List of practice Questions

The direction ratios of two straight lines are \( l, m, n \) and \( l_1, m_1, n_1 \). The lines will be perpendicular to each other if:

The direction ratios of a straight line are \( 1, 3, 5 \). Then its direction cosines are:

The equation of the plane parallel to the plane \( 3x - 5y + 4z = 11 \) is:

Find the value of: \[ \vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times (\vec{c} + \vec{a}) + \vec{c} \times (\vec{a} + \vec{b}) \]

Find the value of the scalar triple product: \[ \hat{i} \cdot (\hat{j} \times \hat{k}) \]

If \[ \vec{a} = \hat{i} + \hat{j} + 2\hat{k}, \] then the corresponding unit vector \( \hat{a} \) in the direction of \( \vec{a} \) is:

If \[ \vec{a} = 3\hat{i} + \hat{j} - 2\hat{k} \quad \text{and} \quad \vec{b} = \hat{i} + \lambda \hat{j} - 3\hat{k} \] are perpendicular to each other, then the value of \( \lambda \) is:

Evaluate the integral: \[ \int \cot^2(x) \, dx \]

The angle between the vectors \[ \vec{a} = 2\hat{i} - 3\hat{j} + 2\hat{k} \quad \text{and} \quad \vec{b} = \hat{i} + 4\hat{j} + 5\hat{k} \] is:

Solve the differential equation: \[ \frac{dx}{dy} = \frac{x}{y} \]

Solve the differential equation: \[ \frac{dx}{dy} + 2y = e^{3x} \]

Find the dot product of the vectors: \[ (4\hat{i} + 3\hat{j} + 3\hat{k}) \cdot (6\hat{i} - 4\hat{j} + \hat{k}) \]

Find the cross product of the vectors: \[ \left( \hat{i} + 3\hat{j} - 2\hat{k} \right) \times \left( -\hat{i} + 3\hat{k} \right) \]

Find the magnitude of the vector: \[ \left| \hat{i} - \hat{j} - \hat{k} \right| \]

Find the dot product of the unit vectors: \[ \hat{j} \cdot \hat{j} \]

Find the cross product of the unit vectors: \[ \hat{k} \times \hat{j} \]

Solve the differential equation: \[ \frac{dx}{dy} = e^{x + y} \]

Solve the equation: \[ \frac{dx}{dy} + 2y = \sin x \]

Solve the system: \[ \begin{pmatrix} x \\ y \end{pmatrix} \quad \Rightarrow \quad \begin{pmatrix} 2x - 1 \\ 9 \end{pmatrix} \] Find the values of \(x\) and \(y\).

Find the derivative of \( \sin^2 x \): \[ \frac{d}{dx} \left( \sin^2 x \right) \]

Find the derivative of \( x^5 + \cos 2x \): \[ \frac{d}{dx} \left( x^5 + \cos 2x \right) \]

Find the derivative of \( \sec^{-1}(x) \): \[ \frac{d}{dx} \left( \sec^{-1}(x) \right) \]

Find the derivative of \( a^x \): \[ \frac{d}{dx} \left( a^x \right) \]

Find the derivative of \( \log(\cos x) \): \[ \frac{d}{dx} \left( \log(\cos x) \right) \]

Find the order and degree of the differential equation: \[ xy \left( \frac{d^2y}{dx^2} \right) + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0 \]