{"id":1928,"date":"2023-10-02T07:25:25","date_gmt":"2023-10-02T07:25:25","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=1928"},"modified":"2023-10-02T07:28:26","modified_gmt":"2023-10-02T07:28:26","slug":"week-6-and-7-ss1-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-6-and-7-ss1-first-term-further-mathematics-notes\/","title":{"rendered":"Week 6 and 7 &#8211; SS1 First Term Further Mathematics Notes"},"content":{"rendered":"<p><strong>WEEK SIX<br \/>\n<\/strong><strong>First Half  Term Revision Questions<br \/>\n<\/strong>1 . Evaluate the following (a) 32<sup>3\/5<\/sup>       (b) 25<sup>1.5<\/sup>         (c) (0.000001)<sup>2     <\/sup>  (d) 343<sup>2\/3\u00a0      <\/sup> (e) 19<sup>0<\/sup><br \/>\n\t\t2 . Solve the following exponential equations (a) 2<sup>x<\/sup> = 0.125 (b) 3<sup>-x<\/sup> = 243 (c) 25<sup>x<\/sup> = 625  (d) 10<sup>x <\/sup> = 1\/0.001<br \/>\n     (e) 4\/2<sup>x<\/sup> =64<sup>x<\/sup><br \/>\n\t\t3 . Solve the following exponential equations (a) 2<sup>2x<\/sup> -6(2<sup>x<\/sup>) + 8 = 0        (b) 2<sup>2x+1<\/sup> -5(2<sup>x<\/sup>) + 2 = 0<br \/>\n     (c) 3<sup>2x<\/sup> \u2013  4(3<sup>x+1<\/sup>) + 27 = 0         (d) 3<sup>2x <\/sup> \u2013 9 = 0        (e) 7<sup>2x<\/sup> \u2013 2( 7<sup>x<\/sup>) + 1 = 0<br \/>\n4 . Change each of the following index form to their logarithmic form (a) 2<sup>6<\/sup> = 64  (b) 3<sup>-3<\/sup> =1\/27<br \/>\n      (c) 25<sup>1\/2<\/sup> =5 (d) 3<sup>5<\/sup> = 243 (e) (0.01)<sup>2<\/sup> = 0.0001<br \/>\n5 . Change the following logarithmic form into index form (a) log<sub>2<\/sub>128 = 7 (b) log<sub>1\/2<\/sub>(1\/4) = 2<br \/>\n     (c) log<sub>7<\/sub>49 = 2 (d) log<sub>5 <\/sub>1\/125 = -3 (e) log<sub>5<\/sub>1 = 0<br \/>\n6 . Simplify each of the following (a) log<sub>3<\/sub>27 + 2log<sub>3<\/sub>9 \u2013log<sub>3<\/sub>54 (b) 1\/2log<sub>4<\/sub>8 + log<sub>4<\/sub>32 \u2013 log<sub>4<\/sub>2<br \/>\n     (c) log<sub>2<\/sub>\u221a8 + log<sub>3<\/sub>\u221a3 (d) log<sub>x<\/sub>x<sup>9<\/sup> (e) log<sub>5<\/sub>12.5 + log<sub>5<\/sub>2<br \/>\n7 . Solve the following logarithmic equations (a) log<sub>10<\/sub>(x<sup>2<\/sup> \u2013 4x + 7) = 2    (b) log<sub>8<\/sub>(x<sup>2<\/sup> \u2013 8x + 18) = 1\/3<br \/>\n     (c) log<sub>5<\/sub>(x<sup>2<\/sup> &#8211; 9) = 0 (d) log<sub>4<\/sub>(x<sup>2<\/sup> + 6x + 11) = \u00bd<br \/>\n8. If log<sub>x<\/sub>27 + log<sub>y<\/sub>4= 5 and log<sub>x<\/sub>27 \u2013 log<sub>y<\/sub>4 = 1.find x and y<br \/>\n9 . Use logarithm table to evaluate the following (a) (3.68)<sup>2<\/sup> x 6.705   (b) \u221a0.897 x 3.536<br \/>\n\t\t<img decoding=\"async\" align=\"left\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi1.png\" alt=\"\"\/>                                                                                        \u221a0.3581                       0.00249<br \/>\n\t\t\t10. 83.67   x 3  0.07124<br \/>\n\t\t352.18  <\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<strong>WEEK SEVEN<br \/>\n<\/strong><strong>TOPIC:  BINARY OPERATIONS: BASIC CONCEPT OF BINARY OPEATIONS<br \/>\n<\/strong><strong>CONTENT<br \/>\n<\/strong><\/p>\n<ul>\n<li>\n<div>Concept of binary operations,\n<\/div>\n<\/li>\n<li>\n<div>Closure property\n<\/div>\n<\/li>\n<li>\n<div>Commutative property\n<\/div>\n<\/li>\n<li>\n<div>Associative property and\n<\/div>\n<\/li>\n<li>\n<div>Distributive property.