1 SULIT SM015 SM015 Mathematics Matematik Semester I Semester I Session 2023/2024 Sesi 2023/2024 PEPERIKSAAN SEMESTER PROGRAM MATRIKULASI MATRICULATION PROGRAMME EXAMINATION SILA BUKA DAN JAWAB BUKU SOALAN INI TANPA PERLU DIBERITAHU Please open and answer this questions booklet without being told to do so INSTRUCTIONS TO CANDIDATE: This Questions booklet consist of 5 set questions. Answer all questions. The full marks for each question or section are shown in bracket at the end of each of the question or section. All steps must be shown clearly. Only non-programmable scientific calculators can be used. Numerical answers can be given in the form of π, e , surd, fractions or up to three significant figures, where appropriate, unless stated otherwise in the question. / KOLEJ MATRIKULASI PAHANG NAME : CLASS : NAME: CLASS:
2 MODEL PSPM SM015 MATHEMATICS SESSION 2023/24 LIST OF MATHEMATICAL FORMULAE SENARAI RUMUS MATEMATIK Quadratic equation 2 ax bx c + += 0: Persamaan kuadratik 2 ax bx c + += 0: a b b ac x 2 4 2 − ± − = Differentiation Pembezaan f x( ) f x'( ) dy dy dt dx dt dx = × 2 2 d dy d y dt dx dx dx dt = sin x cos x cos x -sin x tan x 2 sec x cot x 2 −cos ec x sec x sec tan x x cos ecx −cos cot ec x x Sphere Sfera 3 π 3 4 V = r 2 S = 4 π r Right circular cone Kon membulat tegak 1 2 π 3 V rh = 2 S r rs = + π π Right circular cylinder Silinder membulat tegak V r h 2 = π 2 S r rh = + 2 2 π π
3 MODEL PSPM SET A SM015 MATHEMATICS SESSION 2023/24 SECTION A (25 MARKS) ANSWER ALL QUESTIONS BELOW 1. Evaluate the following limits: (a) 3 3 3 limx 27 x →− x + + 4 marks (b) 5 7 lim → ∞ 6 5 + x − x x 3 marks (c) 2 2 1 lim 4 → − + x − x x 2 marks 2. Find the first derivatives of : (a) (i) 7 6 8 x y + = [2 ] marks (ii) ( ) 2 1 ln 1 − = − x f x x 3 marks (b) Find the value of dy dx for y xx x = − sin 2 cos , when 3 π x = 4 marks 3. Given a curve ( ) 2 3 fx x x x =+− − 27 9 3 . (a) Find the coordinates of the stationary points of the curve. 5 marks (b) Determine whether the stationary points are maximum or minimum. 2 marks
4 MODEL PSPM SET A SM015 MATHEMATICS SESSION 2023/24 SECTION B (75 MARKS) ANSWER ALL QUESTIONS BELOW 1. Given that 1z i = −3 3 and 2 z i = +3 2 , where i = −1 . Express ( ) 3 1 2 2 13 z z i z + − in the form of a bi a b + ∈ , , 6 marks 2. (a) Solve the equation ln 2 3 x x e − = 6 marks (b) Obtain the solution set for 2 x x xx −≤ + ≤ + 13 3 6 marks 3. Given ( ) x a f x x a + = − and g x bx ( ) = + 3, where a b and are constants. Find the possible values of a b and if fg (1 2 ) = and gf (1 1. ) = − 6 marks 4. Given the function ( ) 1 . 2 5 g x x = − (a) Find the domain and range for g x( ). 2 marks (b) Show that g x( ) is a one to one function. Hence find ( ) 1 g x . − 5 marks (c) On the same axis, sketch the graph of g x( ) and ( ) 1 g x . − 3 marks (d) Show that ( ) 1 gg x x. − = 2 marks 5. Functions ( ) 2 2, 2 4, 2 x x f x x xa x + ≥ = −+ < and ( ) , 2 4, 2 b x g x x x x ≥ = − < are continuous at x = 2 (a) (i) Find the values of a and b 4 marks (ii) Show that the function f g + is continuous at x = 2 4 marks (b) Find the horizontal asymptote(s) for the function 2 x + 9 x , if they exists 6 marks
