2022 SGSS AMath Prelims P1

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

2 SGSS/AMP1/4E5N/Prelim/2022 1 Integrate 33(1 ) 2 1 x x    with respect to x. [3] 2 If 3 2 22 1 2 p q    , where p and q are rational numbers, find the value of p and of q. [4]

3 [Turn over SGSS/AMP1/4E5N/Prelim/2022 3 Air is being pumped into a spherical balloon at a constant rate of 20 cm 3 per second. However, when the radius of the balloon stretches beyond 7 cm, air starts to leak out from the balloon at a rate of 4 cm3 per second. Find the rate of change of the radius of the balloon when the radius is 11 cm. [4] 4 Find the coordinates of the points of intersection of the curve 2 22 5 48 0x xy y    and the line 8 2y x  . [5]

4 SGSS/AMP1/4E5N/Prelim/2022 5 Express 2 2 8 3 1 (2 1) x x x x    in partial fractions. [5]

5 [Turn over SGSS/AMP1/4E5N/Prelim/2022 6 (a) Express the equation 1 24 3 2x x   as a quadratic equation in 2x . [2] (b) Hence find the value(s) of x, giving your answer(s) correct to 2 decimal places. [4]

6 SGSS/AMP1/4E5N/Prelim/2022 7 A manufacturing company modelled the average cost per chair, in dollars, using the function 2C( ) 40 480 1442x x x   , where x is the number of chairs produced, in thousands. (a) Express 2C( ) 40 480 1442x x x   in the form 2( )a x h k  , where a, h and k are constants. [2] (b) Hence find the minimum value of C(x) and the corresponding value of x. [2] (c) Interpret the meaning of your answers to part (b). [2]

7 [Turn over SGSS/AMP1/4E5N/Prelim/2022 8 In the diagram, AC is a common chord of the two circles, ABC and ACDE. AD is a tangent to the circle ABC at A. AD and CE intersect at F. (a) Prove that AB is parallel to ED. [3] (b) Prove that triangle ACF is similar to triangle BAF. [2] (c) Hence show that 2AF BF CF  . [1] B A C D E F

8 SGSS/AMP1/4E5N/Prelim/2022 9 (a) Prove that cos 2 1 2(1 sin ) 3 3sin 3      . [3] (b) Hence solve the equation cos 2 1 1 3 3sin 2     for o o0 360  . [4]

9 [Turn over SGSS/AMP1/4E5N/Prelim/2022 10 (a) Differentiate ln(sin )x with respect to x. [2] (b) The diagram shows part of the curve coty x , cutting the x-axis at π , 02     . The line 2y  intersects the curve at point A. Using your result from part (a), find the area of the shaded region. [5] y x O A coty x 2y  π 2

10 SGSS/AMP1/4E5N/Prelim/2022 11 The function f ( )x is defined by 3 2f ( ) 8 18x ax x bx    for all real values of x. It is given that 2x is a factor of f ( )x and when f ( )x is divided by 1x , the remainder is 4. (a) Find the value of a and of b. [5] (b) Factorise f ( )x completely. [3]