Mathematics and Statistics 2016-2017 HSC Science (General) 12th Standard Board Exam Question Paper Solution

If the points A(2, 1, 1), B(0, -1, 4) and C(k, 3, -2) are collinear, then k 

(A) 0

(B) 1

(C) 4

(D) -4

The inverse of the matrix `[[-1,5],[-3,2]]` is _______

1/13`[[2,-5],[3,-1]]`

1/13 `[[-1,5],[-3,2]]`

1/13 `[[-1,-3],[5,2]]`

1/13 `[[1,5],[3,-2]]`

In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.

(A) `1/5`

(B) `sqrt(1/5)`

(C) `4/5`

(D) `2/5`

Find the volume of the parallelopiped whose coterminus edges are given by vectors

`2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

In Δ ABC, prove that, a (b cos C - c cos B) = b² - c².

If from a point Q (a, b, c) perpendiculars QA and QB are drawn to the YZ and ZX planes respectively, then find the vector equation of the plane QAB.

Find the cartesian equation of the line passing throught the points A(3, 4, -7) and B(6,-1, 1).

Write the following statement in symbolic form and find its truth value:
∀ n ∈ N, n² + n is an even number and n² - n is an odd number.

Using truth tables, examine whether the statement pattern (p ∧ q) ∨ (p ∧ r) is a tautology, contradiction or contingency.

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` 

Find the general solution of the equation sin 2x + sin 4x + sin 6x = 0

Solve the following equations by method of reduction: 

x-y + z = 4,

2x + y - 3z = 0,

x + y + z = 2

If θ is the measure of acute angle between the pair of lines given by `ax^2+2hxy+by^2=0,` then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|,a+bne0`

Using vector method, find incentre of the triangle whoose vertices are P(0, 4, 0), Q(0, 0, 3)
and R(0, 4, 3).

Construct the switching circuit for the statement (p ∧ q) ∨ (~ p) ∨ (p ∧ ~ q).

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.

Show that:

`cos^(-1)(4/5)+cos^(-1)(12/13)=cos^(-1)(33/65)`

If l, m, n are the direction cosines of a line, then prove that l² + m² + n² = 1. Hence find the
direction angle of the line with the X axis which makes direction angles of 135° and 45° with Y and Z axes respectively.

Find the vector and cartesian equations of the plane passing through the points A( 1, 1, -2), B(1, 2, 1) and C(2, -1, 1).

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Derivatives of  tan³θ with respect to sec³θ at θ=π/3 is

(A)` 3/2`

(B) `sqrt3/2`

(C) `1/2`

(D) `-sqrt3/2`

The equation of tangent to the curve y = 3x² - x + 1 at the point (1, 3) is 

(a) y=5x+2

(b)y=5x-2

(c)y=1/5x+2

(d)y=1/5x-2

The expected value of the number of heads obtained when three fair coins are tossed simultaneously is

(A) 1

(B) 1.5

(C) 0

(D) -1

Find dy/dx if x sin y + y sin x = 0.

Test whether the function is increasing or decreasing. 

f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0, 

Evaluate: `intsinsqrtx/sqrtxdx`

Form the differential equation by eliminating arbitrary constants from the relation `y=Ae^(5x)+Be^(-5x)`

The probability that a bomb will hit a target is 0.8. Find the probability that out of 10 bombs dropped, exactly 4 will hit the target.

Solve: dy/dx = cos(x + y)

If u and v are two functions of x then prove that

`intuvdx=uintvdx-int[du/dxintvdx]dx`

Hence evaluate, `int xe^xdx`

If `f(x) =(e^(x^2)-cosx)/x^2`, for x= 0, is continuous at x = 0, find f(0).

If y = f(x) is a differentiable function of x such that inverse function x = f^–1 (y) exists, then prove that x is a differentiable function of y and `dx/dy=1/((dy/dx)) " where " dy/dx≠0`

A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.

Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`

Discuss the continuity of the following function, at x = 0.

`f(x)=x/|x|,for x ne0`

`=1,`for `x=0`

If the population of a country doubles in 60 years, in how many years will it be triple under
the assumption that the rate of increase in proportional to the number of inhabitants?
[Given : log 2 = 0.6912 and log 3 = 1.0986.]

A fair coin is tossed 8 times. Find the probability that it shows heads exactly 5 times.

A fair coin is tossed 8 times. Find the probability that it shows heads at least once

Find: `I=intdx/(sinx+sin2x)`

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

Given the p. d. f. (probability density function) of a continuous random variable x as :

 `f(x)=x^2/3, -1`

         = 0 , otherwise

Determine the c. d. f. (cumulative distribution function) of x and hence find P(x < 1), P(x ≤ -2), P(x > 0), P(1 < x < 2)