Mathematics 65/3/1 2018-2019 Commerce (English Medium) Class 12 Question Paper Solution

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^-1x find  `(dy)/(dx)`.

Write the order and degree of the differential equation `((d^4"y")/(d"x"^4))^2 =  [ "x" + ((d"y")/(d"x"))^2]^3`.

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line `(x+3)/3=(4-y)/5=(z+8)/6`

If * is defined on the set R of all real number by *: a * b = `sqrt(a^2 + b^2)` find the identity element if exist in R with respect to *

If A = `[[0 , 2],[3, -4]]` and kA = `[[0 , 3"a"],[2"b", 24]]` then find the value of k,a and b.

Find `int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `

Find `int_  sin ("x" - a)/(sin ("x" + a )) d"x"`

Find `int_  (log "x")^2 d"x"`

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

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. 

Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (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.

If tan^-1 x - cot^-1 x = tan^-1 `(1/sqrt(3)),`x> 0 then find the value of x and hence find the value of sec^-1 `(2/x)`.

Using properties of determinants, prove that

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

If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`

If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`

`"If y" = (sec^-1 "x")^2 , "x" > 0  "show that"  "x"^2 ("x"^2 - 1) (d^2"y")/(d"x"^2) + (2"x"^3 - "x") (d"y")/(d"x") - 2 = 0`

Find the equation of a tangent and the normal to the curve `"y" = (("x" - 7))/(("x"-2)("x"-3)` at the point where it cuts the x-axis

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

Prove that `int_a^b ƒ ("x") d"x" = int_a^bƒ(a + b - "x") d"x" and "hence evaluate" int_(π/6)^(π/3) (d"x")/(1+sqrt(tan "x")`

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

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

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|`.

Find the value of λ for which the following lines are perpendicular to each other `("x"-5)/(5λ+2) = (2 -"y")/(5) = (1 -"z")/(-1); ("x")/(1) = ("y"+1/2)/(2λ) = ("z" -1)/(3)`

hence, find whether the lines intersect or not

If A = `[(1, 1, 1),(0, 1, 3),(1, -2, 1)]`,find A^-1

hence, solve the following system of equations

x + y + z = 6
y + 3z =11
x- 2y + z = 0

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

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

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. 

Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using integration. 

Find the area of the region bounded by the curves (x -1)² + y² = 1 and x² + y² = 1, using integration.

Find the vector and cartesian equation of the plane passing through the point (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 `vecr . (hati + hatj + hatk)` and `vecr.(2hati + 3hatj - hatk) + 4 = 0` and parallel to the x-axis. Hence, find the distance of the plane from the x-axis.

There are two boxes I and II. Box I contains 3 red and 6 Black balls. Box II contains 5 red and black balls. One of the two boxes, box I and box II is selected at random and a ball is drawn at random. The ball drawn is found to be red. If the probability that this red ball comes out from box II is ' a find the value of n 

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.