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

Algebra, Calculus, Geometry and Probability

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

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All 40 questions

Solve the equation $3 x - 5 = 16$ .
What are the roots of $x^{2} - 5 x + 6 = 0$ ?
How many real roots does $x^{2} - 4 x + 7 = 0$ have?
Find the distance between the points $A (1, 2)$ and $B (4, 6)$ .
An arithmetic progression has first term 3 and common difference 4. What is the sum of its first 10 terms?
A geometric progression has $b_{1} = 2$ and ratio $q = 3$ . Find $b_{5}$ .
Evaluate $lo g_{2} 32$ .
Evaluate $lo g_{3} 81 - lo g_{3} 9$ .
At which value of $x$ does $f (x) = x^{3} - 3 x$ have a local minimum?
Evaluate $\int_{0}^{2} 3 x^{2} d x$ .
The legs of a right triangle are 6 and 8. What is the hypotenuse?
A circle has circumference $10 π$ . What is its area?
A cone has base radius 3 and height 4. What is its volume?
What is the length of the space diagonal of a cube with edge 2?
Two fair dice are rolled. What is the probability that the sum is 7?
In how many ways can a committee of 2 people be chosen from 5 people?
If $sin α = \frac{3}{5}$ and $α$ is an acute angle, what is $cos α$ ?
Solve $2 sin x = 1$ for $x$ in the interval $[0, \frac{π}{2}]$ .
Solve the inequality $∣ x - 2∣ < 3$ .
Solve $x^{2} - 4 > 0$ .
The vectors $a = (2, 3)$ and $b = (k, - 4)$ are perpendicular. Find $k$ .
A price rises by 20% and then falls by 20%. How does the final price compare with the original?
Solve the system $x + y = 10$ , $x - y = 2$ .
What is the domain of $f (x) = x - 3$ ?
Solve $2^{x + 1} = 32$ .
Find the sum of the infinite geometric series $6 + 2 + \frac{2}{3} + \dots$ .
What are the coordinates of the vertex of the parabola $y = x^{2} - 6 x + 5$ ?
Find the equation of the tangent line to $y = x^{2}$ at the point where $x = 1$ .
What is the largest value of $f (x) = - x^{2} + 4 x + 1$ ?
In a triangle two sides are 5 and 8 and the angle between them is $6 0^{\circ}$ . Find the third side.
A regular square pyramid has a base edge of 6 and a height of 4. What is its volume?
A regular square pyramid has a base edge of 6 and a height of 4. What is the area of its four lateral faces?
A bag holds 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
Find the area of the region between the graphs of $y = x$ and $y = x^{2}$ for $0 \leq x \leq 1$ .
Solve $lo g_{2} x + lo g_{2} (x - 2) = 3$ .
In an arithmetic progression $a_{3} = 7$ and $a_{7} = 19$ . Find $a_{10}$ .
Simplify $\frac{a ^{2} - b ^{2}}{a - b}$ for $a \neq = b$ .
Let $f (x) = 2 x + 1$ and $g (x) = x^{2}$ . Find $f (g (3))$ .
Worker A can finish a job in 6 hours and worker B in 3 hours. How long will they take working together?
Differentiate $y = ln (x^{2} + 1)$ .

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