Mathematics and Statistics 2015-2016 HSC Science (General) 12th Standard Board Exam Question Paper Solution

Inverse of the statement pattern (p ∨ q) → (p ∧ q) is 

(A) (p ∧ q) → (p ∨ q)

(B) ∼ (p ∨ q) → (p ∧ q)

(C) (∼ p ∨ ∼ q) → (∼ p ∧ ∼ q)

(D) (∼ p ∧ ∼ q) → (∼ p ∨ ∼ q)

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to

(A) `1/sqrt5`

(B) `1/sqrt10`

(C) `1/sqrt15`

(D) `1/(2sqrt5)`

Find the angle between the lines `barr=3hati+2hatj-4hatk+lambda(hati+2hatj+2hatk)` and `barr=5 hati-2hatk+mu(3hati+2hatj+6hatk)`

If p, q, r are the statements with truth values T, F, T, respectively then find the truth value of (r ∧ q) ↔ ∼ p

If `A =[[2,-3],[3,5]]` then find A^-1 by adjoint method. 

By vector method show that the quadrilateral with vertices A (1, 2, –1), B (8, –3, –4), C (5, –4, 1), D (–2, 1, 4) is a parallelogram.

Find the general solution of the equation sin x = tan x.

Find the joint equation of pair of lines passing through the origin and perpendicular to the lines represented by ax²+ 2hxy + by²= 0

Find the principal value of `sin^-1(1/sqrt2)`

Find the cartesian form of the equation of the plane `bar r=(hati+hatj)+s(hati-hatj+2hatk)+t(hati+2hatj+hatj)`

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

A line makes angles of measures 45° and 60° with positive direction of y and z axes respectively. Find the d.c.s. of the line and also find the vector of  magnitude 5 along the direction of line.

Solve the following LPP by graphical method:

Maximize: z = 3x + 5y
Subject to:  x + 4y ≤ 24
                  3x + y ≤ 21
                  x + y ≤ 9
                  x ≥ 0, y ≥ 0

Find the shortest distance between the lines `(x+1)/7=(y+1)/(-6)=(z+1)/1 and (x-3)/1=(y-5)/(-2)=(z-7)/1`

Show that the points (1, –1, 3) and (3, 4, 3) are equidistant from the plane 5x + 2y – 7z + 8 = 0

In any triangle ABC with usual notations prove c = a cos B + b cos A

Find p and k if the equation px² – 8xy + 3y² +14x + 2y + k = 0 represents a pair of perpendicular lines.

The cost of 4 pencils, 3 pens and 2 erasers is Rs. 60. The cost of 2 pencils, 4 pens and 6 erasers is Rs. 90 whereas the cost of 6 pencils, 2 pens and 3 erasers is Rs. 70. Find the cost of each item by using matrices.

Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

Order and degree of the differential equation `[1+(dy/dx)^3]^(7/3)=7(d^2y)/(dx^2)` are respectively 

(A) 2, 3

(B) 3, 2

(C) 7, 2

(D) 3, 7

`∫_4^9 1/sqrtxdx=`_____

(A) 1

(B) –2

(C) 2

(D) –1

If the p.d.f. of a continuous random variable X is given as

`f(x)=x^2/3` for -1< x<2

       =0   otherwise

then c.d.f. fo X is

(A) `x^3/9+1/9`

(B) `x^3/9-1/9`

(C) `x^2/4+1/4`

(D) `1/(9x^3)+1/9`

If `y = sec sqrtx` then find dy/dx.

Evaluate : `∫(x+1)/((x+2)(x+3))dx`

Find the area of the region lying in the first quandrant bounded by the curve y²= 4x, X axis and the lines x = 1, x = 4

For the differential equation, find the general solution:

sec² x tan y dx + sec² y tan x dy = 0

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

If the function `f(x)=(4^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

Evaluate : `∫1/(3+2sinx+cosx)dx`

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 point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

The probability mass function for X = number of major defects in a randomly selected
appliance of a certain type is 

Find the expected value and variance of X.

Prove that `int_0^af(x)dx=int_0^af(a-x) dx`

hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`

If y = e^{tan x}+ (log x)^{tan x} then find dy/dx

If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)¹⁹ = 0⋅1348]

Find a and b, so that the function f(x) defined by

f(x)=-2sin x,       for -π≤ x ≤ -π/2

     =a sin x+b,  for -π/2≤ x ≤ π/2

     =cos x,        for π/2≤ x ≤ π

is continuous on [- π, π]

Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5

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

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

Hence evaluate, `int xe^xdx`

Find the approximate value of log₁₀ (1016), given that log₁₀e = 0⋅4343.