List of practice Questions

The integrating factor of

\[ (1 + 2e^{-x}) \frac{dy}{dx} - 2e^{-x} y = 1 + e^{-x} \] is:
The solution of \( \frac{e^y}{dx} = x + 2 \) is:
The solution of \( \frac{dy}{\cos y} = dx \) is:
The solution of \( (y \cos y + \sin y) \, dy = (2x \log x + x) \, dx \) is:
The area enclosed by the curve \[ x = 3 \cos \theta, \quad y = 5 \sin \theta, \quad 0 \leq \theta \leq 2\pi, \] is equal to:

The limit: \[ \lim_{x \to 0} \frac{\sin \left( \pi \sin^2 x \right)}{x^2} \] is equal to:

If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]
The integral

\[ \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] is equal to:
The value of \[ \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] is equal to:
The dihedral angles in gaseous and solid phases of H\(_2\)O\(_2\) molecule respectively are
Identify the compound which gives CO\(_2\) more readily on heating.
The major components of cement are:

Consider the following (unbalanced) reactions:

\[ \text{BF}_3 + \text{NaH} \xrightarrow{450\,\mathrm{K}} X + \text{NaF} \]

\[ X + \text{H}_2\text{O} \rightarrow Y + \text{H}_2 \uparrow \]

The correct statements about \(X\) and \(Y\) are:

  1. \(X\) is an electron-deficient molecule
  2. In \(X\), a B--B bond is present
  3. \(Y\) is a weak tribasic acid
  4. \(Y\) acts as a Lewis acid
Which of the following does not exist?
Methemoglobinemia is due to:

The IUPAC name of the following compound is: 

 The functional groups present in the product ‘X’ of the reaction given below are:

The set of elements which obey the general electronic configuration \( (n-1)d^{10} ns^2 \) is:}

Identify the set of molecules which are not in the correct order of their dipole moments.

Match the following molecules with their respective molecular shapes:

The coefficient of \( x^3 \) in the expansion of \[ \frac{1}{(1 + 2x)^{-10}} \] is:

Let \( a, b, c \) be positive numbers such that \( abc = 1 \). Then the minimum value of \( a + b + c \) is:

Let \[ f(x) = |\sin x| + |\cos x|, \quad x \in \mathbb{R}. \] The period of \( f(x) \) is:
If

\[ R = \{(x, y) : x, y \in \mathbb{Z}, \, x^2 + 3y^2 \leq 7 \} \] is a relation on the set of integers \( \mathbb{Z} \), then the range of the relation \( R \) is:

Match the following elements with their respective blocks in the periodic table: