{"id":1931,"date":"2023-10-02T07:26:36","date_gmt":"2023-10-02T07:26:36","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=1931"},"modified":"2023-10-02T07:28:26","modified_gmt":"2023-10-02T07:28:26","slug":"week-8-ss1-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-8-ss1-first-term-further-mathematics-notes\/","title":{"rendered":"Week 8 – SS1 First Term Further Mathematics Notes"},"content":{"rendered":"

\u00a0
\n\u00a0WEEK EIGHT:
\n<\/strong>TOPIC: BINARY OPERATIONS: IDENTITY AND INVERSE ELEMENTS
\n<\/strong>Identity element and Inverse element
\nCONTENT:
\n<\/strong>Identity Element:
\n<\/strong> Given a non- empty set S which is closed under a binary operation * and if there exists an element e \u20ac S such that a*e = e*a = a for all a \u20ac S, then e is called the IDENTITY or NEUTRAL element. The element is unique.
\nExample:<\/strong> The operation * on the set R of real numbers is defined by a*b = 2a-1 \u253c b
\n 2
\nfor all a, b \u20ac R. Determine the identity element.
\nSolution:
\n<\/strong> a*e= e*a = a
\n a*b= 2a-1 \u253c b
\n 2
\n a*e = 2a-1 \u253c e = a
\n 2
\n 2a-1+ 2e = 2a
\n 2e = 2a-2a +1
\n e = \u00bd.<\/p>\n

\u00a0Evaluation
\n<\/strong>Find the identity element of the binary operation a*b = a +b+ab
\nInverse Element;
\n<\/strong> If x \u20ac S and an element x-1 <\/sup>\u20ac S such that x*x-1<\/sup> = x-1<\/sup>*x= e where e is the identity element and x-1<\/sup> is the inverse element.
\nExample: An operation * is defined on the set of real numbers by x*y = x + y -2xy. If the identity element is 0, find the inverse of the element.
\nSolution;
\n<\/strong> X *y = x+ y- 2xy
\n x*x-1<\/sup> = x-1*x= e, e = 0
\n x + x-1<\/sup>– 2xx-1<\/sup> = 0
\n x-1<\/sup> -2xx-1<\/sup>= -x
\n x-1<\/sup>(1-2x) = -x
\n x-1<\/sup> = -x\/ (1-2x)
\nThe inverse element x-1<\/sup> = -x\/ (1-2x)<\/p>\n

\u00a0Evaluation:
\n<\/strong> The operation \u2206 on the set Q of rational numbers is defined by: x\u2206 y = 9xy for x,y \u20ac Q
\n Find under the operation \u2206 (I) the identity element (II) the inverse of the element a \u20ac Q<\/p>\n

\u00a0General Evaluation
\n<\/strong><\/p>\n

    \n
  1. \n
    An operation on the set of integers defined by a*b = a2<\/sup> + b2<\/sup> \u2013 2a,find 2*3*4\n<\/div>\n<\/li>\n
  2. \n
    Solve the pair of equations simultaneously\n<\/div>\n
      \n
    1. \n
      2x + y = 3, 4x2<\/sup> \u2013 y2<\/sup> + 2x + 3y= 16\n<\/div>\n<\/li>\n
    2. \n
      22x \u2013 3y <\/sup>= 4, 33x + 5y <\/sup>\u2013 18 = 0\n<\/div>\n<\/li>\n<\/ol>\n

      \u00a0<\/li>\n<\/ol>\n

      Reading Assignment<\/strong>: Read Binary Operation, Further Mathematics Project II, page 16 \u2013 22<\/p>\n

      \u00a0
      \n\u00a0
      \n\u00a0
      \n\u00a0
      \n\u00a0Weekend Assignment<\/strong><\/p>\n

        \n
      1. \n
        Find the identity element e under this operation if the binary operation* is defined by c * d = 2cd+ 4c+ 3d for any real number.\n<\/div>\n<\/li>\n<\/ol>\n

        A. -3 B. -2C+3 C. X-3
        \n 2C+3 2C
        \n 2. An operation is defined by x*y = Logy<\/sup>x <\/sub>, evaluate 10* 0.0001
        \n A. 4 B. -4 C. 3
        \n 3. The binary operation * is defined by x*y= xy<\/sup>– 2x -15, solve for x if x*2= 0
        \n A.x= -3 or -5 B. x= -3 or 5 C. x = 3 or 5
        \n 4. A binary operation * is defined on the set R of real numbers by
        \n m*n = m + n2 <\/sup>for all m, n \u20ac R. If k*3 = 7*4, find the value of k
        \n A. 8 B.28\/3 C.14
        \n 5 .Find the inverse function a-1<\/sup> in the binary operation \u2206 such that for all a,b \u20ac R
        \n a \u2206 b = ab\/ 5
        \n A. 25\/a B.-25\/a C. a\/5
        \n\t\t\t<\/sup>
        \n\u00a0
        \n\t\t\t\t<\/sup> Theory
        \n<\/strong><\/p>\n

          \n
        1. \n
          A binary operation * is defined on the set R of real numbers by\n<\/div>\n<\/li>\n<\/ol>\n

          x*y = x2<\/sup> + y2<\/sup>+
          \n\t\t\t<\/sup>xy for all x, y \u20ac R. Calculate (a) ( 2*3)* 4
          \n (b) Solve the equation 6*x = 27
          \n2. Draw a multiplication table for modulo 4.
          \n (b) Using your table or otherwise evaluate (2X3) X (3X2)<\/p>\n

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          \n\t\t\t<\/strong>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

          \u00a0 \u00a0WEEK EIGHT: TOPIC: BINARY OPERATIONS: IDENTITY AND INVERSE ELEMENTS Identity element and Inverse element…<\/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-1931","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\/1931","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=1931"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1931\/revisions"}],"predecessor-version":[{"id":1932,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/1931\/revisions\/1932"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=1931"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=1931"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=1931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}