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Algebra, Functions, Calculus and Probability

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

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Solve the equation $4 x - 9 = 15$ .
Write $\frac{5 ^{6}}{5 ^{2}}$ as a single power of 5.
What is the distance between the points $(2, 1)$ and $(6, 4)$ ?
What is the area of a triangle with base 10 and height 7?
What is the value of $tan 4 5^{\circ} + sin 9 0^{\circ}$ ?
Solve $x^{2} - 8 x + 15 = 0$ .
An arithmetic sequence has $a_{1} = 4$ and $d = 5$ . What is the sum of its first 8 terms?
A geometric sequence has $a_{1} = 2$ and ratio $q = 3$ . What is $a_{5}$ ?
Evaluate $lo g_{3} 27 + l g 100$ .
What is the volume of a cylinder with radius 5 and height 2?
What is the volume of a cone with base radius 3 and height 4?
Find the derivative of $f (x) = x^{3} - 6 x^{2} + 2$ .
Find the derivative of $g (x) = ln (3 x)$ .
Find the derivative of $h (x) = x e^{x}$ .
Evaluate $\int_{1}^{3} 2 x d x$ .
Two fair dice are thrown. What is the probability that the sum is 8?
In how many ways can 3 students be chosen from a group of 7?
Solve $2^{x + 1} = 32$ .
What is the mean of the numbers $6, 8, 9, 11, 16$ ?
Solve $∣ x + 2∣ = 5$ .
What are the coordinates of the vertex of $y = x^{2} - 2 x - 3$ ?
Solve the system $x + 2 y = 8$ and $3 x - y = 3$ .
A coat costs 80 euros. It is sold with a discount of 30 percent. What is the new price?
Calculate $(3 + 2 i) (3 - 2 i)$ , where $i^{2} = - 1$ .
Solve $cos x = \frac{1}{2}$ for $0 \leq x \leq 2 π$ .
Solve the inequality $x^{2} - 4 x - 5 > 0$ .
Solve $ln (x + 1) = 2$ .
What is the domain of $f (x) = \frac{1}{x - 2} + x + 3$ ?
A sum of 2000 euros earns 4 percent interest per year, added to the account each year. How much is there after 3 years?
A car drives 120 km at 60 km/h and then another 120 km at 40 km/h. What is its average speed for the whole trip?
What is the sum of the infinite series $9 + 3 + 1 + \frac{1}{3} + \dots$ ?
What is the area of the region between the graphs of $y = 2 x$ and $y = x^{2}$ ?
A fair coin is tossed 4 times. What is the probability of getting at least one head?
The vectors $a = [2, - 1]$ and $b = [3, k]$ are parallel. What is $k$ ?
Square corners of side $x$ are cut from a square sheet of side 12, and the sides are folded up to make an open box. What is the largest possible volume?
What is the equation of the tangent line to $y = x^{3}$ at the point where $x = 1$ ?
In mathematics, what is the solution to the linear equation $3 (2 x - 5) = 27$ in real numbers?
In mathematics, what is the simplified value of the exponent expression $\frac{4 ^{5} \cdot 4 ^{3}}{4 ^{6}}$ ?
In mathematics, what are the real roots of the quadratic polynomial equation $x^{2} - 7 x + 12 = 0$ ?
In mathematics, what is the exact value of the logarithmic expression $lo g_{3} (81)$ ?
In mathematics, what is the evaluated value of $lo g_{2} (32) - lo g_{2} (4)$ using logarithmic laws?
In mathematics, what is the numerical discriminant of the quadratic equation $2 x^{2} - 5 x + 3 = 0$ ?
In mathematics, what is the natural domain of the real-valued function $f (x) = 2 x - 8$ ?
In mathematics, what are the coordinates of the vertex of the parabola $y = x^{2} - 6 x + 5$ ?
In mathematics, what is the solution set to the quadratic inequality $x^{2} - 9 < 0$ on the real number line?
In mathematics, what is the value of the finite summation $\sum_{k = 1}^{5} (2 k + 1)$ ?
In mathematics, what is the 10th term of an arithmetic progression with first term $a_{1} = 4$ and difference $d = 3$ ?
In mathematics, what is the sum of the first 20 terms of an arithmetic progression where $a_{1} = 2$ and $a_{20} = 60$ ?
In mathematics, what is the 5th term of a geometric sequence where $b_{1} = 3$ and common ratio $q = 2$ ?
In mathematics, what is the sum of the convergent infinite geometric series $8 + 4 + 2 + 1 + \dots$ ?
In mathematics, what is the common ratio of a geometric progression where $b_{2} = 6$ and $b_{4} = 54$ with positive terms?
In mathematics, what is the common difference $d$ of an arithmetic progression where $a_{3} = 11$ and $a_{7} = 27$ ?
In mathematics, what is the 8th term of an arithmetic progression with general term formula $a_{n} = 5 n - 3$ ?
In mathematics, what is the third term of a geometric sequence where $b_{1} = 12$ and common ratio $q = - \frac{1}{2}$ ?
