RATIO AND PROPORTION (BASIC)

RATIO

1) Express each of the given ratios in its simplest form:

a) 22:66 

b) 1.5 : 2.5 

c) 25/4.: 25/2 

d) 40kg,  1 quintals

e) 10 paise, Rs1

f) 200m , 5km

g) 3 hours , one day

h) 6 months 4/3 years 

i) 4/3:9/4:5/2

2) Divide 64cm long string into two parts in the ratio 5:3.

3) Rs720 is divided between x and y in the ratio 4:5. How many rupees will each get.

4) The angle of a triangle are in the ratio 3:2:7. Find each angle.

5) A rectangular field is 100 by 80m, Find the ratio of 

a) length to its breadth 

b) breadth to its perimeter.

6) The sum of three numbers , whose ratios are 10/3, 21/5, 49/8 is 4917. Find the numbers .

7) The ratio between two quantities is 3: 4. If the first is Rs810, find the second.

8) Two numbers are in the ratio 5:7. Their difference is 10.  Find the numbers.

9) Two numbers in the ratio 10:11. Their sum is 168. Find the numbers.

10) A line is divided into two parts in the ratio 2.5 : 1.3. If the smaller one is 35.1 cm, find the length of the line.

11) In a class, the ratio of boys in the girls is 7 : 8. What part of the whole class are girls?

12) The population of a town is 180000, out of which males are 1/3 of the whole population. Find the number of females. Also, find the ratio of the number of females to the whole.

13) 10 grams of an alloy of metals A and B contains 7.5 gram of metal A and the rest is metal B. Find the ratio between

a)  The weight of the metal A and B in the alloy.

b) The weight of metal B and the weight of alloy.

14) The ages of two boys A and B are 6 years 8 months and 7 years 4 months respectively . Divide Rs3150 in the ratio of their ages.

15) 3 persons start a business and spend Rs25000, Rs 15000 and Rs40000 respectively . Find the share of each out of a profit of Rs14400 in a year.

16) A plot of land, 600 sq. m in area, is divided between two persons such that the first person gets three-fifths of what the second gets. Find the share of each.

17) Two poles of different heights are standing vertically on a horizontal field. At a particular time, the ratio between the lengths of their shadows is 2:3. If the height of smaller pole is 7.5m, find the height of the other pole.

18) Two numbers are in the ratio 4:7. If the LCM is 168, find the numbers.

19) Rs300 is divided between A and B in such a way that A gets half of B. Find 

a) the ratio between shares of A and B.

b) The share of A and the share of B

20) The ratio between two numbers is 5:9. Find the numbers, if their HCF is 16.

21) A bag contains Rs1600 in the form of Rs10 and Rs20. If the ratio between numbers of Rs10 and Rs20 notes is 2:3; find the total number of notes in all.

22) The ratio between the prices of a scooter and a refrigerator is 4:1, if the scooter' cost Rs 45000 more than the refrigerator, find the price of the refrigerator.

PROPORTION

1) Check whether the following quantities form a proportion or not:

a) 3x, 7x, 24 and 56

b) 0.8,3,2.4 and 9

c) 3/2, 13/4, 9/2, 39/4

d) 0.4, 0.5, 2.9 and 3.5

e) 5/2, 11/2, 3.0 and 6.0

2) Find the fourth proportional of:

a) 3, 12 and 4

b) 5, 9 and 45.

c)  2.1, 1.5 and 8.4.

d) 1/3, 2/5 and 8.4 

e) 4 hours 40 minutes, 1 hour 10 minutes and 16 hours 

3) Find the third proportional of:

a) 27 and 9

b) 2m 40cm and 40cm

c)  1.8 and 0.6

d)  1/7 and 3/14 

e) 1.6 and 0.8

4) Find the mean proportional between:

a) 16 and 4

b) 3 and 27

c)!0.9 and 2.5

d) 0.6 and 9.6 

e) 1/4 and 1/16

5) If A: B= 3 : 5 and B: C = 4 : 7 find A : B : C

6) If x: y= 2:3 and y: z =5:7, find x: y: z.

7) If m: n = 4:9 and n: s = 3: 7, find m: s

8) If P: Q= 1/2 : 1/3 and Q: R= 3/2 : 4/3, find P: R

9) if a: b= 5:4 and 2: x= 3:8, find the value of y.

10) Find the value of x when 2.5 :4= x: 7.5.

11) Show that 2, 12 and 72 are in continued proportion.

MULTIPLE CHOICE QUESTIONS 

MULTIPLE CHOICE QUESTIONS (INCREASING & DECREASING)

1) The interval on which the function f(x)= 2x³+ 9x²+ 12x -1 is decreasing is

a) [1, ∞) b) ( -∞, -2] c) [-2,-1] d) [-1, 1]

2) The interval in which the function f(x)= 2x³+ 3x²- 12x +1 is strictly increasing is

a) [-2,1] 

b) ( -∞, -2] U [1,∞)

c) ( -∞, 1]

d) ( -∞, -1] U [2, -∞)

3) The function f(x)= x² e⁻ˣ strictly increases on

a) [0,2]  b) [0, ∞) c) ( -∞, 0] U [2, ∞) d) none

4) The function f(x)= x⁴ - 4x is strictly 

a) decreasing in [1,∞)

b) increasing in [1,∞)

c) increasing in (-∞, 1]

d) increasing in [-1,1]

5) The function f(x)= x²- 2x is strictly decreasing in the interval 

a) (-∞,1] b) [1, ∞) c) [-1, ∞) d) none

6) The function f(x)= x(x -3)² decreases for the values of x given by 

a) 1≤x ≤3 b) x≤0 c) x≥ 0 d) 0≤ x ≤ 3/2

7) The function f(x)= xˣ, x> 0, is increasing on the interval 

a) (,e] b) (0, 1/e) c) [1/e, ∞) d) none

8) The function f(x)= aˣ is increasing on R if 

a) a>0 b) a<0 c) 0<a<1 d) a > 1

8) P(x)= - 5x²+ 125x + 37500 is total profit function of a company where x is the production of the company.

Find the interval in which the profit is strictly increasing.

9) The shape of a toy is given as f(x)= 6(2x⁴ - x²). To make the toy beautiful 2 sticks which are perpendicular to each other were placed at a point (2,3), above the toy. 

Based on the above information, Find the interval where f(x) is increasing?

