List of top Mathematics Questions asked in Maharashtra Board Class XII Exam

Find \( \frac{dy}{dx} \), if \( y = (\log x)^x \).
Find the vector equation of the line passing through the point having position vector \[ 4\hat{i} - \hat{j} + 2\hat{k} \] and parallel to the vector \[ -2\hat{i} - \hat{j} + \hat{k}. \]
If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are the position vectors of the points \( A, B, C \) respectively and \[ 5\mathbf{a} - 3\mathbf{b} - 2\mathbf{c} = \mathbf{0}, \] then find the ratio in which the point \( C \) divides the line segment \( BA \) externally.
Find \( k \), if the sum of the slopes of the lines represented by \[ x^2 + kxy - 3y^2 = 0 \] is twice their product.
In \( \triangle ABC \), if \( a = 18 \), \( b = 24 \), and \( c = 30 \), then find the value of \[ \sin \left(\frac{A}{2} \right). \]

Check whether the matrix 


 is invertible or not.

Construct the truth table for the statement pattern: \[ [(p \rightarrow q) \land q] \rightarrow p \]
A particle is moving along the X-axis. Its acceleration at time \( t \) is proportional to its velocity at that time. Find the differential equation of the motion of the particle.
Evaluate: \[ \int \frac{1}{x^2 + 25} dx. \]
If the vectors \(2\hat{i} - 3\hat{j} + 4\hat{k}\) and \( p\hat{i} + 6\hat{j} - 8\hat{k} \) are collinear, then find the value of \( p \).
Write the compound statement ‘Nagpur is in Maharashtra and Chennai is in Tamilnadu’ symbolically.

If the mean and variance of a binomial distribution are \( 18 \) and \( 12 \) respectively, then the value of \( n \) is __________.

The integrating factor of the linear differential equation \[ x \frac{dy}{dx} + 2y = x^2 \log x \] is __________.

If \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin^4 x \, dx = k \), then \( k \) is ____________.

The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.

The perpendicular distance of the plane \( r \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78 \) from the origin is __________.

If \( \alpha, \beta, \gamma \) are direction angles of a line and \( \alpha = 60^\circ, \beta = 45^\circ \), then \( \gamma \) is _________.

The principal solutions of the equation \( \cos\theta = \frac{1}{2} \) are _________.

The dual of statement \( t \lor (p \lor q) \) is _________.