Week 6 and 7 – SS1 First Term Further Mathematics Notes

WEEK SIX
First Half Term Revision Questions
1 . Evaluate the following (a) 32^3/5 (b) 25^1.5 (c) (0.000001)² (d) 343^2/3 (e) 19⁰
2 . Solve the following exponential equations (a) 2^x = 0.125 (b) 3^-x = 243 (c) 25^x = 625 (d) 10^x = 1/0.001
(e) 4/2^x =64^x
3 . Solve the following exponential equations (a) 2^2x -6(2^x) + 8 = 0 (b) 2^2x+1 -5(2^x) + 2 = 0
(c) 3^2x – 4(3^x+1) + 27 = 0 (d) 3^2x – 9 = 0 (e) 7^2x – 2( 7^x) + 1 = 0
4 . Change each of the following index form to their logarithmic form (a) 2⁶ = 64 (b) 3^-3 =1/27
(c) 25^1/2 =5 (d) 3⁵ = 243 (e) (0.01)² = 0.0001
5 . Change the following logarithmic form into index form (a) log₂128 = 7 (b) log_1/2(1/4) = 2
(c) log₇49 = 2 (d) log₅1/125 = -3 (e) log₅1 = 0
6 . Simplify each of the following (a) log₃27 + 2log₃9 –log₃54 (b) 1/2log₄8 + log₄32 – log₄2
(c) log₂√8 + log₃√3 (d) log_xx⁹ (e) log₅12.5 + log₅2
7 . Solve the following logarithmic equations (a) log₁₀(x² – 4x + 7) = 2 (b) log₈(x² – 8x + 18) = 1/3
(c) log₅(x² – 9) = 0 (d) log₄(x² + 6x + 11) = ½
8. If log_x27 + log_y4= 5 and log_x27 – log_y4 = 1.find x and y
9 . Use logarithm table to evaluate the following (a) (3.68)² x 6.705 (b) √0.897 x 3.536
√0.3581 0.00249
10. 83.67 x 3 0.07124
352.18

WEEK SEVEN
TOPIC: BINARY OPERATIONS: BASIC CONCEPT OF BINARY OPEATIONS
CONTENT

Concept of binary operations,
Closure property
Commutative property
Associative property and
Distributive property.
Definition:
Binary operation is any rule of combination of any two elements of a given non empty set. The rule of combination of two elements of a set may give rise to another element which may or not belong to the set under consideration.
It is usually denoted by symbols such as, *, Ө e.t.c.

Properties:
A. Closure property: A non- empty set z is closed under a binary operation * if for all a, b € Z.
Example; A binary operation * is defined on the set S= {0, 1, 2, 3, 4} by
X*Y = x + y –xy. Find (a) 2 * 4 (b) 3* 1 (c) 0* 3. Is the set S closed under the operation *?
Solution

2 * 4, i.e, x= 2,y=4

2+ 4 – (2×4) = 6-8 = -2.

3* 1 = 3+1-( 3x 1) = 4 – 3= 1
0*3 = 0 + 3 –( 0 x3) = 3

Since -2€ S, therefore the operation * is not closed in S.

B. Commutative Property: If set S, a non empty set is closed under the binary operation *, for all a,b€ S. Then the operation * is commutative if a*b= b*a
Therefore, a binary operation is commutative if the order of combination does not affect the result.
Example; The operation * on the set R of real numbers is defined by:
p*q= p³+ q³-3pq. Is the operation commutative?

Solution
p*q= p³ + q³ -3pq
Commutative condition p*q= q*p
To obtain q*p, use the same operation q*p, use the same operation p*q but replace p by q and q by p.
Hence, q*p= p³+ q³-3qp
In conclusion p*q= q*p, the operation is commutative.

