List of practice Questions

If $P = \sin\frac{2\pi}{7} + \sin\frac{4\pi}{7} + \sin\frac{8\pi}{7}$ and $Q = \cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{8\pi}{7}$, then the point (P,Q) lies on the circle of radius

If a, b are real numbers and $\alpha$ is a real root of $x^2+12+3\sin(a+bx)+6x=0$ then the value of $\cos(a+b\alpha)$ for the least positive value of $a+b\alpha$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + \frac{a}{2}x + b = 0$ and $(\alpha-\beta)(\alpha-\gamma)$, $(\beta-\alpha)(\beta-\gamma)$, $(\gamma-\alpha)(\gamma-\beta)$ are the roots of the equation $(y+a)^3 + K(y+a)^2 + L = 0$, then $\frac{L}{K}= $

All the letters of the word MOTHER are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word MOTHER is

The number of positive integral solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{2025}$ is

The number of positive integral solutions of $xyz = 60$ is

Numerically greatest term in the expansion of $(3x-4y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is

Let K be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[6]{3})^{6144}$. If the coefficient of $x^P (P \in N)$ in the expansion of $\frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})}$ is $a_P$, then $a_K - a_{K+1} - a_{K-1} =$

If $|Z_1 - 3 - 4i| = 5$ and $|Z_2| = 15$ then the sum of the maximum and minimum values of $|Z_1 - Z_2|$ is

If $Z=r(\cos\theta+i\sin\theta)$, $(\theta \neq -\pi/2)$ is a solution of $x^3 = i$, then $r^9(\cos(9\theta)+i\sin(9\theta)) =$

If $f(x) = x^2 - 2(4K-1)x + g(K)>0$ $\forall x \in \mathbb{R}$ and for $K \in (a,b)$, and if $g(K) = 15K^2 - 2K - 7$, then

If local maximum of $f(x) = \frac{ax+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$

If $1+2i$ is a root of the equation $x^4 - 3x^3 + 8x^2 - 7x + 5 = 0$, then sum of the squares of the other roots is

The number of real values of 'a', for which the system of equations $2x+3y+az = 0$, $x+ay-2z=0$ and $3x+y+3z = 0$ has nontrivial solutions is

If the eight vertices of a regular octagon are given by the complex numbers $\frac{1}{x_j-2i}$ ($j=1,2,3,4,5,6,7,8$), then the radius of the circumcircle of the octagon is

If $2^{4n+3} + 3^{3n+1}$ is divisible by P for all natural numbers $n$, then P is

A is a $3 \times 3$ matrix satisfying $A^3 - 5A^2 + 7A + I = 0$. If $A^5 - 6A^4 + 12A^3 - 6A^2 + 2A + 2I = lA + mI$, then $l + m =$

If $A = \begin{pmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1 \end{pmatrix}$, $A^{-1} = \frac{1}{2} \begin{pmatrix} 1 & -1 & 1 \\ -8 & 6 & 2y \\ 5 & -3 & 1 \end{pmatrix}$ then the point $(x,y)$ lies on the curve

Consider a homogeneous system of three linear equations in three unknowns represented by $AX=O$. If $X = \begin{pmatrix}l \\ m \\ 0 \end{pmatrix}, l \neq 0, m \neq 0, l, m \in \mathbb{R}$ represents an infinite number of solutions of this system, then rank of A is

The reaction of benzene with CO and HCl in the presence of anhydrous AlCl\(_3\) gives a compound X. X can also be obtained from which of the following reaction?

What is the product 'Z' in the given sequence of reactions?

The ratio of \(\sigma\) bonds to \(\pi\) bonds in Q is

What is the major product 'Z' in the given reaction sequence?

Polymer X is an example of polyester and Y is an example of polyamide. X and Y are respectively

The general structure of alpha amino acid can be represented as

. Which amino acid is not correctly matched with R-given?