Mathematics 2014-2015 ISC (Commerce) Class 12 Question Paper Solution

Find the value of k if M  = `[(1,2),(2,3)]` and `M^2 - km - I_2 = 0`

Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`

Solve `cos^(-1)(sin cos^(-1)x) = pi/2`

Using L'Hospital's rule, evaluate : `lim_(x->0) (x - sinx)/(x^2 sinx)`

Evaluate: `int (2y^2)/(y^2 + 4)dx`

Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`

The two lines of regressions are 4x + 2y- 3 = 0 and 3x + 6y + 5 =0. Find the correlation co-efficient between x and y.

A card is drawn from a well-shuffled pack of playing cards. What is the probability that it is either a spade or an ace or both? 

If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega +  b omega^2) =  omega^2`

Solve the differential equation `sin^(-1) (dy/dx) = x + y`

Using properties of determinants, prove that:

`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^2+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1 + a^2 + b^2)^3`

Given two matrices A and B 

`A = [(1,-2,3),(1,4,1),(1,-3, 2)]  and B  = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`

find AB and use this result to solve the following system of equations:

x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1

Solve the equation for x: `sin^(-1)  5/x + sin^(-1)  12/x = pi/2, x != 0`

A, Band C represent switches in 'on' position and A', B' and C' represent them in off position. Construct a switching circuit representing the polynomial ABC + AB'C· + A'B'C. Using Boolean Algebra, prove that the .given 'polynomial can be simplified to C(A + B'). Construct an equivalent switching circuit

Verify Lagrange's Mean Value Theorem for the following function:

`f(x ) = 2 sin x +  sin 2x " on " [0, pi]`

Find the equation of the hyperbola whose foci are `(0,+- sqrt10)` and passing through the point (2,3)

if `y = e^(mcos^(-1)x)`, prove that `(1 - x^2) (d^2y)/(dx^2) - x dy/dx = m^2y`

Show that the rectangle of the maximum perimeter which can be inscribed in the circle of radius 10 cm is a square of side `10sqrt2` cm.

Evaluate: `int (sec x)/(1 + cosec x) dx`

Given that the observations are: (9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8). Find the two lines of regression and estimate the value of y when x = 13·5.

In a contest, the competitors are awarded marks out of 20 by two judges. The scores of the 10 competitors are given below. Calculate Spearman's rank correlation.

An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white, it is not replaced into the urn. Otherwise, it is replaced with another ball of the same colour. The process is repeated. Find the probability that the third ball is drawn is black.

Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

1) Exactly two persons hit the target.

2) At least two persons hit the target.

3) None hit the target.

if z = x + iy, `w = (2 -iz)/(2z - i)` and |w| = 1. Find the locus of z and illustrate it in the Argand Plane.

Solve the differential equation:

`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1

Prove by vector method, that the angle subtended on semicircle is a right angle.

Find the volume of a parallelopiped whose edges are represented by the vectors:

`vec a = 2 hat i - 3 hat j - 4 hat k`, `vec b  = hat i + 2 hat j - hat k` and `vec c = 3 hat i +  hat j +  2 hatk`

Find the equation of the plane passing through the intersection of the planes: x + y + z + 1 = 0  and 2x -3y + 5z -2 = 0 and the point ( -1, 2, 1 ).

Find the shortest distance between the lines `vec r = hat i + 2hat j + 3 hat k +  lambda(2 hat i +  3hatj +  4hatk)` and `vec r =  2hat i +  4 hat j + 5 hat k +  mu (4hat i + 6 hat j +  8 hat k)`

Box I contains two white and three black balls. Box II contains four white and one black balls and box III contains three white ·and four black balls. A dice having three red, two yellow and one green face, is thrown to select the box. If red face turns up, we pick up the box I, if a yellow face turns up we pick up box II, otherwise, we pick up box III. Then, we draw a ball from the selected box. If the ball is drawn is white, what is the probability that the dice had turned up with a red face?

Five dice are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of maximum three successes.

Mr. Nirav borrowed Rs 50,000 from the bank for 5 years. The rate of interest is 9% per annum compounded monthly. Find the payment he makes monthly if he pays back at the beginning of each month.

A dietician wishes to mix two kinds ·of food X· and Y in such a way that the  mixture contains at least 10 units of vitamin A, 12 units of vitamin B arid 8 units of vitamin C. The vitamin contents of one kg food is given below:

Orie kg of food X costs Rs 24 and one kg of food Y costs Rs 36. Using Linear Programming, find the least cost of the total mixture. which will contain the required vitamins.

A bill for Rs 7,650 was drawn on 8th March 2013, at 7 months. It was discounted on 18th May 2013 and the holder of the bill received Rs 7,497. What is the rate of interest charged by the bank?

The average cost function, AC for a commodity is given by AC = `x + 5 + 36/x` in terms of output x. Find

1) The total cost, C and marginal cost, MC as a function of x.

2) The outputs for which AC increases

Calculate the index number. for the year 2014, with 2010 as the base year by the weighted aggregate method from the following data:

The quarterly profits of a small scale industry (in thousands of rupees)· is as  follows:

Calculate four quarterly moving averages. Display these and the original figures graphically on the same graph sheet.