Unified Tertiary Matriculation Examination · Mathematics

Calculus, Statistics and Probability

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Find the first derivative $\frac{d y}{d x}$ of the polynomial function: $y = 3 x^{4} - 5 x^{2} + 7 x - 2$ .
Determine the gradient of the tangent to the curve $y = 2 x^{3} - 4 x + 1$ at the point where $x = 2$ .
Find the equation of the tangent line to the parabola $y = x^{2} - 3 x + 2$ at the point $(3, 2)$ .
Find the equation of the normal to the curve $y = 2 x^{2} + 1$ at the point $(1, 3)$ .
Find the coordinates of the turning points (stationary points) of the cubic curve $y = x^{3} - 3 x$ .
Classify the stationary point of the quadratic function $y = 4 x - x^{2}$ .
Differentiate $y = x^{2} sin x$ with respect to $x$ .
Find the derivative $\frac{d y}{d x}$ of the rational function: $y = \frac{x}{x + 1}$ .
Using the chain rule, find the derivative of $y = (2 x + 3)^{5}$ .
The radius of a circular oil slick increases at a constant rate of $0.5 cm/s$ . Find the rate at which the area is increasing when the radius is 6 cm.
Evaluate the indefinite integral: $\int (6 x^{2} - 4 x + 5) d x$ .
Evaluate the definite integral: $\int_{1}^{3} (3 x^{2} - 2 x) d x$ .
Calculate the area enclosed by the curve $y = x^{2}$ , the $x$ -axis, and the vertical lines $x = 0$ and $x = 3$ .
Evaluate the integral using substitution: $\int 2 x (x^{2} + 1)^{3} d x$ .
Calculate the determinant of the $2 \times 2$ matrix: $M = (43 - 2 5)$ .
Find the value of $k$ for which the matrix $(23 k 6)$ is singular.
Evaluate the matrix product: $(1324) (2 - 1)$ .
Find the inverse of the matrix $A = (3512)$ .
Calculate the magnitude of the position vector $v = 6 i - 8 j$ .
Given vectors $a = 3 i + 2 j$ and $b = 4 i - 5 j$ , evaluate the scalar dot product $a \cdot b$ .
Determine the value of $m$ if the vectors $p = 2 i + 3 j$ and $q = m i - 4 j$ are perpendicular.
The mean of the numbers $4, 7, x, 11$ , and $14$ is 9. Find the value of $x$ .
Determine the median of the following ungrouped data set: $12, 5, 8, 19, 14, 7, 10$ .
Find the mode of the numbers: $3, 5, 7, 5, 8, 3, 5, 9$ .
Calculate the population variance of the numbers: $2, 4, 6, 8, 10$ .
If the variance of a statistical distribution is 25, what is its standard deviation?
A fair six-sided die is rolled once. What is the probability of obtaining a prime number?
Two fair dice are thrown simultaneously. What is the probability that the sum of the two numbers is equal to 7?
If $A$ and $B$ are independent events such that $P (A) = 0.4$ and $P (B) = 0.5$ , evaluate $P (A \cap B)$ .
If $E$ and $F$ are mutually exclusive events with $P (E) = \frac{1}{3}$ and $P (F) = \frac{1}{4}$ , find $P (E \cup F)$ .
A box contains 5 red balls and 3 green balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?
In how many distinct ways can the letters of the word NIGERIA be arranged?
In how many ways can a delegation of 3 student representatives be chosen from a group of 8 eligible prefects?
If the permutation equation $^{n} P_{2} = 30$ holds for integer $n > 2$ , find the value of $n$ .
In a diagnostic test, event probabilities are $P (A) = 0.60$ and $P (A \cap B) = 0.18$ . Find the conditional probability $P (B ∣ A)$ .

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Calculus, Statistics and Probability

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