Question

1. (a) Let A = 1, 3, 5, 7 and B = X : X is the first three multiple of 7 . Find (A−B).

(b) Find the remainder when f (x)=3x^3 −11x−48 is divided by x−1 .
(c) Let f (x)= x3 , g(x)= x+5. Find (f •g)(x)
(d) Find the derivative of y = 3+ x^2+ x^3y^3.

Answer 1

a) A-B = { 1 , 3, 5 }

b) The remainder is 50

c) f + g)(x) = x⁴+ 5x³

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

Explanation:

Step(I):-

1) a)

Let A = { 1, 3, 5, 7 } and B = { 7 , 14 , 21 }

A - B = {1,3,5,7} - { 7,14,21}

A - B = { 1 , 3, 5}

b)

Given f (x)=3x³−11x−48

x-1 ) 3 x³ - 11x -48 (- 3x²-3x -2

-3x³ +3x²

3x²-11x-48

-3x²+9x

2x -48

-2x +2

50

The remainder is 50

c) Given f(x) = x³ and g(x) = x + 5

(f. g)(x) = f(x) .g(x) = x³ . (x+5) = x³(x) + 5 x³ = x⁴+ 5x³

d)

Given function y = 3 + x²+x³y³ ...(I)

By using derivative formulas

y = xⁿ

`(dy)/(dx) = n x^(n-1)`

Apply d/dx ( UV) = U V¹ + V U¹



Differentiating equation (I) with respective to 'x' , we get

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

`(dy)/(dx) - ( x^3 3y^2((dy)/(dx) )= 2x- y^3(3x^2))`

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