CESS & Examen d'entrée (Belgium) · Mathematics

Advanced Algebra, Functions, Trigonometry, and Vectors

45 practice questions, each with the answer and an explanation.

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All 45 questions

Find the value of $k$ such that the polynomial $P (x) = 2 x^{3} - 3 x^{2} + k x - 6$ is divisible by $(x - 2)$ .
Solve the inequality $\frac{x - 3}{x + 2} \leq 0$ for $x \in R$ .
Find the real solution to the equation $4^{x} - 3 \cdot 2^{x} - 4 = 0$ .
Solve the logarithmic equation $lo g_{2} (x + 3) + lo g_{2} (x - 1) = 5$ for $x \in R$ .
Given that $sin (θ) = \frac{3}{5}$ , calculate the exact value of $cos (2 θ)$ .
An arithmetic progression has first term $a_{1} = 4$ and common difference $d = 3$ . Compute the sum of the first 20 terms, $S_{20}$ .
Evaluate the sum of the convergent infinite geometric series $\sum_{n = 1}^{\infty} 12 (\frac{1}{3})^{n}$ .
Calculate the perpendicular distance from the point $P (2, - 1)$ to the line $3 x - 4 y + 5 = 0$ .
Determine the radius $R$ of the circle defined by the Cartesian equation $x^{2} + y^{2} - 6 x + 8 y = 0$ .
Find the coefficient of $x^{3}$ in the binomial expansion of $(2 x - 1)^{5}$ .
Determine the inverse function $f^{- 1} (x)$ for $f (x) = \frac{2 x + 1}{x - 3}$ where $x \neq = 3$ .
Find the cosine of the angle $θ$ between vectors $u = (3, 4)$ and $v = (5, 12)$ .
Evaluate the determinant of matrix $A = (53 - 2 4)$ .
Find the inverse of the matrix $M = (3512)$ .
Determine the modulus $r$ and principal argument $θ \in (- π, π]$ of the complex number $z = 1 - i 3$ .
Compute $(1 + i)^{8}$ in standard Cartesian form.
How many distinct committees of 4 members can be selected from a pool of 9 candidates?
In a diagnostic test, disease prevalence is $P (D) = 0.02$ . The test sensitivity is $P (+ ∣ D) = 0.95$ and false positive rate is $P (+ ∣ D^{c}) = 0.05$ . Calculate $P (D ∣ +)$ .
Find the equation of the tangent line to the parabola $y = x^{2} - 4 x + 3$ at the point where $x = 3$ .
Determine the partial fraction decomposition constants $A$ and $B$ such that $\frac{5 x - 1}{( x - 1 ) ( x + 2 )} = \frac{A}{x - 1} + \frac{B}{x + 2}$ .
Given $tan (A) = \frac{1}{2}$ and $tan (B) = \frac{1}{3}$ , compute $tan (A + B)$ .
Find the cross product $u \times v$ of vectors $u = (1, 2, 3)$ and $v = (4, 5, 6)$ .
Find the area of the triangle with vertices $A (1, 2)$ , $B (4, 6)$ , and $C (5, 1)$ in the Cartesian plane.
Determine the domain of the real-valued function $f (x) = \frac{4 - x ^{2}}{x + 1}$ .
Evaluate the determinant of the $3 \times 3$ matrix $B = 100240356$ .
Solve the equation $∣2 x - 5∣ = 3 x + 1$ for $x \in R$ .
Compute the eccentricity $e$ of the ellipse given by the Cartesian equation $\frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1$ .
Find the equations of the asymptotes of the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ .
Solve the linear system for $x$ and $y$ :
$ ${2 x + 3 y = 7 5 x - y = 9$ $
Find all solutions to $2 sin^{2} (x) - sin (x) - 1 = 0$ in the interval $[0, 2 π)$ .
Find the scalar projection of vector $u = (4, 2)$ onto vector $v = (3, - 4)$ .
Determine the period $T$ and amplitude $A$ of the function $f (x) = - 4 sin (6 x - π) + 3$ .
Maximize $Z = 4 x + 3 y$ over the feasible polygon with vertices $(0, 0)$ , $(4, 0)$ , $(3, 3)$ , and $(0, 5)$ .
Solve the inequality $ln (3 x - 2) < ln (x + 6)$ for $x \in R$ .
How many distinct 7-letter anagrams can be formed using all the letters of the word "BELGIUM"?
Find the Cartesian equation of the plane passing through $P (2, - 1, 3)$ with normal vector $n = (3, 4, - 2)$ .
Find the unit vector in the direction of $w = (2, - 3, 6)$ .
Compute the population variance $σ^{2}$ of the dataset ${2, 4, 6, 8, 10}$ .
A discrete random variable $X$ takes values $1, 2, 3, 4$ with probabilities $0.1, 0.4, 0.3, 0.2$ respectively. Find the expected value $E [X]$ .
Find all real roots of the cubic equation $x^{3} - 4 x^{2} + x + 6 = 0$ .
For what value of $m$ are the three points $A (1, 2)$ , $B (3, 8)$ , and $C (m, 14)$ collinear?
Express $cos (3 θ)$ strictly in terms of powers of $cos (θ)$ .
Find the value of constant $c$ for which the linear system has no solution:
$ ${2 x - 4 y = 5 3 x - 6 y = c$ $
Find the solution set of the quadratic inequality $2 x^{2} - 5 x - 3 > 0$ .
Given $f (x) = x^{2} - 2$ and $g (x) = 2 x + 3$ , find $(f \circ g) (- 1)$ .

Advanced Algebra, Functions, Trigonometry, and Vectors

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