Mathematics and Statistics 2012-2013 HSC Science (General) 12th Standard Board Exam Question Paper Solution

The principal solution of the equation cot x=`-sqrt 3 ` is

`pi/6`

`pi/3`

`(5pi)/6`

`(-5pi)/6`

If the vectors `-3hati+4hatj-2hatk, hati+2hatk, hati-phatj` are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

If the line y = x + k  touches the hyperbola 9x² -16y² = 144, then k = _________

7

-7

`+-sqrt7`

`+-sqrt19`

Write down the following statements in symbolic form :

(A) A triangle is equilateral if and only if it is equiangular.
(B) Price increases and demand falls

If `A=[[2,-2],[4,3]]` then find `A^-1` by adjoint method.

Find the separate equations of the lines represented by the equation 3x² – 10xy – 8y² = 0.

Find the equation of the director circle of a circle ` x^2 + y^2 =100.`

Find the general solution of the equation `4cos^2x=1`

Without using the truth table show that P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)

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`

Show that the line x+ 2y + 8 = 0 is tangent to the parabola y² = 8x. Hence find the point of contact

The sum of three numbers is 9. If we multiply third number by 3 and add to the second number, we get 16. By adding the first and the third number and then subtracting twice the second number from this sum, we get 6. Use this information and find the system of linear equations. Hence, find the three numbers using matrices.

Find the general solution of cos x +sin x =1.

If `bara and barb` are any two non-zero and non-collinear vectors then prove that any vector `barr` coplanar with  `bara` and `barb` can be uniquely expressed as `barr = t_1bara + t_2barb` , where t₁ and t₂ are scalars.

Using truth table examine whether the following statement pattern is tautology, contradiction or contingency `(p^^~q) harr (p->q)`

Find k if the length of the tangent segment from (8,-3) to the circle ` x^2+y^2-2x+ky-23=0` is `sqrt10 ` units.

Show that the lines given by `(x+1)/-10=(y+3)/-1=(z-4)/1`  and `(x+10)/-1=(y+1)/-3=(z-1)/4` intersect. Also find the co-ordinates of the point of intersection.

Find the equation of the locus of the point of intersection of two tangents drawn to the hyperbola `x^2/7-y^2/5=1` such that the sum of the cubes of their slopes is 8. 

Solve the following L.P.P graphically:

Maximize: Z = 10x + 25y
Subject to: x ≤ 3, y ≤ 3, x + y ≤ 5, x ≥ 0, y ≥ 0

Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2  units from the point (1,1, 2)

Function ` f (x)= x^2 - 3x +4` has minimum value at

(A) 0

(B)-3/2

(C) 1

(D)3/2

`int1/xlogxdx=...............`

(A)log(log x)+ c

(B) 1/2 (logx )²+c

(C) 2log x + c

(D) log x + c

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

If x=at², y= 2at , then find dy/dx.

Find the approximate value of ` sqrt8.95 `

Find the area of the region bounded by the parabola y² = 16x and the line x = 3.

For the bivariate data r = 0.3, cov(X, Y) = 18, σ_x = 3, find σ_y .

triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.

A function f (x) is defined as
f (x) = x + a, x < 0
= x,       0 ≤x ≤ 1
= b- x,   x ≥1
is continuous in its domain.
Find a + b.

If `x=a(t-1/t),y=a(t+1/t)`, then show that `dy/dx=x/y`

Evaluate : `int1/(3+5cosx)dx`

An insurance agent insures lives of 5 men, all of the same age and in good health. The probability that a man of this age will survive the next 30 years is known to be 2/3 . Find the probability that in the next 30 years at most 3 men will survive.

The surface area of a spherical balloon is increasing at the rate of 2cm²/sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?

The slope of the tangent to the curve at any point is equal to y+ 2x. Find the equation of the curve passing through the origin.

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

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

Hence evaluate, `int xe^xdx`

The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between 25 and 35 minutes with p.d.f 

`f(x) = {{:(1/10",", 25 ≤ x ≤ 35),(0",", "otherwise"):}`

What is the probability that preparation time exceeds 33 minutes? Also, find the c.d.f. of X.

The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

If `ax^2+2hxy+by^2=0` , show that `(d^2y)/(dx^2)=0`

Find the area of the region common to the circle x² + y² =9 and the parabola y² =8x

For 10 pairs of observations on two variables X and Y, the following data are available:

`sum(x-2)=30, sum(y-5)=40, sum(x-2)^2=900, sum(y-5)^2=800, sum(x-2)(y-5)=480`

Find the correlation coefficient between X and Y.