SIMPLIFICATION

1 1 1
1. + (𝑥−2)(𝑥−1) +
(𝑥−2)(𝑥−3) (𝑥−3)(𝑥−1)

𝑥−1 3 1
2. + (𝑥+2)(𝑥−1) −
(2𝑥−1)(𝑥+2) (1−𝑥)(1−2𝑥)

2𝑎−6 𝑎−1 𝑎−2

3. − −
𝑎2 − 9𝑎 + 20 𝑎2 − 7𝑎 + 12 𝑎2 − 8𝑎 + 15

𝑥+𝑦 𝑥−𝑦 4𝑥𝑦

4. − +
𝑥−𝑦 𝑥+𝑦 𝑥 2 + 𝑦2

𝑥+1 𝑥−1 2𝑥 2
5. + −
𝑥−1 𝑥+1 𝑥2 − 1
1 1 2𝑥
6.
1+𝑥+𝑥 2 − 1−𝑥+𝑥 2 − 1+𝑥 2 +𝑥 4

𝑥−𝑦 𝑥+𝑦 2𝑦 3
7.
𝑥 2 − 𝑥𝑦 + 𝑦 2 + 𝑥 2 + 𝑥𝑦 + 𝑦 2 − 𝑥 4 + 𝑥 2𝑦2 + 𝑦4

3𝑥−1 3𝑥+1 54𝑥 3

8. − +
9𝑥 2 − 3𝑥 + 1 9𝑥 2 + 3𝑥 + 1 81𝑥 4 + 9𝑥 2 + 1

𝑎2 −(𝑏−𝑐)2 𝑏2 −(𝑎−𝑐)2 𝑐 2 −(𝑎−𝑏)2

9.
(𝑎+𝑐)2 −𝑏2
+ (𝑎+𝑏)2 + (𝑏+𝑐)2
−𝑐 2 −𝑎2

2 3 5√𝑥−√𝑦
10. + −
√𝑥+√𝑦 √𝑥−√𝑦 𝑥−𝑦

1 1 2√𝑥
11. + +
8(√𝑥−1) 8(√𝑥+1) 8(𝑥−1)

1 2 1 2
12. − + −
𝑥−𝑎 2𝑥+𝑎 𝑥+𝑎 2𝑥−𝑎

𝑥3 𝑥2 𝑥 1
13. − − +
𝑥−1 𝑥+1 𝑥−1 𝑥+1

𝑥4 𝑥4 1 1
14. + − −
𝑥2 + 1 𝑥2 − 1 𝑥2 + 1 𝑥2 − 1

1 1 2 4
15. + + +
1−𝑎 1+𝑎 1+𝑎2 1 + 𝑎4

𝑥+𝑦 𝑥−𝑦 4𝑥𝑦 8𝑥 3 𝑦

16. - + −
𝑥−𝑦 𝑥+𝑦 𝑥 2 +𝑦2 𝑥 4 +𝑦 4
 INDICES
1. Simplify:
2𝑥 ×3 − 2𝑥 33𝑎+2 − 33𝑎+1 112𝑥+1 − 6 ×121𝑥
a) b) c)
2𝑥+2 −2𝑥+1 6 ×27𝑎 5 ×121𝑥
2𝑛
(243) 5 . 32𝑛+1 2 3 4
d) e) (64𝑥 3 ÷ 27𝑎−3 )− 3 f) √27𝑎12 𝑏9 ÷ √16𝑎16 𝑏2
9𝑛+1 × 32(𝑛−2)
1
3 3 3
g) √3𝑥 7 𝑧 −1 𝑦 11 × √72𝑥 −1 𝑦𝑧 4 h) √(𝑎 + 𝑏)−7 . (𝑎 + 𝑏)3

1 1 1
2. Simplify : + +
1+ 𝑥 𝑎−𝑏 +𝑥 𝑐−𝑏 1+ 𝑥 𝑏−𝑐 +𝑥 𝑎−𝑐 1+ 𝑥 𝑐−𝑎 +𝑥 𝑏−𝑎
1 1 1
3. Simplify : + +
1+ 𝑥 𝑎−𝑏 +𝑥 𝑎−𝑐 1+ 𝑥 𝑏−𝑎 +𝑥 𝑏−𝑐 1+ 𝑥 𝑐−𝑎 +𝑥 𝑐−𝑏

4. Simplify: If 𝑥 3 + 𝑦 3 + 𝑧 3 =1.
𝑎𝑥 𝑥 2 −𝑥𝑦+𝑦 2 𝑎𝑦 𝑦 2 −𝑦𝑧+𝑧 2 𝑎𝑧 𝑧 2 −𝑧𝑥+𝑥 2
Show that, ( ) (𝑎−𝑧) (𝑎−𝑥 ) = 𝑎2
𝑎−𝑦

1 1 1
5. If pqr = 1, show that + + =1
1+𝑝+𝑞−1 1+𝑞+𝑟 −1 1+𝑟+𝑝−1

1 1 1
6. If abc + 1 = 0, show that + + =1
1−𝑎−𝑏−1 1−𝑏−𝑐 −1 1−𝑐−𝑎−1
1 1 1
7. If a + b + c = 0, show that
1+ 𝑥 𝑎 +𝑥 −𝑏 + 1+ 𝑥 𝑏 +𝑥 −𝑐 + 1+ 𝑥 𝑐 +𝑥 −𝑎
=1
1 1
−
8. If 𝑥 = 33 +3 3, then prove that 3𝑥 3 - 9x = 10
2 2
−
9. If 𝑥 2 + 2 = 23 +2 3, then prove that 2𝑥(𝑥 2 +3) = 3

