Practice Question On Differential (calculus) For IITJEE, NDA and Airforce

Differential (calculus)

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Question no. 1

If f(x)= tanx + e^-2x -7x³ , then what is the value of f'(0)?

looks_one -2

looks_two -1

looks_3 0

looks_4 3

Option looks_two -1

Solution :
f(x)= tanx + e^-2x -7x³
f'(x) = sec²x - 2e^-2x - 21x²
f'(0) = 1 - 2 = -1

Question no. 2

looks_one ( 3^x+y - 3^x ) / 3^y

looks_two (3^x-y)( 3^y - 1 ) / ( 1 - 3^x)

looks_3 ( 3^x + 3^y ) /(3^x - 3^y)

looks_4 ( 3^x + 3^y ) /( 1 + 3^x+y)

option looks_two (3^x-y)( 3^y - 1 ) / ( 1 - 3^x)

Solution :
3^x + 3^y = 3^x+y
diferentiate with respect to x
3^xlog3 + 3^ylog3(dy/dx) = 3^x+ylog3( 1 + dy/dx)
3^x + 3^y(dy/dx) = 3^x+y( 1 + dy/dx)
dy/dx = (3^x+y - 3^x )/(3^y - 3^x+y )
dy/dx = (3^x)( 3^y - 1 ) /((3^y)) ( 1 - 3^x)
dy/dx = (3^x-y)( 3^y - 1 ) / ( 1 - 3^x)

Question no. 3

If f(x)= sin²x² , then what is f(x) equal to ?

looks_one 4xsin(x²)cos(x²)

looks_two 2sin(x²)cos(x²)

looks_3 4sin(x²)sin²x

looks_4 2xcos²(x²)

option looks_one 4xsin(x²)cos(x²)

Solution :
f(x)= sin²x²
f'(x) = 2 sin x² * cosx² * 2x
f'(x) = 4xsin(x²)cos(x²)

Question no. 4

The derivative of sec²x with respect to tan²x is

looks_one 1

looks_two 2

looks_3 2 secx tanx

looks_4 2sec²xtanx

option looks_one 1

Solution :
derivative of sec²x w.r.t x :

Question no. 5

What is the differential coefficient of log_xx?

looks_one 0

looks_two 1

looks_3 1/x

looks_4 x

option looks_one 0

Solution :
log_xx = 1
differention of constant is zero.

Question no. 6

If 2x² -3y² = 7 , what is equal to ?

looks_one

looks_two

looks_3

looks_4 none of these

option looks_4 none of these

Solution :
2x² -3y² = 7
4x - 6y = 0
= 2x/3y

Question no. 7

If y=cost and x=sint , then what is equal to?

looks_one xy

looks_two x/y

looks_3 -y/x

looks_4 -x/y

option looks_4 -x/y

Solution :
y = cost , x = sint
Differentiate w.r.t t
dy/dt = -sint .......[1]
dx/dt = cost ........[2]
divide [2] by [1]
= -sint/cost
= -x/y

Question no. 8

If f(x)= x² -6x + 8 and there exists apoint c in the interval [2,4] such that f'(c), then what is value of c?

looks_one 2.5

looks_two 2.8

looks_3 3

looks_4 3.5

option looks_3 3

Solution :
f(x)= x² -6x + 8
f'(x) = 2x - 6
0 = 2x -6
x = 3

Question no. 9

If f(x)=2^x , then what is f"(x) is equal to ?

looks_one 2^x (ln2)²

looks_two 2^x+1 (ln2)

looks_3 x(x-1)2^x-2

looks_4 2^x(log₁₀2)²

option looks_one 2^x (ln2)²

Solution :
y = f(x)=2^x
ln y = x ln 2
Differentiate w.r.t x
(1/y)y' = ln 2
y' = y ln 2
Differentiate w.r.t x
y" = y' ln 2
y" = y (ln2)²
y" = 2^x (ln2)²

Question no. 10

If x= cos2t and y =sin²t , then what is equal to?

looks_one 0

looks_two sin(2t)

looks_3 -cos2t

looks_4 -1/2

option looks_one 0

Solution :
x= cos2t
dx/dt = -sin2t . 2 = -4 sint cost .....[1]
y = sin²t
dy/dt = 2 sint cost ......[2]
divide [2] by [1]
dy/dx = -1/2
= 0

Question no. 11

If √x + √y = 2 , then what is at y=1 and x=1 equal to ?

looks_one 5

looks_two 2

looks_3 4

looks_4 -1

option looks_4 -1

Solution :
√x + √y = 2
(1/2)x^-1/2 + (1/2)y^-1/2 = 0
Put x= 1 , y = 1
(1/2) = -1/2
= -1

Question no. 12

What is the derivative of sin²x with respect to cos²x ?

looks_one tan²x

looks_two cot²x

looks_3 -1

looks_4 1

option looks_3 -1

Solution :

Question no. 13

If x=t² , y=t³ , what is equal to ?

looks_one 1

looks_two 3/2t

looks_3 3/4t

looks_4 3/2

option looks_4 3/2

Solution :
x=t²
dx/dt = 2t .......[1]
y=t³
dy/dt = 3t² ......[2]
= (3/2)t²
= (3/2)x
= 3/2

Question no. 14

what is the derivative of log_xx with respect to lnx ?

looks_one 0

looks_two 1

looks_3 1/x

looks_4 x

option looks_one 0

Solution :
log_xx = 1
derivative of constant is zero.

Question no. 15

If e^y + xy= e, then what is the value of at x=0 ?

looks_one e^-1

looks_two e^-2

looks_3 e

looks_4 1

option looks_two e^-2

Solution :
e^y + xy= e
At x = 0
e^y + (0)y= e
e^y = e
y = 1
Differentiate w. r . t x
e^y(dy/dx) + y + x(dy/dx)= 0
dy/dx = -y/(e^y + x)
Differentiate w. r . t x
dy/dx at x = 0 = -1/e
= [-dy/dx(e^y + x)+ y(e^ydy/dx + 1)] / (e^y + x)²
= [(1/e)(e) + 1(e*(-1/e) + 1 )]/e²
= 1/e²

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