity is \( \dfrac{1}{2} \). Reason (R): Length of the perpendicular drawn from focus of an ellipse to its corresponding directrix is \( \dfrac{a(1 - e^2)}{e} \)
Then which one of the following is correct?
A hyperbola passes through the point \( P(\sqrt{2}, \sqrt{3}) \) and has foci at \( (\pm 2, 0) \). Then the point that lies on the tangent drawn to this hyperbola at \( P \) is
The coordinate axes are rotated about the origin in the counterclockwise direction through an angle \( 60^\circ \). If \( a \) and \( b \) are the intercepts made on the new axes by a straight line whose equation referred to the original axes is \( x + y = 1 \), then \( \dfrac{1}{a^2} + \dfrac{1}{b^2} = \, ? \)
If one of the lines given by the pair of lines \( 3x^2 - 2y^2 + axy = 0 \) is making an angle \( 60^\circ \) with the x-axis, then \( a = \)
A circle is drawn with its centre at the focus of the parabola \( y^2 = 2px \) such that it touches the directrix of the parabola. Then a point of intersection of the circle and the parabola is
A circle touches both the coordinate axes and the straight line \( L \equiv 4x + 3y - 6 = 0 \) in the first quadrant. If this circle lies below the line \( L = 0 \), then the equation of that circle is
For three events \( A, B, \) and \( C \) of a sample space, if \[ P(\text{exactly one of A or B occurs}) = P(\text{exactly one of B or C occurs}) = P(\text{exactly one of C or A occurs}) = \frac{1}{4} \] and the probability that all three events occur simultaneously is \( \frac{1}{16} \), then the probability that at least one of the events occurs is
A bag P contains 4 red and 5 black balls, another bag Q contains 3 red and 6 black balls. If one ball is drawn at random from bag P and two balls are drawn from bag Q, then the probability that out of the three balls drawn two are black and one is red, is
On every evening, a student either watches TV or reads a book. The probability of watching TV is \( \frac{4}{5} \). If he watches TV, the probability that he will fall asleep is \( \frac{3}{4} \), and it is \( \frac{1}{4} \) when he reads a book. If the student is found to be asleep on an evening, the probability that he watched the TV is:
Let \( X \) be the random variable taking values \( 1, 2, \dots, n \) for a fixed positive integer \( n \). If \( P(X = k) = \frac{1}{n} \) for \( 1 \leq k \leq n \), then the variance of \( X \) is:
The locus of the third vertex of a right-angled triangle, the ends of whose hypotenuse are \( (1, 2) \) and \( (4, 5) \), is:
If the position vectors of A, B, C, D are \( \vec{A} = \hat{i} + 2\hat{j} + 2\hat{k}, \vec{B} = 2\hat{i} - \hat{j}, \vec{C} = \hat{i} + \hat{j} + 3\hat{k}, \vec{D} = 4\hat{j} + 5\hat{k} \), then the quadrilateral ABCD is a:
Let \( \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \), \( \vec{b} = 3\hat{i} + 3\hat{j} + \hat{k} \), and \( \vec{c} = \hat{i} - 2\hat{j} + 3\hat{k} \) be three vectors. If \( \vec{r} \) is a vector such that \( \vec{r} \times \vec{a} = \vec{r} \times \vec{b} \) and \( \vec{r} . \vec{c} = 18 \), then the magnitude of the orthogonal projection of \( 4\hat{i} + 3\hat{j} - \hat{k} \) on \( \vec{r} \) is:
If \( \sum\limits_{i=1}^{9} (x_i - 5) = 9 \) and \( \sum\limits_{i=1}^{9} (x_i - 5)^2 = 45 \), then the standard deviation of the nine observations \( x_1, x_2, \ldots, x_9 \) is
Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is \( \frac{1}{4} \) and the probability that the second student gets qualified in the same exam is \( \frac{2}{5} \), then the probability that at least one of them gets qualified in that exam is
The equation \[ \cos^{-1}(1 - x) - 2 \cos^{-1} x = \frac{\pi}{2} \] has:
In \( \triangle ABC \), if A, B, C are in arithmetic progression, then \[ \sqrt{a^2 - ac + c^2} . \cos\left(\frac{A - C}{2}\right) =\ ? \]