CSEC Additional Mathematics Paper 2 May/June 2012 - Free CXC Past Paper with Answers

The quadratic equation $x^2 - 4x + 6 = 0$ has roots $\alpha$ and $\beta$. Calculate the value of $\alpha^2 + \beta^2$.

Find the range of values of $x$ for which $\frac{2x - 5}{3x + 1} > 0$.

A customer repays a loan monthly by increasing the payment each month by $5x$. If the customer repaid $50 in the $5^{th}$ month and $70 in the $9^{th}$ month, calculate the TOTAL amount of money repaid at the end of the $$ month.

The equation of a circle is given by $x^2 + y^2 - 4x + 6y = 87$.

A line has equation $x + y + 1 = 0$. Show that this line passes through the centre of the circle.

Find the equation of the tangent to the circle at the point $A(-6, 3)$.

Given $\vec{OA} = \mathbf{a}$, $\vec{OB} = \mathbf{b}$, , where and .

Write $\vec{BP}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

Find $|\vec{BP}|$.

The diagram shows a sector of a circle centre $O$ with an adjoining square. The radius of the circle is 4 m.

If the sector $AOC$ subtends an angle $\frac{\pi}{3}$ at $O$, calculate, giving your answer in terms of $\pi$

the area of the shape $OACMN$.

the perimeter of the shape $OACMN$.

Given that $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$, and , evaluate without using calculators, the value of .

Prove the identity $\frac{1}{\cos \theta} + \tan \theta \equiv \frac{1 - \sin \theta}{\cos \theta}$.

Differentiate the following expression with respect to $x$, simplifying your answer. $\frac{3x + 4}{x - 2}$

The point $P(2, 10)$ lies on the curve $y = 3x^2 + 5x - 12$. Find equations for (i) the tangent to the curve at . (ii) the normal to the curve at .

the tangent to the curve at $P$.

the normal to the curve at $P$.

The length of the side of a square is increasing at a rate of $4 \text{ cms}^{-1}$. Find the rate of increase of the area when the length of the side is $5 \text{ cm}$.

Evaluate $\int_{1}^{3} (16 - 7x) \text{ dx}$.

The point $Q(4, 8)$ lies on a curve for which $\frac{\text{dy}}{\text{dx}} = 3x - 5$. Determine the equation of the curve.

Calculate the area between the curve $y = 2 \cos x + 3 \sin x$ and the $x$-axis from $x = 0$ to .

Calculate the volume of the solid formed when the area enclosed by the curve $y = x^2 + 2$ and the $x$-axis, from $x = 0$ to , is rotated through about the -axis. [Leave your solution in terms of ].

A survey carried out in a town revealed that $25%$ of the households surveyed owned a laptop computer and $70%$ owned a desktop computer. In addition, it was found that $12%$ owned both a laptop and a desktop computer.

If a sample of households from the town is selected at random, determine the proportion that own NEITHER a laptop NOR a desktop computer.

A bag contains 4 red marbles, 3 black marbles and 3 blue marbles. Three marbles are drawn at random without replacement from the bag.

Find the probability that the marbles

drawn are ALL of the SAME colour

contain EXACTLY 1 red marble.

The probability of hiring a taxi from garage $A$, $B$ or $C$ is $0.3$, $0.5$ and $0.2$ respectively. The probability that the taxi ordered will be late from $A$ is $0.07$, from $B$ is $0.1$ and from $C$ is $0.2$.

Illustrate this information on a tree diagram showing the probability on all branches.

A garage is chosen at random, determine the probability that

A car starting from rest at a point $A$, moves along a straight line reaching a velocity of $24 \text{ ms}^{-1}$ by a constant acceleration of $6 \text{ ms}^{-2}$. The car maintains this constant velocity of $24 \text{ ms}^{-1}$ for 5 seconds and is then brought to rest again by a constant acceleration of $-3 \text{ ms}^{-2}$.

Using the graph sheet provided, draw a velocity-time graph to illustrate the motion of the car.

Determine the TOTAL distance travelled by the car.

A second car, moving at a constant velocity of $32 \text{ ms}^{-1}$ drives past point $A$, 3 seconds after the first car left point $A$. Calculate the length of time after the first car started that this second car meets it.

[Assume that the cars meet during the time when the first car is moving at a constant velocity.]

A particle moves in a straight line with acceleration given by $a = (5t - 1) \text{ ms}^{-2}$ at any time $t$ seconds. When $t = 2$ seconds, the particle has velocity and is from a fixed point . Determine

its velocity when $t = 4$

its displacement from $O$ when $t = 3$.