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Maths PMR Form 3 Note

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  NOTES AND FORMULAEPMR MATHEMATICS 1. SOLID GEOMETRY (a)Area and perimeter Triangle  A  = 21 ×  base ×  height = 21 bh Trapezium  A = 21 (sum of two  parallel sides) ×  height = 21 ( a + b) ×    h CircleArea = π r  2 Circumference = 2 π r  Sector Area of sector = ! θ   ×   π r  2 #ength of arc = ! θ     ×  2 π r  C$linder Cur%e surface area = 2 π rh SphereCur%e surface area = & π r  2 (b)Solid and 'olumeCube' =  x   ×    x   ×    x  =  x  Cuboid' = l    ×   b   ×   h  = lbh C$linder  V   = π   r  2 h Cone V   = 1 π   r  2 h Sphere V   = & π r   $ramid V   = 1 ×  base area ×  heightrism V   = Area of cross section ×  length2. CIRCLE THEOREM Angle at the centre = 2 * angle at the circumference  x = 2y Angles in the same segment are e+ual  x = y Angle in a semicircle ∠  ACB  = , o Sum of opposite angles of a c$clic +uadrilateral = 1- o a + b  = 1- o The eterior angle of a c$clic +uadrilateral is e+ual to the interior opposite angle. b = a Angle between a tangent and a radius = , o ∠ OPQ  = , o The angle between atangent and a chord is e+ual to the angle in the alternate segment.  x = y Note prepared by Mr. Sim KY  /f   PT   and  PS   are tangents to a circle0  PT = PS  ∠ TPO  = ∠ SPO ∠ TOP   = ∠ SOP  . POLYGON (a)The sum of the interior angles of a n sided pol$gon= ( n   2) ×  1- o (b)Sum of eterior angles of a pol$gon = ! o (c)ach eterior angle of a regular n sided pol$gon = n ! (d)3egular pentagon ach eterior angle = 42 o ach interior angle = 1 - o (e)3egular heagon ach eterior angle = ! o ach interior angle = 12 o (f)3egular octagonach eterior angle = &5 o ach interior angle = 15 o &. FACTORISATION (a)  xy + xz = x(y + z) (b)  x 2  – y 2  = (x – y)(x + y) (c)  xy + xz + ay + az = x (y + z) + a (y + z)= (y + z)(x + a) (d)  x 2  6 &  x  6  = (  x  6 )(  x  6 1)5. EXPANSION OF ALGERBRAIC EXPRESSIONS (a)2 2   ! 6    = 2 2   5 7 (b)( 6 ) 2  =  2  6 2 *  *  6  2  =  2  6 ! 6 ,(c)(  $)( 6 $) =  2  6 $  $  $ 2  =  2   $ 2 !. LAW OF INDICES (a)  x m   ×    x   n  =  x m 6 n (b)  x m   ÷    x n  =  x m  n (c)(  x m ) n  =  x  m ×  n (d)  x - n  = n  x 1 (e)  n  x x n = 1 (f)  mn  x x nm )( = (g)  x 0  = 14. ALGEBRAIC FRACTION press 2 11 2! k k  k  −− as a fraction in its simplest form.Solution 22 11 1(1 )2!! k k k k  k k  − × − −− = = 2222 1 &1 2(5)5!!! k k k k k k k k k  − + − − −= = = -. LINEAR EQUATION 8i%en that 15 (n 6 2) = n  20 calculate the %alue of n.Solution 15 (n 6 2) = n  25 * 15 (n 6 2) = 5(n  2)n 6 2 = 5n  1 2 6 1 = 5n  n2n = 12n = !,. SIMULTANEOUS LINEAR EQUATIONS (a)Substitution 9ethod$ = 2  5 ::::::::(1)2 6 $ = 4 ::::::::(2)Substitute (1) into (2)2 6 2  5 = 4& = 12 = Substitute  =  into (1)0$ = !  5 = 1(b)limination 9ethodSol%e 6 2$ = 5 ::::::::::(1)   2$ = 4 ::::::::::(2)(1) 6 (2)0 & = 120 = Substitute into (1), 6 2$ = 52$ = 5  , = 7&$ = 721 . ALGEBRAIC FORMULAE 8i%en that ;  (m 6 2) = m0 epress m in terms of ;.Solution;  (m 6 2) = m;  m  2 = m;  2 = m 6 m = &mm = 2& k   − Note prepared by Mr. Sim KY  11. LINEAR INEQUALITIES 1.Sol%e the linear ine+ualit$   2 < 1 .Solution  2 < 1  < 1 6 2 < 12 < &2.#ist all integer %alues of  which satisf$ the linear ine+ualit$ 1   6 2 > &Solution1   6 2 > &Subtract 201 7 2   6 2  2 > &  2 71   > 2 ∴   = 710 0 1.Sol%e the simultaneous linear ine+ualities &p    p and p 6 2 ≥   12  pSolution&p    p&p  p  p   p  1 p 6 2 ≥   12  p* 202p 6 & ≥  p2p  p ≥  7&p ≥  7& ∴  The solution is 7&  p  1.12. STATISTICS 9ean = sum of datanumber of data 9ean = sum of(fre+uenc$ data)sum of fre+uenc$ × 0 when the data has fre+uenc$.9ode is the data with the highest fre+uenc$9edian is the middle data which is arranged in ascending?descending order.1.0 0 &0 !0 -9ean = &!-&.-5 + + + += 9ode = 9edian = &2.&0 50 !0 -0 ,0 1 0 there is no middle number0 the median is the mean of the two middle numbers.9edian = !-2 + = 42 . A pictograp  uses s$mbols to represent a set of data. ach s$mbol is used to represent certain fre+uenc$ of the data.@anuar$ebruar$9arch3epresents 5 boo;s.A !ar cart  uses horizontal or %ertical bars to represent a set of data. The length or the height of each bar represents the fre+uenc$ of each data.&.A pi cart  uses the sectors of a circle to represent the fre+uenc$?+uantiti$ of data.A pie chart showing the fa%ourite drin;s of a groupof students.1. TRIGONOMETRY   TBA SB CA1&. GRAPHS OF FUNCTIONS  (i) #inear function. (ii)Duadratic function. (iii) Cubic function. (i%) 3eciprocal    y   = a x   15. GEOMETRICAL CONSTRUCTIONS Note prepared by Mr. Sim KY E F&5F! Fsin EG 2221 = 2 cos E 2 2221 = Gtan E 1 = 1H adIacent side θ  h$potenuse opposite side  A BC   tan θ   = sideadIacent sideopposite  sin  θ   = h$potenusesideopposite  cos  θ   = h$potenusesideadIacent  y  =  x    y  =    x    y  =  x 2    y  =   x 2    y  =  x 3    y  =   x 3    x y O  x y O  y   = : a x  1!. SCALE DRAWINGS Scale of a drawing = The length of drawingThe length of the actual obIect 14. LINES AND ANGLES 1-. COORDINATES 1. Jistance ( ) ( ) 212212 $$  −−−= 2. 9idpoint0 ( )        ++= 22 2121  yy,xxy,x 1,. TRANSFORMATIONS 1. Translation  x y    ÷    2. 3eflection  . 3otation (i) centre of rotation (ii) angle of rotation (iii) direction of rotation : eample ,  cloc;wise ? ,  anticloc;wise &. nlargement(i) centre of enlargement (ii) scale factor k   = obIectof sideingcorrespondtheof length imageof sideaof length k  2  = area of imagearea of obIect Note prepared by Mr. Sim KY  x y x = y !  = !ab x + y = #0 $

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