ALGEBRA 1

Calculate the value of 4x + k + 80 + y , when x = 8, k = 10 and y =− 33 .

Solution

=4x + k + 80 + y

=4(8) + 10 + 80 + (-33)

=4(8) + 10 + 80 + (-33)

=32 + 10 + 80 – 33

= 58 – 33

= 25 answer

TRY THIS.............................. 

Calculate the value of 8x + k +- 30 + y, when x = 11, k = 17 and y =− 10 .

A. P. 2

The 1st term of an A.P. is 72 and the common difference is 80. Find the 10th term.

Solution

A₁= 72, d = 80

A_n = A₁ + (n-1)d

A₁₀ = A₁ + (10-1)d

A₁₀ = A₁ + 9d

A₁₀ = 72 + (9x80) [ after substituting A₁= 72, d = 80 as given above]

∴ A₁₀ = 72 + 720

A₁₀ = 792

Hence the 10th term is 792.

  

TRY THIS……………

The 1st term of an A.P. is 160 and the common difference is 219. Find the 5th term.

EXPONENTIALS 3

Simplify 9a⁶b⁹ × 6a⁵b¹⁶ .

Solution

= 9a⁶b⁹ × 6a⁵b¹⁶ .

= (9 x 6)( a⁶x a⁵)( b⁹ × b¹⁶)

= 54a⁶⁺⁵b⁹⁺¹⁶.

= 54a¹¹b²⁵.

TRY THIS……………………… 

 Simplify 11c⁹d¹⁷ × 5c⁷d¹³ .

LOG 1

If Log_ax = 4, find Log_a(¹/x¹⁵)

Solution

= Log_a(¹/x¹⁵)

= Log_ax^-15          [since 1/aⁿ = a^-n]

= -15 x Log_ax    [since Log_axⁿ   = nLog_ax ]

= -15 x 4

= -60 answer

TRY THIS………..

If Log_ab= 6, find Log_a(¹/b⁴⁰)

SIMULTANEOUS 1

Solve the following equations by elimination method.

4x - 2y = 6

3x - y = 7

Solution

4x - 2y = 6 ………..….(i)

3x - y = 7……………….(ii)

By using elimination method,

1(4x - 2y = 6

2(3x - y = 7

Then we add to eliminate y first by subtraction

   4x  - 2y = 6

   6x - 2y = 14

   -2x  + 0  = -8

-2x = -8                     [Dividing by 11 both sides] 

-2      -2

x = 4

From equation (1),

4x - 2y = 6

4(4) - 2y = 6

16 - 2y = 6

16-6 = 2y

10 =2y

10 =2y

2     2

y = 5 [after dividing by 2 both sides]

Hence x=4 and y=5.

TRY THIS........

Solve the following equations by elimination method.

5x + y = 20

2x + 3y = 21

SIMPLIFY 1

Simplify the expression r² + t²/ (r + t)

Solution

From difference of two squares p² - q² = (p - q)(p + q)

= r² + t²/ (r + t)

= (r + t)(r - t)

      r + t

= (r + t)(r - t)

      r + t

=  r - t    answer

      

TRY THIS.........................

Simplify the expression 

u² + y²/ (u + y)

EXPONENTIALS 2

If y⁵ = 11, find y¹⁰.

Solution

= y¹⁰

= (y⁵)²

= (11)²      remember y² = 11

= 11 x 11

= 121

Hence y¹⁰ = 121.

TRY THIS……………….

If  y⁹ = 6, find y²⁷.

GEOMETRY 1

If A and B are complementary angles such that A = 29° and B = x + 25° , find the value of x .

Solution

A + B = 90⁰. [Since complementary angles add up to 90⁰]

29⁰ + x + 25⁰ = 90⁰.

x + 54⁰ = 90⁰.

x = 90⁰ - 54⁰

x = 36⁰  answer.

TRY THIS.............................. 

If A and B are complementary angles such that A = 39° and B = x + 37° , find the value of x .

FUNCTIONS 3

PROBABILITY 3

A jar contains 10 blue balls and 11 red balls. Two balls are drawn without replacement. What is the probability of getting two red balls?

Solution

Total number of balls = 10 + 11 = 21

Let P(A) = Probability of getting first red ball and
      P(B) = Probability of getting second red ball

Therefore:

P(A) = 11/21

After first withdraw we are left with 20 balls.

So P(B) = 10/20

Probability of 2 red balls = P(A) x P(B)

                                            = 11/21 × 10/20
                                            = 110/420

                                            = 11/42

TRY THIS...............

A jar contains 4 green balls and 8 red balls. Two balls are drawn without replacement. What is the probability of getting two green balls?

FUNCTIONS 2

A P 1

The first term of an AP is 13 and the last term is 57.If the arithmetic progression consists of 40 terms, calculate the sum of all the terms. 

Solution

A₁ = 13, n=40, A_n = 57

S_n = n(A₁ + A_n)

       2

S₂₀ = 40(13 + 57)

         2

S₂₀ = 20 x 70

S₂₀ = 1400

Hence the sum of all 20 terms is 1400.

TRY THIS………….

The first term of an AP is 11 and the last term is 163.If the arithmetic progression consists of 60 terms, calculate the sum of all the terms. 

SETS 1

In a certain school, 140 students take either Chemistry or physics. 160 take chemistry and 100 take both subjects. Find those who take physics.

Solution

In most cases, OR stands for union whereas AND/BOTH, stands for intersection.

Let Chemistry = n(C), physics = n(P).

n(C)= 160 ,

n(P)= ?

n(CuP) = 140,

n(CnP)= 100

n(CuP) = n(C) + n(P) - n(CnP)

140  =  160 + n(P)  – 100

140  =  160 – 100 + n(P) 

140  =  60+ n(P) 

140 - 60  =  n(P)

n(P)= 80

Hence those taking  physics are 80.    

TRY THIS………….

In a certain school, 185 students take either Chemistry or French. 160 take chemistry and 130 take both subjects. Find those who take French.

PERIMETER 1

Area of rectangle is 60m². Find its perimeter if width is 4m. 

Solution

A = L x w

60 = L x 4

60 = 4L                                                

4      4

15 = L

Now; P =2(L + w)

= 2(15 + 4)

= 2 x 19

= 38m

Hence perimeter is 38m

TRY THIS………..

Area of rectangle is 80m². Find its perimeter if width is 4m. 

PROBABILITY 2

A certain man takes 2 cards from a standard deck of 52 cards.

What will be the probability of getting a pair of Kings?

Solution

(This is a problem with replacement)

n(E) = 4, n(S) = 52

P(E) = n(E)

            n(S)

1st pick = 4/52

2nd pick = 4/52 as well.

P(2 Kings) =   4      x    4

                        52           52

P(2 Kings) =    16    =      1 

                       2704        169  

Therefore Probability of drawing 2 spade cards is 1/169     

TRY THIS............

A certain man takes 2 cards from a standard deck of 52 cards.

What will be the probability of getting a pair of diamonds?