5 MODEL PSPM SET A SM015 MATHEMATICS SESSION 2023/24 6. a) A curve is defined by the parametric equations 1 x t = − 3 t and 3 y t = + t , where t ≠ 0. (c) Show that 2 2 3 3 1 − = + dy t dx t . [4 ] marks (d) Find 2 2 d y dx . Hence, evaluate 2 2 d y dx when t = 2 [9 ] marks b) Find the value of dy dx for y xx x = − sin 2 cos , when 3 π x = 4 marks 7. An open rectangular box is to be made from a piece of paper of width 8 cm and length 15 cm. A square of width x cm is cut from each corner of the paper and the sides are folded up as shown in the figure below. Find the width, length and height of the box that gives the maximum volume. x 10 marks END OF QUESTION PAPER x
6 MODEL PSPM SET A SM015 MATHEMATICS SESSION 2023/24 ANSWERS SECTION A 1. (a) 1 27 (b) 30 6 (c) 2. (a)(i) 7 6 7(ln8)8 dy x dx + = (ii) ( ) ( ) 2 2 1 ' 12 1 x f x x x = − − − (b) −0.5931 3. (a) (−3,0 , 1,32 ) ( ) (b) (−3,0 min, 1,32 max ) ( ) SECTION B 1. (a) 5 12 13 13 i 2. (a) 3 1 xe xe , − = = (b) {x x :3 1 − ≤ ≤− } 3. 1 , 1, 2, 0 3 a ab b = = =− = 4. (a) 5 5 , , ,0 0, 2 2 D R g g (b) DIY, 1 1 5 2 2 g x x (c) (d) DIY 5. a) (i) a b = = 8, 4 (ii) DIY b) HA y y = = − 1, 1 6. (i)DIY (ii) ( ) 3 3 2 20 3 1 t t + , 160 2197 7. 14 35 5 width ,length , height = 3 33 = =
7 MODEL PSPM SET B SM015 MATHEMATICS SESSION 2023/24 SECTION A (25 MARKS) ANSWER ALL QUESTIONS BELOW 1. Evaluate. (a) 4 2 3 lim x 6 x x →+∞ x + − [4 marks] (b) 3 2 5 lim 2 3 + − − →− x x x [5 marks] 2. (a) If sin( 1) 2 y = x + , show that 4 0 3 2 2 − + x y = dx dy dx d y x [5 marks] (b) Find the first derivative of 3 11 5 xy e xy = − [4 marks] 3. By considering the first and second derivatives of f x( ) , show that there is only one maximum point on the graph ln( 1) ( ) 1 x f x x − = − . [7 marks] SECTION B (75 MARKS) ANSWER ALL QUESTIONS BELOW 1. Solve for p q pq and where , such that ≠ ( ) ( ) 3 3 16 . 3 p qi i i + = +− − [6 marks] 2. (a) If 7 35 , − =− x y determine the values of x y and . [6 marks] (b) Find the solution set for the inequality 4 4, 3 x x x − − ≥ + − [6 marks]
8 MODEL PSPM SET B SM015 MATHEMATICS SESSION 2023/24 3. Given ( ) 3 gx x = and ( ) 3 1 h x . x = Find f x( ) such that ( )( ) 1 = + x f gh x x . [4 marks] 4. Given ( ) ( )( ) ( ) 2 2 1 3; ln 2 . 2 x x fx e g f x e − − =+ = + (a) Find domain and range of f x( ). [3 marks] (b) Find g x( ). [5marks] (c) Hence, find ( ) 1 g x . − Sketch the graph of g x( ) and ( ) 1 g x − on the same axes. [6 marks] 5. Given that ( ) 2 1 2 1 − = + x f x x . (a) Show that the f x( ) has one vertical asymptote and state the equation. [3 marks] (b) Find the horizontal and vertical asymptotes of the function ( ) ( ) 1 g x f x = . [8 marks] (c) Sketch the graph of g x( ). [3 marks] 6. (a) If ln 1 , ( ) x ye x = + show that ( ) 2 2 2 1 . d y dy x x xe dx dx + −= [6 marks] (b) Determine whether the function 2 , 1 ( ) 1, 1 x x f x x x ≤ = + > is differentiable at x =1. [6 marks] (c) Find dy sin x for y x dx = . [3 marks]
9 MODEL PSPM SET B SM015 MATHEMATICS SESSION 2023/24 7. (a) Use the derivative to find the maximum area of a rectangle that can be inscribed in a semicircle of radius 10 cm. [5 marks] (b) A cone-shaped tank as shown below. Water flows through a hole A at rate of 6 cm3 per second. Find the rate of change in height of the water when the volume of water in the cone is 3 24 . π cm [5 marks] END OF QUESTION PAPER
