MATHS HOLIDAY HOMEWORK:- Inverse Trigonometric Functions

Q.1. Find sin^-1(sin(3π/5)).

Q.2. Using principle value, evaluate                                                                                                                cos^-1(cos(2π/3))+sin^{- 1}(sin(2π/3))

Q.3. Evaluate sin^-1(sin(4π/5))

Q.4. Prove that sin^-1(4/5)+sin^-1(5/13)+sin^-1(16/65)=π/3

Q.5. Prove that Tan^-1(√x)=[cos^-1((1-x)/(1+x))]/2, where x€[0,1]

Q.6. Prove that 2Tan^-1(1/2)+Tan^-1(1/7)=Tan^-1(31/17)

Q.7. Prove that Tan^-1x+Tan^-1[2x/(1-x2)]=Tan^-1(3x-x3)/1-3x2]

Q.8. Prove that cos^-1x=sin^-1(1-x)/2)

Q.9. If Tan^-1a+Tan^-1b+Tan^-1c=π, Prove that a+b+c=abc

MATHS HOLIDAY HOMEWORK:- Relations and Functions

1-mark

Q.1.  If f(x)=x+7 and g(x)=x-7, then find fog(7)

Q.2.  Write fog if f:R->R and g:R->R, defind by f(x)=8x³ and      g(x)=x1/3

Q.3.   If * is a Binary Operation on the set Z on integers defined by a*b=a+b-5, then write the identity element of * on Z

2-MARK

Q.4.  Let * be any Binary Operation on Z defined by a*b=(3ab)/5, Show that * is commutative as well as associative. Also find the identity if it exists.

Q.5.  Consider f : [0,π/2]->R defined by f(x)=sinx and g : [0,π/2]->R and g(x)=cos x, Show that f and g are one-one, but f+g is not one-one

Q.6.  Find fog and gof if 

          i) f(x)=[x], and g(x)=sinx

         ii) f(x)=x²+2 and g(x)=1-(1/(1-x))

The last question is not clear, I will sent it to later

The next chapter's holiday homework will posted by me in a day or two. Try to complete this before them.