Math questions and answers: school and DİM topics

Question 1

How can I learn math from zero?

Accepted Answer

Start at the very base and don't move on until each topic feels solid: arithmetic, negative numbers, fractions and percentages first, then expressions, equations and functions. Math works like a staircase, so a missing step low down makes everything above it harder; going back to lower-grade textbooks is time well spent. Practice 30–45 minutes a day and, after reading each rule, solve a few examples by hand. Explaining a solution out loud, as if to a friend, quickly shows where your understanding is thin.

Question 2

I don't understand math: what should I do?

Accepted Answer

First pin down exactly where you got lost: each new topic in math builds on earlier ones, so the feeling of “I understand nothing” usually comes from one or two forgotten topics. Go back to them, find a different explanation (another textbook, a video, a teacher or an AI) and work from easy examples to harder ones. When you read a worked solution, ask “why?” at every step, then solve a problem of the same type with no help. In OrujovAI's Solve the steps open one at a time, so you can try each next step yourself before revealing it.

Question 3

How do I prepare for DİM math?

Accepted Answer

Start with the math syllabus on dim.gov.az: it is built on the school course, so turn it into a list of topics and mark your gaps. In DİM's 2026 model, which also applies to 2027 admissions, the math paper in the block (second) stage has 30 tasks: 22 multiple-choice, 5 open tasks with a coded answer and 3 with a written solution, so multiple-choice practice alone is not enough. Do one full timed practice test a week, spend the other days on weak topics and log every mistake in an error notebook. The model can change, so check the official rules for your exam year on the DİM website.

Question 4

How can I get a high score in math?

Accepted Answer

High scores come less from the hardest problems than from not slipping on the easy and medium ones: make the basics automatic and make checking every answer a habit. In the block exam, four wrong multiple-choice answers cancel one correct answer, so a pure guess gains nothing on average, while ruling out even one option makes the guess worth the risk. Wrong answers on open tasks cost nothing, so never leave them blank: write down your best attempt. Practice pacing on mock tests: mark a problem you're stuck on, skip it and come back at the end.

Question 5

Can AI teach me math?

Accepted Answer

Yes, it can be a helpful study partner: it explains a topic at your level, rephrases what you didn't get, writes fresh practice problems and points out the mistake in your working. But AI also slips on arithmetic or misreads a problem, so plug answers back in and check doubtful points against your textbook. You learn more by asking for a hint than for the finished answer. OrujovAI's Live Tutor teaches a whole lesson out loud on a board and checks your understanding with quick questions along the way.

Question 6

How do I add, subtract, multiply and divide fractions?

Accepted Answer

To add or subtract, bring the fractions to a common denominator, then add or subtract the numerators: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. To multiply, multiply numerator by numerator and denominator by denominator: 2/3 × 3/5 = 6/15 = 2/5. To divide, multiply by the reciprocal of the second fraction: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Turn mixed numbers into improper fractions first and simplify the answer at the end.

Question 7

How do I simplify a fraction?

Accepted Answer

Divide the numerator and denominator by the same number, ideally their greatest common divisor (GCD): for 18/24 the GCD is 6, so 18/24 = 3/4. If you can't see the GCD right away, keep dividing by small numbers such as 2, 3 and 5 while a common factor remains. In algebraic fractions, factor the top and bottom first and cancel only common factors: (x² − 4)/(x − 2) = x + 2 for x ≠ 2. You can never cancel terms that are added, so crossing out the 3s in (x + 3)/3 is a mistake.

Question 8

How do I calculate a percentage?

Accepted Answer

A percent is one hundredth, so p% of a number a is a × p/100: 15% of 240 is 240 × 0.15 = 36. To find what percent one number is of another, divide the first by the second and multiply by 100%: 18 out of 72 is 18/72 × 100% = 25%. To find the whole from a part and its percentage, divide: if 30 is 20% of a number, the number is 30 ÷ 0.2 = 150.

Question 9

How do I solve percentage problems?

