2025 DSS PRELIMS P2 Q7

2025 DSS PRELIMS P2 Q7

Secondary 4
10 marks
  1. The diagram shows part of a number grid. A cross outlining five numbers, as shown, can be placed anywhere on the grid.
    1. If \(n\) represents the number in the top left corner of the cross, write down an expression, in terms of \(n\), for the number in the bottom right corner of the cross.[1]
    2. Show that the difference between the products of the numbers in the opposite corners of the cross is always 36.[2]
    3. Show that the sum of the five numbers in the cross cannot be 1715.[3]
    \(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)
    \(10\)\(11\)\(12\)\(13\)\(14\)\(15\)\(16\)\(17\)\(18\)
    \(19\)\(20\)\(21\)\(22\)\(23\)\(24\)\(25\)\(26\)\(27\)
    \(28\)\(29\)\(30\)\(31\)\(32\)\(33\)\(34\)\(35\)\(36\)
    \(37\)\(38\)\(39\)\(40\)\(41\)\(42\)\(43\)\(44\)\(45\)
    \(46\)\(47\)\(48\)\(49\)\(50\)\(51\)\(52\)\(53\)\(54\)
  2. The \(n\)th term of a sequence is given by \(T_n=2n^2-n+3\).
    1. Find the value of \(T_{10}\).[1]
    2. The difference, \(D\), between two consecutive terms of the sequence is \(T_{n+1}-T_n\). Show that \(D=kn+1\), where \(k\) is an integer to be found.[2]
    3. Using the expression from part (ii), explain, without doing any further calculation, how the difference between two consecutive terms of the sequence changes as n increases by 1 each time.[1]

Solution:

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Answer:(a)(i) \(n+20\) (ii) \(36\) (iii) impossible (b)(i) \(193\) (ii) \(D=4n+1,k=4\) (iii) increases by 4

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