CSEC Additional Mathematics Paper 2 May/June 2013 - Free CXC Past Paper with Answers

Let $f(x) = 3x^2 + 6x - 1$.

Express $f(x)$ in the form $a(x + h)^2 + k$ where $h$ and $k$ are constants.

State the minimum value of $f(x)$.

Determine the value of $x$ for which $f(x)$ is a minimum.

State ONE value of $x$ for which $2x^2 + 3x - 5 \ge 0$.

Find the sum to infinity of the following series: $\frac{1}{4} + \frac{2}{4^2} + \frac{1}{4^3} + \frac{2}{4^4} + \dots$ This series can be rewritten as the sum of two geometric series.

A circle, C, has centre with coordinates A $(2, 1)$ and passes through the point B $(10, 7)$.

Express the equation of the circle in the form $x^2 + y^2 + hx + gy + k = 0$, where $h$, $g$ and $k$ are integers to be determined.

The line $l$ is a tangent to the circle C at the point B. Find an equation for $l$.

The position vectors of two points, P and Q, relative to a fixed origin, O, are $10\mathbf{i} - 8\mathbf{j}$ and $\lambda\mathbf{i} + 10\mathbf{j}$ respectively, where $\lambda$ is a constant. Find the value of $\lambda$ such that and are perpendicular.

The position vectors of A and B with respect to a fixed origin, O, are given by $\vec{OA} = -2\mathbf{i} + 5\mathbf{j}$ and $\stackrel{}{O}$ respectively. Find the unit vector in the direction of AB.

The diagram shows a sector cut from a circle of centre O. The angle at O is $\frac{\pi}{6}$. If the perimeter of the sector is $\frac{5}{6}(12 + \pi)$ cm, what is its area?

Solve the equation $2 \cos^2 \theta + 3 \sin \theta = 0$ for $0 \le \theta \le 360^\circ$.

Given that $\tan (\theta - \alpha) = \frac{1}{2}$ and that $\tan \theta = 3$, use the appropriate compound angle formula to find the value of the acute angle .

Given that $y = x^3 - 3x^2 + 2$. Find

the coordinates of the stationary points of $y$

the second derivative of $y$ and hence determine the nature of EACH of the stationary points.

Differentiate $y = (5x + 3)\sin x$ with respect to $x$, simplifying your result as far as possible.

Find $\int (5x^2 + 4) dx$.

Evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} (3 \sin x - 5 \cos x) dx$.

A curve passes through the points P$(0, 8)$ and Q$(4, 0)$ and is such that $\frac{dy}{dx} = 2 - 2x$. Find the area of the finite region bounded by the curve in the first quadrant.

Of the persons buying petrol at a service station, 40 per cent are females. Of the females, 30 per cent pay for their petrol with cash, and of the males, 65 per cent pay for their petrol with cash.

Copy and complete the following tree diagram, by putting in all the missing probabilities, to show this information.

What is the probability that a customer pays for petrol with cash?

Determine which is the more likely event: Event T: Customer is female, GIVEN that the petrol is paid WITH cash. Event V: A male customer does NOT pay for petrol with cash.

The marks obtained by 30 students on an English exam are given as 58 92 41 89 72 66 51 63 80 40 69 45 83 76 53 56 75 50 99 50 85 63 58 75 66 56 81 74 51 94

State ONE advantage of using a stem and leaf diagram versus a box and whiskers plot to display the data.

Construct a stem and leaf diagram to show the data.

Determine the median mark.

Calculate the semi inter-quartile range of the marks.

Two students are chosen at random from the class. Determine the probability that both scored less than 50 on the exam.

A particle starts from rest and accelerates uniformly to $20 \text{ m s}^{-1}$ in 5 seconds. It continues at this velocity for 10 seconds. It then accelerates uniformly until it comes to rest, 15 seconds later.

On the graph paper provided, draw a velocity-time graph to illustrate the motion of the particle.

From your graph determine a) the total distance, in metres, travelled by the particle b) the average velocity of the particle for the entire journey.

A particle travels in a straight line in such a way that after $t$ seconds its velocity, $v$, from a fixed point, O, is given by the function $v = 3t^2 - 18t + 15$.

Calculate

the values of $t$ when the particle is instantaneously at rest

the distance travelled by the particle between 1 second and 3 seconds

the value of $\frac{dv}{dt}$ when a) $t = 2$ seconds b) $t = 3$ seconds.

Give an interpretation for the value in a) 8 (b) (iii) a) b) 8 (b) (iii) b).