\n<\/div>\n<p><strong>Definition:<br \/>\n<\/strong>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.<br \/>\n It is usually denoted by symbols such as, *, \u04e8 e.t.c.<br \/>\n\u00a0\u00a0\u00a0\u00a0<br \/>\nP<strong>roperties:<br \/>\n<\/strong><strong>A. Closure property: A<\/strong> non- empty set z is closed under a binary operation * if for all a, b \u20ac Z.<br \/>\nExample; A binary operation * is defined on the set S= {0, 1, 2, 3, 4} by<br \/>\n X*Y = x + y \u2013xy. Find (a) 2 * 4 (b) 3* 1 (c) 0* 3.  Is the set S closed under the operation *?<br \/>\nSolution\n<\/li>\n<\/ul>\n<ol>\n<li>\n<div>2 * 4, i.e, x= 2,y=4\n<\/div>\n<\/li>\n<\/ol>\n<p>\u00a0\u00a0\u00a0\u00a02+ 4 \u2013 (2&#215;4)       = 6-8 = -2.        <\/p>\n<ol>\n<li>\n<div>3* 1 = 3+1-( 3x 1)    = 4 \u2013 3= 1\n<\/div>\n<\/li>\n<li>\n<div>0*3  = 0 + 3 \u2013( 0 x3) = 3\n<\/div>\n<\/li>\n<\/ol>\n<p> Since -2\u20ac S, therefore the operation * is not closed in S.<\/p>\n<p>\u00a0<strong>B. Commutative Property: <\/strong>If set S, a non empty set is closed under the binary operation *, for all a,b\u20ac S. Then the operation * is commutative if a*b= b*a<br \/>\nTherefore, a binary operation is commutative if the order of combination does not affect the result.<br \/>\n\t\t\tExample; The operation * on the set R of real numbers is defined by:<br \/>\np*q= p<sup>3 <\/sup>+ q<sup>3<\/sup>-3pq. Is the operation commutative?<\/p>\n<p>\u00a0<strong>Solution<br \/>\n<\/strong>p*q= p<sup>3<\/sup> + q<sup>3<\/sup> -3pq<br \/>\nCommutative condition p*q= q*p<br \/>\nTo obtain q*p, use the same operation q*p, use the same operation p*q but replace p by q and q by p.<br \/>\nHence, q*p= p<sup>3<\/sup>+ q<sup>3 <\/sup>-3qp<br \/>\nIn conclusion p*q= q*p, the operation is commutative.<strong><br \/>\n\t\t\t<\/strong><br \/>\n\u00a0<strong>C. Associative Property: <\/strong>If a non \u2013 empty set S is closed under a binary operation *, that is a*b \u20acS. Then a binary operation is associative if (a*b) * c= a*(b*c)<br \/>\nSuch that C also belongs to S.<br \/>\nExample: The operation \u04e8 on the set Z of integers is defined by; a \u04e8 b = 2a +3b -1. Determine whether or not the operation is associative in Z.<br \/>\n<strong>Solution<br \/>\n<\/strong>Introduce another element C<br \/>\n<strong>Associative condition: (a\u04e8b<\/strong>) \u04e8c = a \u04e8 (b \u04e8c)<br \/>\n<strong>(a\u04e8b<\/strong>)\u04e8c = (2a+ 3b- 1) \u04e8 C<br \/>\n                = 2(2a +3b -1) + 3c -1<br \/>\n                = 4a + 6b- 2+ 3c- 1<br \/>\n= 4a +6b+3c- 3.<br \/>\nAlso, the RHS, a \u04e8 (b \u04e8 c) = a \u04e8 (2b+3c- 1)<br \/>\n                                           = 2a+ 3(2b +3c- 1) &#8211; 1<br \/>\n                                           = 2a + 6b +9c -3 -1<br \/>\n                       a \u04e8 (b \u04e8 c)  = 2a+ 6b+ 9c -4<br \/>\nSince,   (a \u04e8 b) \u04e8 c \u2260 a \u04e8 (b \u04e8 c), the operation is not associative in Z.