Mathematics and Statistics Official 2022-2023 HSC Science (General) 12th Standard Board Exam Question Paper Solution

If p ∧ q is F, p → q is F then the truth values of p and q are ________.

T, T

T, F 

F, T

F, F

Select the correct option from the given alternatives:

In ΔABC if c² + a² – b² = ac, then ∠B = ____.

`π/4`

`π/3`

`π/2`

`π/6`

The area of the triangle with vertices (1, 2, 0), (1, 0, 2) and (0, 3, 1) in sq. unit is ______.

`sqrt(5)`

`sqrt(7)`

`sqrt(6)`

`sqrt(3)`

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 ______.

(2, 2)

(2, 2)

(0, 10)

(0, 10)

(4, 0)

(4, 0)

(3, 4)

(2, 4)

If y is a function of x and log (x + y) = 2xy, then the value of y'(0) = ______.

2

0

–1

1

`int cos^3x  dx` = ______.

`1/12 sin 3x + 3/4 sin  x + c`

`1/12 sin 3x + 1/4 sin x + c`

`1/12 sin 3x - 3/4 sin x + c`

`1/12 sin 3x - 1/4 sin x + c`

The solution of the differential equation `dx/dt = (xlogx)/t` is ______.

x = e^ct

x + e^ct = 0

x = e^t + t

xe^ct = 0

Let the probability mass function (p.m.f.) of a random variable X be P(X = x) = `""^4C_x (5/9)^x xx (4/9)^(4 - x)`, for x = 0, 1, 2, 3, 4 then E(X) is equal to ______.

`20/9`

`9/20`

`12/9`

`9/25`

Write the joint equation of co-ordinate axes.

Find the values of c which satisfy `|"c"overline"u"|` = 3 where `overline"u" = hat"i" + 2hat"j" + 3hat"k"`.

Write `int cotx  dx`.

State the degree of differential equation `"e"^((dy)/(dx)) + (dy)/(dx)` = x

Write converse, inverse and contrapositive of the following statement.

If x < y then x² < y² (x, y ∈ R)

If A = `[("x",0,0),(0,"y",0),(0,0,"z")]` is a non-singular matrix, then find A⁻¹ by using elementary row transformations. Hence, find the inverse of `[(2,0,0),(0,1,0),(0,0,-1)]`

Find the Cartesian co-ordinates of the point whose polar co-ordinates are:

`(sqrt(2), pi/4)`

If ax² + 2hxy + by² = 0 represents a pair of lines and h² = ab ≠ 0 then find the ratio of their slopes.

If `overlinea, overlineb, overlinec` are the position vectors of the points A, B, C respectively and `5overlinea + 3overlineb - 8overlinec = overline0` then find the ratio in which the point C divides the line segment AB.

Find the feasible solution of the following inequation:

2x + 3y ≤ 6, x + y ≥ 2, x ≥ 0, y ≥ 0

Show that the function f(x) = x³ + 10x + 7 for x ∈ R is strictly increasing

Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x)  "d"x`

Find the area of the region bounded by the curve y² = 4x, the X-axis and the lines x = 1, x = 4 for y ≥ 0.

Solve the following differential equation:

cos x . cos y dy − sin x . sin y dx = 0

Find the mean of number randomly selected from 1 to 15.

Find the area of the region bounded by the curve y = x² and the line y = 4.

Find the general solution of sin θ + sin 3θ + sin 5θ = 0

If –1 ≤ x ≤ 1, the prove that sin^–1 x + cos^–1 x = `π/2`

If θ is the acute angle between the lines represented by ax² + 2hxy + by² = 0 then prove that tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`

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 the lines `(x-1)/2=(y-2)/3=(z-3)/4 and (x-2)/3=(y-4)/4=(z-5)/5` 

Lines `overliner = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `overliner = (4hati - 3hatj + 2hatk) + μ(hati - 2hatj + 2hatk)` are coplanar. Find the equation of the plane determined by them.

If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... +  ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.

Find `dy/dx` at x = 0.

Find the approximate value of sin (30° 30′). Give that 1° = 0.0175^c and cos 30° = 0.866

Evaluate the following : `int x tan^-1 x .dx`

Find the particular solution of the differential equation `dy/dx` = e^2y cos x, when x = `π/6`, y = 0

For the following probability density function of a random variable X, find P(X < 1).

`{:(f(x) = (x + 2)/18,";"  "for" -2 < x < 4),(               = 0,","  "otherwise"):}`

For the following probability density function of a random variable X, find P(|X| < 1).

`{:(f(x) = (x + 2)/18,";"  "for" -2 < x < 4),(               = 0,","  "otherwise"):}`

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 = `[(1, 2),(3, 4)]` verify that A (adj A) = (adj A) A = |A| I

Prove that the volume of a tetrahedron with coterminus edges `overlinea, overlineb` and `overlinec` is `1/6[(overlinea, overlineb, overlinec)]`.

Hence, find the volume of tetrahedron whose coterminus edges are `overlinea = hati + 2hatj + 3hatk, overlineb = -hati + hatj + 2hatk` and `overlinec = 2hati + hatj + 4hatk`.

Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line `overliner = (7hati + 7hatj + 6hatk) + λ(-2hati + 2hatj + 3hatk)`

If y = cos(m cos^–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

Verify Lagrange’s mean value theorem for the function f(x) = `sqrt(x + 4)` on the interval [0, 5].

Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`

Prove that: `int_0^(2a) f(x)dx = int_0^a f(x)dx + int_0^a f(2a - x)dx`