DHS Math 2 MYE 2021

L Name Register Number DUNMAN HIGH SCHOOL Mid-Year Examination Year 4 DHP MATHEMATICS 2 Additional Materials: Answer Sheet Insert A READ THESE INSTRUCTIONS FIRST Write your name, class and register number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid/tape. Answer all questions. Write your answers on the separate answer paper provided. If working is needed for any question, it must be shown with the answer. Omission of essential working will result in loss of marks. Class 11 May 2021 2 hours 30 minutes Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved scientific calculator is expected, where appropriate. You are reminded on the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. This document consists of 6 printed pages.

1. ALGEBRA Quadratic Equation For the equation ax2 +bx+ c = 0, -b± ✓b 2 -4ac x=------ 2a Binomial Expansion ( a + b )" = a' + (: }•- 1 b + (; }•- 2 b 2 + ... + (;) r b' + .. • + b' , (n) n I n( n-1) ... ( n-r + 1) where n is a positive integer and = { · ) = , · r r!n-r! r. Identities Formulae for MBC 2. TRIGONOMETRY sin2 A+cos 2 A=l sec 2 A = I+ tan 2 A cosec2 A = I + cot 2 A sin (A± B) = sin A cos B ± cos A sin B cos(A±B) = cosAcosB+sinAsinB tan(A±B)= tanA±tanB I+tanAtanB sin2A = 2sinAcosA cos2A=cos 2 A-sin 2 A=2cos 2 A-l=l-2sin 2 A tan 2A = _2_t_an_A_ I-tan 2 A a b C --=--=--sin A sin B sin C a2 = b 2 + c 2 - 2bc cos A Area oft:,.= ½ab sin C

j Answer all the questions 1 Find the range of values of m for which the line 4 y = ( m - 2) x intersects the curve y = -2-. x+l [4] 2 Given that the curve y = ax2 + Sx- c lies completely above the line y = c, find the condition(s) that must be applied to constants a and c. [4] 3 Find the x-coordinates of the points on the curve y = ex+ sin2x, 0::; x::; 1t, at which the gradient 1s zero. [4] 4 2x 2 +x-5 Express 2 in partial fractions. (x-l) [4] 5 Solve log5 x 2 = 3 log✓x 5 -1. [5] 6 (i) Explain why 12x - 3x2 -16 < 0 for all real values of x. [3] (ii) (7 x-3)(x+ 5) Hence, solve 2 > 0 . 12x-3x -16 [2] 7 (i) 5x dy 5 ( ax 2 + b) Given that y = 3 , find the values of a and b such that - = 5 • [3] ( ✓1 - x 2 ) dx ( ✓1 - x 2 ) ( 2) dy y (ii) Hence, show that 1- x dx - --:; = 2,;\,y . [1]

4 8 Mary collected some data from a science experiment to verify an equation 1- by = axy · She plotted a graph of.!. against x and obtained the straight line graph below. 9 y 1 y (4, 19) (l,y --0-=+---------x (i) Find the value of a and of b. [4] (ii) (i) (ii) 1 John claimed that he used the same data set from Mary and plotted the graph of xy against .!_ below. Assuming that Mary's graph is correct, comment if John's graph is X correct. 1 xy (2, 11) ----:-+-,-------- 1 (- 1, - 1) X . sin(2A + B) 3 G1Ven that ---- = - , show that tan 2A = 5 tan B sin(2A - B) 2 . [2] [3] Hence, find tan A without evaluating the values of A and B · , given that cos B = - ~ , where B is an obtuse angle. v241 [3] 2021 Year 4 Mid-Year Examination Mathematics 2

5 10 The temperature, r· C, of a piece of meat left to thaw for t hours after removing from a chiller, is given by T = 20-Ae-"', where A and k are constants . (i) The temperature of the chiller is-15° C . Show that A= 35. [1] (ii) The temperature of the meat is 1.5° C after one hour. Find the value of k. [3] (iii) In order to reduce the risk of bacterial contamination in thawed food, the temperature of the food should not exceed 5° C during the thawing process. Determine, with working, whether the meat has a high risk of bacterial contamination, 3 hours after removing from the chiller. [2] 11 (a) Given that I:~ y ~ 6, solve 2cos( 2(Y/ l) )+ ✓3 = 0, leaving your answers in terms of 1t. (3] (b) Find all the angles between - 180° and 180° which satisfy the equation sec 2 x = 4secx-3tan 2 x . 12 (a) Given that g(x) + 3 is a polynomial with factor x- 3, find the remainder when f(x)=(x2 -3x+4)g(x) is divided by x-3. (b) ( 1 ) 1 Show that I+ . (1 + tan 2 x) = - - - . cosxcotx+smx 1-sinx 13 P ( x) = 3x3 - px2 - qx - 8 has a factor of x2 + 4 . [4] [3] [4] (i) Find the value of p and of q, and the other factor. [3] (ii) P(x) canalsobeexpressedas P(x) = (x 2 -3)g(x)+h(x),where g(x) and h(x) are functions of x. (a) Student Z claims that if ( x 2 - 3) is the divisor, the degr ee of h ( x) can only (b) be 1. Explain whether his claim is true. Find g(x) and h(x) . 2021 Year 4 Mid-Year Examination Ma thematics 2 [I] [3]

