CSEC Additional Mathematics Paper 2 May/June 2018 - Free CXC Past Paper with Answers

Given that $g(x) = -x^2 + x - 3$, express $g(x)$ in the form $a(x + h)^2 + k$ where $a$, $h$ and $k$ are constants.

On the grid provided below, sketch the graph of $g(x)$, showing the maximum point and the $y$-intercept.

In a geometric progression, the 3rd term is 25 and the sum of the 1st and 2nd terms is 150. Determine the sum of the first four terms, given that $r > 0$.

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 5x + 3 = 0$, determine the value of .

SECTION II

Answer BOTH questions.

ALL working must be clearly shown.

Determine the equation of the circle that has centre $(5, -2)$, and passes through the origin.

Determine whether the following pair of lines is parallel. $x + y = 4$ $3x - 2y = -3$

The position vectors of two points, $A$ and $B$, relative to a fixed origin, $O$, are given by $\vec{OA} = 2\mathbf{i} + \mathbf{j}$ and , where and represent the unit vectors in the and directions respectively. Calculate

the magnitude of $\vec{AB}$

the angle $A\hat{O}B$, giving your answer to the nearest whole number.

A wire in the form of a circle with radius 4 cm is reshaped in the form of a sector of a circle with radius 10 cm. Determine, in radians, the angle of the sector, giving your answer in terms of $\pi$.

Solve the equation $\sin^2 \theta + 3 \cos 2\theta = 2$ for $0 \le \theta \le \pi$. Give your answer(s) to 1 decimal place.

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

Given that $y = x^3 + 2x^2 - 1$, determine

the coordinates of the stationary points

the nature of EACH stationary point.

Differentiate $y = 2x \sqrt{(4 - 8x)}$ with respect to $x$, simplifying your answer.

Show, using integration, that the finite area of the curve $y = \sin x$ in the first quadrant bounded by the line $x = \frac{4\pi}{9}$ is smaller than the finite region of $y = \cos x$ in the same quadrant and bounded by the same line.

The finite region in the first quadrant bounded by the curve $y = x^2 + x + 3$, the $x$-axis and the line $x = 4$ is rotated completely about the -axis. Determine the volume of the solid of revolution formed.

A curve which has a gradient of $\frac{dy}{dx} = 3x - 1$ passes through the point $A(4, 1)$. Find the equation of the curve.

SECTION IV

Answer only ONE question.

ALL working must be clearly shown.

The number of runs scored by a cricketer for 18 consecutive innings is illustrated in the following stem-and-leaf diagram.

0 | 2 3 6 7
1 | 0 3 5 8 9
2 | 4 4 6 8
3 | 1 4 5
4 | 7

Key $0|6$ means 6

Determine the median score.

Calculate the interquartile range of the scores.

In the space below, construct a box-and-whisker plot to illustrate the data and comment on the shape of the distribution.

Insecticides $A$, $B$ or $C$ are applied on lots $Q$, $R$ and $S$. The same crop is planted on each lot and the lots are of the same size. The probability that a group of farmers will select $A$, $B$ or $C$ is $40%$, $25%$ and $35%$ respectively. The probability that insecticide $A$ is successful is $0.8$, that $B$ is successful is $0.65$, and that $C$ is successful is $0.95$.

Illustrate this information on a tree diagram showing ALL the probabilities on ALL branches.

An insecticide is selected at random, determine the probability that it is unsuccessful.

A regular six-sided die is tossed 2 times.

Calculate the probability of obtaining a 5 on the 2nd toss, given that a 5 was obtained on the 1st toss.

Determine the probability that a 5 is obtained on both tosses.

Explain why the answers in (c) (i) and (c) (ii) are different.

Determine

its velocity when $t = 2$

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

the distance between the rest points

the time at which the maximum velocity occurs.

A bus starts from rest at Station A and travels a distance of 80 km in 60 minutes to Station B. Since the bus arrived at Station B early, it remained there for 20 minutes then started the journey to Station C. The time taken to travel from Station B to Station C was 90 minutes at an average speed of 80 kmh$^{-1}$.

On the grid provided on page 27, draw a distance-time graph to illustrate the motion of the bus.

Determine the distance from Station B to Station C.

Determine the average speed from Station A to Station B, in kmh$^{-1}$.