CSEC Additional Mathematics Paper 2 May/June 2021 - Free CXC Past Paper with Answers

Given that $\log_2 (6 + \sqrt{12}) - \log_2 (3 + \sqrt{a}) = \log_2 10$, find the value of $a$.

Determine the set of values of $x$ for which $\frac{2 - x}{4x - 9} < 0$.

Alice deposited $4000 in her new savings account at Bank of Fortune, which pays interest at 8% per annum. The bank's compounded interest is represented by the geometric progression $A = P \left(1 + \frac{R}{100}\right)^T$ where is the amount of money accumulated after years, , the percentage rate of interest per annum and , a positive integer, the time in years. Determine the number of years it would take Alice's money to .

The coordinates for the centre of a circle is $(2, 1)$ and the coordinates for a point on its circumference is $(3, 3)$.

Determine the equation of the circle in the form $x^2 + y^2 + ax + by + c = 0$.

The circle intersects the $x$ and $y$-axes at three points. Determine the coordinates of the three points of intersection.

Determine the equation of the tangent to the circle at the point $(3, 3)$.

The position vectors of two points, $P$ and $Q$, relative to a fixed origin, $O$, are given by $\vec{OP} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and . Determine the unit vector in the direction of , giving your answer in simplest surd form.

Given that $\cos M = -\frac{24}{25}$ and that angle $M$ is acute, determine the value for $\tan 2M$.

SECTION III

Answer ALL questions.

ALL working must be clearly shown.

Differentiate $\sin x + \cos 4x$ with respect to $x$.

Differentiate $\frac{2x^2 + 2}{2x + 1}$ with respect to $x$.

Use the principles of differentiation to compute the stationary value of the function $y = x^2 - 4x + 2$.

A motorist starts from a point $X$ and travels 60 m due north to a point $Y$ at a constant speed of $4 \text{ ms}^{-1}$. He stays at $Y$ for 25 seconds and then travels at a constant speed of $10{}^{}$ for 100 m due south to a point, . Calculate

the average speed for the whole journey

the average velocity of the whole journey.

Determine the following integrals, giving each answer in its simplest form.

$\int 2x^2 + 3x \ dx$

$\int 2 \sin 3x \ dx$

Using an integration method, calculate the area of the region in the first quadrant under the graph $y = 3 \sin x$.

A particle starting from rest travels in a straight line with an acceleration, $a$, given by $a = t^2$ where $t$ is the time in seconds.

Determine the velocity, $v$, of the particle in terms of time, $t$.

Calculate the displacement, $s$, of the particle in the interval $t = 0$ to $t = 2$.

Two fair tetrahedral dice with faces numbered 1, 2, 3, 4 are rolled. The numbers obtained on the turned-down face of each dice are noted.

Create a sample space table listing ALL possible outcomes for the two dice.

Using your sample space table created in (a), or otherwise, determine the probability of obtaining a 4

on both dice

on at least one dice

on exactly one dice.

Show that obtaining a 4 on both dice are independent events.

Determine the probability of obtaining a 4 on both dice, given that a 4 was obtained on at least one dice.

The scores of a class of 30 students on a Mathematics test were used to draw the box plot below. (The total score possible is 20 marks.)

Using the box plot, determine the following:

The median score

The range of the scores

The semi-interquartile range of the scores

Comment on the shape of the distribution of the scores.

State ONE reason why it would be impossible to determine the mean score from the box plot.

State what additional piece of information would be needed to determine the mean score.

Given that the sum of the 30 scores for the class is 354 and the sum of the squares of the scores is 4994, determine the standard deviation for the data set.