Accepted Answer

First decide what counts as 100%, because most mistakes come from mixing up the base. An increase of p% means multiplying by 1 + p/100 and a decrease means multiplying by 1 − p/100; for several changes in a row you multiply these factors, so a price that rises 20% and then falls 20% ends at 1.2 × 0.8 = 0.96, which is 4% below where it started. Percentages can't simply be added or subtracted, since each change is taken from the new base.

Question 10

How do I solve an equation?

Accepted Answer

Solving an equation means finding every value that makes the equality true. In a linear equation, move the unknown terms to one side and the numbers to the other (a term changes sign when it crosses over), then divide both sides by the coefficient of the unknown: 3x − 5 = 10 → 3x = 15 → x = 5. If there are brackets or fractions, expand the brackets and multiply the whole equation by the common denominator first. Finally, substitute your root into the original equation to check it.

Question 11

How do I solve a system of equations?

Accepted Answer

The two main methods are substitution and elimination. In substitution you express one unknown from one equation and plug it into the other; in elimination you scale the equations so that adding them cancels a variable: for x + y = 7 and x − y = 1, adding gives 2x = 8, so x = 4 and then y = 3, a pair worth checking in both equations. The graphing method finds where the two graphs intersect, but it usually gives only an approximate answer. For SAT-style systems, OrujovAI's Live Tutor has a free example lesson in English that anyone can replay.

Question 12

How do I solve a quadratic equation?

Accepted Answer

In general, solve ax² + bx + c = 0 (a ≠ 0) with the discriminant: D = b² − 4ac and x = (−b ± √D)/(2a). For x² − 5x + 6 = 0, D = 25 − 24 = 1, so the roots are (5 ± 1)/2, that is 3 and 2. Incomplete equations are quicker: x² − 9 = 0 gives x = ±3, and x² − 4x = 0 factors as x(x − 4) = 0, so x = 0 or x = 4.

Question 13

What is the discriminant and how do I find it?

Accepted Answer

The discriminant is the number that tells you how many real roots a quadratic has; for ax² + bx + c = 0 it is D = b² − 4ac. If D > 0 there are two different roots, if D = 0 there is one (a repeated root), and if D < 0 there are no real roots. For 2x² + 3x − 2 = 0, D = 9 + 16 = 25 and the roots are (−3 ± 5)/4, that is 1/2 and −2. Square a negative coefficient inside brackets: with b = −3, b² = (−3)² = 9, not −9.

Question 14

What is Vieta's theorem?

Accepted Answer

Vieta's theorem links the roots of a quadratic to its coefficients: for ax² + bx + c = 0, the sum of the roots is x₁ + x₂ = −b/a and the product is x₁x₂ = c/a. For a monic equation x² + px + q = 0 it is simply x₁ + x₂ = −p and x₁x₂ = q, so the roots of x² − 7x + 12 = 0 are the two numbers with sum 7 and product 12: 3 and 4. It lets you find and check roots quickly and compute expressions such as x₁² + x₂² = (x₁ + x₂)² − 2x₁x₂ without solving the equation. Use it only when real roots exist (D ≥ 0).

Question 15

How do I solve inequalities?

Accepted Answer

A linear inequality is solved like an equation with one key difference: when you multiply or divide both sides by a negative number, the sign flips: −2x > 6 → x < −3. For quadratic and rational inequalities, first find the roots of the related equation, then work out the signs from the shape of the parabola or with the interval method. Write the answer as an interval and watch the endpoints: square brackets for ≤ and ≥, round brackets for < and >.

Question 16

What is the interval method?

Accepted Answer

It is a way to solve inequalities whose expression is factored or written as a fraction. Move everything to one side, factor, mark the zeros of the numerator and denominator on a number line, and find the sign on each interval with a test point. For (x − 1)(x + 3) < 0 the zeros are −3 and 1 and the expression is negative only between them, so the answer is −3 < x < 1. Zeros of the denominator are never included, and a factor raised to an even power, such as (x − 2)², does not change the sign.

Question 17

How do I solve equations with absolute value?

