Mathematics and Statistics 2013-2014 HSC Science (General) 12th Standard Board Exam Question Paper Solution

Which of the following represents direction cosines of the line :

(a)`0,1/sqrt2,1/2`

(b)`0,-sqrt3/2,1/sqrt2`

(c)`0,sqrt3/2,1/2`

(d)`1/2,1/2,1/2`

`A=[[1,2],[3,4]]` ans A(Adj A)=KI, then the value of 'K' is

2

- 2

10

-10

The general solution of the trigonometric equation tan² θ = 1 is ____________

`theta =npi+-(pi/3),n in z`

`theta =npi+-pi/6, n in z`

`theta=npi+-pi/4, n in z`

`0=npi, n in z`

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment  AB.

The Cartestation equation of  line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

Equation of a plane is `vecr (3hati-4hatj+12hatk)=8`. Find the length of the perpendicular from the origin to the plane.

Find the acute angle between the lines whose direction ratios are 5, 12, -13 and 3, - 4, 5.

Write the dual of the following statements: (p ∨ q) ∧ T

Write the dual of the following statements:

Madhuri has curly hair and brown eyes.

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Prove that three vectors `bara, barb and barc ` are coplanar, if and only if, there exists a non-zero linear combination `xbara + ybarb + z barc = bar0`.

Using truth table prove that ∼p ˄ q ≡ (p ˅ q) ˄ ∼p

In any ΔABC, with usual notations, prove that b² = c² + a² – 2ca cos B.

Show that the equation `x^2-6xy+5y^2+10x-14y+9=0 ` represents a pair of lines. Find the acute angle between them. Also find the point of intersection of the lines.

Express the following equations in the matrix form and solve them by method of reduction :

2x- y + z = 1, x + 2y + 3z = 8, 3x + y - 4z =1

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.

Find the symbolic form of the following switching circuit, construct its switching table and interpret it.

If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Find the equation of the plane passing through the line of intersection of planes 2x – y + z = 3 and 4x – 3y + 5z + 9 = 0 and parallel to the line `(x + 1)/2 = (y + 3)/4 = (z - 3)/5`

Minimize :Z=6x+4y

Subject to : 3x+2y ≥12

x+y ≥5

0 ≤x ≤4

0 ≤ y ≤ 4 

Show that:

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

If y =1 - cos θ , x = 1 - sin θ , then ` dy/dx  at " "θ =pi/4`  is ________

The integrating factor of linear differential equation `dy/dx+ysecx=tanx` is

(a)secx- tan x

(b) sec x · tan x

(c)sex+tanx

(d) secx.cotx

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

Examine the continuity of the function
f(x) =sin x- cos x, for x ≠ 0

      =- 1 ,forx=0

at the poinl x = 0

Verify Rolle's theorem for the function  

f(x)=x²-5x+9 on [1,4]

Evaluate : `intsec^nxtanxdx`

The probability mass function (p.m.f.) of X is given below:

 find E(X²)

Given that X~ B(n = 10, p), if E(X) = 8. find the value of p.

Ify y=f(u) is a differentiable function of u and u = g(x) is a differentiable function of x then prove that y = f (g(x)) is a  differentiable function of x and

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

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

y = A cos (log x) + B sin (log x)

Evaluate : `int x^2/((x^2+2)(2x^2+1))dx` 

An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

Prove that:

`int_0^(2a)f(x)dx = int_0^af(x)dx + int_0^af(2a - x)dx`

If the function f (x) is continuous in the interval [-2, 2],find the values of a and b where

`f(x)=(sinax)/x-2, for-2<=x<=0`

`=2x+1, for 0<=x<=1`

`=2bsqrt(x^2+3)-1, for 1<x<=2`

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

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

If x^py^q = (x + y)^p+q then Prove that `dy/dx = y/x`

Find the area of the sector of a circle bounded by the circle x² + y² = 16 and the line y = x in the ftrst quadrant.

Prove that:

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

A random variable X has the following probability distribution :

(a) Find k
(b) find P(O <X< 4)
(c) Obtain cumulative distribution function (c. d. f.) of X.