Mathematics and Statistics Official 2024-2025 HSC Science (General) 12th Standard Board Exam Question Paper Solution

If A = {1, 2, 3, 4, 5} then which of the following is not true?

∃ x ∈ A such that x + 3 = 8

∃ x ∈ A such that x + 2 < 9

∀ x ∈ A, x + 6 ≥ 9

∃ x ∈ A such that x + 6 < 10

In ΔABC, (a + b) cos C + (b + c) cos A + (c + a) cos B is equal to ______.

a − b + c

a + b − c

a + b + c

a − b − c

If `|bar a|` = 5, `|bar b|` = 13 and `|bara xx barb|` = 25 then `|bar a * bar b|` is equal to ______.

30

60

40

45

The vector equation of the line passing through the point having position vector `4 hat i - hat j + 2hat k` and parallel to vector `-2 hat i - hat j + hat k` is given by ______.

`(4hat i - hat j - 2hat k) + lambda (-2hat i - hat j + hat k)`

`(4hat i - hat j + 2hat k) + lambda (2hat i - hat j + hat k)`

`(4hat i - hat j + 2hat k) + lambda (-2hat i - hat j - hat k)`

`(4hat i - hat j + 2hat k) + lambda (-2hat i - hat j + hat k)`

Let f(1) = 3, f'(1) = `-1/3`, g(1) = −4 and g'(1) = `-8/3`. The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.r. x at x = 1 is ______.

`-29/25`

`7/3`

`31/15`

`29/15`

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

36

54

18

27

The value of `intx^x (1 + logx)dx` is equal to ______.

`1/2(1 + logx)^2 + c`

`x^(2x) + c`

`x^x.logx + c`

`x^x + c`

The area bounded by the line y = x, X-axis and the lines x = −1 and x = 4 is equal to ______ (in square units).

`2/17`

8

`17/2`

`1/2`

Write the negation of the statement.

‘∃ n ∈ N such that n + 8 > 11’

Write unit vector in the opposite direction to `baru = 8hati + 3hatj - hatk`.

Write the order of the differential equation `sqrt(1 + (dy/dx)^2) = ((d^2y)/dx^2)^(3/2)`.

Write the condition for the function f(x) to be strictly increasing for all x ∈ R.

Using truth table, prove that the statement patterns p ↔ q and (p ∧ q) ∨ (~ p ∧ ~ q) are logically equivalent.

Find the adjoint of the matrix `[(2,-2),(4,3)]`.

Find the general solution of tan²θ = 1.

Find the coordinates of the points of intersection of the lines represented by x² − y² − 2x + 1 = 0

A line makes angles of measure 45° and 60° with the positive directions of the Y and Z axes respectively. Find the angle made by the line with the positive direction of the X-axis. 

Find the vector equation of the plane passing through the point having position vector `2hati + 3hatj + 4hatk` and perpendicular to the vector `2hati + hatj - 2hatk`.

Divide the number 20 into two parts such that sum of their squares is minimum.

Integrate the following function w.r.t. x:

x⁹.sec²(x¹⁰)

Evaluate the following:

`int (1)/(25 - 9x^2)*dx`

Evaluate:

`int_(-pi/4)^(pi/4) (1)/(1 - sinx)*dx`

Find the area of the region bounded by the parabola y² = 16x and its latus rectum.

Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = `1/5`, for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is between 1 and 3.

Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by f(x) = `1/5`, for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that the waiting time is more than 4 minutes.

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

Prove that `2 tan^(-1) (1/3) + cos^(-1) (3/5) = pi/2`.

In ΔABC if a = 13, b = 14, c = 15, then find the value of sec A.

In ΔABC if a = 13, b = 14, c = 15, then find the value of `"cosec" A/2`.

A line passes through the points (6, −7, −1) and (2, −3, 1). Find the direction ratios and the direction cosines of the line. Show that the line does not pass through the origin.

Find the cartesian and vector equations of the line passing through A(1, 2, 3) and having direction ratios 2, 3, 7.

Find the vector equation of the plane passing through points A(1, 1, 2), B(0, 2, 3) and C(4, 5, 6).

Find the n^th order derivative of log x.

The displacement of a particle at time t is given by s = 2t³ − 5t² + 4t − 3. Find the velocity and displacement at the time when the acceleration is 14 ft/sec².

Find the equations of tangent and normal to the curve y = 2x³ − x² + 2 at point `(1/2, 2)`.

Three coins are tossed simultaneously; X is the number of heads. Find the expected value and variance of X.

Solve the differential equation:

`x dy/dx = x·tan(y/x)+y`

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that all the five cards are spades.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards; find the probability that none is a spade.

Find the inverse of A = `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)]` by elementary row transformations.

Prove that homogeneous equation of degree two in x and y, ax² + 2hxy + by² = 0 represents a pair of lines passing through the origin if h² − ab ≥ 0. Hence show that equation x² + y² = 0 does not represent a pair of lines.

Let `bara` and `barb` be non-collinear vectors. If vector `barr` is coplanar with `bara` and `barb`, then show that there exist unique scalars t₁​ and t₂​ such that `barr = t_1 bara + t_2 barb`. For `barr = 2hati + 7hatj + 9hatk, bara = hati + 2hatj, barb = hatj + 3hatk`, find t₁, t₂.

Solve the Linear Programming problem graphically:

Maximize z = 3x + 5y subject to x + 4y ≤ 24, 3x + y ≤ 21, x + y ≤ 9, x ≥ 0, y ≥ 0 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 if `(dx)/(dt)` ≠ 0 then prove that `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.

Hence, find the derivative of 7^x w.r.t. x⁷.

Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`

Prove that: `int_a^b f(x) dx = int_a^b f(a + b - x)dx`

Hence evaluate: `int_0^3 sqrtx/(sqrtx + sqrt(3 - x)) dx`

If a body cools from 80°C to 50°C at room temperature of 25°C in 30 minutes, find the temperature of the body after 1 hour.