Number Theory
Given that x ≡ 7 (mod 27), find x mod 3
1
If \(\tan x = m\), find \(\sin x\) in terms of \(m\).
Answer: \[ \frac{m}{\sqrt{m^2 + 1}} \]
Find the derivative of ln(1+x2).
(2x)/(1+x2)
The value of n!/n is 24. What is the value of n?
5
Let \( f(x) = \frac{8x^2 + 9x + 8}{(1 - x)(2x + 3)^2} \). Express this in partial fractions.
\[ \frac{1}{1-x} + \frac{-4x-1}{(2x+3)^2} \]
Given that x and y are relatively prime integers, find the LCM and HCF of x and y in terms of x and y.
LCM: xy, HCF: 1
The depth, \(D\) metres, of the water at the end of a jetty in the afternoon can be modelled by
\[ D = 5.5 + A \sin\!\big(30(t - K)\big), \]
where \(t\) hours is the number of hours after midday, and \(A\) and \(K\) are constants.
Yesterday, the low tide was at 3 pm, and the depth of water at low tide was \(3.5\) m.
\[ \text{Find the values of } A \text{ and } K. \]
\(A = \pm 2\)
\(K = 6\)
Find the derivative of x2025e2025x.
2025x2024e2025x+2025x2025e2025x
If \( \binom{2025}{x} = 2025 \) and \( x \ne 1 \), find the value of \( x \).
2024
Simplify: √4 × ∛2
(2)4/3
Find 264 mod 7.
2
Simplify the expression below in terms of tan(θ):
\[ \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)} \]
<p>\[
\tan^2(\theta)
\]</p>
Find the following limit:
\[ \lim_{n \to \infty} \frac{2n^2 + 7n + 16}{3n^2 + 16n + 4} \]
2/3
In a round table, there are 5 equally spaced seats. In how many ways can 5 members be seated?
24
If x2 + y2 = 25 and x + y = 7, find xy.
The LCM of two integers \(x\) and \(y\) is 180 and the HCF is 12. It is given that \(x \ne 12\) and \(y \ne 12\) and \(y > x\). Find \(x\) and \(y\).
(x,y)=(36,60)
Solve for \(0 \le x \le 180\), the equation:
\[ \sin(x + 10) = \frac{\sqrt{3}}{2} \]
50 and 110
Find the derivative:
\[ \frac{d}{dx}\left(e^{e^{e^{x}}}\right) \]
\[ \left(e^{e^{e^{x}}}\right)\left(e^{e^{x}}\right)\left(e^{x}\right) \]
A game is won when a player gets 3 points ahead. Players A and B are playing, and currently A is 1 point ahead. Each player has equal probability of winning each point. What is the probability that A wins the game?
2/3
The cubic equation \(x^3 + 2x^2 + 3x + c = 0\) has roots \(\alpha, \beta, \gamma\).
\[ \text{Find } \alpha^3 + \beta^3 + \gamma^3. \]
1
Write 3.4 as a fraction in simplest form.
31/9
There is a real root, \(0^\circ \ll a \ll 180^\circ\), for:
\[ (3\sin(a)) x^2 - (4\cos(a)) x + 2 = 0 \]
Find the range of \(\sin a\).
\(\sin a \le \frac{1}{2}\)
Q1. Evaluate the following integral in terms of π (Integration – 500 pts):
\[ \int_{-45}^{45} \sqrt{2025 - x^2}\, dx \]
2025pi/2
A random pizza is made by independently flipping a fair coin for pepperoni, sausage, mushrooms, and onions. The probability that two random pizzas have at least one topping in common is m/n . Find m + n.
431
If x2 + 3xy + y2 = 60, where x and y are real, then determine the maximum possible value of xy.
60