Accepted Answer

The absolute value of a number is its distance from zero, so it is never negative. The equation |f(x)| = a has no solution when a < 0; when a ≥ 0 it splits into f(x) = a or f(x) = −a, so |2x − 3| = 5 gives x = 4 or x = −1. With several absolute values, split the number line at the points where each expression inside is zero and open the absolute values with the right signs on each interval. If the variable also appears on the right-hand side, check every root you find.

Question 18

How do I solve irrational equations?

Accepted Answer

Isolate the root on one side, square both sides, solve the new equation and check every root in the original, because squaring can create extraneous roots. For example, √(x + 3) = x − 3 → x + 3 = x² − 6x + 9 → x² − 7x + 6 = 0, so x = 6 or x = 1. Checking x = 1 gives 2 on the left and −2 on the right, so it is extraneous and the only answer is x = 6. An expression under an even root, and the root itself, can't be negative, so writing these conditions down first also helps.

Question 19

How do I solve exponential equations?

Accepted Answer

The main trick is to write both sides as powers of the same base: when a > 0 and a ≠ 1, equal powers mean equal exponents, so 3²ˣ⁻¹ = 27 = 3³ gives 2x − 1 = 3 and x = 2. If the same power keeps appearing in different forms, substitute a new variable: in 4ˣ − 3 × 2ˣ − 4 = 0, setting t = 2ˣ > 0 gives t² − 3t − 4 = 0, so t = 4 or t = −1, the negative value is rejected and x = 2. If the bases can't be matched, take logarithms of both sides.

Question 20

What is a logarithm?

Accepted Answer

The logarithm of b to base a (logₐb) is the power you must raise a to in order to get b: log₂8 = 3 because 2³ = 8. It requires a > 0, a ≠ 1 and b > 0, so negative numbers and zero have no logarithm. The common logarithm (base 10) is written lg or log, and the natural logarithm (base e ≈ 2.718) is written ln. Logarithms are what you use when the unknown sits in an exponent and you need to bring it down.

Question 21

How do I solve logarithmic equations?

Accepted Answer

Start with the domain: every expression under a logarithm must be positive, and the base must be positive and not equal to 1. Then either use the definition (log₂(x + 1) = 3 → x + 1 = 2³ → x = 7) or rewrite both sides as logarithms to the same base and set the arguments equal. For example, log₃(x² − 4) = log₃(3x) → x² − 3x − 4 = 0 → x = 4 or x = −1, but x = −1 makes 3x negative, so the only answer is x = 4. In harder equations a substitution such as t = logₐx makes the work easier.

Question 22

What are the main logarithm rules?

Accepted Answer

The core rules (for x, y > 0) are logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy and logₐ(xⁿ) = n logₐx. Add logₐ1 = 0, logₐa = 1 and the basic identity: a raised to the power logₐb equals b. The change-of-base formula logₐb = lg b / lg a helps both with calculations and with comparing logarithms that have different bases. A classic mistake: logₐ(x + y) is the logarithm of a sum and is not equal to logₐx + logₐy.

Question 23

What is trigonometry?

Accepted Answer

Trigonometry is the branch of math that studies the relationships between the angles and sides of triangles, and the functions that grow out of them: sine, cosine and tangent. At school it starts with right triangles, then moves on to the unit circle, identities, equations and graphs. Angles are measured in radians as well as degrees: 180° = π radians. Trigonometry is used throughout geometry, physics (oscillations, waves) and engineering.

Question 24

How do I solve trigonometric equations?

Accepted Answer

The goal is to reduce the equation, using identities, factoring or a substitution, to basic ones like sin x = a, cos x = a or tan x = a. Their general solutions are x = (−1)ⁿ arcsin a + πn for sine, x = ±arccos a + 2πn for cosine and x = arctan a + πn for tangent, where n is any integer; sine and cosine equations have no solution when |a| > 1. For example, 2sin x − 1 = 0 → sin x = 1/2 → x = π/6 + 2πn or x = 5π/6 + 2πn. Don't forget the period, and if the problem asks for roots in an interval, try consecutive values of n and keep the ones that fit.

Question 25

What are sine, cosine and tangent?

Accepted Answer

In a right triangle, the sine of an acute angle α is the opposite side divided by the hypotenuse, the cosine is the adjacent side divided by the hypotenuse, and the tangent is the opposite side divided by the adjacent side. On the unit circle, the point for angle α has coordinates (cos α, sin α), and tan α = sin α / cos α. The values you need most: sin 30° = cos 60° = 1/2, sin 45° = cos 45° = √2/2 and tan 45° = 1. The identity sin²α + cos²α = 1 lets you find one function from another.

Question 26

What is an arithmetic progression?

Accepted Answer

An arithmetic progression is a sequence where each term is the previous one plus the same number d (the common difference), for example 2, 5, 8, 11, ... with d = 3. The nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = (a₁ + aₙ)n/2: here a₁₀ = 2 + 9 × 3 = 29 and S₁₀ = (2 + 29) × 10/2 = 155. Its key property is that every term except the first and last is the average of its two neighbors. That makes it easy to check whether three numbers form an arithmetic progression.

Question 27

What is a geometric progression?

Accepted Answer

A geometric progression is a sequence of nonzero numbers in which you multiply by the same number q (the common ratio) to get from one term to the next, for example 3, 6, 12, 24, ... with q = 2. The nth term is bₙ = b₁qⁿ⁻¹, and for q ≠ 1 the sum of the first n terms is Sₙ = b₁(qⁿ − 1)/(q − 1): here b₆ = 3 × 2⁵ = 96. When |q| < 1, the infinite decreasing progression has the sum S = b₁/(1 − q), so 1 + 1/2 + 1/4 + ... = 2.

Question 28

What is a function?

Accepted Answer

A function is a rule that assigns exactly one value y to each x in its domain, written y = f(x). All the allowed x values form the domain, and all the resulting y values form the range. A function can be given by a formula, a table or a graph: y = 2x + 1 is linear and y = x² is quadratic. To tell whether a curve is the graph of a function, use the vertical line test: every vertical line must cross it at most once.

Question 29

How do I draw the graph of a function?

Accepted Answer

Find the domain, the intercepts with the axes and, if you can, where the function increases or decreases, then make a table of a few points and join them with a smooth curve. Learn the basic graphs (line, parabola, hyperbola, y = √x, y = |x|) and use transformations: for a, b > 0, f(x) + b shifts the graph b units up and f(x − a) shifts it a units right. For a parabola, find the vertex first: x₀ = −b/(2a). In the OrujovAI chat, typing “graph y = x² − 2x − 3” is enough to get a graph you can zoom and drag.

Question 30

What is a derivative?

Accepted Answer

A derivative measures how fast a function changes: f'(x) is the limit of the change in the function divided by the change in x, as the change in x approaches zero. Geometrically it is the slope of the tangent line to the graph at that point; physically, the derivative of position with respect to time is velocity. Where the derivative is positive the function increases, where it is negative it decreases, and maxima and minima sit where it changes sign. That is why derivatives are the main tool for finding largest and smallest values.

Question 31

How do I find a derivative?

Accepted Answer

You need a short table and a few rules: (c)' = 0, (xⁿ)' = nxⁿ⁻¹, (sin x)' = cos x, (cos x)' = −sin x, (eˣ)' = eˣ and (ln x)' = 1/x. The derivative of a sum is the sum of the derivatives, constant factors come out front, the product rule is (uv)' = u'v + uv', and the quotient rule is (u/v)' = (u'v − uv')/v². For example, f(x) = 3x⁴ − 2x + 5 gives f'(x) = 12x³ − 2. To check your own work, type the problem into OrujovAI's Solve and compare its steps with yours.

Question 32

How do I find the derivative of a composite function?

Accepted Answer

Use the chain rule: the derivative of f(g(x)) is the derivative of the outer function evaluated at the inner one, times the derivative of the inner function, (f(g(x)))' = f'(g(x)) × g'(x). For example, ((3x + 1)⁵)' = 5(3x + 1)⁴ × 3 = 15(3x + 1)⁴ and (sin 2x)' = 2cos 2x. First decide which function is outer and which is inner, differentiate the outer one while leaving the inner one unchanged, then multiply by the inner derivative. The most common mistake is forgetting that final multiplication.

