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HKDSE Maths M1 study materials
HKDSE Maths M1 study resources

HKDSE Maths M1 practice questions

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Question 1Calculus

Evaluate the limit: lim (x to 0) of (sin 3x)/x.

  1. 1
  2. 0
  3. 3 Correct answer.
  4. infinity

Explanation

Using lim (u to 0) (sin u)/u = 1, write (sin 3x)/x = 3 * (sin 3x)/(3x). As x to 0, this tends to 3 * 1 = 3.

Question 2Calculus

By first principles, the derivative of f(x) = x^2 is defined as the limit of which expression as h tends to 0?

  1. (x^2 - (x+h)^2)/h
  2. (x+h)^2 - x^2
  3. (x^2)/h
  4. ((x+h)^2 - x^2)/h Correct answer.

Explanation

The derivative from first principles is lim (h to 0) [f(x+h) - f(x)]/h = lim [(x+h)^2 - x^2]/h = lim (2xh + h^2)/h = 2x.

Question 3Calculus

Differentiate y = x^3 ln x with respect to x.

  1. 3x^2 ln x + x^2 Correct answer.
  2. 3x^2 ln x
  3. x^2
  4. 3x^2 + 1/x

Explanation

Using the product rule with u = x^3, v = ln x: dy/dx = 3x^2 ln x + x^3 * (1/x) = 3x^2 ln x + x^2.

Question 4Calculus

Find dy/dx if y = e^(2x).

  1. e^(2x)
  2. 2e^(2x) Correct answer.
  3. 2x e^(2x-1)
  4. e^(2x)/2

Explanation

By the chain rule, d/dx of e^(u) is e^(u) * du/dx. With u = 2x, du/dx = 2, so dy/dx = 2e^(2x).

Question 5Calculus

Differentiate y = (2x + 1)/(x - 3) using the quotient rule.

  1. 7/(x-3)^2
  2. 2/(x-3)
  3. -7/(x-3)^2 Correct answer.
  4. -5/(x-3)^2

Explanation

Quotient rule: [(2)(x-3) - (2x+1)(1)]/(x-3)^2 = (2x - 6 - 2x - 1)/(x-3)^2 = -7/(x-3)^2.

Question 6Calculus

A spherical balloon's radius increases at 2 cm/s. At what rate is its volume increasing when r = 5 cm? (V = (4/3)pi r^3)

  1. 100 pi cm^3/s
  2. 200 pi cm^3/s Correct answer.
  3. 50 pi cm^3/s
  4. 400 pi cm^3/s

Explanation

dV/dt = 4 pi r^2 (dr/dt) = 4 pi (25)(2) = 200 pi cm^3/s by the chain rule for related rates.

Question 7Calculus

The function f(x) = x^3 - 3x has a local maximum at which value of x?

  1. 1
  2. 0
  3. -1 Correct answer.
  4. 3

Explanation

f'(x) = 3x^2 - 3 = 0 gives x = 1 or x = -1. f''(x) = 6x; at x = -1, f'' = -6 < 0, so x = -1 is a local maximum.

Question 8Calculus

For the curve y = x^3, the point (0, 0) is which type of point?

  1. A local maximum
  2. A local minimum
  3. A vertical asymptote
  4. A point of inflexion Correct answer.

Explanation

y' = 3x^2 = 0 at x = 0, and y'' = 6x changes sign through x = 0 (from negative to positive). Since the concavity changes, (0,0) is a point of inflexion.

Question 9Calculus

Find the equation of the tangent to y = x^2 at the point (2, 4).

  1. y = 4x - 4 Correct answer.
  2. y = 4x + 4
  3. y = 2x
  4. y = 4x

Explanation

dy/dx = 2x; at x = 2 the slope is 4. Using point-slope: y - 4 = 4(x - 2), so y = 4x - 4.

Question 10Calculus

Evaluate the indefinite integral of (6x^2 + 4) dx.

  1. 12x + C
  2. 2x^3 + 4x + C Correct answer.
  3. 6x^3 + 4x + C
  4. 2x^3 + 4 + C

Explanation

Integrating term by term: integral of 6x^2 is 2x^3, integral of 4 is 4x. So the result is 2x^3 + 4x + C.

Question 11Calculus

Evaluate the integral of e^(3x) dx.

  1. 3e^(3x) + C
  2. e^(3x) + C
  3. (1/3)e^(3x) + C Correct answer.
  4. (1/3)e^(3x-1) + C

Explanation

Integral of e^(ax) dx = (1/a)e^(ax) + C. With a = 3, the result is (1/3)e^(3x) + C.

Question 12Calculus

Use the substitution u = x^2 + 1 to evaluate the integral of 2x(x^2 + 1)^3 dx.

  1. (x^2 + 1)^4 + C
  2. (1/3)(x^2 + 1)^3 + C
  3. (1/2)(x^2 + 1)^4 + C
  4. (1/4)(x^2 + 1)^4 + C Correct answer.

