Exames Nacionais Secundário (Portugal) · Mathematics A

Mathematics A: Functions, Calculus, Complex Numbers and Probability

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

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 98 questions

What is the derivative of $f (x) = x^{3} - 4 x$ ?
Solve the equation $2^{x} = 32$ .
What is the value of $lo g_{2} 8$ ?
An arithmetic progression has first term $u_{1} = 3$ and common difference $r = 4$ . What is $u_{10}$ ?
A geometric progression has $u_{1} = 2$ and common ratio $r = 3$ . What is $u_{5}$ ?
What is the modulus of the complex number $z = 3 + 4 i$ ?
Compute $(2 + i) (1 - i)$ , where $i^{2} = - 1$ .
In how many different orders can $5$ distinct books be placed on a shelf?
A committee of $2$ students is chosen from a group of $6$ students. How many different committees are possible?
Evaluate $x \to 3 lim \frac{x ^{2} - 9}{x - 3}$ .
Let $f (x) = e^{2 x}$ . What is the slope of the tangent line to the graph of $f$ at $x = 0$ ?
What is the equation of the line through the point $(1, 2)$ with slope $3$ ?
Which line is the vertical asymptote of $f (x) = \frac{1}{x - 2} + 3$ ?
For events $A$ and $B$ , $P (A \cap B) = 0.2$ and $P (B) = 0.5$ . What is $P (A ∣ B)$ ?
Solve $lo g_{3} (x + 1) = 2$ .
What is the exact value of $cos (\frac{2 π}{3})$ ?
What is the principal argument of the complex number $z = 1 + i$ ?
The first $10$ terms of the arithmetic progression $1, 3, 5, \dots$ are added. What is the sum?
A variable has a normal distribution. Approximately what proportion of values lie within two standard deviations of the mean?
What is the domain of $f (x) = ln (x - 3)$ ?
For $f (x) = x^{3} - 3 x^{2}$ , at which value of $x$ does the graph have a point of inflection?
At which value of $x$ does $f (x) = x^{2} - 6 x + 5$ have its minimum?
What is the scalar product of the vectors $u = (1, 2)$ and $v = (3, - 1)$ ?
Which equation describes the circle with centre $(1, - 2)$ and radius $3$ ?
What is the coefficient of $x^{2}$ in the expansion of $(1 + x)^{5}$ ?
How many different $3$ -digit codes can be made from the digits $1, 2, 3, 4, 5$ if no digit may be repeated?
What is $x \to + \infty lim \frac{2 x + 1}{x - 3}$ ?
How many solutions does $sin x = \frac{1}{2}$ have in the interval $[0, 2 π]$ , and what are they?
The infinite geometric series $6 + 2 + \frac{2}{3} + \dots$ converges. What is its sum?
Compute $(1 + i)^{4}$ in the form $a + bi$ .
A bag holds $4$ red and $6$ blue balls. Two balls are drawn at random without replacement. What is the probability that both are red?
A rectangle has a perimeter of $40$ cm. What is the largest possible area of the rectangle?
Let $f (x) = x e^{- x}$ for $x \in R$ . What is the maximum value of $f$ ?
What is $n \to + \infty lim (1 + \frac{3}{n})^{n}$ ?
Which of these is a non-real solution of the equation $z^{3} = 8$ ?
What is the equation of the tangent line to the graph of $f (x) = ln x$ at the point where $x = e$ ?
Machine A makes $60%$ of a factory's items and $2%$ of them are defective. Machine B makes the other $40%$ and $5%$ of them are defective. An item is found to be defective. What is the probability…
How many solutions does $2 cos^{2} x - cos x - 1 = 0$ have in the interval $[0, 2 π [$ ?
In complex analysis, what is the polar trigonometric form of the complex number $z = 1 + i 3$ ?
In complex analysis, what is the polar trigonometric form of the complex number $z = - 1 + i$ ?
