Maths Assignment 1 ( Onam Exam) 

Answer any 4 questions from 1 to 5. Each carries 3 scores.

1. (i) A ∩ A' = .................. 

      (A) U  (B) A  (C) ϕ  (D) A'

               Answer:  (C) ϕ 

   (ii) Consider the following sets

   A ={x:x is a factor of 6} and B={x:xis an even integer,2≤x< 9}.

(a) Write A and B in roster form 

(b) Find A ∩ B. 

     Answer : A = { 1,2,3,6 } 

                     B =  { 2,4,6,8 } 

                    A ∩ B.=   { 2 } 

2. (i)The domain of the function f(x)=(1)/(x-1) is .................. 

    (A) {1} (B) R (C) R - {1} (D) R - {-1}

           Answer :  (C) R - {1} 

   (ii) A relation R on the set of natural numbers is defined    by R = {(x,y): y = x + 5; x, y ∈ N, x < 4}

   (a) Write the relation in Roster form. 

   (b) Write the domain and range of the relation.

         Answer: 

              R = { ( 1,6) , (2,7) , (3,8) }

              Domain  =  {1,2,3 }

               Range = { 6,7, 8 }

3. (i) The range of f(x) = cosx is 

       (A) R (B) (-1, 1) (C) R - ((2n+1)π)/(2) (D) [-1, 1]

            Answer: (D) [-1, 1]

    (ii) If cos x = -(4)/(5), x lies in third quadrant, find   cosec x and tan x      

 

4. (i) Solve the inequality 37 - (3x + 5) ≥ 9x - 8(x - 3) 

          Answer:

   

(ii) Show the graph of the solution on number line. 

 

5. (i) If (x + 1, y - 2) = (3, 1), write the values of x and y.

      Answer: 

            x + 1 = 3                               y - 2 = 1

            x    = 3 -1 = 2                         y =  1 +2 = 3 

   (ii) Find the domain and range of the function f(x)=  √x² - 25

                 Answer: Domain = (-∞, -5] U [5, ∞)

                                Range = [0, ∞)

       

Answer any 3 questions from 6 to 9. Each carries 4 scores.

6. Let A = {2, 4, 6, 8} B = {2,3,5,7} and U={1,2,3,4,5,6,7,8}

  (i) Find A' and B'. 

  (ii) Verify that (A ∪ B)' = A' ∩ B' 

Answer:

  (i) A' = U - A

     =  { 1,3, 5, 7 } 

      B ' = U - B 

    =  { 1,4,6,8 } 

(ii)    (A ∪ B)'  = A'∩ B'           

         A = { 2,4,6,8}                A' ∩ B'

         B = { 2,3,5,7}               A' = { 1,3,5,7}

        (A ∪ B)'  = {1}              B' =  { 1,4,6,8} 

                                          

                                       (A ∪ B)'  = A'∩ B' = {1}

7. (i) sin 765⁰ = .................. 

answer 

Sin (2 × 360 + 45)

Sin 45⁰ 

    

 (ii) Prove that cos4x = 1 - 8sin²xcos²x 

L.H.S = Cos 4x

Cos 2(2x)

1 - 2 Sin² 2x

1 - 2 [2 Sin x Cos x]²

1 - 2 [4 Sin² x Cos² x]

1 - 8 Sin² x Cos² x

8. Consider the function f (x) = |x| − 3

(i) Draw the graph of f (x)

(ii) Write the domain and range of f (x)

Answer

 

       

Domain = R

Range=[-3,∞)

9. (i)Express the complex number z = i¹⁹+i⁶ in the form (a+ib).

   (ii) Write the conjugate of the complex number 1 + i√3

   (iii) Find the multiplicative inverse of 1 + i√3

Answer:

  9) (i) Z = i¹⁹ + i⁶

  i³ + i²

-i + -1

-1 - i

(ii) 1 + i√3

       1 - i√3

(iii) 

Answer any 1 question from 10 to 11. Each carries 6 scores.

10.(i) Find sin 75°

sin 75°

(ii) Prove that 

Answers: 

 

(iii) Prove that 2sin² (π/6) + cosec² (7π/6) = (3/2)

11. Let A={x: x ∈ N,x < 5}, B={x : x is a prime number less than 5}        

    and C = {x : x ∈ Z, 3 < x ≤ 5}

   (i) Write A, B, C in roster form.

     A  = { 1,2,3,4}

   B = { 2,3} 

   C = {4,5} 

   (ii) Prove that A X (B ∩ C) = (A X B) ∩ (A X C)

A X (B ∩ C)

A = {1, 2, 3, 4}

B ∩ C = {  }

A X B ∩ C = ø

(A X B) ∩ (A X C)

A = {1, 2, 3, 4}

C = {4, 5}

A X C = {(1,4) (1,5) (2,4) (2,5) (3,4) (3,5) (4,4) (4,5)}

A X B = {(1,2) (1,3) (2,2) (2,3) (3,2) (3,3) (4,2) (4,3)}

(A X B) ∩ (A X C)

= ø

L.H.S. = R.H.S.