Differentiation-Quiz-6

 

 

Differentiation Quiz

Dear Readers,

Important topics for Maths has been designed in such a way that it offers very practical and application-based learning to further make it easier for students to understand every concept or topic by correlating it with the day-to-day experiences.

Q1. 

•  1/2
•  2/3
•  3
•  1

Solution
(a)

 

Q2.If y=cos^(-1)⁡((5cos⁡x-12sin⁡x)/13) where x∈(0,π/2), then dy/dx is

•  1
•  -1
•  0
•  None of these

Solution
(a) Let cos⁡a=5/(13,) then sin⁡a=12/13 So, y=cos^(-1)⁡(cos⁡a.cos⁡x-sin⁡a.sin⁡x) ⇒y=cos^(-1)⁡{cos⁡(x+a)}=x+a (∵x+a is in the first or the second quadrant) ⇒dy/dx=1

Q3.  If y=tan^(-1)⁡√((x+1)/(x-1),) then dy/dx is

•   (-1)/(2|x| √(x^2-1))
•  (-1)/(2x√(x^2-1))
•  1/(2x√(x^2-1))
•  None of these

Solution

 

Q4. The nth derivative of the function f(x)=1/(1-x^2 ) (where x∈(-1,1)) at the point x=0 where n is even is

•  0
•  n!
•  
  

  
  
  
  
  
  

•  

   

  
  
  

Solution
(b) f(x)=1+x^2+x^4+x^6+⋯∞, where |x|≤1 ⇒f^n (0)=n!, where n is even

Q5. If y=√((1-x)/(1+x),) then (1-x^2 ) dy/dx is equal to

•  y^2
•  1/y
•  -y
•  -y/x

Solution
 (c) We have y=√((1-x)/(1+x)) Differentiating w.r.t. x, we get dy/dx=1/2 ((1-x)/(1+x))^(1/2-1) d/dx ((1-x)/(1+x)) =1/2 √((1+x)/(1-x) )×((1+x)(-1)-(1-x)(1))/(1+x)^2 =-√((1+x)/(1-x)) 1/(1+x)^2 ⇒(1-x^2 ) dy/dx=-√((1+x)/(1-x)) 1/(1+x)^2 (1-x^2) ⇒(1-x)^2 dy/dx=-√((1-x)/(1+x) ) ⇒(1-x)^2 dy/dx=-y ⇒(1-x^2 ) dy/dx+y=0

Q6. 

 

•  2
•  1
• 0
•  -1

Solution

 

Q7. if f(x)=x+tan⁡x and f is inverse of g, then g’(x) equals

•  1/(1+[g(x)-x]^2)
•  
•  1/(2+[g(x)-x]^2)
•  None of these

Solution
(c) f(x)=x+tan⁡x f(f^(-1)(y))=f^(-1) (y)+tan⁡f^(-1)(y) y=g(y)+tan⁡g(y) x=g(x)+tan⁡g(x) Differentiating, we get 1=g'(x)+sec^2⁡g(x)g'(x) ⇒g'(x)=1/(1+sec^2⁡g(x))=1/(2+[g(x)-x]^2 )

Q8.If g is the inverse function of f and f' (x)=sin⁡x, then g'(x) is

•  cosec⁡{g(x)}
•  sin⁡{g(x)}
•  -1/sin⁡{g(x)}
•  None of these

Solution
(a) Since g is the inverse function of f, we have f{g(x)}=x ⇒d/dx (f{g(x)})=1 ⇒f'{g(x)}.g'(x)=1 ⇒sin⁡{g(x)} g'(x)=1 ⇒g'(x)=1/sin⁡{g(x)}

Q9.If f(x)=|log_e⁡|x||, then f'(x) equal

•  1/|x| , where x≠0
•  1/x for |x|>1 and -1/x for |x|<1
•  -1/x for |x|>1 and 1/x for |x|<1
•  1/x for x>0 and -1/x for x<0

Solution

 

Q10. 

 

•  e^x
•  -e^x/(1+e^x)^3
•  -e^x/(1+e^x)^2
• (-1)/(1+e^x)^3

Solution

 

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