List of practice Questions

The area (in square units) bounded by \[ x = 4, \quad y = -4, \quad \text{and} \quad y = x \quad \text{is:} \]
The particular solution of the differential equation \[ (1 + y^2) \, dx - xy \, dy = 0, \quad y(1) = 0 \quad \text{represents:} \]
If \( c \) and \( d \) are arbitrary constants, then \[ y = e^{2x} \left( \cosh \sqrt{2} x + d \sinh \sqrt{2} x \right) \] is the general solution of the differential equation:
The function \( f(x) = x^2 + \dfrac{54}{x} \)
If \( (a^2 - 1)x + ay + (3 - a) = 0 \) is a normal to the curve \( xy = 1 \), then the interval in which ‘a’ lies is:
The maximum value of \( x^4 y^4 \) when \( a^2 x^4 + b^2 y^4 = c^6 \) is:
Evaluate: \[ \int e^{\sin x} \cdot \frac{x \cos^3 x - \sin x}{\cos^2 x} \, dx \]
If \( f'(x) = a \cos x + b \sin x \), \( f'(0) = 4 \), \( f(0) = 3 \), and \( f\left( \frac{\pi}{2} \right) = 5 \), then \( f(x) = \)
If \( \int f(x)\,dx = \Psi(x) \), then \( \int x^5 f(x^3)\,dx = \):
If \[ \int \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R}, \text{ then} \] \[ \frac{\sin^2 \alpha - \sin^2 x}{\cos x - \cos \alpha} \, dx = f(x) + Ax + B, \, B \in \mathbb{R} \]
If \( f(x) = \begin{vmatrix} 1 & 6 + x & 36 + x^2 \\ 0 & x - 3 & 3x^2 - 27 \\ 0 & 2x - 4 & 8x^2 - 32 \end{vmatrix} \), then \( \lim\limits_{x \to 1} \frac{f(x)}{f(-x)} \) is:
Let \( f(x) \) be differentiable on \( \mathbb{R} \) and \( f'(m) \ne 0, m \in \mathbb{R} \). If \( \lim\limits_{x \to m} \frac{x f(m) - m f(x)}{x - m} + f'(m) = f(m) \), then \( m = \):
Let \( f(x) \) be a differentiable function such that \( f(1) = 2 \), \( f(2) = 6 \), and \[ f(x+y) = f(x) + kxy + \frac{4}{3} y^2, \quad \forall x, y \in \mathbb{R}. \] Then \( f(x) = \) ?
Let \[ f(x) = \begin{cases} \frac{5e^{|x|} + 2}{3 - e^{|x|}}, & x \ne 0 \\ 0, & x = 0 \end{cases} \] Then at \( x = 0 \), the function \( f(x) \) is:
The ordinates of the points on the curve \( y = \tan^{-1}(\sin(\sqrt{x})) \), \( 0 \leq x \leq 8\pi^2 \), at which the tangent is parallel to the X-axis are:
The rationale behind \underline{\hspace{2cm} therapy is that the client's distress has its origins in the biological, psychological, and social realms.}
Identify the major product of the following reaction:
3-bromo benzoic acid
In the given reactions, 'X' and 'Y' respectively are: \[ \ce{C6H5CH2NH2 ⇔[X] C6H5CONH2 ⇒[Y] C6H5NH2} \]
Addison’s disease is characterized by A. Hyperglycemia B. Weakness C. Increased susceptibility to stress D. Hypertension
Least reactive alkyl chloride in an \( S_N2 \) reaction is:
Which of the following alkyl halides is most reactive towards substitution by \( S_N1 \) mechanism?
What is the major product formed when p-cresol is heated with zinc dust?
What are 'X' and 'Y' respectively in the following reaction sequence? \[ \ce{C6H5CH3→[X] C6H5CO2H →[Y] C6H5CH2OH} \]
The graph drawn between \( \log \left( \frac{x}{m} \right) \) and \( \log (p) \) is shown below with intercept OA. What is the value of \( \frac{x}{m} \) at a pressure of 0.3 atm? (log 2 = 0.30, log 3 = 0.477)
Observe the following complexes \[ [\text{Mn(CN)}_6]^{3-},\quad [\text{Fe(CN)}_6]^{3-},\quad [\text{Co(C}_2\text{O}_4)_3]^{3-},\quad [\text{MnCl}_6]^{3-},\quad [\text{Fe(CN)}_6]^{4-},\quad [\text{CoF}_6]^{3-} \] From the above, the number of inner orbital complexes with paramagnetic nature is: