Luyện Toán IB

The common difference is 4, so the 6th term is 3 + 5(4) = 23.

a10 = 2 + 9(5) = 47. In an arithmetic sequence, the nth term is a1 + (n-1)d, where a1 is the first term and d is the common difference. Substituting n = 10, a1 = 2, and d = 5 gives 2 + 45 = 47.

S12 = (12/2)(2(5) + 11(4)) = 6(10 + 44) = 6 x 54 = 324.

Multiply the powers: x^(2 x 5) = x^10. When raising a power to another power, keep the base and multiply the exponents. This exponent rule applies to all real exponents, so (x^2)^5 is x^(2*5) = x^10.

Add the powers: 2^7 = 128. When multiplying powers with the same base, keep the base and add the exponents: 2^(3+4) = 2^7. Evaluating 2^7 gives 128, not 2^(3*4) = 2^12.

2^6 = 64, so x = 6. The equation asks for the exponent that makes 2^x equal 64. Since 2^5 = 32 and 2^6 = 64, x must be 6; the other choices are results of arithmetic mistakes.

x = 10^3 = 1000. The equation log10(x) = 3 means 10 raised to the power 3 equals x. Since 10^3 = 1000, the solution is 1000, not 10 or 300.

x^2 = 9 gives x = 3 or x = -3. Taking the square root of both sides produces two possibilities because both 3^2 and (-3)^2 equal 9. The equation therefore has two real solutions, so the single positive answer is incomplete.

b^2 - 4ac = 16 - 16 = 0, so there is exactly one solution.

g(3) = 9, then f(9) = 2(9) + 1 = 19.

Solve y = (x + 3)/2 for x: x = 2y - 3.

The radicand must be non-negative: x - 2 >= 0. A square root is defined for real numbers only when the expression inside is at least zero. Solving x - 2 >= 0 gives x >= 2, which includes 2 itself.

Vertex form gives the vertex at (3, 5). In y = a(x - h)^2 + k, the vertex is (h, k). Since the expression has (x - 3)^2 + 5, h = 3 and k = 5, so the vertex is (3, 5).

A left shift replaces x with x + 4. Shifting y = x^2 four units left moves every point to x - 4, so the new equation is y = (x + 4)^2. A right shift would use x - 4, and adding 4 outside would shift upward.

Adding gives 2x = 10, so x = 5 and y = 2.

Power rule: 4x^3. The power rule says d/dx(x^n) = n x^(n-1). For x^4, bring down 4 and reduce the exponent by 1, giving 4x^3.

Differentiate term by term: 15x^2 - 2. The derivative of 5x^3 is 15x^2, and the derivative of -2x is -2. Applying the power rule to each term gives 15x^2 - 2, not 5x^2 - 2.

The derivative of sin(x) is cos(x). This is one of the standard trigonometric derivatives. The other options are derivatives of cos(x), tan(x), or unrelated functions, so cos(x) is correct.

Product rule: 2x e^x + x^2 e^x. For f(x) = x^2 and g(x) = e^x, the product rule gives f'g + fg' = 2x e^x + x^2 e^x. Since e^x is its own derivative, both terms keep e^x.

Chain rule: (1/(3x)) x 3 = 1/x. The derivative of ln(u) is 1/u times du/dx. With u = 3x, du/dx = 3, so the result is (1/(3x))(3) = 1/x.

dy/dx = 2x, so at x = 3 the slope is 6.

Integrate each term: x^3 + 2x + C. The antiderivative of 3x^2 is x^3, and the antiderivative of 2 is 2x. Add the constant of integration C because indefinite integrals represent a family of functions.

Antiderivative x^2, so 1^2 - 0 = 1. The definite integral from 0 to 1 of 2x equals the antiderivative evaluated at the upper limit minus the lower limit. That gives F(1) - F(0) = 1 - 0 = 1.

Integral 0 to 3 x dx = 9/2 = 4.5.

f prime = 3x^2 - 3 = 0 gives x = 1 or x = -1.

Set 2t - 4 = 0, so t = 2. A particle stops when its velocity is zero. Solving 2t = 4 gives t = 2, the only time at which the velocity equals zero.

First derivative is 3x^2; second is 6x. Differentiate x^3 once to get 3x^2, then differentiate again to get 6x. The second derivative measures the rate of change of the first derivative.

