List of top Mathematics & Statistics Questions asked in Maharashtra Class XII

Solve \( x + y \frac{dy}{dx} = \sec (x^2 + y^2) \)
If \( y = e^{m \tan^{-1} x} \), then show that \( (1 + x^2) \left( \frac{dy}{dx} \right)^2 + (2x - 2m) \frac{dy}{dx} = 0 \).
Find the vector equation of the plane passing through the point \( A(-1, 2, -5) \) and parallel to the vectors \( 4\hat{i} - \hat{j} + 3\hat{k} \) and \( \hat{i} + \hat{j} - \hat{k} \).
If \( A(\vec{a}) \) and \( B(\vec{b}) \) be any two points in space and \( R(\vec{r}) \) be a point on the line segment \( AB \) dividing it internally in the ratio \( m:n \), then prove that \( \vec{r} = \frac{m \vec{b} + n \vec{a}}{m + n} \).
If \( A(5, 1, p) \), \( B(1, q, p) \), and \( C(1, -2, 3) \) are vertices of a triangle and \( G \left( -\frac{4}{3}, \frac{1}{3}, -\frac{4}{3} \right) \) is its centroid, then find the values of \( p \), \( q \) by vector method.
If angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \) is equal to the angle between the lines represented by \( 2x^2 - 5xy + 3y^2 = 0 \), then show that \( 100 (h^2 - ab) = (a + b)^2 \).
If \( 2 \tan^{-1} (\cos x) = \tan^{-1} (2 \csc x) \), then find the value of \( x \).
If \( X \sim B(n, p) \) and \( E(X) = 6 \) and \( \text{Var}(X) = 4.2 \), then find \( n \) and \( p \).
Form the differential equation of all lines which make intercept 3 on x-axis.
Solve the differential equation \( y \frac{dy}{dx} + x = 0 \)
Evaluate: \( \int \sec^3 x \, dx \)
Find the equation of tangent to the curve \( y = 2x^3 + x^2 + 2 \) at \( \left( \frac{1}{2}, 2 \right) \).
Find the cartesian equation of the plane passing through \( A(1, 2, 3) \) and the direction ratios of whose normal are 3, 2, 5.
Find the value of \( k \), if \( 2x + y = 0 \) is one of the lines represented by \( 3x^2 + kxy + 2y^2 = 0 \)
Find the principal solutions of \( \cot \theta = 0 \)

Find the cofactors of the elements of the matrix \( \begin{bmatrix} -1 & 2 \\ -3 & 4 \end{bmatrix} \)

Write the separate equations of lines represented by the equation \( x^2 - 9y^2 = 0 \)
The maximum value of the function \( f(x) = \frac{\log x}{x} \) is
If \( f(x) = x^5 + 2x - 3 \), then \( (f^{-1})'(-3) = \)
Equation of line passing through the points (0, 0, 0) and (2, 1, -3) is
In \( \triangle ABC \), if \( c^2 = a^2 + b^2 = ac \), then \( \angle B = \)
Without using truth table prove that \( (p \wedge q) \vee (\sim p \wedge q) \vee (p \wedge \sim q) \equiv p \vee q \)
If a fair coin is tossed 10 times. Find the probability of getting at most six heads.
If \( x = f(t) \) and \( y = g(t) \) are differentiable functions of t so that y is a differentiable function of x and \( \frac{dx}{dt} \neq 0 \), then prove that \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). Hence find \( \frac{dy}{dx} \) if \( x = \sin t \) and \( y = \cos t \).
If u and v are differentiable functions of x, then prove that \( \int u \, dv = uv - \int v \, du \). Hence evaluate \( \int \log x \, dx \).