List of practice Questions

Let \( \left\lfloor x \right\rfloor \) be the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to 0^-} \frac{x \left( \left\lfloor x \right\rfloor + |x| \right)}{|x|} \] is equal to:

The integral

\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \] is:

If

\[ \int x e^{-x} \, dx = M e^{-x} + C, \quad \text{where } C \text{ is an arbitrary constant, then } M \text{ is equal to:} \]

Evaluate the integral \[ \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx \]

When \( y = vx \), the differential equation \[ \frac{dy}{dx} = \frac{y}{x} + \frac{f\left( \frac{y}{x} \right)}{f'\left( \frac{y}{x} \right)} \] reduces to:

The integrating factor of

\[ (1 + 2e^{-x}) \frac{dy}{dx} - 2e^{-x} y = 1 + e^{-x} \] is:

The solution of \( \frac{dy}{\cos y} = dx \) 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:

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:

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