Explanation

With u = x^2 + 1, du = 2x dx, so the integral becomes integral of u^3 du = u^4/4 + C = (1/4)(x^2 + 1)^4 + C.

Question 13Calculus

Evaluate the integral of 1/x dx for x > 0.

  1. ln x + C Correct answer.
  2. x^(-2) + C
  3. -1/x^2 + C
  4. ln|x|^2 + C

Explanation

The integral of 1/x is ln|x| + C, which for x > 0 is ln x + C. This is a standard result in M1 calculus.

Question 14Calculus

Evaluate the definite integral from 0 to 2 of 3x^2 dx.

  1. 12
  2. 8 Correct answer.
  3. 6
  4. 4

Explanation

Integral of 3x^2 is x^3. Evaluating from 0 to 2: 2^3 - 0^3 = 8.

Question 15Calculus

Find the area of the region bounded by y = x^2, the x-axis, x = 0 and x = 3.

  1. 27
  2. 3
  3. 9 Correct answer.
  4. 18

Explanation

Area = integral from 0 to 3 of x^2 dx = [x^3/3] from 0 to 3 = 27/3 - 0 = 9 square units.

Question 16Calculus

The area between the curves y = x and y = x^2 from x = 0 to x = 1 equals which value?

  1. 1/2
  2. 1/3
  3. 1/12
  4. 1/6 Correct answer.

Explanation

On [0,1], x is above x^2. Area = integral of (x - x^2) dx = [x^2/2 - x^3/3] from 0 to 1 = 1/2 - 1/3 = 1/6.

Question 17Calculus

Using the trapezoidal rule with 2 sub-intervals, estimate the integral from 0 to 2 of x^2 dx (with ordinates at x = 0, 1, 2).

  1. 3 Correct answer.
  2. 2.6667
  3. 4
  4. 2

Explanation

Trapezoidal rule: (h/2)[y0 + 2y1 + y2] with h = 1, y values 0, 1, 4. Estimate = (1/2)[0 + 2(1) + 4] = (1/2)(6) = 3.

Question 18Calculus

The trapezoidal rule generally gives an OVERESTIMATE of a definite integral when the curve is which shape over the interval?

  1. Concave down
  2. Concave up (convex) Correct answer.
  3. A straight line
  4. Always exact

Explanation

When a curve is concave up, the chord (trapezoid top) lies above the curve, so the trapezoidal rule overestimates the area.

Question 19Calculus

Differentiate y = ln(2x + 1).

  1. 1/(2x + 1)
  2. 2/(x + 1)
  3. 2/(2x + 1) Correct answer.
  4. 1/(2x)

Explanation

Chain rule: d/dx ln(u) = (1/u)(du/dx). With u = 2x + 1, du/dx = 2, so dy/dx = 2/(2x + 1).

Question 20Calculus

A rectangle has perimeter 20 cm. What dimensions maximise its area?

  1. 4 cm by 6 cm
  2. 2 cm by 8 cm
  3. 10 cm by 0 cm
  4. 5 cm by 5 cm Correct answer.

Explanation

Let width be x, so length is 10 - x. Area A = x(10 - x); A' = 10 - 2x = 0 gives x = 5. Thus a 5 by 5 square maximises area at 25 cm^2.

Question 21Calculus

Evaluate the definite integral from 1 to e of (1/x) dx.

  1. 1 Correct answer.
  2. e - 1
  3. e
  4. 0

Explanation

Integral of 1/x is ln x. Evaluating from 1 to e: ln e - ln 1 = 1 - 0 = 1.

Question 22Calculus

The marginal cost function is C'(x) = 4x + 10 dollars per unit. What is the increase in total cost when production rises from x = 0 to x = 5 units?

  1. 30
  2. 100 Correct answer.
  3. 50
  4. 150

Explanation

Increase in cost = integral from 0 to 5 of (4x + 10) dx = [2x^2 + 10x] from 0 to 5 = 50 + 50 = 100 dollars.

Question 23Calculus

Find the equation of the normal to the curve y = x^2 at the point (1, 1).

  1. y = 2x - 1
  2. y = -2x + 3
  3. y = -x/2 + 3/2 Correct answer.
  4. y = x/2 + 1/2

Explanation

dy/dx = 2x; at x = 1 the tangent slope is 2, so the normal slope is -1/2. Normal: y - 1 = (-1/2)(x - 1), i.e. y = -x/2 + 3/2.

Question 24Calculus

A particle's displacement is s = t^3 - 6t^2 + 9t. At what time(s) is the particle momentarily at rest?

  1. t = 1 and t = 3 Correct answer.
  2. t = 0 only
  3. t = 2 only
  4. t = 3 only

Explanation

Velocity v = ds/dt = 3t^2 - 12t + 9 = 3(t - 1)(t - 3). Setting v = 0 gives t = 1 and t = 3, the times when the particle is at rest.

