Luyện Toán A-Level

Factorise to (x - 2)(x - 3) = 0, so x = 2 or x = 3.

The discriminant is b^2 - 4ac, so substitute a = 1, b = 4, and c = 5: 4^2 - 4(1)(5) = 16 - 20 = -4. A negative discriminant means the quadratic has no real roots, so option a is correct.

One solution requires discriminant 0: k^2 - 64 = 0, so k = 8.

x^2 + 6x = (x + 3)^2 - 9, so x^2 + 6x + 11 = (x + 3)^2 + 2.

The denominator cannot be zero, so x cannot equal 3.

Solve 2x + 1 = 7 to get x = 3, so f^-1(7) = 3.

Evaluate from the inside out: g(2) = 2 + 1 = 3, then substitute that into f, giving f(3) = 3^2 = 9. Option a is correct; 5 is g(4) and 4 is 2^2.

x + 1 = x^2 - 1 gives x^2 - x - 2 = 0, so x = 2 or x = -1; the points are (2, 3) and (-1, 0).

Split 50 into a square factor and the remaining factor: 50 = 25 x 2, so sqrt(50) = sqrt(25) x sqrt(2) = 5 sqrt(2). Option a is correct because the square factor is 25, not 10.

Use the difference of two squares formula: (a + b)(a - b) = a^2 - b^2, so (3 + sqrt(2))(3 - sqrt(2)) = 9 - 2 = 7. Option a is correct.

f(-1) = -1 + 2 + 5 - 6 = 0, so x + 1 is a factor and the remainder is 0.

C(5,2) = 10, so the x^2 term is 10x^2.

A translation 3 units up changes every output y by adding 3, so the new equation is y = f(x) + 3. Adding inside the bracket would move the graph horizontally, so option a is correct.

Roots are x = -1 and x = 5/2; the upward parabola is negative between them.

For ax^2 + bx + c, the sum of roots is -b/a = 6.

Distance = sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = 5.

Power rule: multiply by 5 and reduce the power by 1.

Differentiate term by term: 12x^3 - 4x, and the constant vanishes.

Apply the product rule: differentiate x^2 to get 2x and keep e^x, then keep x^2 and differentiate e^x to get e^x. The result is 2x e^x + x^2 e^x, matching option a.

Quotient rule: (2x(x+1) - x^2)/(x+1)^2 = (x^2 + 2x)/(x+1)^2.

Use the chain rule: differentiate the outer sine to get cos(3x), then multiply by the derivative of 3x, which is 3. This gives 3 cos(3x), so option a is correct.

Chain rule: (1/(x^2+1)) x 2x = 2x/(x^2 + 1).

First derivative is 3x^2 - 6x; differentiating again gives 6x - 6.

dy/dx = 2x, so at x = 1 the slope is 2.

The derivative is 3x^2 - 12x + 9 = 3(x - 1)(x - 3), so x = 1 and x = 3.

Integrate each term by increasing the power by one and dividing by the new power: 4x^3 becomes x^4 and 2x becomes x^2. Adding the constant C gives x^4 + x^2 + C, matching option a.

An antiderivative is x^3, so 2^3 - 1^3 = 8 - 1 = 7.

Integral 0 to 2 x^2 dx = 8/3 - 0 = 8/3.

sin(pi/3) is one of the standard exact values: sqrt(3)/2.

tan(pi/4) = sin(pi/4)/cos(pi/4) = 1.

Sine is positive in the first and second quadrants: pi/6 and 5pi/6.

Multiply the degree measure by pi/180: 135 x pi/180 = 3pi/4. This simplifies because 135 and 180 share a factor of 45, leaving 3pi/4, so option a is correct.

Arc length = r theta = 6 x pi/3 = 2pi cm.

Area = (1/2) r^2 theta = (1/2)(16)(pi/4) = 2pi cm^2.

Sine rule: b/sin(30) = 8/sin(60), so b = 8 x (1/2)/(sqrt(3)/2) = 8/sqrt(3).

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

This is the Pythagorean identity: sin^2(x) + cos^2(x) = 1.

By definition, tan(theta) = sin(theta)/cos(theta), so the quotient is tan(2x).

The period of sin(kx) is 2pi divided by the absolute value of k. Here k = 3, so the period is 2pi/3, which matches option a; pi/3 would come from dividing by 6.

Amplitude is the absolute value of the coefficient multiplying the cosine function. Since y = 4 cos(x) has coefficient 4, the amplitude is 4, making option a correct.

The product rule for logarithms: ln(ab) = ln(a) + ln(b).

x = ln(20) = 2.9957, which rounds to 3.00.