<\/p>\n<p>\u00a0<strong>Evaluation<br \/>\n<\/strong>1. <strong>A<\/strong>n operation* defined on the set R of real numbers is<br \/>\n x* y = 3x+ 2y- 1, x,y \u20acR. Determine (a) 2*3 (b) -4* 5 (c) 1 * 1<br \/>\n\t\t\u00a0\u00a0\u00a0\u00a0                                                                                      3    2<br \/>\nis the operation closed.<\/p>\n<p>\u00a0<strong>D.   Distributive Property:<\/strong> If a set is closed under two or more binary operations<br \/>\n(* \u04e8) for all a, b and c \u20ac S, such that:<br \/>\n              a*(b\u04e8 c) = (a*b )\u04e8( a*c &#8211; Left distributive<br \/>\n              (B\u04e8c) *a = (b*a) \u04e8(c*a) &#8211; Right distributive over the operation \u04e8<\/p>\n<p>\u00a0<strong>Example:<\/strong> Given the set R of real numbers under the operations * and \u04e8 defined by:<br \/>\n        a*b = a+ b- 3, a\u04e8b= 5ab for all a, b \u20ac R. Does * distribute over \u04e8.<br \/>\n<strong>Solution <\/strong>Let a, b,c \u20ac R<br \/>\na* ( b\u04e8c) = (a*b) \u04e8 (a*c)<br \/>\na* (b\u04e8c) = a* (5ab)<br \/>\n               = a+ 5ab -3.<\/p>\n<p>\u00a0(a*b) \u04e8 (a*c) = (a+ b -3) \u04e8 ( a+ c-3)<br \/>\n                      = 5(a +b-3)(a +c -3)<br \/>\n  From the expansion, it&#8217;s obvious that, a* ( b\u04e8c) \u2260 (a*b) \u04e8 (a*c)  therefore * does not distribute over \u04e8.<\/p>\n<p>\u00a0<strong>Evaluation:<br \/>\n<\/strong>1.  A binary operation * is defined on the set R of real numbers by x*y= x +y + 3xy for all x, y\u025bR.<br \/>\n     determine whether or not * is:<\/p>\n<ol>\n<li>\n<div>Commutative?\n<\/div>\n<ol>\n<li>\n<div>Associative?\n<\/div>\n<\/li>\n<\/ol>\n<p>2. The operation <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi2.png\" alt=\"\"\/> on the set R of real numbers is defined by a <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi3.png\" alt=\"\"\/>b = a+b   + ab for ab\u03f5R.<br \/>\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0      2\u00a0\u00a0\u00a0\u00a0<br \/>\nShow that the operation <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi4.png\" alt=\"\"\/>is commutative but not associative on R.<br \/>\n<strong>General Evaluation<br \/>\n<\/strong>1. The operation * on the set R of real numbers is defined by: x * y = 3x + 2y \u2013 1, x, y\u03f5R.<br \/>\n\u00a0\u00a0\u00a0\u00a0Determine (i) 2 * 3 (ii) 1\/3 * \u00bd (iii) -4*5<br \/>\n2. The operation * on the set R, of real numbers is defined by; p*q = p<sup>3<\/sup> + q<sup>3<\/sup> \u2013 3pq; p,q \u03f5R. Is the operation * commutative in R?<br \/>\n3. The operation * and <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi5.png\" alt=\"\"\/>are defined on the set R of natural numbers by a*b = ab and a <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi6.png\" alt=\"\"\/>b = a\/b for all a,b\u03f5R  (a) Does * distribute over <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi7.png\" alt=\"\"\/>? (b) Does <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi8.png\" alt=\"\"\/>distribute over *?