Chain rule: 4(2x + 1)^3 x 2 = 8(2x + 1)^3.

The antiderivative of 1/x is the natural logarithm. For x > 0, integral 1/x dx = ln|x| + C, often written ln(x) + C. The other options come from applying the power rule incorrectly to x^-1.

At a maximum the slope changes from increasing to decreasing.

sin(pi/6) = 1/2, a standard exact value. The sine of 30 degrees is one half, so pi/6 radians also gives 1/2. sqrt(3)/2 is sin(pi/3), and sqrt(2)/2 is sin(pi/4).

cos(pi/3) = 1/2. The cosine of 60 degrees is one half. cos(0) is 1 and cos(pi/2) is 0, so the standard exact value for pi/3 is 1/2.

tan(pi/4) = sin/cos = 1. At pi/4 radians, sine and cosine are both sqrt(2)/2. Dividing them gives 1, which is the standard tangent value for 45 degrees.

90 x pi/180 = pi/2. To convert degrees to radians, multiply by pi/180. Since 90/180 simplifies to 1/2, the result is pi/2 radians.

5pi/6 x 180/pi = 150 degrees. To convert radians to degrees, multiply by 180/pi. The pi factors cancel, leaving 5 x 180 / 6 = 150 degrees.

The sine rule applies to AAS, ASA and SSA cases.

Cosine rule: c^2 = 49 + 25 - 70 cos(60) = 74 - 35 = 39, so c = sqrt(39).

Period = 2pi/2 = pi. For y = sin(bx), the period is 2pi divided by |b|. Since b = 2, the period is 2pi/2 = pi, which means the graph repeats every pi units.

Amplitude is the coefficient 3. For y = A cos(x), the amplitude is |A|, the maximum distance from the midline. Here the coefficient is 3, so the graph reaches 3 above and below its center.

Pythagorean identity: equals 1. The identity sin^2(x) + cos^2(x) = 1 holds for every real x. The other choices are different trigonometric expressions, so the simplified value is exactly 1.

The double-angle identity is 2 sin(x) cos(x). This formula follows from the sine addition identity and is used to simplify trigonometric expressions. The other options do not match the standard double-angle result.

Cosine is zero at pi/2 and 3pi/2. On the interval 0 to 2pi, cos(x) crosses zero at these two points. pi/4 and 5pi/4 give cos values of sqrt(2)/2 and -sqrt(2)/2, and 0 gives 1.

Arc length = r theta = 6 x pi/3 = 2pi. For a sector with radius r and angle theta in radians, arc length is r times theta. Here 6 x pi/3 simplifies to 2pi.

A vertical shift adds a constant outside. Shifting y = sin(x) up 2 units adds 2 to every output, giving y = sin(x) + 2. Adding 2 inside the argument would shift the graph horizontally, not vertically.

sin(30) = opposite/hypotenuse = 4/h, so h = 8. In a 30-60-90 triangle, the side opposite 30 degrees is half the hypotenuse. Since the opposite side is 4, the hypotenuse is 2 x 4 = 8.

cos(0) = 1. At an angle of 0 radians, the point on the unit circle is (1, 0), so the cosine value is 1. The other choices are values at other standard angles.

Mean = 40/5 = 8. Add the five values 4 + 6 + 8 + 10 + 12 to get 40, then divide by 5. The mean is 8, while 9 and 10 come from miscounting or summing incorrectly.

The middle value is 9. With five ordered values, the median is the third value. The list 2, 5, 9, 12, 15 is already ordered, so the median is 9.

5 appears three times, more than any other value. The mode is the value with the highest frequency. In 3, 3, 4, 5, 5, 5, 6, the value 5 occurs three times, while 3 occurs twice and 4 and 6 occur once.

Range = 20 - 3 = 17. The range measures the spread by subtracting the smallest value from the largest. For 3, 8, 12, 20, that is 20 - 3 = 17.

Mean is 4; squared deviations are 4, 0, 4, so variance = 8/3.

There are 3 favourable outcomes (HHT, HTH, THH) out of 8.

E(X) = np = 10 x 0.2 = 2. The expected value of a binomial distribution is the number of trials times the success probability. With n = 10 and p = 0.2, the expected value is 2.

z = (130 - 100)/15 = 2. The z-score measures how many standard deviations a value is above or below the mean. Here 130 is 30 units above the mean, and 30/15 = 2.

