Mathematics and Statistics 2014-2015 HSC Science (General) 12th Standard Board Exam Question Paper Solution

if  `A=[[2,0,0],[0,2,0],[0,0,2]]` then A6=  ......................

6A

12A

16A

32A

The principal solution of `cos^-1(-1/2)` is :

`pi/3`

`pi/6`

`(2pi)/3`

`(3pi)/2`

If an equation hxy + gx + fy + c = 0 represents a pair of lines, then __________

fg = ch         

gh = cf

Jh = cg     

hf= - eg

Write the converse and contrapositive of the statement -
“If two triangles are congruent, then their areas are equal.”

Find 'k' if the sum of slopes of lines represented by equation x² + kxy − 3y² = 0 is twice their product.

Find the angle between the planes `bar r.(2bar i+barj-bark)=3 and bar r.(hati+2hatj+hatk)=1`

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

If `bara=bari+2barj, barb=-2bari+barj,barc=4bari+3barj`,  find x and y such that `barc=xbara+ybarb`

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.

Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

[~(~p ∧ ~q)] v q

If point C `(barc)` divides the segment joining the points A(`bara`) and  B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1 

In any ΔABC if  a² , b² , c² are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.

If θ is the acute angle between the lines represented by equation ax² + 2hxy + by² = 0  then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|, a+b!=0`

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

Construct the switching circuit for the following statement : [p v (~ p ∧ q)] v [(- q ∧ r) v ~ p]

Find the general solution of : cos x - sin x = 1.

Find the equations of the planes parallel to the plane x-2y + 2z-4 = 0, which are at a unit distance from the point (1,2, 3).

A diet of a sick person must contain at least 48 units of vitamin A and 64 units of vitamin B. Two foods F ₁ and F₂ are available. Food F₁ costs Rs. 6 per unit and food F₂ costs Rs. 10 per unit. One unit of food F₁ contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F₂ contains 8 units of vitamin A and 12 units of vitamin B.Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutritional requirements.

A random variable X has the following probability distribution:

then E(X)=....................

0.8

0.9

0.7

1.1

If `int_0^alpha3x^2dx=8` then the value of α is :

(a) 0

(b) -2

(c) 2 

(d) ±2

The differential equation of `y=c/x+c^2` is :

(a)`x^4(dy/dx)^2-xdy/dx=y`

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

(c)`x^3(dy/dx)^2+xdy/dx=y`

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

Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`

If   `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

If y=e^ax ,show that  `xdy/dx=ylogy`

A fair coin is tossed five times. Find the probability that it shows exactly three times head.

Integrate : sec³ x w. r. t. x.

If y = (tan^-1 x)², show that

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

If `f(x)=[tan(pi/4+x)]^(1/x), `

           = k                        ,for x=0

is continuous at x=0 , find k.

Find the co-ordinates of the points on the curve y=x-(4/x) where the tangents are parallel to the line y=2x

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`

Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`

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]

If  `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that"  dy/dx = [-99x^2]/[101y^2]`

Let the p. m. f. (probability mass function) of random variable x be

`p(x)=(4/x)(5/9)^x(4/9)^(4-x), x=0, 1, 2, 3, 4`

         =0 otherwise

find E(x) and var (x)

Examine the maxima and minima of the function f(x) = 2x³ - 21x² + 36x - 20 . Also, find the maximum and minimum values of f(x). 

Solve the differential equation (x² + y²)dx- 2xydy = 0

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)