<\/p>\n<p>\u00a0<strong>Weekend Assignment<br \/>\n<\/strong><\/p>\n<ol>\n<li>\n<div>Two binary operation * and \u04e8 are defined as m * n = mn \u2013 n -1 and m \u04e8 n = mn + n -2 for al real\n<\/div>\n<p>number m n find the value of 3 \u04e8 (4 * 5) (a) 60 (b) 57 (c) 54 (d) 42\n<\/li>\n<li>\n<div>If x * y = x + y \u2013xy, find x, when (x*2) + (x*3) = 63 (a) 24 (b) 22 (c) -12 (d) -21\n<\/div>\n<\/li>\n<li>\n<div>A binary operation * is defined by a * b = a<sup>b<\/sup>. If a * 2 = 2 \u2013 a, find the possible values of a (a) 1, -1\n<\/div>\n<p>(b) 1, 2  (c) 2, -2 (d) 1, -2\n<\/li>\n<li>\n<div>The binary operation * is defined on the set of integers p and q by p*q = pq + p + q. Find 2*(3*4)\n<\/div>\n<p>(a) 59 (b) 19 (c) 67 (d) 38\n<\/li>\n<li>\n<div>A binary operation <img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi9.png\" alt=\"\"\/>on real numbers is defined by x<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi10.png\" alt=\"\"\/>y = xy + x + y for any two real numbers and y. The value of (-3\/4)<img decoding=\"async\" src=\"https:\/\/ecolebooks.com\/nigeria\/wp-content\/uploads\/9jalessonsimages\/100223_0725_Week6SS1Fi11.png\" alt=\"\"\/>6 is (a) 3\/4 (b) -9\/2 (c) 45\/4 (d) -3\/4\n<\/div>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<strong>Theory<br \/>\n<\/strong>1. \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0The operation * is defined on the set R of real numbers by a* b=   a+b   _ 1<br \/>\n      \u00a0\u00a0\u00a0\u00a0for all a, b \u20acR  .                                                                                       5<br \/>\n  \u00a0\u00a0\u00a0\u00a0Is the operation * commutative in R?.<br \/>\n2. \u00a0\u00a0\u00a0\u00a0The operation * is defined on the set R of real numbers by x*y = x + y + xy\/2 for all x,y \u20acR<br \/>\n     \u00a0\u00a0\u00a0\u00a0(a) is the operation * commutative? (b) is the operation * associative over the set R?\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<br \/>\n<strong>Reading Assignment<\/strong>: Read Binary Operation, Further Mathematics Project II, page 13 \u2013 22<\/p>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<\/li>\n<\/ol>\n<p>\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<br \/>\n\u00a0<strong><br \/>\n\t\t\t\t\t<\/strong>\u00a0<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>WEEK SIX First Half Term Revision Questions 1 . Evaluate the following (a) 323\/5 (b)&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,173],"tags":[],"class_list":["post-1928","post","type-post","status-publish","format-standard","hentry","category-posts","category-first-term-ss1-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1928","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=1928"}],"version-history":[{"count":2,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1928\/revisions"}],"predecessor-version":[{"id":1930,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1928\/revisions\/1930"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=1928"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=1928"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=1928"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}