Mathematics and Statistics Official 2023-2024 HSC Science (General) 12th Standard Board Exam Question Paper Solution

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

c ∧ (p ∨ q)

c ∧ (p ∧ q)

t ∧ (p ∧ q)

t ∧ (p ∨ q)

The principle solutions of the equation cos θ = `1/2` are ______.

`π/6, (5π)/6`

`π/3, (5π)/3`

`π/6, (7π)/6`

`π/3, (2π)/3`

If α, β, γ are direction angles of a line and α = 60°, β = 45°, then γ = ______.

30° or 90°

45° or 60°

90° or 130°

60° or 120°

The perpendicular distance of the plane `bar r. (3 hat i + 4 hat j + 12 hat k) = 78` from the origin is ______.

4

5

6

8

The slope of the tangent to the curve x = sin θ and y = cos 2θ at θ = `π/6` is ______.

`-2sqrt3`

`(-2)/sqrt3`

−2

`-1/2`

If `int_((-pi)/4) ^(pi/4) x^3 * sin^4 x  dx` = k then k = ______.

1

2

4

0

The integrating factor of linear differential equation `x dy/dx + 2y = x^2 log x` is ______.

`1/"x"`

k

`1/"n"^2`

x²

x

`1/x^2`

If the mean and variance of a binomial distribution are 18 and 12 respectively, then n = ______.

36

54

18

27

Write the following compound statement symbolically.

Nagpur is in Maharashtra and Chennai is in Tamil Nadu.

Evaluate:

`int1/(x^2 + 25)dx`

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 the truth table for the statement pattern:

[(p → q) ∧ q] → p

Check whether the following matrix is invertible or not:

`[(cos theta, sin theta),(-sin theta, cos theta)]`

In ΔABC, if a = 18, b = 24, c = 30 then find the values 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 `bara, barb, barc` are the position vectors of the points A, B, C respectively and `5 bar a - 3 bar b - 2 bar c = bar 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 `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.

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

Evaluate:

`int log x dx`

Evaluate the definite integral:

`int_0^(pi/2) cos^2 xdx`

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.

Find the probability distribution of number of heads in two tosses of a coin.

Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.

Prove the following:

`tan^-1(1/2) + tan^-1(1/3) = pi/(4)`

In ΔABC, prove the following:

`(cos A)/a + (cos B)/b + (cos C)/c = (a^2 + b^2 + c^2)/(2abc)`

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

Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj -2hatk) + μ(hati + 4hatj - 5hatk)`

Find the angle between the line `bar r = (hat i + 2hat j + hat k) + lambda(hat i + hat j + hat k)` and the plane `bar r *(2hat i + hat j + hat k) = 8`.

If y = sin^–1x, then show that `(1 - x^2) (d^2y)/(dx^2) - x * dy/dx` = 0

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]

Prove that:

`int 1/(a^2 - x^2) dx = 1/2 a log ((a +x)/(a-x)) + c`

Solve the following differential equation:

`x * dy/dx - y + x * sin(y/x) = 0`

Find k, if the following function is p.d.f. of r.v.X:

f(x) = `{:(kx^2(1 - x)",", "for"  0 < x < 1),(0",", "otherwise"):}`

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 tanθ = `|(2sqrt(h^2 - ab))/(a + b)|` Hence find the condition that the lines are coincident.

Find the volume of the parallelopiped 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 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 box is 147 sq. cm. What should be its dimensions in order that the volume is largest?

Evaluate:

`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`

Prove that:

`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`