List of practice Questions

Let A = (2, 0, -1), B = (1, -2, 0), C = (1, 2, -1), and D = (0, -1, -2) be four points. If \(\theta\) is the acute angle between the plane determined by A, B, C and the plane determined by A, C, D, then \(\tan\theta =\)
Let \([x]\) represent the greatest integer function. If \(\lim_{x \to 0^+} \frac{\cos[x] - \cos(kx - [x])}{x^2} = 5\), then \(k =\)
If the angle between the pair of lines $2x^2 + 2hxy + 2y^2 - x + y - 1 = 0$ is $\tan^{-1}\left(\frac{3}{4}\right)$ and $h$ is a positive rational number, then the point of intersection of these two lines is
If the equation of the circle passing through the point $(8, 8)$ and having the lines $x + 2y - 2 = 0$ and $2x + 3y - 1 = 0$ as its diameters is $x^2 + y^2 + px + qy + r = 0$, then $p^2 + q^2 + r =$
If a unit circle $S = x^2 + y^2 + 2gx + 2fy + c = 0$ touches the circle $S' = x^2 + y^2 - 6x + 6y + 2 = 0$ externally at the point $(-1, -3)$, then $g + f + c =$
A($a$, 0) is a fixed point, and $\theta$ is a parameter such that $0<\theta<2\pi$. If P($a \cos \theta$, $a \sin \theta$) is a point on the circle $x^2 + y^2 = a^2$ and Q($b \sin \theta$, $-b \cos \theta$) is a point on the circle $x^2 + y^2 = b^2$, then the locus of the centroid of the triangle APQ is
If $(h, k)$ is the image of the point $(2, -3)$ with respect to the line $5x - 3y = 2$, then $h + k =$
A basket contains 5 apples and 7 oranges, and another basket contains 4 apples and 8 oranges. If one fruit is picked out at random from each basket, then the probability of getting one apple and one orange is
Two cards are drawn from a pack of 52 playing cards one after the other without replacement. If the first card drawn is a queen, then the probability of getting a face card from a black suit in the second draw is
In a school there are 3 sections A, B, and C. Section A contains 20 girls and 30 boys, section B contains 40 girls and 20 boys, and section C contains 10 girls and 30 boys. The probabilities of selecting section A, B, and C are 0.2, 0.3, and 0.5, respectively. If a student selected at random from the school is a girl, then the probability that she belongs to section A is
In a triangle ABC, if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$, then $\cos A : \cos B : \cos C =$
In a triangle ABC, if A, B, and C are in arithmetic progression, $r_3 = r_1 r_2$, and $c = 10$, then $a^2 + b^2 + c^2 =$
In a right-angled triangle, if the position vector of the vertex having the right angle is $-3\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}$ and the position vector of the midpoint of its hypotenuse is $6\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}$, then the position vector of its centroid is
If the position vectors of the vertices A, B, C of a triangle are $3\mathbf{i} + 4\mathbf{j} - \mathbf{k}$, $\mathbf{i} + 3\mathbf{j} + \mathbf{k}$, and $5(\mathbf{i} + \mathbf{j} + \mathbf{k})$ respectively, then the magnitude of the altitude drawn from A onto the side BC is
If the vectors $2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$, $-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, and $p\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ are coplanar, then the unit vector in the direction of the vector $9p\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$ is
Assertion (A): For the lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$, if $(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar. Reason (R): $|(\mathbf{a} - \mathbf{p}) \cdot (\mathbf{b} \times \mathbf{q})|$ is $|\mathbf{b} \times \mathbf{q}|$ times the shortest distance between the lines $\mathbf{r} = \mathbf{a} + t \mathbf{b}$ and $\mathbf{r} = \mathbf{p} + s \mathbf{q}$.
$\cos(13^\circ)\sin(17^\circ)\sin(21^\circ)\cos(47^\circ) =$
The sum of the solutions of $\cos x \sqrt{16 \sin^2 x} = 1$ in $(0, 2\pi)$ is
The number of integers between 10 and 10,000 such that in every integer every digit is greater than its immediate preceding digit, is
All letters of the word `AGAIN' are permuted in all possible ways, and the words so formed (with or without meaning) are written as in a dictionary. Then the $50^{th}$ word is
If $y = \frac{3}{4} + \frac{3 \cdot 5}{4 \cdot 8} + \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + ... \infty$, then
Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $(1 + x - x^2)^6$ is
The set of all real values of $x$ for which $\frac{x^2-1}{(x-4)(x-3)} \ge 1$ is
If $\alpha$, $\beta$, and $\gamma$ are the roots of the equation $2x^3 + 3x^2 - 5x - 7 = 0$, then $\frac{1}{\alpha^2} + \frac{1}{\beta^2} + \frac{1}{\gamma^2} =$

What is \( A \) in the following reaction? 

\[ \mathrm{CH}_3-\mathrm{CH}= \mathrm{CH}-\mathrm{CH}_2-\mathrm{CH}_2-\mathrm{CN} \xrightarrow{(1)\, \text{A} (\mathrm{i-Bu})_2} \xrightarrow{(2)\, \mathrm{H}_2\mathrm{O}} \rightarrow A \]