MULTIPLE CHOICE QUESTIONS (EXPONENTS)

MULTIPLE CHOICE QUESTIONS:

1) (-2/3)⁻³ is equal to:

a) 27/8 b) -8/27 c) -27/8 d) 8/27

2) (-3)² ÷ (-1/2)³ is equal to:

a) -9/8 b) 9/8 c) -72 d) 72

3) The reciprocal of (-2)⁵ is

a) -32 b) -1/32 c) 32 d) 1/32

4) If (1/5)³ x (1/5)ˣ⁺³ = 5⁻², then x is equal to:

a) 4 b) 1/4 c) -1/4 d) -4

5) 4⁰ + 6⁰ - 8⁰ is equal to:

a) 0 b) 1 c) 2 d) -1/2

6) (2³)² ÷ (2²)⁴ is equal to:

a) 4 b) -4 c) -1/4 d) 1/4

7) If (-2/3)ˣ = - 243/32, then x is equal to:

a) -5 b) 5 c) 1/5 d) -1/5

8) (-4)³ ÷ (4)⁴ is equal to:

a) 4 b) -4 c) -1/4 d) 1/4

9) 7⁰ x 5 - (-2)³ - 8⁰ is equal to:

a) 12 b) -12 c) 1/12 d) -1/12

10) If (9/10)² x (10/9)⁵ = (9/10)¹⁻ⁿ, then n is equals to:

a) 4 b) -4 c) 1/4 d) -1/4

11) (3/5)⁻¹ ÷ (-5/2)⁻¹ is equal to:

a) 25/6 b) -25/6 c) 6/25 d) -6/25

12) If (-3)ˣ⁻¹ = - 243, then (243)ˣ⁻⁶ is equal to:

a) (243)⁻¹² b) 243 c) 0 d) 1

13) {(5²)³ x 5⁴} ÷ 5³ is equal to:

a) 5¹³ b) 5⁷ c) 5²¹ d) none

14) If (3⁻¹) . X = (6)⁻¹, then the value of X is

a) 1/2 b) -1/2 c) 2 d) -2

15) If (-9)⁻¹ ÷ X = (18)⁻¹, then the value of X is 

a) 2 b) -2 c) 1/2 d) -1/2

16) (2⁰ + 7⁰)/5⁰ is 

a) 2 b) 0 c) 9/5 d) 1/5

17) On Simplifying: (2³⁰ + 2²⁹)/(2³¹ - 2³⁰), we get 

a) 1 b) 2 c) 2/3 d) 3/2

18) The value of √(3⁻²) 

a) 1/9 b) 9 c) -3 d) 1/3

18) (256/625)⁻³⁾⁴ in its simplified form is equal to 

a) 25/64 b) 64/125 c) 125/64 d) 64/25

19) (32)¹⁾⁵ x (125)⁻¹⁾³ in its simplified form is equal to 

a) 16/25 b) 4/5 c) 2/5 d) 2/25

20) (5ⁿ⁺² - 6x 5ⁿ⁺¹)/(13 x 5ⁿ - 2 x 5ⁿ⁺¹) equal to 

a) 5/3 b) -5/3 c) 3/5 d) -3/5

21) The value of (8⁻⁴⁾³ ÷ 2⁻²)¹⁾² is 

a) 1/2 b) 2 c) 1/4 d) 4

22) If x is a positive real number, then ⁴√³√x² is 

a) x¹⁾²⁴ b) x¹⁾⁶ c) x¹⁾¹² d) x¹⁾²⁰

23) If x= 2 and y= 3, then the value of xʸ + yˣ is 

a) 15 b) 17 c) 19 d) 21

24) Which of the following is equal to a ?

a) a¹³⁾⁷ ⁻⁵⁾⁷  b) ¹²√(a⁴)¹⁾³ c) (√a⁵)²⁾⁵ d) a¹³⁾⁷ x a⁷⁾¹³.

25) If a= cᶻ, b= aˣ and c= bʸ, then show that xyz= 1.

26) Find the value of x, if √(5⁰ + 2/3)= (0.6)²⁻³ˣ.

27) Solve: {√(3/5)}ˣ⁺¹ = 125/27.

28) Find the value of 64⁻¹⁾³ (64¹⁾³ - 64²⁾³).

29) Find the value of x if (3²ˣ⁻⁸)/225 = 5³/5ˣ.

30) Find the value of x, if (2/3)ˣ (3/2)²ˣ = 81/16

31) The value of (9)⁰·⁰⁶ x (9)⁰·⁴⁴ is 

a) 3 b) 81 c) 9.9 d) 729

32) Which of the following is equal to a ?

a) (√³)²⁾³ b) a¹²⁾⁷ - a⁵⁾⁷ c) ¹²√{(a⁴)¹⁾³} d) a¹²⁾⁷ x a⁷⁾¹²

33) Simplify: [5(8¹⁾³ + 27¹⁾³)³]¹⁾⁴.

34) If a= b²ˣ, b= c²ʸ, c= a²ᶻ, find the value of xyz.

35) Simplify: (9¹⁾³ x 27⁻¹⁾²)/(3¹⁾⁶ x 3⁻²⁾³)

MULTIPLE CHOICE QUESTIONS (SURDS)

1) Simplest rationalisation factor of ³√40 is 

a) ³√25 b) ³√5 c) √40 d) √5

2) 2√5 + √5 is equal to 

a) 2 √10 b) 10 c) 3√5 d) 3 √10

3) The perimeter of the given figure is

a) 60√5 b) 12√5 c) 27√5 d) 32√5

4) On simplification of (2√5/3 - √2/2 + 6√11) + (√5/3 + 3√2/2 - √11), we get 

a) √5 + √2 + 5√11) 

b) √5/2 + 2√2 + √11

c) √5 + √2 + 6√11

d)  √5 + 2√2 + 5√11

5) The product of ³√7 and √5 is 

a) ³√35 b) ⁶√35 c) ⁶√6125 d) ⁶√1225

6) The product of √18/6 and √18/3 is

a) 1 b) 1/12 c) 1/3 d) √2

7) √5 x √7 x √15 x  √21 in simplified form is

a) √105 b)  √210 c) 105 d) 210

8) (3 + √3)(3 - √3) on simplification becomes equal to 

a) 18 b) 2√3 c) 6 d) 9

9) The value of (3+ √5)²(3 - √5)² is

a) 15 b) 16 c) 4 d) 14

10)  ³√250 ÷ ³√10 in simplified form is equal to 

a) ³√25 b) 5 c) √5 d) ³√2500

11) 30/(√20 + √5) is equal to 

a) 10/3√5 b) 30/√5 c) 10/√5 d) 12 √5

12) 6/(√12 - √3) is equal to 

a) 1/√3 b) 2/√3 c) 2√3 d) 6√3

13) If x = 9 - 4√5, then x + 1/x is equal to 

a) 8√5 b) - 8√5 c) 18 d) 81

14) If √3= 1.732, then the value of 1/√3 approximately is

a) 0.866 b) 0.433 c) 0.288 d) 0.577

15) If √2= 1.414, then the value of √3 ÷ √6 up to three places of decimal is

a) 0.235 b) 0.707 c) 1.414 d) 0.471

16) Two classmates Salma and Anil simplified two different expressing during the revision hour and explained to each other their simplifications. Salma explains simplification of √2/(√5 + √3) and Anil explain simplification √28 + √98 + √147. 