In mathematics, what is the sum of the first 5 terms of a geometric progression where $b_{1} = 2$ and $q = 3$ ?
In mathematics, what is the first term $a_{1}$ of an arithmetic progression if $a_{10} = 52$ and common difference $d = 5$ ?
In mathematics, what is the exact numerical value of $sin (3 0^{\circ}) + cos (6 0^{\circ})$ ?
In mathematics, what is the value of the trigonometric product $tan (4 5^{\circ}) \cdot cos (0^{\circ})$ ?
In mathematics, which fundamental identity holds true for any real angle $α$ in trigonometry?
In mathematics, what is the fundamental period of the real trigonometric function $f (x) = sin (4 x)$ ?
In mathematics, what is the length of the hypotenuse of a right triangle whose legs measure 9 cm and 12 cm?
In mathematics, what is the geometric area of an equilateral triangle with side length $a = 6$ cm?
In mathematics, what is the volume of a right circular cylinder having base radius $r = 3$ cm and height $h = 7$ cm?
In mathematics, what is the scalar dot product of the Cartesian vectors $u = (3, - 2)$ and $v = (4, 5)$ ?
In mathematics, what is the straight-line distance between points $A (1, 2)$ and $B (4, 6)$ in the coordinate plane?
In mathematics, what is the standard Cartesian equation of a circle centered at origin $(0, 0)$ with radius $R = 5$ ?
In mathematics, what is the first derivative of the polynomial function $f (x) = 4 x^{3} - 5 x^{2} + 7 x - 9$ ?
In mathematics, what is the derivative of the exponential function $f (x) = e^{3 x}$ with respect to variable $x$ ?
In mathematics, what is the evaluated value of the definite integral $\int_{0}^{2} (3 x^{2} + 2 x) d x$ ?
In mathematics, what are the stationary critical points of the cubic polynomial function $f (x) = 2 x^{3} - 9 x^{2} + 12 x - 1$ ?
In mathematics, what is the limiting value of the trigonometric expression $lim_{x \to 0} \frac{s i n ( 3 x )}{x}$ ?
In mathematics, what is the slope of the tangent line drawn to the polynomial curve $y = x^{3} - 2 x + 4$ at point $x = 2$ ?
In mathematics, what is the standard antiderivative $\int \frac{1}{x} d x$ on the positive real domain $x > 0$ ?
In mathematics, what is the second derivative $f^{''} (x)$ of the trigonometric function $f (x) = cos (2 x)$ ?
In mathematics, what is the local minimum value attained by the quadratic function $f (x) = x^{2} - 4 x + 7$ ?
In mathematics, what is the total area bounded between the curve $y = x^{2}$ and the x-axis on interval $[0, 3]$ ?
In mathematics, what is the derivative of the rational function $f (x) = \frac{x}{x + 1}$ for $x \neq = - 1$ ?
In mathematics, what is the value of the definite integral $\int_{0}^{π} sin (x) d x$ ?
In mathematics, what is the horizontal asymptote of the rational function $f (x) = \frac{3 x ^{2} - 5}{x ^{2} + 2}$ as $x \to \pm \infty$ ?
In mathematics, what is the derivative of the product function $f (x) = x \cdot e^{x}$ with respect to $x$ ?
In mathematics, what is the inflection point x-coordinate of the cubic function $f (x) = x^{3} - 6 x^{2} + 9 x$ ?
In mathematics, how many distinct 4-digit PIN access codes can be formed using digits 0-9 if repetition is permitted?
In mathematics, in how many different ways can a committee of 3 delegates be chosen from a group of 8 candidates?
In mathematics, what is the theoretical probability of obtaining a sum of 7 when rolling two fair standard six-sided dice?
In mathematics, if independent events $A$ and $B$ have probabilities $P (A) = 0.4$ and $P (B) = 0.5$ , what is $P (A \cap B)$ ?
In mathematics, what is the statistical median of the ordered numerical dataset ${3, 7, 8, 12, 14, 19, 21}$ ?
In mathematics, what is the probability of randomly drawing an Ace or a King from a standard 52-card playing deck?
In mathematics, what is the sum of all interior geometric angles in a convex regular hexagon (6-sided polygon)?
In mathematics, what is the geometric volume of a three-dimensional sphere having radius $R = 3$ cm?
In mathematics, what is the value of $x$ satisfying the logarithmic equation $lo g_{2} (x + 3) = 4$ ?
In mathematics, what is the first derivative of the composite logarithmic function $f (x) = ln (x^{2} + 1)$ ?
In mathematics, what is the complete set of solutions to $cos (x) = 0$ on the trigonometric interval $[0, 2 π)$ ?
In mathematics, what is the asymptotic limit $lim_{x \to \infty} \frac{5 x ^{2} - 3 x + 2}{2 x ^{2} + 7 x - 1}$ ?

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