CSEC Additional Mathematics Paper 2 May/June 2015 - Free CXC Past Paper with Answers

Given that $f(x) = 3x^2 - 9x + 4$:

Express $f(x)$ in the form $a(x+b)^2 + c$, where $a, b$ and $c$ are real numbers.

State the coordinates of the minimum point of $f(x)$.

The equation $3x^2 - 6x - 4 = 0$ has roots $\alpha$ and $\beta$. Find the value of .

Determine the coordinates of the points of intersection of the curve $2x^2 - y + 19 = 0$ and the line $y + 11x = 4$.

An employee of a company is offered an annual starting salary of $36 000 which increases by $2 400 per annum. Determine the annual salary that the employee should receive in the ninth year.

The equation of a circle is given by $x^2 + y^2 - 12x - 22y + 152 = 0$.

Determine the coordinates of the centre of the circle.

Find the length of the radius.

Determine the equation of the normal to the circle at the point $(4, 10)$.

The position vectors of two points, $A$ and $B$, relative to an origin $O$, are such that $\mathbf{OA} = 3\mathbf{i} - 2\mathbf{j}$ and . Determine

Determine the unit vector $\vec{AB}$.

the acute angle $AOB$, in degrees, to one decimal place.

The following diagram shows a circle of radius $r = 4$ cm, with centre O and sector AOB which subtends an angle, $\theta = \frac{\pi}{6}$ radians at the centre.

If the area of the triangle AOB $$, then calculate the area of the shaded region.

Solve the following equation, giving your answer correct to one decimal place. $8 \sin^2 \theta = 5 - 10 \cos \theta$, where $0^\circ \le \theta \le 360^\circ$

Prove the identity $\frac{\sin \theta + \sin 2\theta}{1 + \cos \theta + \cos 2\theta} = \tan \theta$.

Differentiate the following expression with respect to $x$, simplifying your answer. $(2x^2 + 3) \sin 5x$

Find the coordinates of all the stationary points of the curve $y = x^3 - 5x^2 + 3x + 1$.

Determine the nature of EACH point in (i) above.

A spherical balloon of volume $V = \frac{4}{3}\pi r^3$ is being filled with air at the rate of $200 \text{ cm}^3 \text{ s}^{-1}$. Calculate, , the rate at which the radius is increasing when the radius of the balloon is .

Evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} 3 \cos \theta \, d\theta$.

A curve has an equation which satisfies $\frac{dy}{dx} = kx(x-1)$ where $k$ is a constant. Given that the value of the gradient of the curve at the point is , determine

the value of $k$

the equation of the curve.

Calculate, in terms of $\pi$, the volume of the solid formed when the area enclosed by the curve $y = x^2 + 1$ and the $x$-axis, from $x = 0$ to , is rotated through about the -axis.

There are three traffic lights that a motorist must pass on the way to work. The probability that the motorist has to stop at the first traffic light is $0.2$, and that for the second and third traffic lights are $0.5$ and $0.8$ respectively. Find the probability that the motorist has to stop at

ONLY ONE one traffic light

AT LEAST TWO traffic lights.

Use the data in the following table to estimate the mean of $x$.

Research in a town shows that if it rains on any one day then the probability that it will rain the following day is $25%$. If it does not rain one day then the probability that it will rain the following day is $12%$. Starting on a Monday and given that it rains on that Monday:

Draw a probability tree diagram to illustrate the information, and show the probability on ALL of the branches.

Determine the probability that it will rain on the Wednesday of that week.

A particle moving in a straight line has a velocity of $3 \text{ m s}^{-1}$ at $t = 0$ and $4$ seconds later its velocity is $9 \text{ m s}^{-1}$.

On the answer graph sheet, provided as an insert, draw a velocity–time graph to represent the motion of the particle.

Calculate the acceleration of the particle.

Determine the increase in displacement over the interval $t = 0$ to $t = 4$.

A particle moves in a straight line so that $t$ seconds after passing through a fixed point $O$, its acceleration, $a$, is given by $a = (3t - 1) \text{ m s}^{-2}$. The particle has a velocity, , of when and its displacement, , from is metres when . Find

the velocity when $t = 4$

the displacement of the particle from $O$ when $t = 3$.