Write both the simplification.

What value does it depict ?

17) Varun was facing some difficulty in simplifying 1/(√7 + √3).  His classmate Priya gave him a clue to Rationalise the denominator for sumplifiu. Varun simplified the expression and thanked Priya for his goodwr.

How did Varun simplify 1/(√7 - √3) ?

What value does it indicate ?

18) For rationalising the denominator of the expression 1/(√27 - √18), the simplest rationalisation factor is 

a) √3 - √2 b)  √27 - √18 c)  √3 + √2 d)  √18 - √27

19) The product ³√2 ⁴√2 ¹²√32 equal to 

a) √2 b) ¹²√2 c) 2 d) ¹²√32

20) 14√112 + 28√7 in its simplified form is equal to 

a) 2 b) 1/2 c) 8 d) 1/8

21) Simplify: (5 - √6)/(5+ √6) by rationalising the denominator.

22) Find the value of (2- √3)/√3. It is given that √3= 1.732.

23) Simplify: (2/3) √(144/64) and state whether the result is a rational number or an irrational number.

24) Simplify: 2 ³√40 - 4 ³√320 + 3 ³√625.

25) Evaluate: (4√3 - √2)(3√2 + 2√3).

26) Simplify: (√3 + √7)/(√27 + √63 - √28 - √48).

27) Find the value of a in the following:

6/(3√2 - 2√3) = 3√2 - a√3.

28) If x= (√5 +3)/2, find the value of x²+ 1/x².

SURDS (2)

EXERCISE - A

1) Add:

a) (4√3 + 7√2) and (√3 - 5√2).

b)  (√5 + 2√3) and (2√5 - 5√3).

c)  (6√7 - √2/2  + 2√5/3) and (√5/3 + 3√2/2 - √7).

d)  (2√3 + 5√5 - 7√7) and (3√5 - √3 + √7).

2) Multiply: 

a)  2√3 by  5√27

b) 3√28  by  2√7.

c)  3√8 by 3√2.

d) 4√12  by 7√6.

e)  ³√2 by ⁴√3.

f) 2 ⁴√3 by 5 ⁴√81.

g)  7√6 by 5 √24

h) ³√32  by ³√250.

i)  ³√7 by √2 

3) Simplify :

a) √3 x ³√3 x ⁴√4.

b)  ³√2 x ⁴√2 x ¹²√32.

4) Divide:

a) 12√15 by 4√3.

b)  4√28 by 3√7.

c) 21√384 by 8√96

d)  ⁶√12 by  √3 ³√2.

e)  ³√18 by ³√9.

f) ³√128 by  ⁵√64.

5) Simplify by combining the like terms:

a)  2 ³√40 + 3 ³√625 - 4 ³√320.

b)  √147 -  √108 - √3.

c)  4√3 - 3√12 + 2√75.

d)  3√45 - √125 + √200 -  √500

e) 3√48 - (5/2) √(1/3) + 4√3.

f)  2 ³√4 + 7 ³√32 - ³√500.

g)  2 ³√54 + 3 ³√16 + 5 ³√128.

h) √125 - 4√6 + √294 - 2 √(1/6).

i) ⁴√81 - 8 ³√216 + 15 ⁵√32 + √225.

6) Simplify:

a) (5+ √3)(7+  √5).

b)  (3 + √2)( 4+√3) 

c) (√5  + √2)(√3 + √2).

d)  (√13 + √11)(√13 - √11).

e)  (4 +√3)(4 - √3)

f) (√13 - √6)(√13 + √6)

g) (3√5 + 2√7)(3√5 - 2√7).

h) (√5 + √7)².

i)  (3√5 + 5√2)²

j) (4√3 - 3√5)².

EXERCISE - B

1) Write the simplest rationalisation factor of each of the following:

a) 2 √2

 

b) √18

c) √75

d) √112

e) ³√36

f) ³√72

g) 2 ³√5

h) ⁴√768

i) ⁶√192

j) √3 +  √5.

k) 5 -  √6

l) 2 √2 + 3 √3

m)  √32

n)  ³√49 

o) ⁴√25

p)  ⁴√25.

q)  ⁵√(a²b³c⁴)

r) ⁵√486 

s) 2√5 - √3

2) Rationalise the denominator of each of the following:

a)  4/√5

b)  4√3/3√7.

c) 1/√12

d) 4/³√16

e)  3 ³√5/³√9.

f) 2 ³√5/³√7

g) 2 ⁵√4/⁵√16

h)  (√2 + 1)/√3

i)  2/√5

j)  1/√18

k) (√3 + √5)/√2.

l)  ³√3/2³√5

EXERCISE - C

1) Express each of the following with a rational denominator:

a) 1/(1+ √2)

b) √7/(√5 + 1).

c)  2√3/(√6 + 2)

d) 3/(√5 + √2)

e)  3√2/(√6 + √3) 

f) 1/(√7 - √6). 

g) 2/(√3 - √5).

h)  2/(√7 - √5).

i) 1/(5+ 2√5)

j) 6/(3√2 + 2√3).

k) 1/(7 - 2√3)

l) 6/(2√3 - √6).

m) √7/(√5+ 1)

n)   7√3/(√10 +√3).

o) 10/(7 - 2√3)

p) 5/(4√3 - 3√2).

EXERCISE - C

1) Rationalise the denominator and Simplify: 

a)  (2+√3)/(2 - √3) 

b) (√7 + √2)/(9+ 2√14)

c) (6 - 4√3)/(6+ 4√3)

d)  (√2 + √3)/(√2 - √3).

e) (3√2 + 1)/(2√5 - 3).

f) (13+ 3√5)/(13 - 3√5).

g)  (√7 - √5)/(√7 + √5)

h) (√11 - √7)/(√11+ √7).

i)  (4√3 + 5√2)/(√48 + √18).

j)  (6 - 4√2)/(6 + 4√2).

k) (√7 - √6)/(√7 + √6).

l) (7√3 - 5√2)/(√48 + √18).

