Mathematics and Statistics 2017-2018 HSC Science (General) 12th Standard Board Exam Question Paper Solution

If A = `[(2,-3),(4,1)]`, then adjoint of matrix A is

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

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

(C)  `[(1,3),(4,-2)]`

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

The principal solutions of sec x = `2/sqrt3` are _____ 

`pi/3,(11pi)/6`

`pi/6, (11pi)/6`

`pi/4,(11pi)/4`

`pi/6,(11pi)/4`

The measure of the acute angle between the lines whose direction ratios are 3, 2, 6 and –2, 1, 2 is ______.

Write the negations of the following statements :

1) All students of this college live in the hostel

2) 6 is an even number or 36 is a perfect square.

If a line makes angles α, β, γ with co-ordinate axes, prove that cos 2α + cos2β + cos2γ+ 1 = 0.

Find the distance of the point (1, 2, –1) from the plane x - 2y + 4z - 10 = 0 .

Find the vector equation of the lines which passes through the point with position vector `4hati - hatj +2hatk` and is in the direction of `-2hati + hatj + hatk`

if `bara = 3hati - 2hatj+7hatk`, `barb  = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`

By vector method prove that the medians of a triangle are concurrent.

Using the truth table, prove the following logical equivalence :

p ↔ q ≡ (p ∧ q) ∨ (~p ∧ ~q)

If the origin is the centroid of the triangle whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r), then find the values of p, q, r.

Show that every homogeneous equation of degree two in x and y, i.e., ax² + 2hxy + by² = 0 represents a pair of lines passing through origin if h²−ab≥0.

In `triangle ABC` prove that `tan((C-A)/2) = ((c-a)/(c+a))cot  B/2`

Find the inverse of the matrix `A = [(1,2,-2),(-1,3,0),(0,-2,1)]`using elementary row transformations.

Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by 5x² +2xy- 3y² = 0.

Find the angle between the lines `(x -1)/4 = (y - 3)/1 = z/8`  and `(x-2)/2 = (y + 1)/2 = (z-4)/1`

Write converse, inverse and contrapositive of the following conditional statement :

If an angle is a right angle then its measure is 90°.

Prove that `sin^(-1) (3/5) + cos^(-1) (12/13) = sin^(-1) (56/65)`

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

Solve the following LPP by graphical method:

Minimize Z = 7x + y subject to 5x + y ≥ 5, x + y ≥ 3, x ≥ 0, y ≥ 0

Let the p. m. f. of a random variable X be __

P(x) = `(3-x)/10` for x = -1,0,1,2

= 0                        otherwise

Then E(X ) is ________.

1

2

0

-1

if `int_0^k 1/(2+ 8x^2) dx = pi/16` then the value of k is ________.

(A) `1/2`

(B) `1/3`

(C) `1/4`

(D) `1/5`

Integrating factor of linear differential equation `x (dy)/(dx) + 2y =x^2 log x` is ____________

`1/x^2`

`1/x`

`x`

`x^2`

Evaluate `int e^x [(cosx - sin x)/sin^2 x]dx`

if `y = tan^2(log x^3)`, find `(dy)/(dx)`

Find the area of ellipse `x^2/1 + y^2/4 = 1`

Obtain the differential equation by eliminating the arbitrary constants from the following equation :

`y = c_1e^(2x) + c_2e^(-2x)`

Given X ~ B (n, p)
If n = 10 and p = 0.4, find E(X) and var (X).

Evaluate `int 1/(3+ 2 sinx + cosx) dx`

If `x = acos^3t`, `y = asin^3 t`,

Show that `(dy)/(dx) =- (y/x)^(1/3)`

Examine the continuity of the function:

f(x) = `(log100 + log(0.01+x))/"3x"," for "x != 0 =  100/3 `for x = 0; at x = 0.

Examine the maxima and minima of the function f(x) = 2x³ - 21x² + 36x - 20 . Also, find the maximum and minimum values of f(x). 

Prove that:

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

Prove that: 

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

if  `f(x) = (x^2-9)/(x-3) + alpha`               for x> 3

           =5,                                     for x = 3

          `=2x^2+3x+beta`,             for x < 3

is continuous at x  = 3, find α and β.

Find `dy/dx` if `y = tan^(-1) ((5x+ 1)/(3-x-6x^2))`

A fair coin is tossed 9 times. Find the probability that it shows head exactly 5 times.

Verify Rolle’s theorem for the following function:

f (x) = x² - 4x + 10 on [0, 4]

Find the particular solution of the differential equation:

`y(1+logx) dx/dy - xlogx = 0`

when y = e² and x = e

Find the variance and standard deviation of the random variable X whose probability distribution is given below :