10 MODEL PSPM SET B SM015 MATHEMATICS SESSION 2023/24 ANSWERS SECTION A 1. (a) 3 (b) 3 2 2. (a)DIY (b) 3 3 11 3 3 5 xy xy dy ye dx xe − = + 3. DIY SECTION B 1. p = −15 and q = 9 2. (a) 9 2 x = and 5 2 y = (b) {x x or x : 42 3 ≤− ≤ < } 3. ( ) 1 1 f x x = + 4. (a) , 3, D R f f (b) 1 ln 1 2 gx x 5. (a) VA 1 2 x = − (b) HA: y = 0 VA : x = −1 and x =1 (c) 6. (a)DIY (b) f is not differentiable at x =1 (c) sin 1 sin ln (cos ) dy x x x xx dx x = + 7. (a) 2 100cm (b) π − = − 1 1 2 dh cms dt (c) 1 2 1 x gx e
11 MODEL PSPM SET C SM015 MATHEMATICS SESSION 2023/24 SECTION A (25 MARKS) ANSWER ALL QUESTIONS BELOW 1. Evaluate the following (if exist): (a) 2 2 4 lim x 2 x x [4 marks] (b) 2 2 3 limx 5 1 x x x [5 marks] 2. Find dy dx of the following: (a) sec x y x [4 marks] (b) 2 y x cot 4 1. [5 marks] 3. Given the curve ( ) 3 2 3 18 1 2 fx x x x Use the first derivative test to determine the maximum and minimum points of f x( ). 7 marks SECTION B (75 MARKS) ANSWER ALL QUESTIONS BELOW 1. Given 1 86 . 5 5 z i z Obtain 1 z z and hence determine the values of real numbers and if 2 1 1 iz z . z z [6 marks] 2. (a) Solve the equation 2 2 log 15log 2. x x 5 marks (b)Find the interval for x so that 2 x 1 x 7 marks
12 MODEL PSPM SET C SM015 MATHEMATICS SESSION 2023/24 3. Given that fx x () 3 8 and 2 hx x x ( ) 3 12 2 such that f g x hx ( ) ( ). (a) Determine g x( ) in the from 2 ax b c ( ). 5 marks (b) State the domain and range of g x( ). 2 marks (c) Sketch g x( ). 3 marks 4. Given 1 ( ) . 1 1 1 1 f x x (a) Simplify f x( ) and evaluate 1 . 2 f 3 marks (b) The domain of f x( ) is a set of real number except three numbers. Determine the numbers. 5 marks 5. (a) A function f x is defined by 2 1 , 52 , 23 6, 3 5 a x x fx x b x x x Determine the values of a and b so that f x is continuous in the interval 5,5 5 marks (b) Given () . 1 x g x x Determine its vertical and horizontal asymptote. 9 marks
13 MODEL PSPM SET C SM015 MATHEMATICS SESSION 2023/24 6. A curve with equation 22 2 3 6 y x x y ae by − − = +− where a and b are constants , passes through the point (1,2). (c) Given 1 dy dx = at point (1,2), determine the values of a and b 9 marks (d) Evaluate 2 2 d y dx at (1,2). 6 marks 7. Diagram below shows a front view of a window. Diagram 1 The arc of the window shown in Diagram 1 is a semicircle. The perimeter of the window is 8 metre. (a) Express the front surface area of the window in terms of x and . [4 ] marks (b) Find the length (in metre) of the window when the front surface area is maximum. [6 ] marks END OF QUESTION PAPER metre 4 metre
14 MODEL PSPM SET C SM015 MATHEMATICS SESSION 2023/24 ANSWERS SECTION A 1. (a) 4 (b) 2 5 2. (a) 1 sec tan ln dy y x xx dx x = + (b) 1 2 2 2 2 4 cos (4 1) 4 1 x ec x x − + + 3. 2,23 is maximum point, (3,-39.5) is minimum p oint SECTION B 1. 1 z 2 z , 56 192 25 25 i i 2. (a) 1 8, 32 x x (b) 2, 1 1,2 3. (a) 2 gx x () 2 6 (b) ( ) ( ) , , 6, D R g x g x (c) 4. (a) 1 13 21 24 x fx f x (b) 1 0, 1, 2 x 5. (a) 3 , 6 2 a b (b) 1, 1 1, 1 vertical asymptote is x x horizontal asymptote is y y 6. (a) ∴ = =− a b 5, 5 (b) 2 2 3 4 d y dx = − 7. (a) 2 A xx =− − + ( 2 8) 16 π (b) 2.2404m
