Maharashtra Board 2024 Class 12 Mathematics and Statistics (40-J-862) Question Paper (Available) :Download Solution PDF with Answer Key

Question 1:

Select and write the correct answer for the following multiple choice type of

(1). The dual of statement t ∨ (p ∨ q) is ___.

(a) c ∧ (p ∨ q)  
(b) c ∧ (p ∧ q)  
(c) t ∧ (p ∧ q)  
(d) t ∧ (p ∨ q)

(2). The principal solutions of the equation cos(θ) = 1/2 are ___.

(a) π/6, 5π/6  
(b) π/3, 5π/3  
(c) π/6, 7π/6  
(d) π/3, 2π/3

(3). If α, β, γ are direction angles of a line and α = 60°, β = 45°, then γ is ___.

(a) 30° or 90°  
(b) 45° or 60°  
(c) 90° or 130°  
(d) 60° or 120°

(4). The perpendicular distance of the plane r̅ · (3î + 4ĵ + 12k̂) = 78 from the origin is ___.

• (a)  4 
• (b)  5 
• (c)  6 
• (d)  8 

(5). The slope of the tangent to the curve x = sin(θ) and y = cos(2θ) at θ = π/6 is ___.

(a) -2√3

(b) -2/√3

(c) -2

(d) -1/2

(6). If ∫ from -π/4 to π/4 of x³ sin⁴(x) dx = k, then k is ___.

(a) 1

(b) 2

(c) 4

(d) 0

(7). The integrating factor of the linear differential equation x(dy/dx) + 2y = x^2 log(x) is ___.

(a) x  
(b) 1/x  
(c) x^2  
(d) 1/x^2
 

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

(a) 36  
(b) 54  
(c) 18  
(d) 27
 

Question 2:

Answer the following questions :

(1). Write the compound statement ‘Nagpur is in Maharashtra and Chennai is in Tamilnadu’ symbolically.

(2). If the vectors 2î - 3ĵ + 4k̂ and p î + 6ĵ - 8k̂ are collinear, then find the value of p.

(3). Evaluate: ∫ 1/(x² + 25) dx.

(4). 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.

Construct Truth Table for the Statement Pattern: [(p → q) ∧ q] → p

Check Whether the Matrix is Invertible or Not:
A = [ cos(θ)  sin(θ)
     -sin(θ)  cos(θ) ]

In Triangle ABC, if a = 18, b = 24, and c = 30, then find the value of: sin(A/2)

Find k, if the sum of the slopes of the lines represented by: x² + kxy - 3y² = 0 is twice their product.

If a, b, c are the position vectors of the points A, B, C respectively and 5a - 3b - 2c = 0, then find the ratio in which the point C divides the line segment BA.

Find the vector equation of the line passing through the point having position vector:4î - ĵ + 2k̂ and parallel to the vector: -2î - ĵ + k̂.

Find dy/dx, if y = (log x)^x.

Evaluate: ∫ log(x) dx

Evaluate: ∫ from 0 to π/2 of cos²(x) dx

Find the area of the region bounded by the curve  y = x² , and the lines  x = 1 , x = 2 , and  y = 0 .

Solve: 1 + dy/dx = cosec(x + y); put x + y = u. 

If two coins are tossed simultaneously, write the probability distribution of the number of heads.

Express the following switching circuit in symbolic form of logic. Construct the switching table.

Prove that: tan^-1 (1/2) + tan^-1 (1/3) = π/4. 

In  △ABC, prove that: cos A/a + cos B/b + cos C/c = a² + b² + c² /2abc.

Prove by vector method, the angle subtended on a semicircle is a right angle.

Find the shortest distance between the lines r̅ = (4î - ĵ) + λ( î + 2ĵ - 3k̂ ) and r̅ = ( î - ĵ - 2k̂ ) + μ ( î + ĵ - 5k̂ ).

Find the angle between the line r̅ = ( î + 2ĵ + k̂ ) + λ( î + ĵ + k̂ ) and the plane r̅ · (2î + ĵ + k̂) = 8.

If y = sin^-1x , then show that: (1 - x²)d²y/{dx²} - x(dy/dx) = 0.

Find the approximate value of tan^-1(1.002). Given: π = 3.1416.

Prove that:
∫ 1/(a² - x²) dx = (1/(2a)) log((a + x)/(a - x)) + C

Solve the differential equation: x(dy/dx) - y + x(sin(y/x)) = 0.

Find k , if the probability density function is given by: f(x) = kx²(1 - x), for 0 < x < 1, = 0 otherwise is the p.d.f of random variable X.

A die is thrown 6 times. If "getting an odd number" is a success, find the probability of 5 successes.

Solve the following system of equations by the method of reduction: x + y + z = 6,  y + 3z = 11,  x + z = 2y. 

Prove that the acute angle θ between the lines represented by the equation:ax² + 2hxy + by² = 0 is:
tan(θ) = |(2√(h² - ab)) / (a + b)|. Hence find the condition that the lines are coincident.
 

Find the volume of the parallelepiped whose vertices are
 A(3,2,-1), B(-2,2,-3), C(3,5,-2)  and  D(-2,5,4) .

Solve the following L.P.P. by graphical method:
Maximize:  z = 10x + 25y. 
Subject to:  0 ≤ x ≤ 3,

                    0 ≤ y ≤ 3,

                    x + y ≤ 5. 
Also find the maximum value of z.

If  x = f(t) and y = g(t)  are differentiable functions of  t , so that y  is a function of  x  and (dx/dt) ≠ 0 , then prove that: dy/dx = dy/dt/dx/dt. 

Hence find dy/dx, if  x = at² ,  y = 2at .

A box with a square base is to have an open top. The surface area of the box is 147 cm². What should its dimensions be in order that the volume is largest?

Evaluate: I = ∫ (5e^x) / ((e^x + 1)(e^2x + 9)) dx

Prove that: ∫ from 0 to 2a of f(x) dx = ∫ from 0 to a of f(x) dx + ∫ from 0 to a of f(2a - x) dx.

Hence show that: ∫ from 0 to π of sin(x) dx = 2 ∫ from 0 to π/2 of sin(x) dx.