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

`cos[tan^-1  1/3 + tan^-1  1/2]` = ______

`1/sqrt2`

`sqrt3/2`

`1/2`

`pi/4`

If θ is the angle between any two vectors `bara` and `barb` and `|bara · barb| = |bara xx barb|` then θ is equal to ______.

0

`π/4  "or"  (3π)/4`

`π/2`

`pi  "or"  pi/6`

The angle between the lines `barr = (hati+2hatj - 3hatk) + λ(3hati+2hatj+6hatk) and barr = (5hati-2hatj+7hatk) +μ(hati+2hatj+2hatk)` is ______.

`cos^-1 (17/21)`

`cos^-1 (20/21)`

`cos^-1 (18/21)`

`cos^-1 (19/21)`

The perpendicular distance of the plane `barr.(2hati +3hatj-hatk)=5`, from the origin is ______.

`5/sqrt14` units

`5/14` units

5 units

`sqrt14/5` units

If x = `e^(x/y)` then `dy/dx` = ______. 

`1-y/x`

`1+y/x`

`(x-y)/(xlogx)`

`(x+y)/(xlogx)`

`y = c^2 + c/x` is solution of ______.

`x^4 ((dy)/(dx))^2 - x (dy)/(dx) - y = 0`

`x^2 ((dy)/(dx))^2 + y = 0`

`x^3 ((d^2y)/(dx^2)) - x (dy)/(dx) + y = 0`

`x (d^2y)/(dx^2) = 4y`

Given that X ~ B(n, p). If n = 10 and p = 0.4 then E(X) and Var(X) respectively are ______.

4, 0.24

0.4, 0.24

4, 2.4

3, 0.24

The approximate value of tan (44°30'), given that 1° = 0.0175^c, is ______.

0.8952

0.9528

0.9285

0.9825

Find the combined equation of the following pair of lines:

2x + y = 0 and 3x − y = 0

Find the value of `sin^-1(sin  (5pi)/3)`.

Evaluate:

`int5^x/3^xdx`

Write the integrating factor (I.F.) of the differential equation `(dy)/(dx) + y = e^-x`.

If the statements p, q are true statements and r, s are false statements, then determine the truth value of the statement pattern: 

(q ∧ r) ∨ (∼P ∧ s)

Find the inverse of the following matrix.

`[(2, -3),(-1, 2)]`

Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.

Find the acute angle between the lines represented by xy + y² = 0.

Using the truth table show that the statement pattern p → (q → p) is a tautology.

If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.

Find points on the curve given by y = x³ − 6x² + x + 3, where the tangents are parallel to the line y = x + 5.

Evaluate:

`int (e^x (1 + x))/(sin^2 (xe^x))dx`

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.

Evaluate:

`int_0^(pi/4) sqrt(1 + sin 2x)*dx`

The probability distribution of X is as follows:

Find:

1. k
2. P[X < 2]
3. P[X ≥ 3]
4. P[1 ≤ X < 4]
5. P(2)

Find the particular solution of `r (dr)/(dθ) + cos θ` = 5 at r = `sqrt2` and θ = 0.

In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.

Are the four points A(1, −1, 1), B(−1, 1, 1), C(1, 1, 1) and D(2, −3, 4) coplanar? Justify your answer.

Show that the difference between the slopes of the lines given by (tan²θ + cos²θ)x² − 2xy tan θ + (sin²θ)y² = 0 is two.

Find the vector equation of a line passing through the point `hati + 2hatj + 3hatk` and perpendicular to the vectors `hati + hatj + hatk` and `2hati - hatj + hatk`.

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

Find the equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin.

Find `dy/dx` if `y = tan^-1 (sqrt((3 - x)/(3 + x)))`.

Find the approximate value of f(x) = x³ + 5x² − 2x + 3 at x = 1.98.

Evaluate:

`int sin(x+a)/cos(x-b)dx`

Solve the following differential equation:

dr + (2r cotθ + sin2θ)dθ = 0

Let X ~ B(10, 0.2). Find P(X = 1).

Let X ~ B(10, 0.2). Find P(X ≥ 1).

Find the expected value, variance and standard deviation of random variable X whose probability mass function (p.m.f.) is given below: 

Simplify the following circuit so that the new circuit has minimum number of switches. Also, draw the simplified circuit.

If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` then find A⁻¹ by the adjoint method.

In ΔABC, D and E are points on BC and AC, respectively, such that BD = 2DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find the ratio `(BP)/(PE)` using the vector method.

A firm manufactures two products A and B on which profit earned per unit are ₹ 3 and ₹ 4 respectively. Each product is processed on two machines M₁ and M₂. The product A requires one minute of processing time on M₁ and two minutes of processing time on M₂, B requires one minute of processing time on M₁ and one minute of processing time on M₂. Machine M₁ is available for use for 450 minutes while M₂ is available for 600 minutes during any working day. Find the number of units of products A and B to be manufactured to get the maximum profit.

If y = f(u) is a differentiable function of u and u = g(x) is differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x then prove that `dy/dx=dy/(du)xx(du)/dx`. Hence find `d/dx(1/sqrtsinx)`.

Evaluate the following:

`int x^2 sin 3x  dx`

Prove that:

`{:(int_(-a)^a f(x) dx  = 2 int_0^a f(x) dx",", "If"  f  "is an even function"),(                                       = 0",", "if"  f  "is an odd function"):}`

Hence find the value of `int_-1^1tan^-1x  dx`.

Find the area of the ellipse `x^2/a^2 + y^2/b^2 = 1`.

Hence write area of `x^2/25 + y^2/16 = 1`.