List of top Higher Mathematics Questions asked in MPBSE Class XII Board

Minimize \( Z = 3x + 5y \) subject to the constraints: \[ x + 3y \geq 3, x + y \geq 2, x \geq 0, y \geq 0. \]
An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. Suppose that the probability of drawing each ball is the same. What is the probability that both drawn balls are black?
Maximize \( Z = 3x + 9y \) subject to the constraints: \[ x + 3y \leq 60, x + y \geq 10, x \leq y, x \geq 0, y \geq 0. \]
Find the projection of vector \( \mathbf{a} = \hat{i} + 2 \hat{j} + 3 \hat{k} \) on the vector \( \mathbf{b} = 2 \hat{i} + \hat{k} \).
Show that for any two vectors \( \mathbf{a} \) and \( \mathbf{b} \), always \( | \mathbf{a} \cdot \mathbf{b} | \leq \|\mathbf{a}\| \|\mathbf{b}\| \).
If direction ratios of a line are -18, 12, -4, then find its direction cosines.
Find the value of \( x \) for which \( x ( \hat{i} + \hat{j} + \hat{k} ) \) is a unit vector.
Find local maximum and local minimum values of the function given by \[ f(x) = 3x^4 + 4x^3 - 12x^2 + 12 \]
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector \( 3\hat{i} + 2\hat{j} - 2\hat{k} \).
Find the interval in which the function given by \[ f(x) = x^2 - 4x + 6 \] is decreasing.
Evaluate \[ \int_{-1}^{1} \sin^5x \cos^4x \, dx \]
Evaluate \[ \int_{-1}^{2} \left( x^3 - |x| \right) dx. \]
Differentiate \( X^X \) with respect to \( X \).
If \( y = \sin^{-1} \left( \frac{2x}{1 + x^2} \right) \), then find \( \frac{dy}{dx} \).
Show that the function given by \( f(x) = 12x - 3 \) is increasing on \( R \).
If \( A = [a_{ij}]_{m \times n} \) and \( B = [b_{ij}]_{j \times p} \), then order of \( AB \) will be \( m \times p \).
Integrating factor of differential equation \( \frac{dy}{dx} - y = \cos x \) is \( e^{-x} \).
\( f : x \to y \) is an onto function then range of \( f = y \).
Angle between vectors \( \vec{a} = \hat{i} + \hat{j} - \hat{k} \) and \( \vec{b} = \hat{i} - \hat{j} + \hat{k} \) is 60°.
\( \tan^{-1} 1/2 + \tan^{-1} 1/3 = \tan^{-1} 1/5 \).
What is trivial relation?
Find magnitude of the vector \( \hat{i} + \hat{j} + \hat{k} \)
If \( A \) and \( B \) are two independent events with \( P(A) = 0.3 \) and \( P(B) = 0.4 \), then find \( P(B|A) \)
Find the principal value of \( \sin^{-1} \left( \frac{1}{\sqrt{2}} \right)\)
\( \int e^x [f(x) + f'(x)] dx \)= \(\underline{\hspace{2cm}}\)