List of practice Questions

If \[ A = \begin{bmatrix} 1 & 2 & -2 \\ 2 & -1 & 2\\ -1 & 1 & -2 \end{bmatrix}, \] then find $A + 2A^{-1}$.
If $(3 + 4i)^{2025} = 5^{2023}(x + iy)$, then find $\sqrt{x^2 + y^2}$.
If \[ \left(\frac{\cos \theta + i \sin \theta}{\sin \theta + i \cos \theta}\right)^{2024} + \left(\frac{1 + \cos \theta + i \sin \theta}{1 - \cos \theta + i \sin \theta}\right)^{2025} = x + iy, \] and $x + y$ at $\theta = \frac{\pi}{2}$ is
The coefficient of \(x^3\) in the expansion of \(\frac{x^4 + 1}{(x^2 + 1)(x - 1)}\) when it is expressed in terms of positive integral powers of \(x\), is
If \(x\) is a real number, then the number of solutions of \(\tan^{-1}\left(\sqrt{x(x+1)}\right) + \sin^{-1}\left(\sqrt{x^2 + x + 1}\right) = \dfrac{\pi}{2}\) is
Domain of the real-valued function \(f(x) = \log(x^2 - 1) + x \, \coth^{-1}x\) is
If \(|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 5, |\vec{a} + \vec{b} + \vec{c}| = \sqrt{69}\) and angle between \((\vec{a}, \vec{b}) = \dfrac{\pi}{3}\), then angle between \((\vec{c}, \vec{a}) =\)
One of the latus recta of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ subtends an angle $2 \tan^{-1} \left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2 = 36$ and $e$ is the eccentricity of the hyperbola, then find $\sqrt{a^2 + e^2}$.
If $L$ is the normal drawn to the parabola $y^2 = 8x$ at the point $t = \frac{1}{\sqrt{2}}$, then the foot of the perpendicular drawn from the focus of the parabola onto the normal $L$ is:
If the acute angle between the circles $S \equiv x^2 + y^2 + 2kx + 4y - 3 = 0$ and $S^1 \equiv x^2 + y^2 - 4x + 2ky + 9 = 0$ is $\cos^{-1}\left(\frac{3}{8}\right)$ and the centre of $S^1 = 0$ lies in the first quadrant, then the radical axis of $S = 0$ and $S^1 = 0$ is:

If \(\cos \alpha = \sec h \beta\), then \(\beta =\)

If \((1+x)^n = \sum_{r=0}^n \binom{n}{r} x^r\), then the value of \[ C_0 + (C_0 + C_1) + (C_0 + C_1 + C_2) + \cdots + (C_0 + C_1 + \cdots + C_n) \] is:
Consider the following statements:
Statement-I: CH$_3$NH$_2$ is more basic than NH$_3$, but C$_6$H$_5$NH$_2$ is less basic than NH$_3$.
Statement-II: The order of basic strength of amines in aqueous phase follows
(C$_2$H$_5$)$_2$NH > C$_2$H$_5$NH$_2$ > C$_6$H$_5$NH$_2$
The structure of the nitrogen-containing heterocyclic base shown below represents:
What is the drug used to control depression and hypertension?
What are X and Y respectively, in the given set of reactions?
CH3CH2CH2Cl
In the following sequence of reactions, what is the end product (D)?
C$_2$H$_5$Br $\xrightarrow{\text{KCN}}$ A $\xrightarrow{\text{H$_3$O$^+$}}$ B $\xrightarrow{\text{LiAlH$_4$}}$ C $\xrightarrow{\text{Cu, 573 K}}$ D
The most acidic carboxylic acid is:
A carbonyl compound X(C$_5$H$_8$O) gives yellow precipitate with NaOI. Hemiacetal of X with methanol/dry HCl is:
Which of the following does not involve in Friedel-Crafts reaction?
Consider the following.
Statement-I: Gold sol is prepared by Bredig’s arc method.
Statement-II: Bredig’s arc method involves only dispersion but not condensation.
Which of the following sets are correctly matched?
I) Hg — distillation
II) Cu — poling
III) B — zone refining
IV) Ti — liquation
The oxides of nitrogen obtained by the reaction of nitric acid with
The ion with 4f$^7$ configuration is:
Which of the following is the common monomer for the polymers Bakelite and Melamine?