Mathematics and Statistics Set 1 2021-2022 HSC Science (General) 12th Standard Board Exam Question Paper Solution

The negation of p ^ (q → r) is ______.

∼ p ∧ (∼q →∼r)

p ∨ (∼q ∨∼r)

∼ p ∧ (∼q →r)

p → (q ∧∼r)

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`

Equation of line passing through the points (0, 0, 0) and (2, 1, –3) is ______.

`x/2 = y/1 = z/-3`

`x/2 = y/-1 = z/-3`

`x/1 = y/2 = z/3`

`x/3 = y/1 = z/2`

The value of `hati . (hatj xx hatk) + hatj . (hatk xx hati) + hatk . (hati xx hatj)` is ______ 

0

–1

1

3

If f(x) = x⁵ + 2x – 3, then (f^–1)¹ (–3) = ______.

0

– 3

`-1/3`

`1/2`

The maximum value of the function f(x) = `logx/x` is ______.

e

`1/e`

e²

`1/e^2`

If `int (dx)/(4x^2 - 1)` = A log `((2x - 1)/(2x + 1))` + c, then A = ______.

1

`1/2`

`1/3`

`1/4`

If the p.m.f of a r. v. X is

P(x) = `c/x^3`, for x = 1, 2, 3

        = 0, otherwise

then E(X) = ______.

`216/251`

`294/251`

`297/294`

`294/297`

Find the principal value of `cot^-1 ((-1)/sqrt(3))`

Write the separate equations of lines represented by the equation 5x² – 9y² = 0

If f'(x) = x^–1, then find f(x)

Write the degree of the differential equation (y''')² + 3(y") + 3xy' + 5y = 0

Using truth table verify that:

(p ∧ q)∨ ∼ q ≡ p∨ ∼ q

Find the cofactors of the elements of the matrix

`[(-1, 2),(-3, 4)]`

Find the principal solutions of cot θ = 0

Find the value of k. if 2x + y = 0 is one of the lines represented by 3x² + kxy + 2y² = 0

Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.

Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.

Find the equation of tangent to the curve y = 2x³ – x² + 2 at `(1/2, 2)`.

Evaluate: `int_0^(π/4) sec^4 x  dx`

Solve the differential equation

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

Show that function f(x) = tan x is increasing in `(0, π/2)`.

Form the differential equation of all lines which makes intercept 3 on x-axis.

If X ~ B (n, p) and E(X) = 6 and Var (X) = 4.2, then find n and p.

If 2 tan^–1(cos x) = tan^–1(2 cosec x). then find the value of x.

If the angle between the lines represented by ax² + 2hxy + by² = 0 is equal to the angle between the lines 2x² − 5xy + 3y² = 0, then show that 100(h² − ab) = (a + b)²

Find the distance between the parallel lines `x/2 = y/-1 = z/2` and `(x - 1)/2 = (y - 1)/-1 = (z - 1)/2`

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r = (mbarb + nbara)/(m + n) `

Find the vector equation of the plane passing through the point A(–1, 2, –5) and parallel to the vectors `4hati - hatj + 3hatk` and `hati + hatj - hatk`.

If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0

Evaluate: `int (dx)/(2 + cos x - sin x)`

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

A wire of length 36 metres is bent in the form of a rectangle. Find its dimensions if the area of the rectangle is maximum.

Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.

If a fair coin is tossed 10 times. Find the probability of getting at most six heads.

Without using truth table prove that (p ∧ q) ∨ (∼ p ∧ q) v (p∧ ∼ q) ≡ p ∨ q

Solve the following system of equations by the method of inversion.

x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2

Using vectors prove that the altitudes of a triangle are concurrent.

Solve the following LPP by graphical method:

Minimize z = 8x + 10y, subject to 2x + y ≥ 7, 2x + 3y ≥ 15, y ≥ 2, x ≥ 0, y ≥ 0.

If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost

If u and v ore differentiable functions of x. then prove that:

`int uv  dx = u intv  dx - int [(du)/(d) intv  dx]dx`

Hence evaluate `intlog x  dx`

Find the area of the region lying between the parabolas y² = 4ax and x² = 4ay.

 Show that: `int _0^(pi/4) log (1 + tanx) dx = pi/8 log2`