Set 2

Explanation

Substitute point $(a, b)$ into $(x, y)$ in the equation

Find the first derivative

Remember the concept that the first derivative of a function equals the gradient of the line

Determine the value of $b$ from equation $(1)$ and $a + b$

When $a = -2$

When $a = 4$

Following the available options, the answer is $-4$.

Explanation

Add the two equations

Since we are asked for the value of $x + 3y$, we use the substitution $x + 3y = p$

Apply factorization

Take the square root of both sides

This means $x + 3y = p \rightarrow x + 3y = -2$.

Explanation

The integer $p$ is a root of $f(x) = 0$. We can conclude that $x = p$ so $f(p) = 0$

The factorization above is obtained using Horner's method.

Since the form $p^2 - p - 3 = 0$ cannot be factored further, we only obtain $p = -1$. Thus the function becomes

With $x = p = -1$, the gradient is

Explanation

Given $f(a) = 19$ then

Factor using Horner's method to get

Thus we obtain $a = 2$, and the quadratic function becomes

Vertex point $p$

Therefore the value of $p + 2q + 3a$ is

Explanation

Create the equation of the line from the points it passes through, namely $(0, -2)$ and $\left(\frac{3}{2}, 0\right)$

We cannot directly find the distance from a point to the line. Find the point whose position is closest to the line. Assume the point is $(a, b)$ so that

The point becomes $(a, a^2 - 1)$ as shown in the following illustration

We need to determine the distance from point $(a, a^2 - 1)$ to the line $-4x + 3y + 6 = 0$

The condition for minimum value is $f'(x) = 0$

Distance between line and parabola when $a = \frac{2}{3}$

Explanation

The given sequence can be written as

We can simplify it into a general form where

For odd $n$, the $n$th term is $u_n = -\frac{1}{2^n}$

For even $n$, the $n$th term is $u_n = \frac{3}{2^n}$

Therefore, the $12$th term or even term is

Explanation

Given sequence

Each term if expanded becomes

Therefore, the $n$th term can be formulated as follows

Then the $12$th term is

Explanation

Given sequence

Each term if expanded becomes

Therefore, the $n$th term can be formulated as follows

Then the $12$th term is

Explanation

A geometric sequence has a ratio between its terms, or the value of the ratio between terms

The quadratic equation $2x^2 + kx + 6 = 0$ has roots $x_1 = a = 1$ and $x_2 = b$

Product of roots

Sum of roots

Therefore the geometric sequence is

Then the $4$th term is

Thus the values of $u_4$ and $k$ respectively are $27$ and $-8$.

Explanation

Given quadratic equation

With roots $x_1$ and $x_2$, the product of roots is

So the geometric sequence is $x_1, k, x_2$.

The ratio between terms is

Factorize

Determine the form of the quadratic equation based on the obtained value of $k$.

Check the discriminant value, the roots are not integers.

Therefore the sequence is

The ratio between terms is

Then the $n$th term of the sequence is

Explanation

Explanation

Given

Asked

Let $\frac{1}{h} = a$; $h \to \infty$ so the value $a \to 0$.

Since $f(x) = \sin^2 x$, then

Therefore, the value of

Explanation

First find the derivative of the function

If $y = f(g(x))$ then the first derivative is $y' = g'(x) \cdot f'(g(x))$. Check the limit value

Then we use another method, L'Hospital's rule or the first derivative