NMT National Multi-Subject (Ukraine) · Mathematics (NMT)

Advanced Mathematics for the NMT

54 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 54 questions

In algebra, what are the roots of the quadratic equation $x^{2} - 7 x + 10 = 0$ ?
In algebra, what is the value of $lo g_{3} (81)$ ?
In algebra, what is the solution set of the linear inequality $4 - 2 x \geq 10$ ?
In algebra, what is the 10th term of an arithmetic progression with first term $a_{1} = 3$ and common difference $d = 4$ ?
In algebra, what is the 6th term of a geometric progression with first term $b_{1} = 3$ and common ratio $q = 2$ ?
In algebra, what is the domain of the real-valued function $f (x) = 2 x - 8$ ?
In algebra, what is the sum of the roots of the polynomial equation $2 x^{2} - 12 x + 7 = 0$ ?
In algebra, what is the solution to the exponential equation $4^{x + 1} = 64$ ?
In algebra, what is the solution to the logarithmic equation $lo g_{2} (x - 3) = 4$ ?
In algebra, what is the value of the algebraic expression $(a + b)^{2} - (a - b)^{2}$ ?
In trigonometry, what is the exact value of $sin (3 0^{\circ}) + cos (6 0^{\circ})$ ?
In trigonometry, which trigonometric identity expresses $cos (2 α)$ in terms of $cos (α)$ and $sin (α)$ ?
In trigonometry, what is the fundamental period of the tangent function $f (x) = tan (x)$ ?
In trigonometry, what are the solutions of $sin (x) = 0$ across the real numbers?
In trigonometry, what is the value of $tan (4 5^{\circ})$ ?
In calculus, what is the derivative of the function $f (x) = x^{4} - 3 x^{2} + 5$ with respect to $x$ ?
In calculus, what is the derivative of the natural exponential function $f (x) = e^{3 x}$ ?
In calculus, what is the slope of the tangent line to the parabola $y = x^{2}$ at the point where $x = 3$ ?
In calculus, what is the critical point of the polynomial function $f (x) = x^{2} - 6 x + 8$ ?
In calculus, what is the value of the definite integral $\int_{0}^{3} x^{2} d x$ ?
In calculus, what is the general antiderivative $\int cos (x) d x$ ?
In calculus, what is the value of the definite integral $\int_{1}^{4} \frac{1}{x} d x$ ?
In geometry, what is the area of a right-angled triangle with legs of lengths $6$ and $8$ ?
In geometry, what is the length of the hypotenuse of a right-angled triangle with legs $5$ and $12$ ?
In geometry, what is the sum of the interior angles of a convex hexagon (6-sided polygon)?
In geometry, what is the area of an equilateral triangle with side length $a = 4$ ?
In geometry, what is the circumference of a circle with radius $r = 7$ ?
In geometry, what is the area of a circle with diameter $d = 10$ ?
In geometry, what is the area of a trapezoid with parallel bases $a = 6$ , $b = 10$ , and height $h = 5$ ?
In geometry, what is the area of a rhombus whose diagonals have lengths $d_{1} = 8$ and $d_{2} = 14$ ?
In coordinate geometry, what is the distance between the points $A (1, 2)$ and $B (4, 6)$ in the Cartesian plane?
In coordinate geometry, what are the coordinates of the midpoint of the line segment joining $P (2, - 4)$ and $Q (8, 2)$ ?
In vector algebra, what is the scalar product (dot product) of vectors $u = (2, 5)$ and $v = (3, - 1)$ ?
In vector algebra, under what condition are two non-zero vectors $a$ and $b$ orthogonal?
In stereometry, what is the volume of a right circular cylinder with base radius $r = 3$ and height $h = 5$ ?
In stereometry, what is the volume of a right circular cone with base radius $r = 6$ and height $h = 4$ ?
In stereometry, what is the surface area of a sphere with radius $r = 5$ ?
In stereometry, what is the total surface area of a cube with edge length $a = 3$ ?
In stereometry, what is the volume of a rectangular prism with edge lengths $4$ , $5$ , and $6$ ?
In probability theory, what is the probability of rolling an odd number on a standard fair six-sided die?
In combinatorics, what is the number of combinations $C_{5}^{2}$ (choosing 2 elements from 5 without order)?
In combinatorics, how many distinct permutations can be formed using all 4 letters of the word 'MATH'?
In statistics, what is the median of the ordered dataset ${3, 5, 7, 9, 11, 13}$ ?
In algebra, what is the solution of the radical equation $3 x + 1 = 5$ ?
In algebra, what is the value of the finite geometric series sum $S_{4} = 1 + 2 + 4 + 8$ ?
In trigonometry, what is the value of $cos (0)$ ?
In calculus, what is the second derivative $f^{''} (x)$ of the function $f (x) = x^{3} - 6 x^{2}$ ?
In coordinate geometry, what is the slope of the line given by the equation $2 x - 4 y + 7 = 0$ ?
In geometry, in a right triangle with an acute angle of $3 0^{\circ}$ and hypotenuse of length $10$ , what is the length of the opposite leg?
In algebra, what is the remainder when the polynomial $P (x) = x^{3} - 2 x^{2} + 4 x - 5$ is divided by $(x - 1)$ ?
In algebra, what is the range of the quadratic function $f (x) = (x - 2)^{2} + 3$ across all real numbers?
In calculus, at which $x$ -value does the function $f (x) = x^{2} - 4 x + 7$ attain its global minimum?
In probability theory, if events $A$ and $B$ are mutually exclusive with $P (A) = 0.3$ and $P (B) = 0.4$ , what is $P (A \cup B)$ ?
In algebra, what is the solution set of the absolute value inequality $∣ x - 5∣ < 3$ ?

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