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

If`[ bara  bar b barc ] ≠ 0  and  barp = [ barb xx barc ]/([ bara  bar b barc  ]), barq = [ barc xx bara ]/([ bara  bar b barc  ]), barr = [ bara xx barb ]/[ bara  bar b barc ]`

then `bara . barp + barb . barq + barc . barr` is equal to ______.

0

1

2

3

The inverse of the matrix `[[2,0,0],[0,1,0],[0,0,-1]]`is --------

(a) `[[1/2,0,0],[0,1,0],[0,0,-1]]`

(b) `[[-1/2,0,0],[0,-1,0],[0,0,1]]`

(c) `[[-1,0,0],[0,-1/2,0],[0,0,1/2]]`

(d) `1/2[[-1/2,0,0],[0,-1,0],[0,0,-1]]`

Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........

`+-1/sqrt51,+-5/sqrt51,+-1/sqrt51`

`+-5/sqrt51, +-1/sqrt51, +- (-5)/sqrt51`

`+-sqrt5,+-1,+-5`

`+-sqrt51,+-sqrt51+-sqrt51`

Write truth values of the following statements :`sqrt5` is an irrational number but 3 +`sqrt 5` is a complex number.

True

False

Write truth values of the following statements: ∃ n ∈ N such that n + 5 > 10.

True

False

If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector  `2hati + hatj + 2hatk.`

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

Using truth table, prove the following logical equivalence:

(p ∧ q) → r ≡ p → (q → r)

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

Show that `2sin^-1(3/5) = tan^-1(24/7)`

Solve the following equations by the method of reduction :

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

Prove that the volume of a parallelopiped with coterminal edges as  ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges  `bar i+barj, barj+bark `

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.

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a +   c - b)

If p : It is a day time, q : It is warm, write the compound statements in verbal form
denoted by -
(a) p ∧ ~ q  
(b)  ~ p  → q   
(c)  q  ↔  p

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

Parametric form of the equation of the plane is `bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk)` λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

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 general solution of :  sinx · tanx = tanx - sinx + 1

The differential equation of the family of curves y=c₁e^x+c₂e^-x is......

(a)`(d^2y)/dx^2+y=0`

(b)`(d^2y)/dx^2-y=0`

(c)`(d^2y)/dx^2+1=0`

(d)`(d^2y)/dx^2-1=0`

If X is a random variable with probability mass function

P(x) = kx ,  x=1,2,3

      = 0 ,     otherwise

then , k=..............

(a) 1/5

(b) 1/4

(c) 1/6

(d) 2/3

If `sec((x+y)/(x-y))=a^2. " then " (d^2y)/dx^2=........`

(a) y

(b) x

(c) y/x

(d) 0

If `y=sin^-1(3x)+sec^-1(1/(3x)), `  find dy/dx

Evaluate :`intxlogxdx`

If `int_0^h1/(2+8x^2)dx=pi/16 `then find the value of h.

The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly two of the next four components tested will survive.

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

Examine the continuity of the following function at given point:

`f(x)=(logx-log8)/(x-8) , `

` =8, `

If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`

Solve : 3e^x tanydx + (1 +e^x) sec² ydy = 0

Also, find the particular solution when x = 0 and y = π.

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.

Evaluate : `intlogx/(1+logx)^2dx`

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and  hence, find dy/dx if x=a cost, y=a sint

Show that the function defined by f(x) =|cosx| is continuous function.

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

Given X ~ B(n, p). If n = 20, E(X) = 10, find p_, Var. (X) and   S.D. (X).

A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`