====================================================================
 SURDS
Simplify:
3 3 3
1. √32 + √8 −√72 2. 5√81 − 2 √24 + √375 3. (√3 + √2)(√3 + √2)

2 √𝑥 2 −9 49−5𝑥
4. (√𝑥 + 𝑎 − √𝑥 − 𝑎) 5. 6.
√𝑥−3 7−√5𝑥
3 3
5 √81−2 √24
7. 3 3
2 √48 + 3 √162

Rationalize the denominator and simplify:

√𝑎+𝑏−√𝑎−𝑏 √𝑥+√𝑎 √𝑥−√𝑎 𝑎−√𝑎2 −1 𝑎+√𝑎2 −1

i) ii) − iii) +
√𝑎+𝑏+√𝑎−𝑏 √𝑥−√𝑎 √𝑥+√𝑎 𝑎+√𝑎2 −1 𝑎−√𝑎2 −1

√𝑎+𝑏+√𝑎−𝑏 √𝑎+𝑏−√𝑎−𝑏 4 √5+√3

iv) + v) 3√20 + −
√𝑎+𝑏−√𝑎−𝑏 √𝑎+𝑏+√𝑎−𝑏 √5 √5−√3
=====================================================================

PROBABILITY
1. From a well- shuffled pack of 52 cards. A card is drawn at random. Find the probabilities of the
following events.
a) It is either a king or queen.
b) It is either a non – faced card or a jack
c) It is neither a black jack nor a red queen.

2. From the number cards numbered 1 to 30, a card is drawn at random. Find the probabilities of the
following events.
a) The number is either divisible by 5 or by 7.
b) The number is either a multiple of 7 or a multiple of 8.

3. A box contains 4 yellow, 6 green and 8 red balls. A ball is drawn at random from the box. Find
the probabilities of the following events.
i) The ball is either yellow or red.
ii) The ball is either green or yellow.
iii) The ball is neither red nor green.

4. A number card is drawn randomly from the set of numbered cards, numbered from 6 to 39. Find
the probability that the card may be prime number or a cubed number.

5. When a fair die is thrown, find the probabilities of the following events.
i) Getting either 1 or 6.
 ii) Getting a multiple of 2 or a multiple of 3.

6. A box contains 4 red, 3 blue and 5 white balls. A ball is drawn at random and it is replaced, then
another ball is drawn. Find the probability that
i) The first is white and second is blue.
ii) Both of them are red
iii) Both of them are of the same colour.
iv) Both of them are not blue.

7. A card is drawn randomly from a pack of 52 cards and a die is thrown once. Determine the
probability of not getting a card king as well as 6 on die.

8. A fair coin is tossed three times. Draw a probability tree diagram to show all the possible outcomes
and determine
i) The probability of getting 3 heads.
ii) The probability of getting at least 2 heads.
iii) The probability of getting at most 1 tail.

9. A bag contains 3 red and 5 white balls. A ball is drawn at random and replaced, then another ball
is drawn. Draw a probability tree diagram and find the probability that both of them are not of the
same colour.

10. A bag contains 3 red and 4 white balls. Two balls are drawn one after another without replacement.
Show all the probabilities in a tree diagram.

11. Two cards are drawn from a well- shuffled deck of 52 cards without replacing the first card. By
drawing a tree- diagram, find the probability that both card are jacks.

====================================================================

EQUATIONS
1. A number is twice the other number. If their sum is 30, find the numbers.
2. If the sum of two numbers is 9 and their product is 18, find the numbers.
3. Divide 11 into two parts so that their product will be 24.
4. Find three consecutive numbers such that twice the first, three times the second and four times the
third together make 29.
5. Two numbers are in the ratio of 4:5. If there is 80, find the greater number.
6. The sum of the ages of the father and son is 44 years. After 8 years the age of the father will be
twice the age of the son. Find their present age.
7. A year hence a father will be 5 times as old as his son. Two years ago, he was three times as old
as his son will be four years hence. Find their present age.
8. The sum of the ages of a father and his son is 40 years. If they both live on till the son becomes as
old as the father is now, the sum of their ages will be 96 years. Find their present ages.
9. Two years ago, father was 6 times as old as his son was. Three years hence he will be 5 years
older than 3 times the age of his son. Find their present age.
10. The sum of the digits of a two digit number is 11. The number formed by interchanging the digits
of the number will be 45 more than the original number. Find the original number
11. A certain number of two digits is seven times the sum of the digits. If 36 is subtracted from the
number, the digits will be reversed. Find the number.
12. The present ages of two brothers are respectively 15 years and 22 years. After how many years the
product of their ages will be 408 years?
13. A two digit number is such that the product of the digits is 8. When 18 is added to the number the
digits interchange their places. Find the number.

14. The difference of the ages of two brothers is 4 years and the product of their ages is 221.
Determine the ages of two brothers.

15. A room is 2m longer than its breadth. If the area of the floor of the room is 120 m2, find its length
and breadth.

===============================================================

TAX AND MONEY EXCHANGE

1. A shopkeeper purchased a laptop for Rs. 40,000 and sold at a profit 20% to customer. How
much did the customer pay for it with 13% VAT?
2. The cost of an article is Rs. 10, 000 and it is sold at some profit. If a customer bought it for
Rs. 12,430 with 13% VAT, calculate the profit % on the article.

3. A shopkeeper bought a sunglasses and sold it at a profit of 20% to a customer. If the customer
paid Rs 21,696 for it with 13% VAT, at what price did the shopkeeper buy it?

4.