Photo AI
Question 11 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2014 - Paper 1
Question 11
Question 11 (15 marks) Use a SEPARATE writing booklet. (a) Solve $$ (x + rac{2}{x})^2 - 6igg(x + rac{2}{x}igg) + 9 = 0 $$ (b) The probability that it rains on... show full transcript
Worked Solution & Example Answer:Question 11 (15 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 11 - 2014 - Paper 1
Step 1
Solve $(x + \frac{2}{x})^2 - 6\bigg(x + \frac{2}{x}\bigg) + 9 = 0$
Answer
Let ( y = x + \frac{2}{x} ). Then, the equation simplifies to ( y^2 - 6y + 9 = 0 ). Factoring gives ( (y - 3)^2 = 0 ), thus ( y = 3 ).
Substituting back, we have: [ x + \frac{2}{x} = 3 ] This can be rearranged to: [ x^2 - 3x + 2 = 0 ] Factoring yields ( (x - 1)(x - 2) = 0 ), giving solutions ( x = 1 ) and ( x = 2 ).
Step 2
Write an expression for the probability that it rains on fewer than 3 days in November.
Answer
The number of days it rains, ( X ), follows a binomial distribution ( X \sim Binomial(n = 30, p = 0.1) ). The expression for the probability of it raining on fewer than 3 days is: [ P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) ] Using the binomial formula: [ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]
Step 3
Sketch the graph $y = 6 \tan^{-1} x$, clearly indicating the range.
Answer
The function ( y = 6 \tan^{-1} x ) has a horizontal asymptote. As ( x \to -\infty ), ( y \to -6\frac{\pi}{2} = -6\pi ) and as ( x \to \infty ), ( y \to 6\frac{\pi}{2} = 6\pi ). Thus, the range is ( (-6\pi, 6\pi) ). Sketch must indicate these asymptotic behaviors.
Step 4
Evaluate $\int \frac{x}{2\sqrt{x} - 1} dx$ using the substitution $x = u^2 + 1$.
Answer
Using the substitution ( x = u^2 + 1 ), we find ( dx = 2u , du ). Substituting into the integral: [ \int \frac{u^2 + 1}{2u - 1} (2u , du) = 2 \int \frac{u(u^2 + 1)}{2u - 1} , du ] Proceed to simplify and evaluate this integral using appropriate techniques.
Step 5
Step 6