ALGEBRA - 3

Express as a single fraction m - 5 + m+1

                                               6          4

solution

= m - 5 + m+1

     6         4

= 2(m – 5) + 3(m+1)

           12         

= 2m - 10 + 3m+3

           12         

= 2m + 3m + 3- 10

         12         

= 5m +  -7

      12         

= 5m - 7

     12         

=  5m - 7                 answer

      12       

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

Express as a single fraction 4c - 1 + 2d+1

                                               4           2

FACTORIZATION-5

Factorize completely   (3x-2)² - x²  

solution

= (3x-2)² - x²  

= [(3x-2) - x][ (3x-2) + x ]  --- since a² + b² = (a+b)(a-b) Here a is (x-2) and b is x.

= [3x-2 - x][ 3x-2 + x ] 

= [3x-x-2][ 3x+ x - 2  ] 

= [2x-2][ 4x - 2  ] 

= [2x-2][ 4x - 2  ] 

Hence  (3x-2)² - x²  = (2x-2)(4x - 2) 

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

Factorize completely   (4x-1)² - x²  

SUM OF A G.P - 1

Find the sum of the 1st six terms of the geometrical progression 3+6+12+24+…….

Solution

G₁ = 3, r=2, n=6

S_n = G₁(rⁿ-1)/r-1

               

S₇ = 3(2⁶-1)/2-1

            

S₇ = 3(2⁶-1)

S₇ = 3(64-1)

S₇ = 3(63)

S₇ = 189

Hence the sum of the 1st eight terms is 189.

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

Find the sum of the 1st seven terms of the geometrical progression 5+10+20+…….

EXPONENTIAL EQUATIONS - 1

If a² = 15; find the value of a⁴.

Solution

a⁴ = (a²)².

a⁴ = (15)².   [ since a² = 15]

a⁴ = 15 x 15

a⁴ = 225   answer

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

NECTA FORM II - 2012 QN. 8

If a² = 12; find the value of a⁴.

SLOPE or GRADIENT - 1

A line passes through (9a, 8) and (8, 5a). If its slope is 2, find a.

Solution

x1=9a, y1 = 8, x2 = 8, y2 = 5a

Slope(m) =    y2 – y1

                     x2 – x1

             2 =  5a – 8

                   8 – 29a

    2 =    5a – 8     [cross multiplying]

  1        8 – 9a

2(8 – 9a) = 5a – 8

16 – 18a = 5a – 8

16 – 18a + 8 = 5a

16 + 8 = 5a+ 18a

24 = 23a

24 = 23a

23    23

²⁴/23 = a

Hence a = ²⁴/23

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

A line passes through (3a, 5) and (2, 6a). If its slope is 8, find a.

ABSOLUTE VALUE FUNCTION 1

If f(x) = |x - 3| evaluate f(-27)

solution

f(x) = |x - 3|

f(-27) = |-27 - 3|

f(-27) = |-30| = 30 since ay number out of absolute signs is +ve.

Hence f(-27) = 30

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

NECTA 1997 QN. 3b (i)

If f(x) = |x - 7| evaluate f(-27)

LOGARITHMS - 4

Find x if Log_x1024 = 10

Solution

Log_x1024 = 10

1024 = x¹⁰ .

2¹⁰ = x¹⁰ .    [ since 1024= 2¹⁰]

2¹⁰ = x¹⁰ .

x = 2

Hence x = 2

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

NECTA 1999 QN. 3b (ii)

Find x if Log_x32 = 5

VECTORS - 1

If u = 6i + 2j and v = 3i + 10j find 7u + 3v

solution

= 7u + 3v

= 7(6i + 2j) + 3(3i + 10j)

= 42i + 14j + 9i + 30j

= (12i + 30i) + (14j + 30j)

= 42i + 44j

Hence 7u + 3v = 42i + 44j

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

NECTA 2001 QN. 12a (i)

If u = 4i + 3j and v = 2i + 4j;  find 2u + 3v.

FACTORIZATION 4

Factorize completely cq + cr - nr - n².  

Solution

= cn + cr - nr - n².

= (cn + cr) – (nr - n²).      [Grouping the factors]

= c(n + r) - n(r + n).

 = (n + r)(c - n).

  Hence cn + cr - nr - n². =  (n + r)((c - n).    

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

NECTA 1995 QN. 20a

Factorize completely pq + pr - rq - q².   

ALGEBRA-2

What is x if 6x + 5y = 40 and y=6?

solution

6x + 5y = 40

6x + 5(6) = 40;           [since y=6.]

6x + 30 = 40

6x = 40 - 30

6x = 10

6x = 10

6      6

x = ¹⁰/6

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

NECTA FORM II - 2013 QN. 9

What is x if x + 2y = 16 and y=6?