In complex numbers, what is the value of $(1 - i)^{8}$ evaluated using de Moivre's formula?
In complex analysis, what is the geometric locus in the Argand plane described by $∣ z - 3 + 2 i ∣ = 5$ ?
In complex analysis, what is the geometric locus represented by $∣ z - 2∣ = ∣ z - 4 i ∣$ ?
In complex analysis, what is the imaginary part of the complex quotient $z = \frac{3 + 4 i}{1 - 2 i}$ ?
In complex analysis, if a complex number $w$ in the second quadrant has argument $12 0^{\circ}$ , what is the principal argument of its complex conjugate in radians within $(- π, π]$ ?
In complex numbers, what are all roots of the equation $z^{3} = - 8$ ?
In complex numbers, what is the quotient $\frac{z _{1}}{z _{2}}$ if $z_{1} = 6 cis (\frac{π}{2})$ and $z_{2} = 2 cis (\frac{π}{6})$ ?
In complex analysis, what is the real part of the complex number $z = e^{i \frac{π}{3}}$ ?
In complex numbers, what is the simplified value of the imaginary unit raised to the power $2026$ , namely $i^{2026}$ ?
In complex analysis, what is the modulus of the complex number $z = \frac{3 + 4 i}{1 - 2 i}$ ?
In calculus, what is the horizontal asymptote of the rational function $f (x) = \frac{4 x ^{2} - 7}{2 x ^{2} + 5 x - 1}$ as $x \to + \infty$ ?
In calculus, what is the equation of the vertical asymptote of the rational function $g (x) = \frac{2 x + 1}{x - 3}$ ?
In calculus, what is the value of the limit $lim_{x \to 0} \frac{e ^{3 x} - 1}{2 x}$ ?
In differential calculus, what is the first derivative of the logarithmic function $f (x) = ln (x^{2} + 4)$ ?
In differential calculus, what is the derivative of the product function $f (x) = x^{2} e^{x}$ ?
In differential calculus, at which coordinate $x$ does the cubic function $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ achieve its local maximum?
In differential calculus, what are the coordinates of the inflection point of the cubic curve $f (x) = x^{3} - 3 x^{2} + 4$ ?
In integral calculus, what is the value of the definite integral $\int_{0}^{2} (3 x^{2} - 2 x + 1) d x$ ?
In integral calculus, what is the value of the definite integral $\int_{1}^{e} \frac{l n x}{x} d x$ ?
In differential calculus, what is the equation of the tangent line to the parabola $y = x^{2} - 3 x + 5$ at the point where $x = 2$ ?
In differential calculus, what is the second derivative $f^{''} (x)$ of the trigonometric function $f (x) = cos (2 x)$ ?
In calculus, what is the value of the limit $lim_{x \to + \infty} x (e^{\frac{1}{x}} - 1)$ ?
In calculus, Bolzano's Intermediate Value Theorem guarantees that a continuous function $f$ on $[a, b]$ has at least one root in $(a, b)$ under which condition?
In calculus, what is the value of the limit $lim_{x \to 0} \frac{s i n ( 5 x )}{t a n ( 2 x )}$ ?
In integral calculus, what is the area of the region bounded by the curves $y = 2 x$ and $y = x^{2}$ between their intersection points?
In analytic geometry, what is the Cartesian equation of the plane passing through $P (1, 2, - 1)$ with normal vector $n = (2, - 3, 1)$ ?
In analytic geometry, what is the dot product of the vectors $u = (2, - 1, 4)$ and $v = (3, 5, - 2)$ in $R^{3}$ ?
In analytic geometry, what is the distance between the points $A (1, 0, - 2)$ and $B (4, - 4, 10)$ in $R^{3}$ ?
In analytic geometry, for which value of $k$ are the vectors $u = (k, 2, - 3)$ and $v = (4, - 1, 2)$ perpendicular in $R^{3}$ ?
