2021 JC1 H2 Math Promo-13s

2021 H2 Math JC1 Promo Exam Paper Junior College Type 1 ANGLO-CHINESE JUNIOR COLLEGE EOY 2 ANDERSON SERANGOON JUNIOR COLLEGE EOY 3 CATHOLIC JUNIOR COLLEGE EOY 4 DUNMAN HIGH SCHOOL EOY 5 EUNOIA JUNIOR COLLEGE EOY 6 HWA CHONG INSTITUTION EOY 7 JURONG PIONEER JUNIOR COLLEGE EOY 8 NANYANG JUNIOR COLLEGE EOY 9 RAFFLES INSTITUTION EOY 10 ST ANDREW’S JUNIOR COLLEGE EOY 11 TEMASEK JUNIOR COLLEGE EOY 12 TAMPINES MERIDIAN JUNIOR COLLEGE EOY 13 VICTORIA JUNIOR COLLEGE EOY NIOR CCCCCOOOOOLLLLLLLLLLEEEEEGGGGGE E LLEGE EOY SCHOOOOOOOOOOOOOOOOOOOOOOOOOOOLLLLL EOYYYYYYYYYY UNIORRRRRRRR CCCCCCCCCCCOOOOOOOOOLLEGGGGGGGGGGGEEEEEEEEEEEEEEEEEE EEEEEEOOY A CHONNNNNNNNNNNNNNNNNNGGGGGGGGGGGGGGGG IIIIIIIIIIINNNNNNNNNNNNNNNSSSSSSSSSSSSSSSSSSTTTTTTTTTTTTTTTTTTIIIIIIIIIIIITTTTTTTTTTTTTTTUUUUUUUUUUUUUUUUUUUTTTTTTTTTTTTTTTTIIIIIIIIIIIIIIOOOOOOOOOOOOOOOOOOOOOONNNNNNNNNNNNNNNNNNNNNN JUROOOONNNNNNNNNNNNNNGGGGGGGGGGGGGGGGGG PPPPPPPPPPPPPPPPIIIIIIIIIIOOOOOOOOOOOOOOOOONNNNNNNNNNNNNNNNNEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEERRRRRRRRRRRRRRRRRRRRRRR JJJJJJJJJJJJJJJJUNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIOOOOOOOOOOOOOOORRRRRRRRRRRRRRRRRRR CCCCCCCCCCCCCCCCCOLLLLLLLLLLEEEEEGGGGGGGGEEEEEE 8 NAAAAAAAAANNNNNNNNNNNNNNNYYYYYYYYYYYAAAAAAAAAAAAANNNNNNNNNNNNNNNNGGGGGGGGGGGG JUNNNNNNNNNNNNNNNNNNNNIIIIIIIIIOOOOOOOOOOOOOOORRRRRRRRRRRRRRRRRRR CCCCCCCCCCCCCCCCOOOOOOOOOOOOOOOOOOOLLLLLLLLEEEEEGGGGGEEEEE 9999999999999999 RRRRRRRRRRRRRRRRRRAAAAAAAAAAAAAAAAAFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLEEEEESSSSSSSSS IIIIIIIIIIINNNNNNNNNNNNNNNNSSSSSSSSSSSSSSSSTTTTTTTTTTTTTIIIIIIIIITTTTTTTTTTTTTTTTUUTTIIIIIIOOOOOONNNNN 10 SSSSSSSSSSSSSSSTTTTTTTTTTT AAAAAAAAAAAAAAAANNNNNNNNNNNNNNNNNNNDDDDDDDDDDDDDDDDDDRRRRRRRRRRRRRRRRRRRREEEEEEEEEEEEEEEEEEWWWWWWWWW’SSSSSS JJJJJUUUUUNIO TEMMMMMMMMMMMMMMMMMMASSSSSSEEEEEKKKKKK JJJJJJUN MPIN www.KiasuExamPaper.com 1

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2021 ACJC H2 Maths Promo Paper Duration: 3 hrs Marks: 100 Attempt all questions. 1 State a sequence of transformations that will transform the cur ve with equation 2sin(2 )cos( )yxx D on to the curve with equation 2sin(4 3 )cos(2 )yx x DD , where D is a positive constant. [3] 2 Solve algebraically the inequality 2 3 12 x xx ! . [3] Hence solve the inequality 2 2 3 112 xx xx ! . [2] 3 A curve C has equation 22 22 41 2100 xy xx y , x \ , 8.xz Show that 2d2 d2 1 6 yx y xx y y . [2] Hence, prove that curve C does not have any stationary point. [3] 4 The diagram shows the curve f( )yx . There are two vertical asymptotes with equations 2x and 2x respectively. The curve crosses the x-axis at the point A and has a maximum turning point at B where it crosses the y-axis. The curve also has a minimum turning point at C. The coordinates of A, B and C are (, 0 )a , (0 , 1 0) and (,)pq respectively, where a, p and q are constants. Sketch the following curves and state the equations of the asym ptotes, the coordinates of the turning points and of points where the curve crosses the axes, if any. Leave your answers in terms of a, p or q where necessary. (i) 1 f( )y x , and [3] (ii) f(2 | | )yx . [3] www.KiasuExamPaper.com 3

