NYJC 2026 Prelim P1 QP

This document consists of 6 printed pages and 2 blank pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2026 [Turn Over 1 A curve with equation 1cy ax b x= ++ − , where a, b and c are non-zero constants, has a turning point at ( )3, 7 . Given that the area bounded by the curve for y > 0 , the x -axis, the lines 2x= and 3x= is 4.5 ln16+ units2, find the values of a, b and c. [5] 2 (a) Without using a calculator, solve the inequality 2 51 39 x xx +≤ +− . [4] (b) Hence, solve exactly the inequality 2 5 e1 e 9e 3 x xx +≤ −+ . [2] 3 Referred to the origin O, the points A, B and C have non-zero position vectors a, b and c respectively. It is given that ×=ab c and ,k×=bc a where k is a positive constant. (a) Deduce that a, b and c are mutually perpendicular and find k in terms of .b [3] (b) It is now given that 1=a and 3.=b Using a suitable scalar product, find the exact value of .−bc [3] (c) State the geometrical interpretation of .×bc [1] 4 Do not use a calculator in answering this question. It is given that 1 – 3i is a complex root of the equation ( )32 12 0z pz p z q+ + + += where p and q are real numbers. (a) State another complex root of the equation in the form of a + ib, where a and b are real numbers and b ≠ 0. [1] (b) Find the values of p and q, and find the third root. [4] (c) The point A represents the complex number 1 – 3i on an Argand diagram with respect to the origin O. The point C represents the complex number x + iy, where x and y are real numbers and x > 0. Find the values of x and y such that OA = AC and angle OAC is 90o. [2] 5 The sum, nS , of the first n terms of a sequence 123,,,uu u … is given by 11 2 634nS nn= +− ++ . (a) Give a reason why the series ru∑ converges and write down the value of the sum to infinity. [2] (b) Show that ( )( )( )234n nu nnn= +++ . [2] (c) It is given that

2 NYJC 2026 JC2 Preliminary Examination 9758/01 11 59 k rr rk r uu ∞ = += =∑∑ . Find the value of k. [3] 6 A curve has equation ln(5 ) .5 xy x−= − (a) By differentiation, find the equation of the normal to the curve at the point when the curve cuts the x-axis. [4] (b) Find the exact area of the region bounded by the normal, the curve and the y -axis. [4] 7 It is given that ( )2cos e 1 xy −= − . (a) Show that 2 4 2 dd edd xyy k kyxx −= − , where k is a constant to be found. [3] (b) By differentiation of the result in part (a) , find the first three non- zero terms of the Maclaurin expansion of ( )2cos e 1 x− − . [3] (c) If your expansion in part (b) is used to estimate ( ) 0.7 2 0 cos e 1 dx x− −∫ , explain how you can improve the accuracy of your approximation. [1] (d) The first two non-zero terms of the Maclaurin expansion of ( )2cos e 1 x− − are equal to the first two non-zero terms of the series expansion of 3 2a bx− , where a and b are constants. Using standard series from the List of Formulae (MF27), find the values of a and b. [2] 8 A curve C has parametric equations sec 1x θ= + , tan 2,y θ= − where .22ππ θ−<< (a) Find the equations of the tangents to C at x = 3. Hence find the acute angle between the two tangents. [5] (b) Find the cartesian equation of C. You need not state any restrictions on the values of x. [1] (c) Using your answer to part (b), find the volume generated when the region enclosed by C and the line 3x= is rotated 2π radians about the x-axis. [3]

3 NYJC 2026 JC2 Preliminary Examination 9758/01 [Turn Over 9 A pattern on a T-shirt, shown in Fig. 1, consists of a sequence of similar sectors. The first sector AOB of a circle has centre O, and radii OA and OB. (a) The angle AOB is 1.2 radians. Given that the maximum area of AOB is 57 cm2, f ind the range of values of OA. [2] [The area of a sector of a circle with radius r is 212 r θ , where θ is the angle, in radians, subtended by the arc at the centre.] (b) The area of AOB is now fixed at 60 cm2, and the area of each sector is 35 of the area of the preceding one. (i) Find the area of the 5th sector in the pattern. [2] (ii) Given that the total area of n sectors printed cannot exceed 148.48 cm 2, find the maximum number of sectors that can be printed. [3] (c) The cost of printing the first sector AOB is $C and the cost of printing each sector is $0.15 less than the preceding one. Find the maximum value of C if the total cost of printing the first 5 sectors does not exceed $4.99. [2] Fig. 1 A B O

4 NYJC 2026 JC2 Preliminary Examination 9758/01 10 (a) Use the substitution sinx aθ= to find ( ) 1 2 2 0 d 1 a x x ax− ⌠ ⌡ in terms of a, where a is a positive constant. [5] (b) Hence, or otherwise, find the exact value of 1 2 1 0 cos 2 dx xx −∫ . [3] (c) Given that 1f ( ) cos 2 ,xx x −= find the exact area of the region enclosed by f( )yx= and f( ).yx=− [2] 11 The lines 1l and 2l have equations 21 12 13 λ    = +−  − r and 32 41 pq µ    = +   r respectively, where p and q are constants, and λ and µ are parameters. (a) Suppose that 1l intersects 2.l (i) Find the coordinates of the point of intersection and state a linear relationship between p and q. [3] (ii) Find the acute angle between 1l and 2l when 2.q= [2] (b) By using vector product, show that a vector perpendicular to both 1l and 2l is 23 6 q q m +  − , where m is a constant to be determined. [2] (c) The plane π contains 2l and is parallel to 1.l Find the shortest distance between 1l and π in terms of p and q. [4]

5 NYJC 2026 JC2 Preliminary Examination 9758/01 [Turn Over 12 A group of scientists uses Carbon-11 as a radioactive tracer to study carbon movement in plants. After the tracer has been introduced into the plant, the amount of Carbon-11, C, decreases through radioactive decay at a rate equal to 2.5% of the amount present at time t minutes. (a) It is given that the initial amount of Carbon-11 is 123 M egabecquerel ( MBq). Write down a differential equation for the decay and solve the differential equation, giving your answer in the form f( )Ct= , where f(t) is a function to be determined. [4] (b) Sketch the graph of C against t. Explain the significance of the asymptote. [2] The group of scientists subsequently extended the decay model by incorporating compartmental modelling. In this improved model, the plant is divided into several compartments, such as leaves, stems and roots. After introducing Carbon- 11 as a radioactive tracer, the rate of change of Carbon-11 is now determined by the rate of inflowing Carbon-11, outflowing Carbon-11 and radioactive decay. (c) It is g iven that the rates of inflowing and outflowing Carbon-11 are e tt − and 7 MBq per minute respectively. Show that the differential equation corresponding to the compartmental model can be expressed in the form ( )d g,d tt CC λ+= where λ is a constant and g(t) is a function, both of which are to be determined. [2] (d) By multiplying both sides of the differential equation in part (c) by e tλ and rewriting ( )ed d edd asttCC Ctt λλλ +  , find the general solution to the differential equation in the form h( )Ct= , where h(t) is a function to be determined. [4]

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