Independence is defined by the product rule. Two events A and B are independent when P(A and B) = P(A) x P(B). Equal probabilities, conditional probability 1, and simple addition describe different concepts.

Pearson correlation is always between -1 and 1. A value of 1 is perfect positive correlation, -1 is perfect negative correlation, and 0 means no linear relationship. Values outside this range are impossible for a correlation coefficient.

Graphing functions helps verify answers and explore behaviour. The IB calculator is expected to support graphing, solving, statistics, and table work. Storing files, playing games, or typing essays are not legitimate exam uses.

IB rewards method even when the final answer is wrong.

Units, precision and signs are common marks lost. Before submitting, check that the answer uses the required unit, the correct number of significant figures, and the right sign. Checking only the final line or erasing working can hide errors and lose method marks.

Paper 1 requires fluent non-calculator methods. Practising without a calculator under timed conditions builds the mental arithmetic and algebraic fluency needed for the exam. Watching videos, reading only, or memorising answers do not provide active practice.

This is the standard formula for 1 + 2 + ... + n. The sum of the first n natural numbers is n(n+1)/2, which can be proven by pairing terms. For example, 1 + 2 + 3 = 6 and 3(4)/2 = 6, so the statement is true.

The derivative of a constant is 0. A constant function does not change, so its rate of change is zero. The derivative of x is 1, which may be confused with a constant, so the statement is false.

For non-negative f, the definite integral equals the area. The integral sums infinitely thin slices under the curve, and when f(x) >= 0 those slices have non-negative height. If the function goes below the axis, the integral measures signed area instead.

The normal curve is symmetric around the mean. Its mean, median, and mode coincide at the center, and the tails extend equally in both directions. This symmetry is a defining property of the normal distribution.

This is the multiplication rule for independent events. If A and B are independent, P(A and B) = P(A) x P(B). The rule follows because knowing one event occurs does not change the probability of the other.

sin(pi) = 0. On the unit circle, pi radians corresponds to the point (-1, 0), so sine is 0. The value 1 belongs to sin(pi/2), making the statement false.

Logarithms undo exponentials. If b^x = y, then log_b(y) = x, so the logarithm is the inverse operation of exponentiation. This inverse relationship is fundamental to solving exponential equations.

Arithmetic series diverge unless the common difference is zero. The terms keep growing by a fixed amount, so partial sums increase without bound. Infinite series can converge only when their terms approach zero, as with certain geometric series.

3 and 6 divide 12 exactly. A factor of 12 is a whole number that divides 12 without a remainder. 12/3 = 4 and 12/6 = 2, while 5 does not divide 12 and 24 is a multiple, not a factor.

x = 2 or x = -2. Adding 4 to both sides gives x^2 = 4, and taking square roots gives both 2 and -2. The values 4 and 16 do not satisfy the original equation.

The derivative of x^2 and x^2 + 5 is 2x.

Mean, median and mode summarise the centre; range measures spread.

Normal distributions are symmetric, so mean equals median, and about 68% lies within one standard deviation.

The Pythagorean, double-angle and quotient identities are valid. tan(x) = sin(x)/cos(x), sin(2x) = 2 sin(x) cos(x), and sin^2(x) + cos^2(x) = 1 are standard identities. sin(x) = 2 is impossible because sine is bounded between -1 and 1.

Each equation matches its standard function type. Linear, quadratic, exponential, and logarithmic equations each have a characteristic form. Matching by the highest power, variable exponent, or logarithm identifies the correct pair.

The derivative measures change, the integral accumulates, stationary points have zero slope, and tangents touch curves.

The standard sine values are 1, 1/2, sqrt(3)/2 and sqrt(2)/2 respectively.

These are the standard definitions. The mean is the average, the median is the middle ordered value, the mode is the most frequent value, and standard deviation measures spread. Matching each term to its definition distinguishes the four statistics.

These definitions describe the main probability concepts. Independent events do not affect each other's probabilities, mutually exclusive events cannot occur together, a complement is the set of outcomes not in the event, and conditional probability updates a probability given known information.

Derivatives describe growth and extrema; the second derivative describes concavity; integrals give area.