StudyDojo

Question 1

HLPaper 2

Consider the polynomial $f(x) = 23x^4 - 80.5x^3 + 57.5x^2 - 34.5x + 138$

1.

Given that $(x + 23)$ is a factor of $f(x)$ , find the value of $f(-23)$ .

[2]

2.

Find all the remaining factors of $f(x)$ and write the full factored form.

[4]

3.

Find the sum and product of all roots of $f(x)$ .

[2]

Question 2

SLPaper 1

1.

The quadratic equation $3x^2 + px + q = 0$ has roots $\alpha$ and $\beta$ where $\alpha + \beta = 5$ and $\alpha\beta = 2$ . Find the values of $p$ and $q$ .

[3]

Question 3

SLPaper 1

A ship sails from point $P$ on a bearing of $310^\circ$ for $k \text{ km}$ to point $Q$ . It then changes course and sails on a bearing of $225^\circ$ for $10 \text{ km}$ to point $R$ .

1.

Find the distance $PR$ in terms of $k$ .

[3]

2.

If $PR = 15$ km, find the exact value of $k$ where you answer can include $\cos\alpha^\circ$ for some $\alpha\in(0,360)$

[4]

Question 4

SLPaper 1

Let $f(x) = 0.2e^x + 2 - 4$ for $-3 \leq x \leq 2$

1.

On the following grid, sketch the graph of $y = f(x)$

[3]

2.

The graph of f is reflected in the x-axis, then translated 1 unit in the direction of the positive x-axis and 2 units in the direction of the positive y-axis to obtain the graph of a function g. Find the expression for g(x).

[3]

Question 5

HLPaper 2

The following diagram shows the graph of $y = f\left( x \right)$ , $-3 \leqslant x \leqslant 5$ .

Graph

1.

The value of $(f \circ f)(1)$ .

[2]

2.

Given that ${f^{-1}(b) = 3}$ , determine the value of ${b}$ .

[2]

3.

Given that $h(x) = 2f(x - 1)$ , find the domain and range of $h$ .

[2]

Question 6

SL & HLPaper 2

Consider the rational function $f(x) = \frac{kx - 11}{7x - 4k}$ , where $k$ is a parameter.

1.

Find the values of $k$ for which the vertical asymptote of the function $f(x)$ exists.

[5]

Question 7

SL & HLPaper 1

Consider the quadratic function $f(x) = 2(x - 2)^2 + 4$ .

1.

Describe the transformations applied to the function $y = x^2$ to obtain the function $f(x)$ .

[3]

2.

Given that $x^2\geq f(x)$ under the interval $[2,4]$ , find the area between the curves $y = x^2$ and $f(x) = 2(x - 2)^2 + 4$ within that interval.

[5]

Question 8

HLPaper 2

Consider the function $f(x) = \sin(x)$ .

1.

Graph the function $g(x) = \sin(2x) - 3$ .

[4]

2.

Graph the function $h(x) = \sin\left(\frac{x}{3}\right) + 2$ .

[4]

Question 9

SLPaper 1

the $\tan$ function will be explored for the following section

1.

Express $\tan(θ)$ in terms of sine and cosine.

[1]

2.

Explain why $\tan(3\pi - x) = -\tan(x)$ using the unit circle.

[3]

3.

Validate this identity for $x = \frac{\pi}{4}$ .

[2]

4.

Sketch the tangent function to reflect this property.

[3]

Question 10

SLPaper 2

Consider the exponential equation $9^x = 7x + 4$ .

1.

Sketch the graph of $y = 9^x$ and $y = 7x + 4$ on the same axes.

[2]

2.

Hence find the approximate solution to the equation $9^x = 7x + 4$ .

[2]

3.

Hence, find the area between the curves within the interval of the intersection points

[2]

Functions

Functions