The parametric equations of a line are given below. $x = 1 + 3t$ $y = 4t$ Find the Cartesian equation of this line in the form $y = mx + c$. In how many different ways can all the letters of the word CONDOBOLIN be arranged in a line? Consider the polynomial $P(x) = x^3 + ax^2 + bx - 12$, where $a$ and $b$ are real numbers. It is given that $x + 1$ is a factor of $P(x)$ and that, when $P(x)$ is divided by $x - 2$, the remainder is -18. Find $a$ and $b$.

Photo AI

The parametric equations of a line are given below - HSC - SSCE Mathematics Extension 1 - Question 11 - 2023 - Paper 1

Question 11

The parametric equations of a line are given below. $x = 1 + 3t$ $y = 4t$ Find the Cartesian equation of this line in the form $y = mx + c$. In how many dif... show full transcript

Worked Solution & Example Answer:The parametric equations of a line are given below - HSC - SSCE Mathematics Extension 1 - Question 11 - 2023 - Paper 1

Step 1

The parametric equations of a line are given below. Find the Cartesian equation of this line in the form y = mx + c.

96%

114 rated

Answer

To eliminate the parameter $t$ , we can express $t$ in terms of $x$ using the first equation:

From $x = 1 + 3t$ , we have: $t = \frac{x - 1}{3}$
Now, substitute $t$ in the second equation for $y$ : $y = 4t = 4 \left( \frac{x - 1}{3} \right) = \frac{4(x - 1)}{3} = \frac{4x - 4}{3}$
Rearranging gives: $y = \frac{4}{3}x - \frac{4}{3}$

Thus, the Cartesian equation is $y = \frac{4}{3}x - \frac{4}{3}$ .

Step 2

In how many different ways can all the letters of the word CONDOBOLIN be arranged in a line?

99%

104 rated

Answer

The word CONDOBOLIN has 12 letters in total with some repetitions:

C: 1
O: 2
N: 2
D: 1
B: 1
L: 1
I: 1

Using the formula for permutations of multiset: $ext{Total arrangements} = \frac{n!}{n_1! \cdot n_2! \cdots n_k!}$

Substituting the values: $ext{Total arrangements} = \frac{12!}{2! \cdot 2! \cdot 1! \cdot 1! \cdot 1! \cdot 1! \cdot 1!}$

Calculating gives: $= \frac{479001600}{4} = 119750400$

Therefore, the total number of arrangements is 119750400.

Step 3

Consider the polynomial P(x) = x^3 + ax^2 + bx - 12, where a and b are real numbers. Find a and b.

96%

101 rated

Answer

Using the factor theorem, since $x + 1$ is a factor:

Evaluate $P(-1)$ : $P(-1) = (-1)^3 + a(-1)^2 + b(-1) - 12 = -1 + a - b - 12 = 0$ $a - b - 13 = 0 \\ \Rightarrow a - b = 13 \tag{1}$

Using the remainder theorem, since the remainder when divided by $x - 2$ is -18: 2. Evaluate $P(2)$ : $P(2) = (2)^3 + a(2)^2 + b(2) - 12 = 8 + 4a + 2b - 12 = -18$ Simplifying gives: $4a + 2b - 4 = -18 \\ \Rightarrow 4a + 2b = -14 \\ \Rightarrow 2a + b = -7 \tag{2}$

Now, solve equations (1) and (2) together:

From (1): ( b = a - 13 \ ext{Substituting into (2):}) $3a - 13 = -7 \\ \Rightarrow 3a = 6 \\ \Rightarrow a = 2$$$
Substitute back to find $b$ : $b = 2 - 13 = -11$

Thus, $a = 2$ and $b = -11$ .

The parametric equations of a line are given below - HSC - SSCE Mathematics Extension 1 - Question 11 - 2023 - Paper 1

Worked Solution & Example Answer:The parametric equations of a line are given below - HSC - SSCE Mathematics Extension 1 - Question 11 - 2023 - Paper 1

The parametric equations of a line are given below. Find the Cartesian equation of this line in the form y = mx + c.

In how many different ways can all the letters of the word CONDOBOLIN be arranged in a line?

Consider the polynomial P(x) = x^3 + ax^2 + bx - 12, where a and b are real numbers. Find a and b.

Join the SSCE students using SimpleStudy...