2015 Paper 1

The diagram shows part of the graph of the function $\displaystyle y=p \cos qx+r.$

Write down the values of $\displaystyle p, q$ and $\displaystyle r.$

A function $\displaystyle g$ is defined on $\displaystyle \mathbb{R}$, the set of real numbers, by $\displaystyle g(x)=6-2x$

(a)  Determine an expression for $\displaystyle g^{-1}(x).$
(b)  Write down an expression for $\displaystyle g(g^{-1}(x)).$

ABCD is a rectangle with sides of lengths $\displaystyle x$ centimetres and $\displaystyle (x-2)$ centimetres, as shown.

If the area of ABCD is less than $\displaystyle 15\text{ cm}^{2}$, determine the range of possible values of $\displaystyle x.$

Given that $\displaystyle \tan 2x=\frac{3}{4}$
$\displaystyle 0 \lt x \lt \frac{\pi}{4}$
find the exact value of

(a)  $\displaystyle \cos 2x.$
(b)  $\displaystyle \cos x.$

$\displaystyle T(-2,-5)$ lies on the circumference of the circle with equation
$\displaystyle (x+8)^{2}+(y+2)^{2}=45$

(a)  Find the equation of the tangent to the circle passing through T.
(b)  This tangent is also a tangent to a parabola with equation
$\displaystyle y=-2x^{2}+px+1-p$
where $\displaystyle p \gt 3$
Determine the value of $\displaystyle p.$

The diagram shows part of the graph of $\displaystyle y=a \cos bx.$ The shaded area is $\displaystyle \frac{1}{2}\text{ unit}^{2}.$

What is the value of
$\displaystyle \int_{0}^{\frac{3\pi}{4}}(a \cos bx)dx$?

The function $\displaystyle f(x)=2^{x}+3$
is defined on $\displaystyle \mathbb{R}$, the set of real numbers.
The graph with equation $\displaystyle y=f(x)$ passes through the point $\displaystyle P(1,b)$ and cuts the y-axis at Q as shown in the diagram.

(a)  What is the value of $\displaystyle b$?

(b)  (i) Copy the above diagram. On the same diagram, sketch the graph with equation
$\displaystyle y=f^{-1}(x)$

      (ii) Write down the coordinates of the images of P and Q.

(c)  $\displaystyle R(3,11)$ also lies on the graph with equation $\displaystyle y=f(x).$
Find the coordinates of the image of R on the graph with equation
$\displaystyle y=4-f(x+1)$

The rate of change of the temperature, $\displaystyle T^{\circ}\text{C}$ of a mug of coffee is given by
$\displaystyle \frac{dT}{dt}=\frac{1}{25}t-k$, $\displaystyle 0 \le t \le 50$
•  $\displaystyle t$ is the elapsed time, in minutes, after the coffee is poured into the mug
•  $\displaystyle k$ is a constant
•  initially, the temperature of the coffee is $\displaystyle 100^{\circ}\text{C}$
•  10 minutes later the temperature has fallen to $\displaystyle 82^{\circ}\text{C}$

Express $\displaystyle T$ in terms of $\displaystyle t.$