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

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

(A) `1/5[[3,-1],[-2,1]]`

(B) `1/5[[3,1],[-2,1]]`

(C) `1/5[[-3,1],[-2,1]]`

(D) `1/5[[3,-1],[2,-1]]`

If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`

(A) 100

(B) 101

(C) 110

(D) 109

If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.

(A) `0,1/sqrt2,-1/sqrt2`

(B) `0,-1/sqrt2,-1/sqrt2`

(C) `1,1/sqrt2,1/sqrt2`

(D) `0,-1/sqrt2,1/sqrt2`

`barr=(hati-2hatj+3hatk)+lambda(2hati+hatj+2hatk)` is parallel to the plane `barr.(3hati-2hatj+phatk)=10`, find the value of p.

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

Write the negations of the following statements:

a.`forall n in N, n+7>6`

b. The kitchen is neat and tidy.

Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.

If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.

If `tan^-1(2x)+tan^-1(3x)=pi/4`, then find the value of ‘x’.

Write the converse, inverse and contrapositive of the following statement.
“If it rains then the match will be cancelled.”

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

Find the equation of the plane passing through the intersection of the planes 3x + 2y – z + 1 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r=(mbarb+nbara)/(m+n)` . Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.

The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`

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

Find the vector equation of a line passing through the points A(3, 4, –7) and B(6, –1, 1).

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

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

If `A=[[1,-1,2],[3,0,-2],[1,0,3]]` verify that A (adj A) = |A| I.

A company manufactures bicycles and tricycles each of which must be processed through machines A and B. Machine A has maximum of 120 hours available and machine B has maximum of 180 hours available. Manufacturing a bicycle requires 6 hours on machine A and 3 hours on machine B. Manufacturing a tricycle requires 4 hours on machine A and 10 hours on machine B.
If profits are Rs. 180 for a bicycle and Rs. 220 for a tricycle, formulate and solve the L.P.P. to determine the number of bicycles and tricycles that should be manufactured in order to maximize the profit.

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`

find the acute angle between the lines
x² – 4xy + y² = 0.

Given f (x) = 2x, x < 0
                 = 0, x ≥ 0 
then f (x) is _______ .

discontinuous and not differentiable at x = 0

continuous and differentiable at x = 0

discontinuous and differentiable at x = 0

continuous and not differentiable at x = 0

If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`

(A) 1

(B) 2

(C) –1

(D) –2

The function f (x) = x³ – 3x² + 3x – 100, x∈ R is _______.

(A) increasing

(B) decreasing

(C) increasing and decreasing

(D) neither increasing nor decreasing

Differentiate 3^x w.r.t. log₃x

Check whether the conditions of Rolle’s theorem are satisfied by the function
f (x) = (x - 1) (x - 2) (x - 3), x ∈ [1, 3]

Evaluate: `int sqrt(tanx)/(sinxcosx) dx`

Find the area of the region bounded by the curve x² = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.

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

If the function `f(x)=(5^sinx-1)^2/(xlog(1+2x))`  for x ≠ 0 is continuous at x = 0, find f (0).

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.

Suppose that 80% of all families own a television set. If 5 families are interviewed at  random, find the probability that
a. three families own a television set.
b. at least two families own a television set.

Find the approximate value of cos (60° 30').

(Given: 1° = 0.0175c, sin 60° = 0.8660)

The rate of growth of bacteria is proportional to the number present. If, initially, there were
1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½
hours. 

[Take `sqrt2` = 1.414]

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) is continuous on [–4, 2] defined as 

f (x) = 6b – 3ax, for -4 ≤ x < –2
       = 4x + 1,    for –2 ≤ x ≤ 2

Show that a + b =`-7/6`

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

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

Hence evaluate, `int xe^xdx`

Probability distribution of X is given by

Find P(X ≥ 2) and obtain cumulative distribution function of X

Solve the differential equation `dy/dx -y =e^x`

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/dxne0`

Hence if `y=sin^-1x, -1<=x<=1 , -pi/2<=y<=pi/2`

then show that `dy/dx=1/sqrt(1-x^2)`, where  `|x|<1`

Evaluate: `∫8/((x+2)(x^2+4))dx`