C. Associative Property: If a non – empty set S is closed under a binary operation *, that is a*b €S. Then a binary operation is associative if (a*b) * c= a*(b*c)
Such that C also belongs to S.
Example: The operation Ө on the set Z of integers is defined by; a Ө b = 2a +3b -1. Determine whether or not the operation is associative in Z.
Solution
Introduce another element C
Associative condition: (aӨb) Өc = a Ө (b Өc)
(aӨb)Өc = (2a+ 3b- 1) Ө C
= 2(2a +3b -1) + 3c -1
= 4a + 6b- 2+ 3c- 1
= 4a +6b+3c- 3.
Also, the RHS, a Ө (b Ө c) = a Ө (2b+3c- 1)
= 2a+ 3(2b +3c- 1) – 1
= 2a + 6b +9c -3 -1
a Ө (b Ө c) = 2a+ 6b+ 9c -4
Since, (a Ө b) Ө c ≠ a Ө (b Ө c), the operation is not associative in Z.

Evaluation
1. An operation* defined on the set R of real numbers is
x* y = 3x+ 2y- 1, x,y €R. Determine (a) 2*3 (b) -4* 5 (c) 1 * 1
3 2
is the operation closed.

D. Distributive Property: If a set is closed under two or more binary operations
(* Ө) for all a, b and c € S, such that:
a*(bӨ c) = (a*b )Ө( a*c – Left distributive
(BӨc) *a = (b*a) Ө(c*a) – Right distributive over the operation Ө

Example: Given the set R of real numbers under the operations * and Ө defined by:
a*b = a+ b- 3, aӨb= 5ab for all a, b € R. Does * distribute over Ө.
Solution Let a, b,c € R
a* ( bӨc) = (a*b) Ө (a*c)
a* (bӨc) = a* (5ab)
= a+ 5ab -3.

(a*b) Ө (a*c) = (a+ b -3) Ө ( a+ c-3)
= 5(a +b-3)(a +c -3)
From the expansion, it’s obvious that, a* ( bӨc) ≠ (a*b) Ө (a*c) therefore * does not distribute over Ө.

Evaluation:
1. A binary operation * is defined on the set R of real numbers by x*y= x +y + 3xy for all x, yɛR.
determine whether or not * is:

Commutative?
1. Associative?
2. The operation on the set R of real numbers is defined by a b = a+b + ab for abϵR.
2
Show that the operation is commutative but not associative on R.
General Evaluation
1. The operation * on the set R of real numbers is defined by: x * y = 3x + 2y – 1, x, yϵR.
Determine (i) 2 * 3 (ii) 1/3 * ½ (iii) -4*5
2. The operation * on the set R, of real numbers is defined by; p*q = p³ + q³ – 3pq; p,q ϵR. Is the operation * commutative in R?
3. The operation * and are defined on the set R of natural numbers by a*b = ab and a b = a/b for all a,bϵR (a) Does * distribute over ? (b) Does distribute over *?
Weekend Assignment
1. Two binary operation * and Ө are defined as m * n = mn – n -1 and m Ө n = mn + n -2 for al real
  number m n find the value of 3 Ө (4 * 5) (a) 60 (b) 57 (c) 54 (d) 42
2. If x * y = x + y –xy, find x, when (x*2) + (x*3) = 63 (a) 24 (b) 22 (c) -12 (d) -21
3. A binary operation * is defined by a * b = a^b. If a * 2 = 2 – a, find the possible values of a (a) 1, -1
  (b) 1, 2 (c) 2, -2 (d) 1, -2
4. The binary operation * is defined on the set of integers p and q by p*q = pq + p + q. Find 2*(3*4)
  (a) 59 (b) 19 (c) 67 (d) 38
5. A binary operation on real numbers is defined by xy = xy + x + y for any two real numbers and y. The value of (-3/4)6 is (a) 3/4 (b) -9/2 (c) 45/4 (d) -3/4
  
  Theory
  1.         The operation * is defined on the set R of real numbers by a* b= a+b _ 1
      for all a, b €R . 5
      Is the operation * commutative in R?.
  2.     The operation * is defined on the set R of real numbers by x*y = x + y + xy/2 for all x,y €R
      (a) is the operation * commutative? (b) is the operation * associative over the set R?
  Reading Assignment: Read Binary Operation, Further Mathematics Project II, page 13 – 22