Maharashtra Board Class 12 Mathematics and Statistics Question Paper 2023 with Answer Key pdf is available for download here. The question paper was divided into two sections - Section A for objective questions and Section B for subjective questions.
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If \( p \land q \) is F, \( p \rightarrow q \) is F, then the truth values of \( p \) and \( q \) are
In \( \Delta ABC \), if \( c^2 + a^2 - b^2 = ac \), then \( \angle B = \)
The area of the triangle with vertices \( (1, 2, 0), (1, 0, 2) \), and \( (0, 3, 1) \) in square units is
If the corner points of the feasible solution are \( (0, 10), (2, 2) \), and \( (4, 0) \), then the point of minimum \( z = 3x + 2y \) is
If \( y \) is a function of \( x \) and \( \log(x + y) = 2xy \), then the value of \( y'(0) \) is
The integral of \( \cos^2x \) is
The solution of the differential equation \( \frac{dx}{dt} = x \log \frac{x}{t} \) is
Let the probability mass function (p.m.f.) of a random variable \( X \) be \( P(X = x) = \binom{4}{x} \left( \frac{5}{9} \right)^x \left( \frac{4}{9} \right)^{4-x} \), for \( x = 0, 1, 2, 3, 4 \), then \( E(X) \) is equal to
i. Write the joint equation of coordinate axes.
ii. Find the values of \( c \) which satisfy \( |c\vec{u}| = 3 \) where \( \vec{u} = \hat{i} + 2\hat{j} + 3\hat{k} \).
Write \( \int \cot{x} \, dx \).
Write the degree of the differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} = x \).
Write the inverse and contrapositive of the following statement:
\textit{If \( x < y \) then \( x^2 < y^2 \).
If \[ A = \begin{pmatrix} x & 0 & 0
0 & y & 0
0 & 0 & z \end{pmatrix} \]
is a non-singular matrix, then find \( A^{-1 \) by elementary row transformations.
Find the Cartesian co-ordinates of the point whose polar co-ordinates are \( \left( \sqrt{2}, \frac{\pi}{4} \right) \).
If \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of lines and \( h^2 = ab \neq 0 \), then find the ratio of their slopes.
If \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) are the position vectors of the points A, B, C respectively and \( 5\overrightarrow{a} + 3\overrightarrow{b} - 8\overrightarrow{c} = 0 \), then find the ratio in which the point C divides the line segment AB.
Solve the following inequalities graphically and write the corner points of the feasible region: \[ 2x + 3y \leq 6, \quad x + y \geq 2, \quad x \geq 0, \quad y \geq 0 \]
Show that the function \( f(x) = x^3 + 10x + 7 \), \( x \in \mathbb{R} \), is strictly increasing.
Evaluate: \[ \int_0^{\frac{\pi}{2}} \sqrt{1 - \cos{4x}} \, dx \]
Find the area of the region bounded by the curve \( y^2 = 4x \), the X-axis and the lines \( x = 1 \), \( x = 4 \) for \( y \geq 0 \).
Solve the differential equation: \[ \cos{x} \cos{y} \, dy - \sin{x} \sin{y} \, dx = 0 \]
Find the mean of numbers randomly selected from 1 to 15.
Find the area of the region bounded by the curve \( y = x^2 \) and the line \( y = 4 \).
Find the general solution of \( \sin \theta + \sin 3\theta + \sin 5\theta = 0 \)
If \( -1 \leq x \leq 1 \), prove that \( \sin^{-1}{x} + \cos^{-1}{x} = \frac{\pi}{2} \)
If \( \theta \) is the acute angle between the lines represented by \( ax^2 + 2hxy + by^2 = 0 \), then prove that \[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are \( -2, 1, -1 \) and \( -3, -4, 1 \).
Find the shortest distance between lines: \[ \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}, \quad \frac{x-2}{4} = \frac{y-4}{5} = \frac{z-5}{6} \]
Lines \[ \overrightarrow{r} = (i + j - k) + \lambda(2i - 2j + k), \quad \overrightarrow{r} = (4i - 3j + 2k) + \mu(i - 2j + 2k) \]
are coplanar. Find the equation of the plane determined by them.
If \( y = \sqrt{\tan{x} + \sqrt{\tan{x} + \sqrt{\tan{x} + \dots}}} \), then show that \[ \frac{dy}{dx} = \sec^2{x} \cdot \frac{2y - 1}{y^2 - 1} \]
Find \( \frac{dy}{dx} \) at \( x = 0 \).
Find the approximate value of \( \sin{30^\circ 30'} \).
Given that \( 1^\circ = 0.0175 \) radians and \( \cos{30^\circ} = 0.866 \).
Evaluate: \[ \int x \tan^{-1}{x} \, dx \]
Find the particular solution of the differential equation: \[ \frac{dy}{dx} = e^{2y} \cos{x}, \quad when \quad x = \frac{\pi}{6}, \, y = 0 \]
For the following probability density function of a random variable \( X \), find: \[ f(x) = \frac{x + 2}{18}, \quad for \, -2 < x < 4 \]
(a) \( P(X < 1) \) and (b) \( P(|X| < 1) \)
A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.
Simplify the given circuit by writing its logical expression. Also write your conclusion.
If \[ A = \begin{pmatrix} 1 & 2
3 & 4 \end{pmatrix} \]
verify that \[ A(adjA) = (adjA)A = |A|I \]
Prove that the volume of a tetrahedron with coterminous edges \( \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} \) is \[ Volume = \frac{1}{6} \left| \overrightarrow{a} \cdot (\overrightarrow{b} \times \overrightarrow{c}) \right| \]
Hence, find the volume of the tetrahedron whose coterminous edges are \[ \overrightarrow{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \overrightarrow{b} = -\hat{i} + \hat{j} + 2\hat{k}, \quad \overrightarrow{c} = 2\hat{i} + \hat{j} + 4\hat{k}. \]
Find the length of the perpendicular drawn from the point \( P(3, 2, 1) \) to the line \[ \overrightarrow{r} = (7\hat{i} + 7\hat{j} + 6\hat{k}) + \lambda(-2\hat{i} + 2\hat{j} + 3\hat{k}) \]
If \( y = \cos(m^{-1}x)) \), then show that \[ (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} + m^2y = 0 \]
Verify Lagrange's mean value theorem for the function \( f(x) = \sqrt{x + 4} \) on the interval \([0, 5]\).
Evaluate: \[ \int \frac{2x^2 - 3}{(x^2 - 5)(x^2 + 4)} \, dx \]
Prove that: \[ \int_0^a f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a - x) \, dx \]
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