EXERCISE - D

Rationalise the denominator of each of the following:

a) 1/(√5 + √6 - √11).

b) 3/(√3 - √2 + √5).

c)  1/(√2 + √3 + √5).

d) 1/(3+ √5 - 2√2)

e) 6/(3+ √2 - √5).

f)  1/(√7 + √6 - √13).

g) 1/(1+ √2 - √3).

EXERCISE - E

Find the values of a and b in each of the following:

a)  (√3 -1)/(√3+ 1)= a + b√3.

b) (3 - √5)/(3+ 2√5)= a√5 - b.

c) (√5+ √3)/(√5 - √3)= a + b√15.

d)  (√2 - √5)/(√2 + √5)=  a + b√10.

e) (3+ √7)/(3- √7) = a + b√7.

f) (5 + √6)/(5 - √6) = a + b √6.

g) (1+ √6)/(3+ √6)= a + b √6.

h) (3+ 2√2)/(2 - √2)= a + b√2.

i)  (3 + √7)/(3- 4√7) = a + b √7.

j)  (5 + √3)/(7 - 4√3)= a + b √3.

k)  (√2 + √3)/(3√2 - 2√3)= a + b√6.

l)  (√7 -1)/(√7 +1)  - (√7 +1)/(√7 - 1)= a + b√7.

m) (5 + 2√3)/(7 + 4√3) = a + b √3.

n) (√2 + √3)/(3√2 - 2 √3)= a - b √6.

o)  (7 + √5)/(7 - √5)  - (7 - √5)/(7+ √5)= a + 7 √5 b.

EXERCISE - F

Simplify each of the following by rationalising the denominator:

a)  (√5 + √3)/(√5 - √3) + (√5 - √3)/(√5 + √3) 0.

b) (√5 -2)/(√5 +2)  - (√5 +2)/(√5 -2).

c) (3√2 - 2√3)/(3√2 + 2√3) + √12/(√3 - √2).

d)  2/(2√5 - √3) + 3/(2√5 + √3).

e)  2√6/(√2 + √3) + 6√2/(√6 + √3).

f)  (4 +√5)/(4 - √5) + (4 - √5)/(4+ √5).

g) 1/(1+ √2+√3) + 1/(1 - √2 + √3).

EXERCISE - G

Simplify each of the following by rationalising the denominator:

a) 1/(2 +√3) + 2/(√5 - √3) + 1/(2+ √5).

b) 3√2/(√6- √3) + 2√3/(√6 +2) - 4√3/(√6 - √2).

c) 4√3/(2- √2) - 30/(4√3 - 3√2) - 3√2/(3+ 2√3).

d) 1/(3+ √7) + 1/(√7+ √5)+ 1/(√5 + √3)+ 1/(√3 +1).

e) 1/(2+ √5) + 1/(√5+ √6) + 1/(√6+ √7) + 1/(√7+ √8).

f) 3√2/(√3 +√6) + 4√3/(√2 + √6) + √6/(√2 + √3).

g) 7√3/(√10 + √3) - 2√5/(√6+ √5)- 3√2/(√15 + 3√2).

h) Prove: 1/(1+√2) + 1/(√2+ √3) + 1/(√3 + √4) + .....+ 1/(√8+ √9)= 2

i) Find the value of 2(√2 +√6)/(3√2 + √3).

j) Evaluate: {√(5+ 2√6)} + {√(8 - 2√15)}.

k) Find the value of [√{(√5 +2)} + √{(√5 - 2)}]/√{√5  + 1)}.

EXERCISE - H

A) Find the values of each of the following correct to three places of decimals, it being given that √2= 1.4142, √3= 1.732, √5= 2.236 & √10= 3.162, √6= 2.449

a) 4/√3

b) (5- √2)/√2.

c) 1/(√2 +1).

d) 1/(√2 + √3).

e) 87/(7 - 2√5).

f) 25/(√40 - √80).

g) (2√6 - √5)/(3√5 - 2√6).

h) 1/(√3 - √2 -1).

i) (2+ √3)/(2 - √3) + (2 - √3)/(2 + √3) + (√3 - 1)/(√3 +1).

j) √75) + √48/2 - √192

k) 3/√2

l) 2/(√3 -1).

m) 3/(√5 - √2)

n) (1+ √2)/(√5 + √3) + (1 - √2)/(√5- √3).

EXERCISE - I

1) If a= 9 - 4√5, find the value of √a - 1/√a.

) If a= 7 + √40, find the value of √a + 1/√a.

) If x= 3 + 2√2, find the value of x² + 1/x².

) If x= 9 - 4√5, find the value of x² + 1/x².

) If x= (5 - √21)/2, find the value of (i) x + 1/x (ii) x² + 1/x².

) If x= 1/(2 - √3), find the value of x³- 2x² - 7x + 5.

) If x= 1/(2 - √3), find the value of x³- 2x² - 7x + 4.

) If x= (√3 +1)/(√3 -1) and y= (√3 -1)/(√3 +1), find the value of x² - xy + y².

) If x= (√2 +1)/(√2 -1) and y= (√2 -1)/(√2 +1), find the value of x² + xy + y².

) If x= (√3 - √2)/(√3 + √2) and y= (√3 +√2)/(√3 -√2), find the value of x² + xy + y².

) If x= (√5 + √10)/(√10 -√5) and y= (√10 -√5)/(√10 +√5), find the value of √a - √b - 2 √(ab)= 0.

) If x= {√(p + 2q) + √(p - 2q)}/{√(p + 2q) - √(p - 2q)}. find the value of qx² - px + q= 0.

.

MULTIPLE CHOICE QUESTIONS (COORDINATE GEOMETRY)

