NJC 2026 H3 Math Prelim

* NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 Higher 3 MATHEMATICS 9820/01 Preliminary Examination 18 September 2026 Paper 1 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for improper presentation. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages.

2 1 (a) Find 3 5 .3 1 d x x x +   [3] (b) Find ( ) 2 3 sin2 d . xxx x+ [3] (c) Find the exact value of a if 22 2 ee d d . aa xx aa xxxx = [3] 2 (a) By considering the Maclaurin expansion of 2 1 1 x− , show that the Maclaurin expansion of 1sin x− , up to 7x , is 3 5 7 1 3 1 3 5 .2 3 2 4 5 2 4 71 6x x xx        + + +             [4] Let na be the coefficient of 21nx + in the Maclaurin series of 1sin x− . (b) (i) Show that 2 1 (2 1) .2( 1)(2 3)nna na n n+ += ++ [2] (ii) Hence, for 10 2x , prove that the successive terms of the series satisfy 23 1 21 1 .4nn nnaxax ++ +  [2] (c) Suppose that 19π3 8 640 R= + + + . Show that 5 1792R without using the value of π . [4]

3 3 Consider the differential equation 2( 1)y x p p= + + , where d d yp x= and p is a function of x. (a) Show that, for a non-constant p, d 2dx xpp+ =− . [3] (b) By considering d ( e )d pxp , show that ,2 2 e px p C −=− + + where C is an arbitrary constant. [3] (c) Show that the corresponding non -constant solutions to the differential equation may be represented parametrically by 22 2 e , 2 ( 1)e .ppx p C y p C p −−=− + + =− + + + [1] (d) Determine whether the differential equation can have any solutions for which p is a constant. [2] (e) There are two solutions curves passing through the poin t ( )1, 3 . Find the exact coordinates of the stationary point for each curve. [5] 4 You may use the result that ( )( )2 1 1 2 1 6 n r n n nr = ++= in answering this question. Let Nk denote the set of positive integers from 1 to k, i.e.  N 1, 2, 3, ..., 1,k kk=− . Three numbers a, b and c are randomly selected from Nk with replacement. Let kp denote the probability that ( ),,abc can form the 3 sides of a triangle. (a) Show that 2 5 8p = . [2] (b) By considering the complement event or otherwise, find kp in terms of k. [5] It is given instead that a, b and c are randomly selected from Nk without replacement for 3k  . Let kQ denote the number of ordered triplets ( ),,abc that can form the 3 sides of a triangle. For any integer 1m , (c) show that ( )2 2 2 6 2 1mmQ Q m m+ − = − , and [4] (d) prove by induction that ( )( )2 1 4 5mQ m m m= − − . [4]

4 5 (a) Given that a, b and c are constants such that 0,abc   and that x, y and z are non- negative variables, show that ( ).a yx by cz a x z + +++ [1] In the acute-angled triangle ABC, a, b and c are the lengths of sides BC, CA and AB, respectively, with a  b  c. P is a point inside, or on the sides of, the triangle, and x, y and z are the perpendicular distances from P to BC, CA and AB, respectively. The area of the triangle is S. (b) (i) Find S in terms of a, b, c, x, y and z. [1] (ii) Find both the minimum value of the sum of the perpendicular distances from P to the three sides of the triangle and the values of x, y and z which give this minimum sum, expressing your answers in terms of some or all of a, b, c and S. [2] (c) (i) Show that, for all real a, b, c, x, y and z, ( )( ) 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) .a b c x y z bx ay cy bz az cx ax by cz+ + + + = − + − + − + + + [2] (ii) Find both the minimum value of the sum of the squares of the perpendicular distances from P to the three sides of the triangle and the values of x, y and z which give this minimum sum, expressing your answers in terms of some or all of a, b, c and S. [5] (d) Find both the maximum value of the sum of the squares of the perpendicular distances from P to the three sides of the triangle and the values of x, y and z which give this maximum sum, expressing your answers in terms of sum or all of a, b, c and S. [3]

5 Use the information in the mathematical text to answer Question 6. You should read the whole mathematical text before you start answering the question. Integer Partition While Leibniz appears to be the first person to consider the partitioning of integers into sums, Euler was the first person to make truly deep discoveries. J.J. Sylvester was the next researcher to make major contributions. The twentieth century saw an explosion of research with monumental contributions from Rogers, Hardy, MacMahon, Ramanujan and Rademacher. In number theory and combinatorics, a partition (also called an integer partition ) of a non - negative integer n, is a way of writing n as a sum of positive integers. For example, we may write 4 as sums in the following way: 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1. The numbers that occur in these partitions are called parts. We use ( )pn to denote the number of ways in which n can be expressed as a sum of positive integers, with the order not counting. In the example above, ( )45p = . Using this definition, we have ( )11p = , ( )22p = , ( )33p = . We take ( )00p = . A common diagrammatic method to represent partitions , in descending order, of an integer is its Young diagrams, in which rows of squares are used to represent the partitions. Using English conventions, rows in a Young diagram are aligned in the upper-left corner. For example, the partitions of 4 have the following Young diagrams: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1 Conjugate and self-conjugate partitions If we flip the Young diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14: 6 + 4 + 3 + 1 = 4 + 3 + 3 + 2 + 1 + 1 Such partitions are said to be conjugate of one another. In the example of the number 4, partitions 4 and 1 + 1 + 1 + 1 are a conjugate pair, and partitions 3 + 1 and 2 + 1 + 1 are a conjugate pair, Partition 2 + 2 has itself as conjugate, such partitions are said to be self-conjugate. Odd parts and distinct parts Among the 5 partitions of the number 4, there are 2 that contain only odd (number) parts: 3 + 1 and 1 + 1 + 1 + 1.

6 There are also 2 partitions with only distinct parts, in which no number occurs more than once: 4 and 3 + 1. Restricted part size or number of parts We use ( )kpn to denote the number of partitions of n into at most k parts. For the number 4, • ( )1 41p = (partition 4), • ( )2 43p = (adding partitions 3 + 1 and 2 + 2), • ( )3 44p = (adding partition 2 + 1 + 1), • ( )4 45p = (adding partition 1 +