6 14 The diagram shows the daily jogging path p ABC taken by Peter. He starts from point P and ends at point C. Angle APC = 0 , angle p AB = angle BCP = 90° , PA = AB = 4 km and BC=dkm. dkm A p (i) Show that d = 4 sin 0 - 4 cos 0 , where 0° < 0 < 90° . Express din the form R sin( 0- a) , where R > 0 and 0° < a < 90° . [3] [2] (ii) (iii) By using (ii), comment whether it is possible for Peter to complete a 12-km run using the track P ABC if 0 varies. [2] 15 Answer this question on Insert A. 16 Answer this question on Insert A. 17 Answer thls question on Insert A. ~End of Paper~ 2021 Year 4 Mid-Year Examination Mathematics 2

l Name I Register Number DUNMAN HIGH SCHOOL Mid-Year Examination Year4 DHP MATHEMATICS 2 {Insert A) This insert consists of 5 printed pages. 15 Function g is defined by g : x H _ _!. ..c + x + 4, x E R , - 2 < x ~ 5. 2 I Class I 11 May 2021 (i) Express g(x) in the form of a(x +b)2 +c , where a, band care real numbers. [2] (h") © OHS 2021 Sketch the graph of y = g(x) and state the range of g. , Y - - ~--- _:... ~ -~ • . ~ ~L -:---L - -., _ _ -.t-,-,-___;--, - -f- -~-:- - -;- --;-- i . --1 i -;--- -,- r.---- - -- ::-i ' -1-· -· -, --- - j , ' i ~- -;- -·-~ _:__,_- _::_ , -➔ >- ;- ____:__ - - - :-- I - I --> I I ' __ ' I I 1 i 3 _ I i I l i -~1 .L-~ - --,-!·-•- -j- --· ·r!·•· I - ,- ·-~-•-·-,-~-Y---f-1•-:•--;-- >--ri-- ·-H ~. :---;= .:..-:-·-__ ._,_., _ I I I : I I I · I . I ~· I ' , I ' _ __j__ __ _ j, I t , 1 . ,_ ____ !__ -!c---. -. --'- -.! , ,. ' 1 j I l j • I I ' ' -1- I • i ! [4]

2 A function h is defined by h : x H - .!_ x2 + x + 4 , x e JR , - 2 < x ~ k . 2 (iii) Find the largest value of k for which h has an inverse. , I (iv) For this value of k, find the function h -i and express it in similar form. (v) Sketch the graph of y = h-1(x) on the same axes as (ii). [1] (3] [2]

3 16 The graph of function y = f ( x) for O.::; x.::; 2n is shown below. __ y_ . -. ---- -. - .. . . .. --i - - . - -· . - - . I l ···2 .. -~-------- ~--- --~--~ ---·--··- ---- ~--- ·-·- -·--··--: -·- ... ··---· -·· ....... / I I I i .• ···· -· -- .. .• - - . .• -- •. ~ :• - . i .. - 1- - ------------- ,,-...------+---, ...,.---- 0 X ==44------ --'-----+-- - - ---- --,-- (i) S tat e the period of y = f ( x) . [I] . (ii) State the amplitude of y = f ( x) . [I] (iii) Given that f(x) = asinbx+c, write down the equation of the graph. [I] (iv) 2x On the same axes, sketch the graph of y = 1-- for O ~ x ~ 2n. 7t [2] (v) Hence state the number of solutions, for O ~ x ~ 2n, of the equation n(l-c-asinbx) = 2x. [2]

17 4 80cm An important aspect to having a smooth car ride is to ensure the wheels are well balanced .. To do that, the wheel is first mounted on the machine and rotated. The machine will then automatically calculate the weight and location of the balance counter weights to ensure that there are no heavy spots on the tyre. The diagr