Question 33

What is an integral?

Accepted Answer

Integration is the reverse of differentiation: if F'(x) = f(x), then F(x) is an antiderivative of f(x), and the indefinite integral is written ∫f(x)dx = F(x) + C. A definite integral is a number: when f(x) ≥ 0, it equals the area between the graph and the x-axis over the interval [a, b]. It is calculated with the Newton–Leibniz formula (the fundamental theorem of calculus): ∫ₐᵇ f(x)dx = F(b) − F(a). Integrals give areas and volumes, and in physics quantities such as distance and work.

Question 34

How do I calculate an integral?

Accepted Answer

For indefinite integrals, use the basic table: ∫xⁿdx = xⁿ⁺¹/(n + 1) + C (n ≠ −1), ∫(1/x)dx = ln|x| + C, ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C and ∫eˣdx = eˣ + C. The integral of a sum is the sum of the integrals, and constant factors come outside: ∫(4x³ + 2x)dx = x⁴ + x² + C. For a definite integral, find an antiderivative and subtract its value at the lower limit from its value at the upper limit: ∫₀² x²dx = 2³/3 − 0 = 8/3. Checking is easy: differentiate your answer and you should get back the function you integrated.

Question 35

What is a limit and how do I find it?

Accepted Answer

A limit is the value f(x) approaches as x gets closer to some number, or to infinity. First simply substitute the value of x: if you get a number, that is the answer. If you get 0/0, factor and cancel or multiply by the conjugate: as x → 2, (x² − 4)/(x − 2) = x + 2 → 4. As x → ∞, divide the numerator and denominator by the highest power of x, and as x → 0 remember the standard limit sin x / x → 1.

Question 36

What is the Pythagorean theorem?

Accepted Answer

The Pythagorean theorem says that in a right triangle the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². The hypotenuse is the longest side, opposite the right angle. The converse is also true: if a triangle's sides satisfy c² = a² + b², the triangle is right-angled. The whole-number triples you'll meet most often are 3, 4, 5 and 5, 12, 13, along with their multiples such as 6, 8, 10.

Question 37

How do I solve problems with the Pythagorean theorem?

Accepted Answer

Find a right triangle in the diagram, or create one with an extra line, identify the hypotenuse and solve for the unknown side: a leg is a = √(c² − b²) and the hypotenuse is c = √(a² + b²). For example, if the foot of a 5 m ladder stands 3 m from a wall, its top reaches √(25 − 9) = 4 m up. The diagonal of a rectangle, the height of an isosceles triangle and the distance between two points on the coordinate plane are all found the same way.

Question 38

How do I find the area of a triangle?

Accepted Answer

The most common formula is S = ½ah, where a is a side and h is the height drawn to it: a triangle with base 10 and height 6 has an area of 30. With two sides and the angle between them, use S = ½ab sin C; with all three sides, use Heron's formula S = √(p(p − a)(p − b)(p − c)), where p is half the perimeter. A right triangle's area is half the product of its legs, and an equilateral triangle with side a has area a²√3/4.

Question 39

How do I find the area and circumference of a circle?

Accepted Answer

For a circle of radius r, the area is A = πr² and the circumference is C = 2πr = πd, where d is the diameter. With r = 5 cm, A = 25π ≈ 78.5 cm² and C = 10π ≈ 31.4 cm. Test answers are often left in terms of π, so don't rush to multiply by 3.14 unless the question asks for it. If you're given the diameter, halve it first, since the area formula uses the radius, not the diameter.

Question 40

How do I find the sum of a polygon's interior angles?

Accepted Answer

The interior angles of a convex polygon with n sides add up to 180°(n − 2): 360° for a quadrilateral, 540° for a pentagon and 720° for a hexagon. The reason is simple: the diagonals from one vertex split the polygon into n − 2 triangles, and each triangle's angles add up to 180°. Each angle of a regular polygon is 180°(n − 2)/n, so a regular hexagon has 120° angles. The exterior angles, one at each vertex, always add up to 360°.