15 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 SECTION A (25 MARKS) ANSWER ALL QUESTIONS BELOW 1. Find the limit of the following, if it exists. (a) 3 2 2 8 lim → 2 − x − x x x 4 marks (b) 2 lim 4 2 x xx x →+∞ + − 5 marks 2. (a) Given ( ) 3 4 x y xe− = − . Find dy dx and 2 2 d y dx . 6 marks (b) Given that 3 2 2 3 6, x y xy − += find dy dx 3 marks 3. Given a function 2 fx x x 3 , find the nature of the stationary point(s) and the point(s) of inflection if exist. 7 marks SECTION B (75 MARKS) ANSWER ALL QUESTIONS BELOW 1. If z i 3 , show that 2 z i 2 23 and obtain 2 z in polar form. [6 marks] 2. (a) Solve − • = 1 2 3 72 x x . [3 marks] (b) Solve + = 7 2 5 x x [4 marks] (c) Solve the inequality 1 1 . 6 1 x x < − − [5 marks] 3. Given that f x ax b ( ) = + where a and b are constants, gx x ( ) = − 6 5 and fg x x ( ) = + 3 2 , find the values of a and b. [5 marks]
16 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 4. The functions f and g is defined as follows : 12 x fx e → + − , 2 : ln 1 g x x → − , x >1. Show that g x( ) is the inverse of f x( ). Sketch both graphs on the same axes. Hence, state the domain and range of f x( ) and g x( ). [13 marks] 5. Given 1 , 0 6 () , 0 4 3 4 , x e x x f x x x x C + ≤ + = <≤ − > where c is a constant. (a) Determine whether f x is continuous at x 0. 5 marks (b) Given that f x is discontinuous at x 4, determine the values of 4 marks (c) Find 1 lim . x f x 2 marks (d) Find vertical asymptote of f x . 3 marks 6. (a) Use the definition () () '( ) x a fx fa f a lim→ x a − = − to find f a '( ) for the given value of a. (i) 3 fx x x () 2 = + ; a = 2 3 marks (ii) 1 f x( ) x = ; a = 4 6 marks (b) Differentiate ( ) 2 x x e g x e = − , x ≠ ln 2 with respect to x . Hence, calculate the exact value of x if g x ′( ) = −2 6 marks
17 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 7. Car X is travelling east in a straight line at a speed of 80 km/h and car Y is travelling north in a straight line at 100 km/h as shown in the diagram below. a) Find the distance of the two cars when car X is 0.15 km and car Y is 0.08 km from P. b) Obtain an equation that describes the rate of change of the distance between the two cars. c) Hence, evaluate the rate of change of the distance between the two cars when car X is 0.15 km and car Y is 0.08 km from P, correct to 2 decimal places. Car X P Car Y 10 marks END OF QUESTION PAPER
18 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 ANSWER PART A 1. (a) 8 (b) 1 4 2. (a) ( ) 3 3 13 dy x e x dx − = − , ( ) 2 -3 2 3 14 3 d y x e x dx = − (b) 2 6 6 dy x y dx x y − − = − 3. 1,4max, 3,0min, 2,2inflection PART B 1. (a) DIY , 4 cos sin 3 3 i 2. (a) x 3 (b) x 1 (c) ( ) 7 1, 6, 2 or ∞ 3. 1 9 , 2 2 a b = = 4. DIY , 1, 1, , DR D R ff g g 5. (a) continuous at x 0. (b) C : C < o 10 1 r C> 0 (c) 1.3679 (d) VA : x 3 6. (a)(i) 14 (ii) 1 16 7. 117.65
19 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 SECTION A (25 MARKS) ANSWER ALL QUESTIONS BELOW 1. a) Find the value of m if 2 2 0 3 lim 3. x 4 8 mx x → x x + = − 3 marks b) Evaluate (i) 0 3 3 lim . x x → x − − 3 marks (ii) 2 3 5 6 lim . x 3 x x x → + − + − 3 marks 2. Given ( ) ( )( ) 2 3 1 1 1 x f x x x + = − + . By expressing f x( )in partial fractions, find f x ′( ) and hence, evaluate f ′(3) . [9 ] marks 3. The curve ( ) ( 3) px r f x x x + = − has two stationary points at (− − 1, 1) and at point H. Determine the values of p and r . State the coordinates of H. [7 ] marks SECTION B (75 MARKS) ANSWER ALL QUESTIONS BELOW 1. Find 8 6 i in the form of a bi + where a b, . ∈ [6 marks] 2. (a) Solve 3 log 1 6log 3x x − ≤ . [7 marks] (b) ) Solve the inequality x x −≤ + 12 3 .
