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Can you please help with Question 5 on the 2008 MAT?

Certainly. The nth student (Sn) changes the state of every nth locker - i.e. the multiples of n. If changed an odd number of times, the locker is open -- if even then closed. i. how many closed after 3rd stu...
CA
Answered by Carlo A. • MAT tutor
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Circle the correct letter: The equation x^3 - 30x^2 + 108x - 104 = 0 has a) No real roots; b) Exactly one real root; c) Three distinct real roots; d) A repeated root.

Firstly, the polynomial is a cubic, and so we know what its graph looks like, hence it must have at least one real root, and a) is false. Computing a root would be time consuming, so instead we adopt a diffe...
VC
Answered by Valerio C. • MAT tutor
4095 Views

Show that the inequality x^4 < 8x^2 + 9 is satisfied for when -3 < x < 3 .

(x^2 - 9)(x^2 + 1) &lt; 0 solving the equation to get solutions to the equality (x^2 - 9)(x^2 + 1) = 0 : x = +/- 3 or x = +/- 1 now consider points either side of these x-intercepts... for x&gt;3: equality i...
HT
Answered by Hakkihan T. • MAT tutor
1500 Views

The sequence xn is given by the formula x_n = n^3 − 9n^2 + 631. What is the largest value of n for which x_n > x_(n+1)?

We know that x_n &gt; x_(n+1) is true if and only if x_n - x_(n+1) &gt; 0 is true.So x_n - x_(n+1) = (n^3 − 9n^2 + 631) − ((n + 1)^3 − 9(n + 1)^2 + 631) = (n^3 − n^3 − 3n^2 − 3n − 1) − 9(n^2 − n^2 − 2n − 1) ...
TT
Answered by Tadas T. • MAT tutor
7575 Views

How many distinct real roots does the equation x^3 − 30x^2 + 108x − 104 = 0 have?

We can see that 104 = 2^3 * 13 = 2 2 26, 30 = 2 + 2 + 26, and 108 = 2 2 + 2 26 + 2*26, so the coefficients agree with the Vieta's formulas, so the roots of the equation above are 2, 2, 26. In conclusion, it ...
AI
Answered by Andreea I. • MAT tutor
10306 Views