Question 41

How do I find the volume of a prism?

Accepted Answer

The volume of a prism is the base area times the height: V = S × h. For a rectangular box that becomes V = abc, and for a cube V = a³. For example, if the base is a right triangle with legs 3 and 4 and the prism is 10 tall, S = ½ × 3 × 4 = 6 and V = 6 × 10 = 60. In an oblique prism the height is not the slanted edge but the perpendicular distance between the bases.

Question 42

How do I find the volume of a pyramid?

Accepted Answer

A pyramid's volume is one third of the base area times the height: V = ⅓Sh. For a pyramid with a square base of side 6 and a height of 5, S = 36 and V = ⅓ × 36 × 5 = 60. So a pyramid fills a third of a prism with the same base and height, and the volume of a cone (V = ⅓πr²h) follows the same logic. If the height isn't given, you usually find it with the Pythagorean theorem from a right triangle inside the pyramid.

Question 43

What is a vector?

Accepted Answer

A vector is a quantity with both size (length) and direction, drawn as an arrow; force and velocity are examples from physics. In the plane a vector is given by coordinates: a = (x, y) has length √(x² + y²), and addition and multiplication by a number work coordinate by coordinate. The dot product of a = (x₁, y₁) and b = (x₂, y₂) equals x₁x₂ + y₁y₂ and also |a||b|cos φ, where φ is the angle between them. If the dot product is zero, the vectors are perpendicular.

Question 44

What is probability and how do I calculate it?

Accepted Answer

Probability expresses how likely an event is as a number from 0 to 1: 0 means impossible and 1 means certain. When all outcomes are equally likely, P = m/n, where m is the number of favorable outcomes and n is the number of all outcomes. For example, the probability of rolling an even number on a die is 3/6 = 1/2. The probability that the event does not happen is 1 − P.

Question 45

How do I solve probability problems?

Accepted Answer

Start by carefully counting all possible outcomes and the favorable ones: a list or table works for small problems, combinatorics formulas for bigger ones. For independent events that must all happen, multiply the probabilities; for mutually exclusive events where any one of them will do, add them. With “at least one” problems, the complement is faster: the chance of at least one head when tossing two coins is 1 − 1/4 = 3/4. The answer must always be between 0 and 1, so anything larger means a mistake somewhere.

Question 46

What is combinatorics?

Accepted Answer

Combinatorics counts the ways to choose and arrange things. The number of permutations of n different items is n!, the number of ordered selections of k items from n is n!/(n − k)!, and the number of combinations, where order doesn't matter, is C(n, k) = n!/(k!(n − k)!). For example, a 2-person team can be picked from 5 people in 10 ways, but a president and a vice president in 5 × 4 = 20 ways. The key question in every problem is whether order matters.

Question 47

How do I solve motion (speed and distance) problems?

Accepted Answer

Every motion problem rests on distance = speed × time, so put the distance, speed and time for each moving object into a table and draw a sketch. When objects move toward each other their speeds add (the closing speed), and when they move in the same direction you subtract: two cars 300 km apart driving toward each other at 60 and 90 km/h meet after 300 ÷ 150 = 2 hours. On a river, the speed downstream is v + v₀ and upstream v − v₀, where v₀ is the speed of the current. Keep units consistent by converting minutes to hours and meters to kilometers.

Question 48

How do I solve work problems?

Accepted Answer

Treat the whole job as 1: someone who finishes it in t hours does 1/t of it per hour, and when people work together their rates add. If one worker needs 6 hours and another 3, together they do 1/6 + 1/3 = 1/2 of the job per hour and finish in 2 hours. Pool-and-pipe problems work the same way, with a draining pipe counted as a negative rate. Sanity-check the answer: working together must be faster than the fastest worker alone.

Question 49

How do I solve mixture problems?