Write 16 as a power of 2: 2^4 = 16, so the equation 2^x = 16 has solution x = 4. The other options do not satisfy the equation when substituted into 2^x.

The nth term of an arithmetic sequence is a + (n - 1)d. With a = 3, d = 4, and n = 5, this gives 3 + 4(4) = 19, so option a is correct.

Use the arithmetic sum formula S_n = n/2(2a + (n - 1)d). With n = 10, a = 2, and d = 3, this gives 5(4 + 27) = 155, matching option a.

The nth term of a geometric sequence is ar^(n - 1). With a = 5, r = 2, and n = 4, this gives 5 x 2^3 = 5 x 8 = 40, so option a is correct.

Each term is half of the previous term, so r = 1/2.

For an infinite geometric series with first term a and common ratio r, the sum is a/(1 - r). Here a = 6 and r = 1/2, so the sum is 6/(1 - 1/2) = 12, matching option a.

An infinite geometric series converges only when the common ratio satisfies |r| < 1, so the terms shrink toward zero. Options with r > 1 or r = 2 describe divergent series, so option a is correct.

Raising 10 to both sides gives x = 10^2 = 100.

S5 = 3(2^5 - 1)/(2 - 1) = 3 x 31 = 93.

The logarithm log2(32) asks which power of 2 equals 32. Since 2^5 = 32, log2(32) = 5, making option a correct; 16 is 2^4, not the answer.

The nth term of a geometric sequence is ar^(n - 1). With a = 4, r = 3, and n = 3, this gives 4 x 3^2 = 4 x 9 = 36, so option a is correct.

Subtract consecutive terms: 9 - 5 = 4, 13 - 9 = 4, and 17 - 13 = 4. Because the difference is constant, the sequence is arithmetic with common difference 4, matching option a.

x = ln(100)/ln(3) = 4.1918, which rounds to 4.19.

Continuing the pattern: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, so the 10th term is 55.

A reflection in the x-axis multiplies the function by -1.

The x term is C(4,1)(2x)^1(3)^3 = 4 x 2 x 27 = 216.

Set 3t^2 - 12 = 0, so t^2 = 4 and t = 2 (for t >= 0).

The second derivative is 6x - 12 = 0 at x = 2, where concavity changes.

For f(x) >= 0, the definite integral equals the area under the curve.

That combination identifies a local maximum by the second derivative test.

The cosine function completes one full cycle every 2pi radians, so its period is 2pi. The statement is therefore true.

A geometric series converges when the absolute value of the common ratio is less than 1, because the terms then approach zero. The statement says greater than 1, so it is false.

The natural logarithm ln(1) asks which power of e equals 1. Since e^0 = 1, ln(1) = 0, so the statement is true.

An arithmetic series with non-zero common difference diverges when the number of terms is infinite.

Cubing and cube-rooting undo each other: g(f(x)) = (x^3)^(1/3) = x and f(g(x)) = (x^(1/3))^3 = x. The statement is true.

x^2 decreases for x < 0, so it is not increasing on the whole real line.

Solve x^2 = 4 by taking the square root of both sides, which gives both x = 2 and x = -2. Options 4 and -4 are not roots because 4^2 and (-4)^2 equal 16, so options a and b are correct.

The derivative of x^2 is 2x and the derivative of x^2 + 5 is also 2x; the others give 4x and 1.

Only pi/2 gives sine equal to 1 in that interval.

The common difference is 3, the 10th term is 2 + 9(3) = 29, and the sum is 2 + 5 + 8 + 11 = 26; the sequence is not geometric.

Subtracting logs is the quotient rule, and log(b^-1) = -log(b).

The derivative is 3x^2 - 3 = 0 at x = 1 or x = -1.

A quadratic has degree 2, the discriminant is b^2 - 4ac, stationary points have zero derivative, and asymptotes are approached but not reached.

These are the standard product, quotient, chain and power rules.

Match each standard antiderivative: sin(x) becomes -cos(x) + C, cos(x) becomes sin(x) + C, sec^2(x) becomes tan(x) + C, and e^x becomes e^x + C. These are the standard integrals used in A-Level calculus.

Match each identity: sin(2x) = 2 sin(x) cos(x), cos(2x) = cos^2(x) - sin^2(x), tan(x) = sin(x)/cos(x), and sin^2(x) + cos^2(x) = 1. These are the double-angle and Pythagorean identities.

Arithmetic has a common difference, geometric has a common ratio, convergence needs |r| < 1, and Fibonacci sums previous terms.

Match each logarithm law: the product rule is log(ab) = log(a) + log(b), the quotient rule is log(a/b) = log(a) - log(b), the power rule is log(a^b) = b log(a), and the change of base formula is log_b(a) = log(a)/log(b).