In analytic geometry, what is the center and radius of the sphere defined by $x^{2} + y^{2} + z^{2} - 4 x + 6 y - 2 z - 11 = 0$ ?
In analytic geometry, what is the parametric vector equation of the line passing through point $A (1, - 2, 3)$ with direction vector $v = (2, 1, - 1)$ ?
In analytic geometry, what is the angle between the non-zero vectors $u = (1, 0, 1)$ and $v = (0, 1, 1)$ in $R^{3}$ ?
In analytic geometry, what is the normal vector to the plane defined by $3 x - 4 y + 5 z - 8 = 0$ ?
In analytic geometry, at what point does the line $(x, y, z) = (1 + t, 2 - t, 3 + 2 t)$ intersect the plane $x + y + z = 12$ ?
In analytic geometry, what is the shortest distance from the origin $(0, 0, 0)$ to the plane $2 x - 2 y + z - 9 = 0$ ?
In analytic geometry, for what value of $k$ are the vectors $u = (2, - 4, 6)$ and $v = (1, - 2, k)$ collinear in $R^{3}$ ?
In trigonometry, what is the exact value of the sine function $sin (\frac{7 π}{6})$ ?
In trigonometry, what is the exact value of the cosine function $cos (\frac{5 π}{4})$ ?
In trigonometry, what are all solutions of the equation $cos (2 x) = 0$ in the closed interval $[0, π]$ ?
In trigonometry, which identity is mathematically equivalent to $cos (2 θ)$ for all real numbers $θ$ ?
In trigonometry, what are all solutions of $2 sin (x) - 3 = 0$ in the interval $[0, 2 π]$ ?
In trigonometry, what is the fundamental positive period of the function $f (x) = tan (3 x)$ ?
In trigonometry, what is the fundamental positive period of the function $g (x) = 4 sin (\frac{x}{2})$ ?
In trigonometry, which expression is equivalent to $cos (\frac{π}{2} - x)$ for all real $x$ ?
In trigonometry, if $tan (θ) = - \frac{3}{4}$ and $θ$ belongs to the second quadrant $(\frac{π}{2}, π)$ , what is $cos (θ)$ ?
In trigonometry, which expression is mathematically identical to $sin (x + π)$ for all real $x$ ?
In trigonometry, what is the principal value of the inverse sine function $arcsin (\frac{3}{2})$ in radians?
In combinatorics, how many distinct 4-digit codes can be formed using digits ${1, 2, 3, 4, 5, 6, 7}$ without repetition?
In combinatorics, in how many ways can a delegation of 3 students and 2 teachers be selected from 8 students and 5 teachers?
In probability theory, if $P (A) = 0.6$ , $P (B) = 0.5$ , and $P (A \cup B) = 0.8$ , what is $P (A \cap B)$ ?
In probability theory, if events $A$ and $B$ are statistically independent with $P (A) = 0.4$ and $P (B) = 0.5$ , what is $P (A \cup B)$ ?
In probability theory, when rolling a balanced 6-sided die 3 times, what is the probability of rolling at least one six?
In probability theory, what is the variance of a discrete random variable $X$ with values ${1, 2, 3}$ and probabilities $P (X = 1) = 0.3$ , $P (X = 2) = 0.5$ , and $P (X = 3) = 0.2$ ?
In probability theory, if $P (B) = 0.4$ and $P (A \cap B) = 0.1$ , what is the conditional probability $P (A ∣ B)$ ?
In combinatorics, in how many distinct circular arrangements can 6 people be seated around a round dining table?
In set theory and combinatorics, how many distinct subsets does a finite set containing exactly 6 elements possess?
In probability theory, two cards are drawn consecutively without replacement from a standard 52-card deck. What is the probability that both cards are aces?
In probability theory, what is the expected value of a binomial distribution $X \sim B (n, p)$ with parameters $n = 100$ and $p = 0.25$ ?

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