Mathematics 65/3/3 2018-2019 Commerce (English Medium) Class 12 Question Paper Solution

Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) . 

Find the value of p for which the following lines are perpendicular : 

`(1-x)/3 = (2y-14)/(2p) = (z-3)/2 ; (1-x)/(3p) = (y-5)/1 = (6-z)/5`

Find the integerating factor of the differential equation `xdy/dx - 2y = 2x^2` . 

If A is a square matrix of order 3 with |A| = 4 , then the write the value of |-2A| . 

If `y = sin^-1 x + cos^-1 x , "find"  dy/dx`

If A = `[[3,9,0] ,[1,8,-2], [7,5,4]]` and B =`[[4,0,2],[7,1,4],[2,2,6]]` , then find the matrix `B'A'` .

Find : `∫_a^b logx/x` dx

Form the differential equation representing the family of curves `y² = m(a² - x²) by eliminating the arbitrary constants 'm' and 'a'. 

Find : 

`∫ sin(x-a)/sin(x+a)dx`

Find : 

`∫(log x)^2 dx`

Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk`  and  `vecb = 3hati - 2hatj + 2hatk` . 

Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.

Mother, father and son line up at random for a family photo. If A and B are two events given by
A = Son on one end, B = Father in the middle, find P(B / A).

Let X be a random variable which assumes values  x₁ , x₂, x₃ , x₄ such that  2P (X = x₁) = 3P (X = x₂) = P (X = x₃) = 5P (X = x₄). Find the probability distribution of X.

A coin is tossed 5 times. Find the probability of getting (i) at least 4 heads, and (ii) at most 4  heads. 

If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .

Find the value of x, if tan `[sec^(-1) (1/x) ] = sin ( tan^(-1) 2) , x > 0 `.

If e^y ( x +1)  = 1, then show that  `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`

Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`

Find the intervals in which function f given by f(x)  = 4x³ - 6x² - 72x + 30 is (a) strictly increasing, (b) strictly decresing .

Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation. 

If f (x)  = `(4x + 3)/(6x - 4) , x ≠ 2/3`, show that fof (x) = x for all ` x ≠ 2/3` . Also, find the inverse of f.

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

If y = (sec^-1 x )² , x > 0, show that 

`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`

Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx`  and hence evaluate   `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .   

Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`

Find the value of  λ for which the following lines are perpendicular to each other: 

`(x - 5)/(5 lambda + 2 ) = ( 2 - y )/5 = (1 - z ) /-1 ; x /1 = ( y + 1/2)/(2 lambda ) = ( z -1 ) / 3`

Let `veca` , `vecb` and `vecc` be three vectors such that `|veca| = 1,|vecb| = 2, |vecc| = 3.` If the projection of `vecb` along `veca` is equal to the projection of `vecc` along `veca`; and `vecb` , `vecc` are perpendicular to each other, then find `|3veca - 2vecb + 2vecc|`.

Solve the differential equation:  ` (dy)/(dx) = (x + y )/ (x - y )`

Solve the differential equation: (1 +x² ) dy + 2xy dx = cot x dx 

Find the area of the region. 

{(x,y) : 0 ≤ y ≤ x² , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .

Evaluate `int_1^4 ( 1+ x +e^(2x)) dx` as limit of sums.

Find the mean and variance of the random variable X which denotes the number of doublets in four throws of a pair of dice.

Find the vector and cartesian equations of the plane passing throuh the points (2,5,- 3), (-2, - 3,5) and (5,3,-3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (–1, –3, –1).

Find the equation of the plane passing through the intersection of the planes `vec(r) .(hat(i) + hat(j) + hat(k)) = 1"and" vec(r) . (2 hat(i) + 3hat(j) - hat(k)) +4 = 0 `and parallel to x-axis. Hence, find the distance of the plane from x-axis.

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. 

If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A^-1Hence, solve the system of equations: 

x +y + z = 6

y + 3z = 11

and x -2y +z = 0

Find the inverse of the following matrix, using elementary transformations: 

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A
require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours and 20 minutes available  for cutting and 4 hours available for assembling. The profit is Rs. 50 each for type A and Rs. 60 each  for type B souvenirs. How many souvenirs of each type should the company manufacture in order to  maximize profit? Formulate the above LPP and solve it graphically and also find the maximum profit.