CLASS 12 MATH - REVISION 1 - INVERSE TRIGONOMETRY, RELATIONS AND FUNCTIONS, CONTINUITY AND DIFFERENTIABILITY

 CLASS 12

MATH REVISION

 INVERSE TRIGONOMETRY

RELATIONS AND FUNCTIONS

CONTINUITY AND DIFFERENTIABILITY

1. Let A = {1,2,3}, B = {4,5,6,7} and let f ={(1,4), (2,5), (3,6)}  be a function from A to B.  Show that f is one-one.

2. Let A and B be sets.  Show that f: AxB → BxA such that f(a,b) = (b,a) is bijective function.

3. State whether the function f: R →  R defined by f(x) = 3 - 4x is one-one, onto or bijective. 

4. Check injectivity and surjectivity of f: N → N given by f(x) = x³.

5. Check whether the relation R defined in the set {1,2,3,4,5,6} as R={(a, b): b = a+1} is reflexive, symmetric or transitive.

6. Check whether the relation R in R defined by R = { (a,b) : a ≤ b³} is reflexive, symmetric or transitive.

7.   Find the principal value of:

1. sin⁻¹ (-1/2)

2. cos⁻¹ (√3/2)

3. cot⁻¹ (-1/√3)

4. cosec⁻¹ (-√2)

5. sec⁻¹ (2/√3)

6. tan⁻¹(-1)

8. Find the value of the following:

1. tan⁻¹(1)  +  cos⁻¹ (-1/2)  +  sin⁻¹ (-1/2)

2. tan⁻¹ (√3) - sec⁻¹ (-2)

9. Prove that sine function is everywhere continuous.

10. Examine the continuity of f, where f is defined by f(x) = sin x - cos x, if x≠0   and -1 if x=0.

11. Find k so that f(x) = kx² , if x ≤ 2

                                       3,   if x > 2

is continuous at x = 2.

12. Find a and b such that the function defined by

f(x) = 5  , if x ≤ 2

          ax + b ,  if 2 < x < 10

          21  ,  if x ≥ 10

is a continuous function.

Differentiate the following:

13. cos(sinx)

14. cosx³. sin²(x⁵)

15. 2x + 3y = sin y

16. x y + y²  = tan x  +  y

17. y = sin⁻¹(  (1-x²)/(1+x²)  )

18. cosx / logx

19. (sinx)^x +  sin⁻¹√x

20. y ^ x  =  x ^ y

21. (cosx) ^ y =  (cos y) ^ x

22. (x² - 5x + 8) (x³ + 7x + 9)

23. x = sin t    and  y = cos 2t

 

24. x = cosθ - cos2θ    and y = sinθ - sin 2θ 

25. x = 4t and y = 4 / t

26. y = A sin x + B cos x, then prove that y₂ + y = 0.

27. If y = 3 e^2x + 2 e^3x, prove that y₂ - 5y₁ + 6y = 0.

28. If y = 3 cos ( logx ) + 4 sin ( logx ), show that x²y₂ + xy₁ + y = 0.

29. If e^(x+1) = 1, prove that y₂ = y².

30. If y = (tan⁻¹x)² , show that (x²+1)² y₂ + 2x (x²+1) y₁ = 2.