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2 11 - 8 find the missing number in which order period is the first member of the pear is 4 more than a second member 512 037 / 4 - 8 state whether or not the given order to your number is the solution of the equation plot the following points in the right the name of the figure of plot the following points in check whether their colony or not 12:21 1401223 0 without clotting the points indicate the quadrant in which they will lie coordinates 3 and exercise - 5 at suicide - 3 ordinary is -16 is 5 in the ordinate is -3 ordinary is 5 exercise 3 find the coordinates of the point kuchh ordinance is minus 56 which live on exactly the point at which two co-ordinate accessibility is called the origin abscissa ordinance coordinates if cissor the point is negative the ordinate is positive negative then it lies in the first or second quadrant first or third quadrant second or fourth quadrant second or third quadrant 2011 live in the first quadrant second quadrant third quadrant food 35 32 then the co-ordinate of the point is ordinate is -2 and which lies in the y axis are 40441 44 the coordinates of the mirror image of the point 23 23 if the perpendicular distance of a point P from the exacts with 6 units in the foot of the perpendicular line the negative direction 0530 0752 a plot in the graph paper then a point on the y-axis are co-ordinate of the point lying on the accession satisfying the equation name the quadrant and who is the graph of the point x why lies when x and y x and ygrab the point 32 32 32 32 what must be the coordinate of the point ABCD ABCD tank which of the orders of the number 7392 26 39 is a member of the solution sir what must be the value of X4 lies under line write the verbal sentence again equation plot the point 53 and from it draw is perpendicular to accepted and y-axis respectively write the coordinates of which of the following points lie on the y85 068004 0041 232010 a point like on the positive direction of x-axis at a distance of 6 units from the y-axis water its coordinates what will be the coordinates it lie on a negative direction of y-axis at a distance of 6 minutes write the coordinates of the vertices of a rectangle which is 5 minutes long and three minutes wide if the rectangle is in the first quadrant its longer side lies on the x-axes and one but its at the origin 

MULTIPLE CHOICE QUESTIONS (POLYNOMIAL OR REMAINDER THEOREM)

1) The value of p(1/2) for p(z)= z⁴ - z² + z is 

a) 7/16 b) 5/16 c) 3/16 d) 1/16

2) If p(x)= 2x² - 3x +5, then the value of {p(0)+ p(1)}/p(-1) is 

a) 1/10 b) 4/11 c) 9/10 d) 4/5

3) Zero polynomial p(x), where p(x)= ax +1, a≠ 0 is

a) 1 b) -a c) 0 d) -1/a 

4) Zeroes of the polynomial p(x)= (x+2)(x +5) are

a) 2,5 b) -2,-5 c) 1/2,1/5 d) -1/2,-1/5

5) Zeroes of the polynomial p(x)= x(x -1)(x -2) are

a) 0, -1,2 b) 0,-1-2 c) 0,1,-2 d) 0,1,2

6) Which of the following is a zero of the polynomial x³+ 3x² - 3x -1?

a) -1 b) -2 c) 1 d) 2

7) The number to be added to the polynomial x² - 5x +4, so that 3 becomes its zero, is

a) 4  b) - 4  c) - 2 d) 2

8) The number to be subtracted from the polynomials x² - 16x +30, so that 15 becomes its zero, is 

a) 15  b) 16 c) 30 d) 0

9) A polynomial whose zeros are √2 and - √2 is

a) x²+ 2 b) x - 2  c) x² - 2 d) x  + 2

10) if x= 2 is a zero of the polynomials x²- 2k +2, then the value of k is 

a) 1 b) 2 c) 3 d) 4

11) The value of k for which the polynomial x³ + 3x² - 3x + k has -3 as its zero , is 

a) - 9  b) - 3  c) 9  d) 12

12) The remainder when p(x)= x³+ 1 is divided by x + 1 is

a) - 6 b) 0 c) 1 d) 6

13) The remainder when x⁵¹ + 51 is divided by x + 1, is 

a) 51 b) 50  c) -1 d) 0 

14) The remainder when x²+ 2x +1 is divided by x + 1, is

a) 4 b) 0 c) 1 d) - 2

15) The remainder when f(x)= x³ + 4x² - 3x +1 is divided by x - 2, is

a) 16 b) 12 c) 17 d) 19 

16) if x + 1 is a factor of the polynomials 2x²+ Kx, then the value of k is

a)  -2  b) - 3 c) 4 d) 2

17) if x + a is a factor of x⁴ - a²x² + 3x - 6a, then the value of a is

a)  0 b) 1  c) -1 d) 2 

18) x + 1 is a factor of the polynomial 

a) x³+ x² - x + 1 b) x³+ x² + x + 1 c) x⁴ x³+ x² + 1 d) x⁴+ 3x³+ 3x² + x + 1 

19) For the polynomial (x + 2), (x - 2), the value of p(0), p(1), p(-2) respectively are 

a) 0, 3, - 4 b) - 1, 0, 3  c) - 4 , - 3, 0 d) 1, 4, -3

20) If p(x)= x² - 4x +3, then the value of p(2) -  p(-1) + p(1/2) is 

a) 31/4  b) -31/4  c) 21/4  d) -21/4 

21) If polynomial x³ - 2mx² +16 is divided by x + 2, then the value of m is 

a) -2  b) 2  c) 1 d) - 1

22) If 2x - 1 is a factor of 8x⁴ + 4x³ - 16x² + 10x + a, then the value of a is

a)  -2 b) 2 c) 1 d) - 1

23) Show that (3x -2) is a factor of 3x³ + x² - 20x + 12.

24) Find the remainder when 2x² - x + 1 is divided by (2x +1).

25) The value of polynomial p(x)= 2x³+ 9x² + 10x +3 at 1 is

a) 26 b) 24 c) 25 d) 29

26) Which of the following is a zero of the polynomial x² - 5x + 6 ?

a) 3 b) -3 c) 5 d) 6

27) If -4 is a zero of the polynomial x² + 11x + k, then k is equal to 

a) 28 b) 44 c) 16 d) 60

28) The remainder when p(x)= x³+ 3x²+ 3x +1 is divided by x +1 is

a) -6 b) 0 c) 1 d) 6

29) Find the Zeroes of the polynomial p(x)= (x -2)² - (x +2)².

30) Show that x -1 is a factor of x¹⁰ - 1 and also x¹¹ - 1.

31) Find the value of a so that (2x -1) will be a factor of 2x³ + ax² + 11x + a + 3.

32) 

MULTIPLICATION CHOICE QUESTIONS (EXPANSION)

1) Which one of the following is a polynomial ?

a) x²/3 - 2/x² b) x³ + 4x³⁾²/√x c) √(3y) + 5 d) (x²-1)/(x²+1)

2) The Coefficient of x² in (2x²-5)(4+ 3x²) is 

a) 2  b) 3 c) 8 d) -7

3) √2 is a polynomial of degree 

a) 2 b) 0 c) 1 d) 1/2

4) Degree of polynomial (x³-2)(x²+ 11) is 

a) 0 b) 5 c) 3 d) 2

5) Degree of zero polynomial is 

a) 0  b) any natural number c) 1  d) not defined 

6) Standard form of the polynomial 1/x⁻³ + x/8 + 6x⁵ + √3/5 is

a) x³ + x/8 + 6x⁵ + √3/5

b) 6x⁵ + x³ + x/8 + √3/5

c) 6x⁵ + √3/5 + x/8 + x³

d) x³ + 6x⁵ +√3/5 + x/8

7) x² + 5x - 1/2 is a 

a) quadratic polynomial in x

b) binomial 

c) monomial 

d) cubic polynomial in x.

