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

If A = {2, 3, 4, 5, 6}, then which of the following is not true?

(A) ∃ x ∈ A such that x + 3 = 8

(B) ∃ x ∈ A such that x + 2 < 5

(C) ∃ x ∈ A such that x + 2 < 9

(D) ∀ x ∈ A such that x + 6 ≥ 9

If 2x + y = 0 is one of the lines represented by 3x² + kxy + 2y² = 0, then the value of k is

`1/2`

`11/2`

`5/2`

`(-11)/2`

If a line is inclined at 60° and 30° with the X and Y-axes respectively, then the angle which it makes with Z-axis is

(A) 0

(B) `pi/4`

(C) `pi/2`

(D) `pi/6`

If A = `[(1,2),(3,4)]` and AX = I then find X by using elementary transformations

With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b² − c²

Show that the equation of a tangent to the circle x² + y² = a² at the point P(x₁,y₁) on it is xx₁ + yy₁ = a²

Find k, if the line 2x − 3y + k = 0 touches the ellipse 5x² + 9y² = 45.

Find the co-ordinates of the point, which divides the line segment joining the points A(2, − 6, 8) and B(− 1, 3, − 4) externally in the ratio 1 : 3.

Using truth table, prove that ~ p ∧ q ≡ (p ∨ q) ∧ ~ p

Find the values of p and q, if the following equation represents a pair of perpendicular lines:
px² − 8xy + 3y² + 14x + 2y + q = 0.

Find the equations of tangents to the parabola y² = 12x from the point (2, 5).

The cost of 2 books, 6 notebooks and 3 pens is  Rs 40. The cost of 3 books, 4 notebooks and 2 pens is Rs 35, while the cost of 5 books, 7 notebooks and 4 pens is Rs 61. Using this information and matrix method, find the cost of 1 book, 1 notebook and 1 pen separately.

Prove that `sin^(−1) (-1/2) + cos^(-1) (-sqrt3/2) = cos^(-1) (-1/2)`

Show that the product of lengths of perpendicular segments drawn from the foci to any tangent to the hyperbola `x^2/25 + y^2/16 = 1` is equal to 16.

Construct the new switching circuit for the following circuit with only one switch by simplifying the given circuit:

Find the locus of a point, the tangents from which to the circle x² + y² = a² are mutually perpendicular

Find the shortest distance between the lines

`(x+1)/7 = (y + 1)/(-6) = (z + 1)/1 and (x - 3)/1 = (y - 5)/(-2) = (z - 7)/1`

Find the angle between the line `(x - 1)/3 = (y + 1)/2 = (z + 2)/4` and the plane 2x + y − 3z + 4 = 0.

Solve the following L. P. P. graphically:Linear Programming

Minimize Z = 6x + 2y

Subject to

5x + 9y ≤ 90

x + y ≥ 4

y ≤ 8

x ≥ 0, y ≥ 0

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

If x^y = e^x−y , then `dy/dx` = ______

A) `(1+x)/(1 + log x)`

B) `log x/(1 + log x)^2`

C) `(1 - log x)/(1 + log x)`

D) `(1-x)/(1 + log x)`

`int 1/(1 + cos x)` dx = _____

A) `tan(x/2) + c`

B) `2 tan (x/2) + c`

C) -`cot (x/2) + c`

D) -2 `cot (x/2)` + c

If X ~ B (n, p) and E(X) = 12, Var(X) = 4, then the value of n is _______

(A) 3

(B) 48

(C) 18

(D) 36

Find the equation of tangent to the curve y = 3x² − x + 1 at P(1, 3).

Evaluate: `int 1/(x(x-1)) dx`

Solve the differential equation `y - x dy/dx = 0`

In a bivariate data, n = 10, `bar x` = 25, `bary` = 30 and `sum xy` = 7900. Find cov(X,Y)

A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).

Examine the function for maximum and minimum f(x) = x³ − 9x² + 24x.

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/dx≠0`

The probability distribution of X, the number of defects per 10 metres of a fabric is given by

Find the variance of X

If `sqrt(1-x^2)  + sqrt(1- y^2)` =  a(x − y), show that dy/dx = `sqrt((1-y^2)/(1-x^2))`

Solve the differential equation `cos^2 x dy/dx` + y = tan x

Find the area of the region bounded by the curves y² = 4x and 4x² + 4y² = 9 with x >= 0.

Find the approximate value of tan⁻¹ (1.001).

Examine continuity of the function f(x) at x = 0, where

`f(x) = (10^x + 7^x - 14^x - 5^x)/(1-cos 4x) , " for " x != 0`

`= 10/7 , " for"  x = 0`

The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo similar operations,

(a) None will recover

(b) Half of them will recover.

Prove that:

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

Find the volume of the solid generated, when the area between ellipse 4x² + 9y² = 36 and the chord AB, with A (3, 0), B (0, 2), is revolved about X-axis.

Find Karl Pearson’s coefficient of correlation between the variables X and Y for the following data