CSEC Additional Mathematics Paper 2 May/June 2014 - Free CXC Past Paper with Answers

Given that $f(x) = -2x^2 - 12x - 9$.

(i) Express $f(x)$ in the form $k + a(x+h)^2$, where $a, h$ and $k$ are integers to be determined. (ii) State the maximum value of $f(x)$. (iii) Determine the value of $x$ for which $f(x)$ is a maximum.

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

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

Find the set of values of $x$ for which $3 + 5x - 2x^2 \le 0$.

A series is given by $0.2 + 0.02 + 0.002 + 0.0002 + \dots$.

(i) Show that this series is geometric. (ii) Find the sum to infinity of this series, giving your answer as an exact fraction.

Find the sum to infinity of this series, giving your answer as an exact fraction.

Determine the value of $k$ such that the lines $x + 3y = 6$ and $kx + 2y = 12$ are perpendicular to each other.

A circle of radius $5$ cm has as its centre the point of intersection of the two perpendicular lines in (i). Determine the equation for this circle.

$RST$ is a triangle in the coordinate plane. Position vectors $R, S,$ and $T$ relative to an origin, $O$, are $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$, and respectively.

Determine the length of the hypotenuse.

[Hint: A rough drawing of $RST$ might help].

Figure 1 shows the sector $OAB$ of a circle with centre $O$, radius $9$ cm and angle $0.7$ radians.

(i) Find the area of the sector $OAB$. (ii) Hence, find the area of the shaded region, $H$.

Hence, find the area of the shaded region, $H$.

Given that $\sin \frac{\pi}{6} = \frac{1}{2}$ and , show that

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

The equation of a curve is $y = 3 + 4x - x^2$. The point $P(3, 6)$ lies on the curve.

Find the equation of the tangent to the curve at $P$, giving your answer in the form $ax + by + c = 0$, where $a, b, c \in \mathbb{Z}$.

Given that $f(x) = 2x^3 - 9x^2 - 24x + 7$.

Determine the nature of EACH of the stationary points of $f(x)$.

Evaluate $\int_1^2 x(x^2 - 2) \, dx$.

Evaluate $\int_0^{\frac{\pi}{2}} (4 \cos x + 2 \sin x) \, dx$, leaving your answer in surd form.

A curve passes through the point $P(2, -5)$ and is such that $\frac{dy}{dx} = 6x^2 - 1$.

Find the area of the finite region bounded by the curve, the $x$-axis, the line $x = 3$ and the line $x = 4$.

There are 60 students in the sixth form of a certain school. Mathematics is studied by 27 of them, Biology by 20 of them and 22 students study neither Mathematics nor Biology. If a student is selected at random, what is the probability that the student is studying

both Mathematics and Biology?

Biology only?

Two ordinary six-sided dice are thrown together. The random variable $S$ represents the sum of the scores of their faces landing uppermost.

Copy and complete the sample space diagram below.

$P(S \le 4)$.

Let $D$ be the difference between the scores of the faces landing uppermost. The table below gives the probability of each possible value of $d$.

Find the values of $a, b$ and $c$.

The aptitude scores obtained by 51 applicants for a supervisory job are summarized in the following stem and leaf diagram.

$5|1$ means 51

Find the median and quartiles for the data given.

Construct a box-and-whisker plot to illustrate the data given and comment on the distribution of the data.

Figure 2 below, not drawn to scale, shows the motion of a car with velocity, $V$, as it moves along a straight road from Point $A$ to Point $B$. The time, $t$, taken to travel from Point $A$ to Point $B$ is 90 seconds and the distance from Point $A$ to Point $B$ is 1410 m.

What distance did the car travel from Point $A$ towards Point $B$ before starting to decelerate?

Calculate the deceleration of the car as it goes from 25 m s$^{-1}$ to 10 m s$^{-1}$.

For how long did the car maintain the speed of 10 m s$^{-1}$?

From Point $B$, the car decelerates uniformly, coming to rest at a Point $C$ and covering a further distance of 30 m. Determine the average velocity of the car over the journey from Point $A$ to Point $C$.

A particle travels along a straight line. It starts from rest at a point, $P$, on the line and after 10 seconds, it comes to rest at another point, $Q$, on the line. The velocity $v$ m s$^{-1}$ at time $t$ seconds after leaving is for for

Find the time when the velocity is at its maximum.

Determine the maximum velocity.

Find the distance moved by the particle from $P$ to the point where the particle attains its maximum velocity.