8) A polynomial of degree 5 in x has at most 

a) 5 terms b) 10 terms c) 6 terms d) 4 terms

9) The expansion of (x + y + z)² is 

a) x²+ y²+ z² - 2xy - 2yz - 2zx

b) x²+ y²+ z² + 2xy + 2yz + 2zx

c) x²+ y²+ z² - xy - yz - zx

d) x²+ y²+ z² + xy + yz + zx

10) The expansion of (x - y)³ is 

a) x³+ y³ + 3x²y + 3xy²

b) x³+ y³ - 3x²y + 3xy²

c) x³- y³ - 3x²y + 3xy²

d) x³- y³ + 3x²y - 3xy²

11) The product (x/2 - 3y)(3y + x/2)(x²/4 + 9y²) is equal to 

a) x⁴/16 + 81y⁴ b) x⁴/81 + 16y⁴  c) x⁴/81 -  16y⁴ d) x⁴/16 - 81y⁴ 

12) 75 x 75 + 2 x 75 x 25 + 25 x 25 in simplified form is equal to 

a) 10000 b) 6250 c) 7500 d) 3750

13) (8.83 x 8.83 - 2.17 x 2.17)/6.66 in its simplified form is equal to 

a) 9 b) 10 c) 11 d) 12

14) If x+ y+ z = 0, then x³+ y³+ z³ is equal to 

a) x²+ y²+ z²+ 3xyz b) 3xyz c) 3x²y²z² d) x²+ y²+ z² - xy - yz - zx

15) If 49x² - y = (7x + 1/2)(7x - 1/2), then the value of y is

a) 0 b) 1/4 c) 1/√2 d) 1/2

16) The expanded form of (3a - 5b - c)² is 

a) 9a²+ 25b²+ c² - 30ab + 10bc - 6ac.

b) 9a²+ 25b²+ c² + 30ab - 10bc + 6ac.

c) 9a²+ 25b²+ c² - 30ab - 10bc + 6ac.

d) 9a²+ 25b²+ c² + 30ab + 10bc - 6ac.

17) The product of (x/2 + 2y)(x²/4 - xy + 4y²) is 

a) x³/6 + 6y³ b) x³/8 + 8y³ c) x³/8  - 8y³  d) x³/6 - 6y³ 

18) The expanded form of (x + 1/3)³ is 

a) x³+ 1/27 + 3x² + x/3

b) x³+ 1/27 + x² + x/3

c) x³+ 1/9 + 3x² + 3x

d) x³+ 1/9 + 3x + x/3

19) If x+ 1/x = 8, then the value of x²+ 1/x² is 

a) 62 b) 64 c) 66 d) 60

20) The value of p³ - q³ if p - q = -8, pq = -12

a) -244 b) -240 c) -224 d) -260

21) If 9x²- 30x + k is a perfect square then the value of k is 

a) 25 b) 5 c) 36 d) 81

22) The value of a²+ b²+ c², if a+ b+ c= 13 and ab + bc + ca= 27 is

a) 250 b) 223 c) 115 d) 81

23) Using a suitable identity evaluate 101 x 102.

24) If 4a²+ b²= 40 and ab = 6, find the value of 2a + b.

25) If x+ y= 3, xy = 2, find the value of x³+ y³.

26) Evaluate: x³ - 1/x³ if x - 1/x = 6.

27) Simplify: (0.645) x (0.645)+ 2 x (0.645) x (0.355) + (0.355) x (0.355).

28) Simplify: (x + 2y - 5z)² - (x - 2y + 5z)².

29) Use a suitable identity evaluate (2a - 3/a +1)(2a + 3/a +1).

30) Simplify: (3x + 5y)(9x² - 15xy + 25y²).

31) Find the product (x - 2/x)(x² + 2 + 4/x²).

32) Degree of which polynomial is 0?

a) x b) 15 c) y d) x + x²

33) If x²+ y²+ z²= 40 and xy + yz + zx = 30, then x + y+ z is equals to 

a) only +10 b) only -10 c) ±10 d) ±70

34) Using a suitable identity evaluate (0.99)².

35) Evaluate: (2)³ - (1/5)³.

36) Using suitable identity evaluate 55³ - 25³ - 30³

37) Simplify: (3.59 x 3.59 - 2.41 x 2.41)/(3.59 + 2.41)

38) If 3x + 2y =12 and xy = 6, find the value of 9x²+ 4y².

39) If x²+ 1/x²= 7, find the value of x + 1/x.

40) If x²+ 1/x²= 83, find the value of x³ - 1/x³.

41) Using suitable identity find the following product:

(x+ y + 2z)(x²+ y²+ 4z² - xy - 2yz - 2zx).

42) If a+ b+ c = 6 and ab + bc + ca= 11, find the value of a³+ b³+ c³ - 3abc.

43) Simplify: (2x - 5y)³ - (2x + 5y)³.

MULTIPLE CHOICE QUESTIONS (FACTORIZATION)

Day - 1

1) The factorization of - x²+ 5x - 6 yields 

a) -(x -2)(3-x) b) -(2 - x)(3-x) c) (x -2)(x -3) d) (2+ x)(3-x) 

2) The value of (348)² - (347)² is 

a) (1)² b) 685 c) 695 d) 705

3) If the area of a rectangle of 4x²+ 4x - 3, then its possible dimensions are 

a) 2x -3, 2x+1 b) 2x -1, 2x+3 c) 3x +1,  2x -3 d) 3x - 1, 2x+3

4) The factors of 12y² - y - 6 are

a) (12y -1)(y +6) b) (12y +1)(y -6) c) (3y -2)(4y +3) d) (3y +2)(4y -3) 

5) The factor of 1/2 - x²/50 is

a) (1/2)(1- x/5)(1- x/5) b) (1/2)(1/5 + x)(1/5 - x) c) (1/2)(1+ x/5)(1- x/5) d) (1/2)(1+ x/5)(1+ x/5)

6) The factor of a³ + 27 are

a) (a+3)(a²+ 3a+ 9) b) (a+3)(a²- 3a+ 9) c) (a-3)(a²- 3a+ 9) d) (a-3)(a²+ 3a+ 9) 

7) √{2a² + 2√6 ab + 3b²} in its simplified form is equals to 

a) (√2a - √3b b) (√2a + √3b) c) (√3a + √2b) d) (√3a - √2b)

