Project Report on Maths Questions

Maths

TABLE OF CONTENTS
Question 1........................................................................................................................................1
Question 2........................................................................................................................................1
Question 3........................................................................................................................................2
Question 4........................................................................................................................................2
Question 5........................................................................................................................................2
Question 6........................................................................................................................................3
Question 7........................................................................................................................................3
Question 8........................................................................................................................................3
Question 9........................................................................................................................................4
Question 10 .....................................................................................................................................5
Question 11......................................................................................................................................6
Question 12......................................................................................................................................7
Question 13......................................................................................................................................9
Question 14 .....................................................................................................................................9
Question 15......................................................................................................................................9
Question 1......................................................................................................................................11
Question 2......................................................................................................................................11
Question 3......................................................................................................................................14
Question 4......................................................................................................................................15

Question 1
a) Value of cos(cot^-1(0.8)) = ?
= cos(tan 0.8)
= cos (0.01396)
= 0.999 ~= 1 ( in terms of degree)
b) value of x in (cosh 2x+ sinh 2x)^2 = 5, x is the real number
we will take the square root from both the sides;
cos h (2x) + sin h (2x) = √ 5
cos 2x + sin 2x = log(5)
cos 2x + sin 2x = 0.698
(cos ^2 x – sin ^2 x) + 2sin x cos x = 0.698
Question 2
a) inverse function of f(x) = (x+1)/(x-1) , where (x ≠ 1)
let's say y = x+1/x-1
y( x-1)= x+1
yx – y = x+1
yx – x = y+1
x(y-1)= y+1
x = y+1/y-1
f^-1(x) = y+1/y-1
b) determine, if it exists lim = sin x / π - x
x → π
lim ( x → π ) = (d/dx) [sin x] / d/ dx [ π – x]
= cos x / d/dx [ π – x ]
= cos x / d/dx ( π ) - d/dx ( x)
= cos x / d/dx ( π ) - 1
lim ( x → π ) = cos x / d/dx (π) – 1

Question 3
Y = x ln x + 2-x/ e^x
dy/dx = d/dx[x ln(x)+(2-x) e^-x]
= d/dx (x) * ln(x) + x * d/dx [ln(x)] + d/dx [2-x] * e^-x + (2-x) * d/dx [e^-x]
by taking common from both the sides,
= ln (x)+ (-d/dx [x]) (2-x) e^-x + (0-1)e^-x + 1
= ln (x)-1 (2-x) e^-x – e^-x+1
= ln(x) – (2-x)e^-x – e^-x +1
therefore it is = e^-x [e^x ln(x) + e^x + x – 3 )
Question 4
Y = sin^5 ( e^tan x)
dy / dx = d/dx[sin ^5 (e^tan x)]
= 5 sin ^4 [e^tan x] * d/dx [ sin [e^tan (x)]
= 5 cos [e^tan x] * d/dx [ e^tan x] * sin [e^tan x]
= 5 e^tan x * d/dx tan x * cos [ e^tan x] * sin ^4 ( e^tan x)
dy /dx = 5 e^tan x * sec^2(x) * cos (e^tan x) * sin ^4 (e^tan x)
Question 5
Y = √ tan ^-1 (sin hx)
= d/dx √ tan ^-1 (sin hx)
= ½ (tan^-1)^1/2-1 (sin hx) * d/dx ( tan ^-1 * sin hx)
= (1/sin ^2 (hx) + 1 * d/dx [sin hx] ) / 2 √tan ^-1 (sin hx)
= cos hx * d/dx [hx] / 2 [ sin ^2 (hx) + 1 √ tan ^-1 (sin hx)]
= 1 h cos (hx)/ 2 (sin ^2 (hx) +1 √ tan ^-1 (sin (hx)))
dy/dx = h cos (hx) / 2 [ sin^2 (hx) + 1] * √ tan ^-1 (sin (hx))