2024 NJC Paper 1 H2 Math Prelim (ANS)

Question 1 (Connected Rates of Change) (i) ( )2 2 2 2 kkk r r r rr   = + = +  + =  = − 2 2 211 22 2 2 k krA r r r r = = − = −  d 2d2 Ak rr =− Interestingly, many students could derive the formula 21 2Ar = by considering 2π2πAr = but not many could do so similarly for the arc length formula .sr = As a result, there are many students who could not prove the required result. (ii) 2 dd 2d 2 d 22 3 10 60 A k r rtt k k k k =− = −  =− This part of the question is better performed than the previous part as almost the whole cohort is able to relate the rates of change. However, many could not apply the information that “arc length is equal to the radius” in a useful manner.

Question 2 (Systems of Linear Equations) ( ) ( ) ( ) 32 2 2 2 0 8 4 2 0 4 2 8 --- (1) a b c a b c a b c − + − + − + =  − + − + =  − + = ( ) 322 2 2i π i π i π3 3 3 42i π i πi2 π 33 3e 3e 3e 0 27e 9 e 3 e 0 1 3 1 327 9 i 3 i 02 2 2 2 9 3 9 3 3 327 i 02 2 2 2 a b c a b c a b c a b c a b      − − −              −−   −           + + + =                + + + =    + − + + − − + =           − − + + − =    Comparing real parts, 9327 0 --- (2)22a b c− − + = Comparing imaginary parts, 9 3 3 3 0 3 0 --- (3)22 a b a b− =  − = Solving (1), (2) and (3) with the GC, we get a = 5, b = 15, c = 18. Alternative Method 2i π3 1 3 3 3 33e 3 i i 2 2 2 2 − = − − = − − Since all coefficients of the polynomial are real, then 3 3 3 i22−+ is also a root of the equation by the Conjugate Root Theorem. Therefore, ( ) ( ) ( )( ) 32 2 2 32 3 3 3 3 3 32 i i 2 2 2 2 3 9(3)2 24 2 3 9 5 15 18 z az bz c z z z zz z z z z z z + + +   = + + + + −       = + + + = + + + = + + + Therefore a = 5, b = 15, c = 18 Both methods were fairly common. Many careless mistakes were made in calculations. Common mistakes:  Not stating “Since the coefficients of the polynomial are all real” when using the Conjugate Root Theorem.  Incorrect conversion of complex numbers from exponential form to cartesian form. (Please identify the quadrant the complex number is in.)  Not converting the complex numbers to cartesian form (in the first method).  Writing “ 2i π33ez −= is a root …”. (Should be “ 2i π33e − is a root …” or “ 2i π33ez −= is a solution …”).

Question 3 (Inequalities) (i) Method 1 ( ) ( )( ) ( )( ) 2 2 2 2 2 2 2 2 2 2 6 145 6 1045 6 4 5 045 41 045 41 045 41 04 5 1 4 5 1 0 since 4 1 1 0 for all x xx x xx x x x xx x xx x xx x xx xx xx − +− − − +− − − + −  +− −− +− + +− + +−  + −  +    Method 2 ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( )( ) 2 222 22 22 2 2 6 145 6 4 5 4 5 4 5 6 4 5 0 4 5 4 1 0 4 5 1 4 1 0 4 5 1 0 since 4 1 1 0 for all x xx x x x x x x x x x x x x x x x x xx xx − +−  − + −  + −  + − − − + −