8) (2x + 1/3)² -(x - 1/2)² in its Factorized form is equals to 

a) (x -1/6)(3x + 5/6) b) (3x + 1/6)(x - 5/6) c) (x +1/6)(3x - 5/6) d) (3x - 1/6)(x + 5/6) 

9) Factors of a³ - 2√2 b³ are

a) (a - √2 b)(a²+ √2 ab + 2b²)  b) (a - 2√2 b)(a²- √2 ab + 2b²)  c) (a + √2 b)(a² - √2 ab + 2b²) d) (a + √2 b)(a²+ √2 ab + 2b²) 

10) The value of 10³ - (5)³ - (5)³ is 

a) 750 b) 1000 c) 250 d) 500

11) Factorize 4(x + y)² - 9(x - y)²

a) (5x -y)(5y - x) b) (5x + y)(5y - x) c) (5x -y)(5y + x) d) (5x +y)(5y + x)

12) Factorize x³ - 3x²+ 3x +7

a) (x+1)(x²- 4x+7) b) (x-1)(x²+ 4x+7) c) (x+1)(x²+ 4x+7) d) (x-1)(x²- 4x+7)

13) Evaluate: (2)³ - (1/5)³.

14) Using a suitable identity evaluate 55³ - 25³ - 30³.

15) Factorize: a(a -3) - b(b -3).

16) Factorize: 1- 18x - 63x².

17) Factorize: x²+ x/6 - 1/6.

18) Factorize: 27x⁴ - 8x.

EXPONENTS (BASIC)(2)

EXERCISE - A

1) Find the value of:

a) 6²

b) 7³

c) 4⁴

d) 5⁵

e) 8³

f) 7⁵

2) Evaluate:

1) 2³ x 4²

2) 2² x 3³

3) 3² x 4²

4) 2³ x 5²

5) 3² x 5³

6) (4 x 3)³

7) 3³ x 5²

8) 5³ x 2⁴

9) (5 x 4)²

3) Evaluate:

a) (3/4)⁴

b) (-5/6)⁵

c) (-3/-5)³

d) (2/5)²

e) (-4/5)⁴

f) (-3/7)⁴

g) (-2/5)³

h) (1/6)⁴

i) (-1/4)⁵

j) (3/5)⁵

k) (1/-2)³

4) Evaluate:

a) (2/3)³ x (3/4)².

b) (-3/4)³ x (2/3)⁴.

5) Which is greater:

a) 2³ or 3²

b) 2⁵ or 5²

c) 4³ or 3⁴

d) 5⁴ or 4⁵

6) Express each of the following in exponential form:

a) 512.

b) 3600

c) 1250

d) 1350

e) 1458

f) 1176

g) 1/32

h) -27/125

i) 8/-512

j) 1/1024

k) 32/243

l) 16/625

7) Find the reciprocal of:

a) (-2/3)⁴

b) (3/2)⁻⁴

c) (-2/7)⁴

d) (-2/9)⁴

e) (1/2)⁻¹⁰

f) (4/7)⁻⁵

7) If a= 2 and b= 3, find the value of:

a) (a + b)².

b) (b - a)³

8) Express:

a) 1024 as power of 2.

b) 729 as a power of 3.

9) If 27 x 32 = 3ˣ x 2ʸ; find the values of x and y.

10) If 64 x 625 = 2ᵃ x 5ᵇ; find the value of (i) a and b (ii) 2ᵇ x 5ᵃ

EXERCISE - B

1) Evaluate:

a) 2⁸ ÷ 2³.

b) (3⁰)⁶ 

c) 5⁴ ÷ 5³ x 5⁵

d) 2³ ÷ 2⁸

e) 8³ x 8⁻⁵ x 8⁴

f) 4⁴ ÷ 4³ x 4⁰

g) (2⁶)⁰

h) 5⁴ x 5³ ÷ 5⁵

i) (3⁵ x 4⁷ x 5⁸)⁰

2) Simplify, giving answers with positive index:

a) i) (5/7)⁻²

    ii) (-4/7)⁻³

    iii) (7)⁻¹²

    iv) 2b⁶ . b³ , 5b⁴

b) (-a⁵)(a²)

c) (-3)²(3³)

d) (5a²b)(2ab²)(a³b)

e) 3ʸ . 3² . 3⁻⁴

f) 4x²y² ÷ 9x³y³

g) (a¹⁰)¹⁰ (1⁶)¹⁰

h) - (3ab)² (-5 a²bc⁴)²

i) (2a³)⁴ (4a²)²

j) (1/2x)³ x (6x)²

k) {(5x⁷)³ . (10x²)²}/(2x⁶)⁷

l) x²y³. 6x⁵y. 9x³y⁴

m) (-y²) (-y³)

n) (-4x) (-5x²)

o) x²ᵅ⁺⁷ . x²ᵅ⁻⁸

p) 2⁴ᵅ . 2³ᵅ . 2⁻ᵅ

q) (10²)³ (x⁸)¹²

r) (n²)² (-n²)³

s) (-2)² x (0³) x (3)³

t) (4x²y³)³ ÷ (3x²y³)³

u) (1/4ab²c)² ÷ (3/2a²bc²)⁴

v) (7p²q⁹r⁵)² (4pqr)³/(14p⁶q¹⁰r⁴)²

3) Simplify and express the answer in the positive exponent form:

a) ((-3)³ x 2⁶)/(6 x 2³)

b) -128/2187

c) {(2³)⁵ x 5⁴}/(4³ x 5²)

d) (a⁻⁷ x b⁻⁷ x c⁵ x d⁴)/(a³ x b⁻⁵ x c⁻³ x d⁸)

e) {36 x (-6)² x 3⁶}/(12³ x 3⁵)

f) (a³ b⁻⁵)⁻²

4) Evaluate:

a) 6⁻² ÷ (4⁻² x 3⁻²)

b) 5³ x 3² + (17)⁰ x 7³

c) (2²)⁰ + 2⁻⁴ ÷ 2⁻⁶ + (1/2)⁻³

d) [(5/6)² x 9/4)] ÷ [(-3/2)² x 125/216]

e) 2⁵ x 15⁰ + (-3)³ - (2/7)⁻²

f) 5ⁿ x 25ⁿ⁻¹ ÷ (5ⁿ⁻¹ x 25)ⁿ⁻¹)

5) If m= -2 and n= 2; find the value of:

a) m² + n² - 2mn.

b) 6m⁻³ + 4n².

c) mⁿ + nᵐ

d) 2n³ - 3m

6) State true or false 

a) 8 x 8¹⁵ = 64¹⁶

b) 27⁰ = 54903⁰

c) - 1ⁿ = -1, if n is an odd whole number.

d) (-3)⁻³ = +9

e) 16⁸ ÷ 4² = 4⁶

f) (-1)ⁿ = ¹, if n is an even whole number 

g) 4⁻⁴ = -16

EXERCISE - C

1) Evaluate:

a) (3/2)⁵ x (3/2)⁷

b) 8¹¹ x 8⁻¹³ x 8²

c) (-7/5)⁴ x (-7/5)⁶ x (-7/5).

d) (5/3)⁴ x (5/3)⁶ ÷ (5/3)⁴

2) Simplify:

a) {(-3/4)²}⁻³

b) [{(-2/3)²}⁻³]⁻¹

c) [{4/5}⁻⁸]¹⁾⁴

d) [(2/5)³ x (2/5)⁴ x (2/5)¹¹ ÷ (2/5)⁸]

3) Find the value of n, if (-2)³ⁿ⁺¹ x (-2)⁴= (-2)⁸.