Accepted Answer

The key idea is that when you mix solutions, the amounts of pure substance add up, not the percentages. For each solution, the pure amount is mass times concentration: mixing 200 g of a 10% solution with 300 g of a 20% solution gives 20 + 60 = 80 g of substance in 500 g, a concentration of 80/500 = 16%. Adding water leaves the amount of substance unchanged and only increases the total mass, while evaporation does the opposite. Alloy problems are solved the same way, with metal in place of the dissolved substance.

Question 50

How do I solve a proportion?

Accepted Answer

A proportion is an equality of two ratios, a/b = c/d, and the rule to remember is that the product of the extremes equals the product of the means, ad = bc, which is why you can cross-multiply. For example, x/4 = 15/6 gives 6x = 60 and x = 10. In a word problem, first decide whether the quantities are directly or inversely proportional: if 3 kg of apples cost 6 manats, 5 kg cost 10 manats (direct), but if 4 workers finish a job in 6 days, 8 workers need 3 days (inverse).

Question 51

What is a matrix?

Accepted Answer

A matrix is a rectangular table of numbers arranged in rows and columns; one with m rows and n columns is called an m × n matrix. Matrices of the same size are added entry by entry, multiplying by a number multiplies every entry, and the product of two matrices uses the “row times column” rule and exists only when the first has as many columns as the second has rows. The determinant of a 2 × 2 matrix is ad − bc, where a, b form the first row and c, d the second. Matrices are used to write and solve systems of linear equations compactly, and in computer graphics and data analysis.

Question 52

How can I memorize math formulas quickly?

Accepted Answer

A more reliable way than cramming is to understand where a formula comes from: once you have derived the polygon angle sum from triangles, or the arithmetic progression sum by pairing terms, they are hard to forget. Write your own formula sheet by hand, then try to rewrite it without looking, and use each formula right away in two or three problems. Review at growing intervals: after one day, three days and a week. If you ask the OrujovAI chat to “make flashcards for the logarithm rules”, the cards open right inside the conversation.

Question 53

Where can I find DİM-format math tests?

Accepted Answer

The most reliable source is DİM itself: it publishes test collections and the “Abituriyent” journal with explained tasks and correct answers, and it runs its own mock exams, all listed on dim.gov.az. Tutoring centers and websites write DİM-format tests too, but quality varies, so check the publication year and whether they match the current exam model. In OrujovAI's exam practice you can take short DİM-format math tests at three levels with an explanation for every question; they are practice, not official tasks, and for now multiple-choice only, so practice coded and written tasks with DİM's own materials.

Question 54

How do I solve logic problems?

Accepted Answer

Read the problem carefully, ideally twice, and jot down each fact, because in logic puzzles a single word (“at least”, “only”, “none”) can change the whole solution. When there are many possibilities, draw a grid and cross out the impossible cases; with number sequences, look at the differences or ratios between neighboring terms and for a repeating pattern. Other useful moves are trying small cases, working backward from the end and checking parity (odd and even). Test your answer against every condition: if even one fails, the solution is wrong.

Question 55

How can I learn to solve geometry problems?

Accepted Answer

First learn the definitions and core theorems well: congruence and similarity of triangles, the Pythagorean theorem, angles formed by parallel lines, angles in a circle and the area formulas. For every problem, draw a large, neat diagram, mark the given data on it and look for right or similar triangles, adding an auxiliary line such as a height or a diagonal when needed. Work through typical problems topic by topic and keep a notebook of solution ideas, because the same tricks keep coming back in geometry. When you're stuck, send a photo of the diagram to OrujovAI's Solve, but open only the first step as a hint.

Question 56

How should I write answers to open math questions?

Accepted Answer

In DİM exams, open tasks come in two kinds: for coded-answer tasks only the final answer is checked, so recheck your arithmetic and code it exactly as the sample on the answer sheet shows, while for written tasks the solution itself is graded. In a written solution, state what is given, which formula you use and why, show the calculations and end with a clear answer; DİM's announcement for 2027 admissions says one math task will be a proof, where every step needs a theorem or definition behind it. Written tasks are marked against criteria and in parts, so even if you can't finish, write down the part you got right. Find sample tasks in DİM's publications, and show your own solution to a teacher or an AI with the request “check this like an examiner: which step is missing?”.