2022 SGSS AMath Prelims P2

Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c   , 2 4 2 b b acx a    Binomial expansion   1 2 2 .... ....1 2 n n n n n r r nn n na b a a b a b a b b r                          , where n is a positive integer and ! ( 1) ( 1) !( )! ! n n n n n r r r n r r           2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin ( ) sin cos cos sinA B A B A B   cos ( ) cos cos sin sinA B A B A B   tan tantan ( ) 1 tan tan A BA B A B    sin 2A = 2 sin A cos A cos 2A = cos2 A  sin2 A = 2 cos2 A  1 = 1  2 sin2 A 2 2 tantan 2 1 tan AA A  Formulae for ABC sin sin sin a b c A B C  2 2 2 2 cosa b c bc A    = Abcsin2 1

1 (a) If 2764 3 xm , find the exact value of xm . [2] (b) State the condition for m, in order for 2764 3 xm to be valid. [1]

2 (a) Given that 29 2 x xy   , find x y d d . [2] (b) Hence, find    x x x d )9( 9 22 2 . [2]

3 Show that the equation 10)12( zz ee has only 1 solution and find its value correct to 2 decimal places. [5]

4 The equation of a polynomial is given by f(x) = 17113 23  xxx . (a) Find the remainder when f(x) is divided by x + 2. [1] (b) Solve the equation 017113 23  xxx . [7]

5 The diagram above shows a rectangle ABCD with AB = 3 cm and BC = 5 cm. The side AD of the rectangle makes an acute angle  with the horizontal ground. (a) Show that 3cos 5sinCF    , where CF is the height of C above the ground. [3]

(b) Express CF in the form cos( )R   , where R > 0 and 0 90   . [3] (c) Find the maximum value of CF and the corresponding value of  . [3]

7 The table shows experimental values of two variables, x and y. x 10 20 30 40 y 0.06 0.38 2.37 14.70 It is known that x and y are related by the equation A xby 10  , where A and b are constants. (a) Draw a straight line graph of lg y against x, using a scale of 2 cm to represent 5 units on the x-axis and 2 cm to represent 0.5 units on the lg y-axis. [3]