20 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 [5 marks] 3. Consider functions of ( ) ( ) 2 fx x x =− + > 2 1, 2 and gx x x ( ) =+ > ln 1 , 0. ( ) a) Find 1 f − and 1 g , − and state the domain and range for each of the inverse function. 8 marks b) Obtain ( gf x )( ). 2 marks 4. Given that ( ) 2 () 3 2 . x g x e and f g x x = − = Find f x( ). 4 marks Hence, find 1 f x( ). − State its domain and range. 4 marks 5. a) A function f is defined by 2 1 () . 4 x f x x + = − (i) Find the vertical asymptotes. 3 marks (ii) Determine lim x f x and lim . x f x Hence state the horizontal asymptote(s). 4 marks b) Function f is given by 2 1, 2 ( ) 2, 2 ax x f x xa x − ≤− = + >− F is continuous for all values of x. (i) Find the value of the constant, a. [3 ] marks (ii) Find the 2 ( ) ( 2) limx 2 fx f →− x − − + if it exists. 4 marks
21 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 6. (a) The parametric equations of a curve is given by 2 1 (2 1) , . t t xe ye + −− = = (i) Find dy dx and 2 2 d y dx at point ( ) 3 1 e e, − 6 marks (ii) Given 2 z x xy = − .Express z in terms of t and find . dz dt Hence, deduce the set value of t such that dz dt is positive. [5 marks] (b) Given tan 4 y x π = + , show that 2 2 2 0 d y dy y dx dx − = 4 marks 7. Oil is poured into a hemisphere bowl of radius 5 cm at the rate of 3 4 cm /s π . The volume, 3 V cm , of the oil in the bowl when its depth is h cm is given by 2 3 1 5 3 V hh π = − . At the instant when the depth of the oil in the bowl is 2 cm , find a) the rate of change of h , [3 ] marks b) the rate of change of the radius of the oil’s surface area. [7 ] marks END OF QUESTION PAPER
22 MODEL PSPM SET E SM015 MATHEMATICS SESSION 2023/24 ANSWER PART A 1. (a) m 12 (b) 3 6 (c) 1 2. ( ) ( ) ( ) 2 2 2 2 2 2 22 1 1 x x f x x x − −− ′ = + − + , ( ) 2 3 5 f ′ = − 3. 3 25 1, 5, , 5 9 rpH = = = − PART B 1. z iz i = + =− − 1 3, 1 3 2. (a) { 3 1 : or 0 log 3 9 x x x ≤ <≤ (b) ( ] 5 , 7 ,3 3 − −∞ − ∪ 3. a) ( ) 1 fx x 1 2 − =± − + , 1 1 (1, 2, ) ( ) f f D R − − =∞ = ∞ ( ) 1 1 x gxe − = − , 1 1 ( , 0, ) ( ) g g D R − − = −∞ ∞ = ∞ b) ( )( ) ( ) 2 gf x x x = −+ ln 4 6 4. 1 1 1 ( ) ln( 3), ( ) 3, , ( 3, ) x f f fx x f x e D R − − − = + = − =ℜ = − ∞ 5. a) (i)VA x x 2 and 2 (ii)HA y y 1 and 1 b) (i) a 1 (ii) the limit does not exist 6. (a)(i) 4 t 1 dy e dx − = = − , 2 7 2 1 2 t d y e dx − = = (ii) {tt R : ∈ } (b)DIY 7. a) 1 cm/s 4 dh dt = b) 3 cm/s 16