4) Find 3³+ 4³ + 5³ and give the answer in cube.

5) If (x⁵ . x⁻²)²= 64  what is the value of x⁻⁴ ?

6) Find the value of a²b³ when a= 2 and b = 3.

7) Find the value of (2/3)⁻² + (1/3)⁻² + (1/2)⁻⁵.

8) By what number should (3)⁻⁸ be multiplied to get (3)⁻⁴ ?

9) By what number should (4)⁻⁵ be multiplied to get (4)¹² ?

10) By what number should (-3/2)⁻³ be divided so that the quotient is (4/5)⁻².

11) By what number should (2)⁻⁴ be multiplied to get 2²⁷ ?

12) Verify that: (x . y)ᵐ = xᵐ . yᵐ for x= 2, y= 3 and m= 4

EXERCISE - D

Evaluate:

1) 2⁻⁵

2) (1/4)⁻⁵

3) (-2/7)⁻⁴

4) (2/5)⁻⁴ x (2/5)⁴

5) (1/7)⁵ x (2/3)⁴ x (1/7)⁻⁸

6) (3)⁻⁶ x (6)⁶ x 27 x (8)⁻²

7) (3⁵ ÷ 3⁸)⁵ x 3⁶.

8) {(-3/2)²}⁻³

9) (1/2)⁻² + (1/3)⁻² + (1/4)⁻²

10) {(216)²⁾³}¹⁾²

11) 9³⁾² - 3 x 5⁰ - (1/81)⁻¹⁾²

12) ((3²)³ + (2/3)⁰ + 3⁵ x (1/3)⁴

13) (4⁻³ x 9⁻⁵ x b⁻⁴)/(4⁻⁵ x a⁻⁸ x b³)

14) (49 x t⁻⁵)/(7⁻³ x 10 x t⁻⁹)

15) If 27ʸ = 9/3ʸ, then find y.

16) If (5ⁿ x 5³ x 5⁻²)/5⁻⁵ = 5¹² then find n.

17) Find the value of n if 25pⁿ⁻³ = 1

18) Find the value of n if 7ⁿ = 343.

EXERCISE - E

1) Simplify:

a) 2²⁾³ x 2¹⁾³.

b) 3⁵⁾⁶ x 3²⁾³.

c) 4²⁾⁵ x 4⁻²⁾⁵.

d) 3¹⁾³/3¹⁾⁶.

e) 4⁶⁾⁷/4²⁾³.

f) 7¹⁾⁴/7¹⁾⁵.

g) 2¹⁾⁴ x 3¹⁾⁴.

h) 3⁵⁾⁸ x 5⁵⁾⁸.

i) 7²⁾³ x 3²⁾³.

j) (5³)¹⁾³.

k) (7¹⁾³)⁴.

l) (3¹⁾⁵)¹⁰.

m) 7¹⁾² x 7¹⁾³.

n) 11¹⁾²/11¹⁾⁴.

o) 7¹⁾² x 8¹⁾².

p) (3³)²⁾³.

2) Evaluate:

a) (36)¹⁾²

b) (27)¹⁾³

c) (32)¹⁾⁵.

d) (81)³⁾⁴.

e) (243)³⁾⁵.

f) (16)⁵⁾⁴.

g) (27)⁻¹⁾³.

h) (512)⁻¹⁾⁹.

i) (625)⁻³⁾⁴.

j) (27/64)⁻²⁾³.

k) (64/125)⁻²⁾³.

l) (81/16)⁻³⁾⁴.

m) (64)¹⁾⁶.

n) (81)⁻¹⁾⁴.

o) (256/625)⁻¹⁾⁴.

3) If a= 2 and b= 3, find the value of the following:

a) (aᵇ + bᵃ)⁻¹

b) (aᵃ + bᵇ)⁻¹.

4) Simplify:

a) {(243)³⁾⁵ x (25)³⁾²}/{(625)¹⁾² x (8)⁴⁾³ x (16)⁵⁾⁴}.

b) (3⁴⁰ + 3³⁹)/(3⁴¹ - 3⁴⁰).

c) 27⁻¹⁾³(27¹⁾³ - 27²⁾³).

d) (2¹⁾² x 3¹⁾³ x 4¹⁾⁴)/(10⁻¹⁾⁵ x 5³⁾⁵) ÷ (3⁴⁾³ x 5⁻⁷⁾⁵)/(4⁻³⁾⁵ x 6).

e) 3/(625)⁻¹⁾⁴ + 2/(343)⁻²⁾³ + 4/(243)⁻¹⁾⁵.

f) (64/125)⁻²⁾³ + 1/(256/625)¹⁾⁴ + √25/³√64.

g) (1³ + 2³ + 3³)¹⁾².

h) (5¹⁾³ x 25¹⁾³ x 125²⁾³)/625¹⁾².

i) (2/3)⁴(8/3)⁻¹²(32/3)⁶.

j) [{(81)¹⁾²}¹⁾⁴]².

k) 4/(216)⁻²⁾³  - 1/(256)⁻³⁾⁴.

l) {(2⁻¹ x 3²)/(2² x 3⁻⁴)}⁷⁾² x {(2⁻² x 3³)/(2³ x 3⁻⁵)}⁻⁵⁾² 

5) Find the value of p in each of the following:

a) (2³)⁴ = (2²)ᵖ.

b) 27ᵖ = 9/3ᵖ.

c) 5ᵖ⁻⁴ x 2ᵖ⁻⁵ = 5.

d) (3/5)ᵖ (5/3)²ᵖ= 125/27.

6) Find the values of x if (2/3)ˣ(3/2)²ˣ= 81/16