5 Referred to the origin O, points A and B have position vectors a and b respectively. The modulus of a is 2 and b is a unit vector. The angle between a and b is 60D . Point C lies on AB, between A and B, such that ACk C B , where 0 < k < 1. (i) Express OC JJJG in terms of a and b. [1] (ii) Show that the length of projection of on OC OA JJJGJ J G is given by

4 21 k k

. [3] (iii) Find, in terms of k, the area of triangle OAC. [3] 6 The Cartesian equation of line 1L is 22 3xyz abc , where ,,abc are constants. The line 2L is parallel to the vector 43ij . The line 3L passes through the origin and the point with position vector j k . (i) Given that 1L is perpendicular to 2L , form an equation relating and .ab [1] (ii) Given that 1L intersects 3L , show that 522 0 .ab c [3] (iii) Hence express and in terms of .ab c [1] (iv) Find the acute angle between 1L and 3L . [2] 7 The functions f and g are defined by 2 1f : , 1 x x 6 , 2 1xx d , 2g: ( 2) ,x xk 6 , 0xxt where k is a constant. (i) Sketch on the same diagram the graphs of (a) f yx (b) 1fyx (c) 1ffyx stating the equations of any asymptotes and the coordinates of any endpoints. [3] (ii) Find 1f and state the domain of 1f . [3] (iii) Show that the composite function gf exists and find its range. [2] www.KiasuExamPaper.com 4

8 The figure below shows a cross-section OBCE of a car headlight whose reflective surface is modelled in suitable units by the curve with parametric equations (s i n )xa TT , (1 cos )ya T for 02 TSd d , where a is a positive constant. (i) Find in terms of a (a) the length of OE, [2] (b) the maximum height of the curve OBCE. [1] (ii) Show that d cotd2 y x T . [3] Point B lies on the curve and has parameterE . TS is tangential to the curve at B and BC is parallel to the x-axis. Given that 6TBC S‘ , (iii) show that 2 3 SE . [2] (iv) Show that the equation of normal to the curve at the point B is 2 2ky k x a S , w h e r e k is an exact constant to be determined. [3] 9 (a) Given that 2 1 1 12 16 n r rn n n ¦ , find 1 2 7 2 n r r rr

¦ in terms of n. [4] ( b ) ( i ) Use the method of differences to show that 2 2 3 ,4 1 11 n r AA rn n ¦ where A is a constant to be determined. [3] (ii) Explain why the series 2 2 1 1r r f ¦ converges, and write down its value. [2] (iii) Hence deduce that 222 222 ...234 is less than 3 2 . [2] 10 Referred to the origin O, the points A, B and C have position vectors 42ij , 2D ij k and 7 E ij k respectively, where D and E are constants. (i) Given that A, B and C are collinear, show that D =5, and find the value of E . [3] The plane S contains the line L, which has equation 23 ( 2 )P rij i j k . The plane S is also parallel to the line that passes through the points A and B. (ii) Find the shortest distance from point A to the line L. [2] (iii) Show that the cartesian equation of the plane S is 5xyz . [2] (iv) Find the position vector of the foot of the perpendicular from point A to the plane S . [3] (v) Hence find the reflection of the line that passes through point s A and B about the planeS . [2] B C O E T S x y www.KiasuExamPaper.com 5

11 The figure below shows a container with an open top. The uniform cross section ABCD of the container is a trapezium with AB = BC = CD = 10 cm. AB and CD are each inclined to the line BC at an acute angle of T radians. The length of the container is 50 cm and the container is placed on a horizontal table. (i) Show that the volume V of the container is given by 35000 sin (1 cos )cmV TT . [2] Hence using differentiation, find the exact maximum value of V, proving that it is a maximum. [5] (ii) For the remaining part of the question, T is fixed at 4 S . Water fills the container at a rate of 31100cm s . At time t seconds, the depth of the water is h cm. The surface of the water is a rectangle PQRS. When 3c m ,h find the rate of change of (a) the depth of the water, h, [3] (b) the surface area of the water PQRS. [2] 12 Mrs Tan plans to start a business which requires a start-up ca pital of $700,000. She decided to first save $200,000 by depositing money every month into a savings plan. For the remaining $500,000, she intends to take a loan from a finance company. She deposited $3000 into the savings plan in the first month and on the first day of each subsequent month, she deposited $100 more than the previous mon th. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches $200