Problem Sheet 1

Problem Sheet – 1 1. State what types of numbers of the following: 1 3 3 0, 3, -2, , − , π, 3, 1 , 2.71828, 4 2 4 4 2. Place an appropriate inequality sign ‘>’ or ‘<’ between each pair of real numbers given below: (i)

1 2 and (ii) − 4 and − 5 2 3

(iii) −

3 1 and − 5 2

(iv) −

5 6 and − 9 7

64 (ii) 6 12 + 2 75 − 3 98 (iii) 3 − 8 × 3 64 (iv) (4 3 x ) 2 y 3 3 1 1 ( 4 x 2 ) 3 (6 x 3 ) 2 e x −3 + (v) x − 4 (vi) (2e1.2 x ) 3 (vii) 27 3 × 8 3 (viii) (2 x 3 ) 2 (3 x 2 ) 3 e

3. Simplify:

(i)

(ix) (xii)

p q

m p q

n

x .x . x n

(x)

x − m .x n −1 .x m +1

9(4 x ) 2 16 x +1 − 2 x +1.8 x

(xi)

3 5.27 3.9 4 3(81) 4

2 m +3.3 2m − n .5 m + n +3.6 n +1 6 m +1.10 n +3.15 m

4. (a) If 4 x = 8 y , then find

x x . (b) If 27 x = 9 y , then find . y y

5. Find the value of (i) (81) 0.5 (iv)

(ii) (0.001)1 / 3 (iii)

2 n + 2 n −1 2 n +1 − 2 n

(v)

(.3)1 / 3 .(1 / 27)1 / 4 .(9)1 / 6 .(0.81) 2 / 3 (0.9) 2 / 3 .(3) −1 / 2 .(1 / 3) − 2 .(243) −1 / 4

21 / 2 × 31 / 3 × 41 / 4 10 −1 / 5 × 5 3 / 5

÷

3 4 / 3 × 5 −7 / 5 4 −3 / 5 × 6

6. Factorize the following expressions: (i) y 2 + y − 6

(ii) x 4 y 4 − 4x 2 (iii) m 4 − 81 (iv) 2 x 2 − x − 3 (v) 6 x 3 − 3 x 2 − 3 x

7. Solve the following equations: (i) 2 x 2 + 3 x − 1 = 0 (ii) 2 x 2 + 7 x − 15 = 0 (iii) 3 x 2 + 7 x − 13 = 0 (iv)

7 9 = x−4 x−2

1 1 1 2 x + 1 x x − 3 5x − 1 + − = = − (vi) (vii) 5 x 2 + 9 x + 4 = 0 3 4 2 6 x + 2 x +1 x + 3 (viii) x 4 − 10 x 2 + 9 = 0 (ix) ( x + 1)( x + 3)( x + 4)( x + 6) = 72 (x) 5 x + 5 2−x = 26 (v)

(xi) x + x =

6 25

x+3

x −3

2x − 3

+ = (xii) x 4 − 13x 2 + 36 = 0 (xiii) x+2 x−2 x −1

1

8. Solve the following inequalities and hence show it in a number line: x2 − x (i) 2 x − 16 > 0 (ii) 5 x + 25 < 20 (iii) 16 − 4 x ≤ 0 (iv) < 2x + 3 2 (v) − 2 x + 25 ≥ 21 (vi) x 2 − 2 x − 3 > 0 9. Show that (i)

 xa   xb 

(ii)

(iii)

   

a +b

m+ n

 xb ×  xc 

xm xn

 q+r  r− p x  

   

b+ c

2

2

× n+ p 1

 xc ×  xa 

xn

   

c+a

=1

2

xp

2

×

 p −q  r + p   ×  x p−q     

p+m 1

xp

2

xm

2

=1

 q −r  p + q   ×  x q −r     

2

1

 r− p  =1   