Question 25Calculus

Evaluate the integral of x e^x dx using integration by parts.

  1. x e^x + e^x + C
  2. x e^x - e^x + C Correct answer.
  3. (x^2/2) e^x + C
  4. e^x + C

Explanation

Integration by parts with u = x, dv = e^x dx: integral = x e^x - integral of e^x dx = x e^x - e^x + C.

Question 26Calculus

What is the gradient of the curve y = e^x at the point where x = 0?

  1. 0
  2. e
  3. 1 Correct answer.
  4. undefined

Explanation

dy/dx = e^x. At x = 0, the gradient is e^0 = 1. The curve y = e^x crosses the y-axis at (0,1) with slope 1.

Question 27Calculus

Find the x-coordinate of the stationary point of y = x e^(-x).

  1. 0
  2. -1
  3. e
  4. 1 Correct answer.

Explanation

By the product rule, dy/dx = e^(-x) - x e^(-x) = e^(-x)(1 - x). Setting this to zero (e^(-x) is never 0) gives 1 - x = 0, so x = 1.

Question 28Calculus

Differentiate y = (x^2 + 1)^4 with respect to x.

  1. 4(x^2 + 1)^3
  2. 8x(x^2 + 1)^4
  3. 8x(x^2 + 1)^3 Correct answer.
  4. 2x(x^2 + 1)^3

Explanation

Chain rule: dy/dx = 4(x^2 + 1)^3 * d/dx(x^2 + 1) = 4(x^2 + 1)^3 * 2x = 8x(x^2 + 1)^3.

Question 29Calculus

Find the integral of (3x^2 + 2x) e^(x^3 + x^2) dx... actually simpler: integrate (2x + 2)/(x^2 + 2x + 5) dx.

  1. 2 ln|x^2 + 2x + 5| + C
  2. (x^2 + 2x + 5)^2 + C
  3. 1/(x^2 + 2x + 5) + C
  4. ln|x^2 + 2x + 5| + C Correct answer.

Explanation

The numerator 2x + 2 is exactly the derivative of the denominator x^2 + 2x + 5, so the integral has the form integral of f'(x)/f(x) dx = ln|f(x)| + C.

Question 30Calculus

Find the coordinates of the turning point of y = x^2 - 6x + 11.

  1. (3, 11)
  2. (-3, 2)
  3. (3, 2) Correct answer.
  4. (6, 11)

Explanation

dy/dx = 2x - 6 = 0 gives x = 3. Then y = 9 - 18 + 11 = 2, so the turning point is (3, 2), a minimum.

Question 31Calculus

Evaluate the integral from 0 to 1 of e^(2x) dx. (4 d.p.)

  1. 6.3891
  2. 1.5973
  3. 7.3891
  4. 3.1945 Correct answer.

Explanation

Integral = [(1/2)e^(2x)] from 0 to 1 = (1/2)(e^2 - 1) = (1/2)(7.389056 - 1) = (1/2)(6.389056) = 3.1945.

Question 32Calculus

The volume of a solid of revolution is found by rotating a region about the x-axis. The volume equals the integral of which expression?

  1. pi * integral of y^2 dx Correct answer.
  2. integral of y dx
  3. 2 pi * integral of y dx
  4. pi * integral of y dx

Explanation

Rotating y = f(x) about the x-axis gives volume V = pi * integral of [f(x)]^2 dx = pi * integral of y^2 dx (the disc method).

Question 33Calculus

Differentiate y = ln(x^2 + 3x).

  1. 1/(x^2 + 3x)
  2. (2x + 3)/(x^2 + 3x) Correct answer.
  3. (2x + 3)
  4. (2x)/(x^2 + 3x)

Explanation

Chain rule for logarithms: dy/dx = (1/(x^2 + 3x)) * (2x + 3) = (2x + 3)/(x^2 + 3x).

Question 34Calculus

Find dy/dx if y = x^x (x > 0).

  1. x * x^(x-1)
  2. x^x ln x
  3. x^x (ln x + 1) Correct answer.
  4. x^x

Explanation

Take logs: ln y = x ln x. Differentiating: (1/y)(dy/dx) = ln x + 1, so dy/dx = y(ln x + 1) = x^x(ln x + 1).

Question 35Calculus

Find the absolute maximum value of f(x) = -x^2 + 4x + 1.

  1. 4
  2. 5 Correct answer.
  3. 1
  4. 9

Explanation

f'(x) = -2x + 4 = 0 gives x = 2. f(2) = -4 + 8 + 1 = 5. Since the parabola opens downward, this is the absolute maximum.

Question 36Calculus

Evaluate the integral of (2x + 1)/x dx for x > 0.

  1. ln(2x + 1) + C
  2. x^2 + ln x + C
  3. 2x + ln x + C Correct answer.
  4. 2 + ln x + C

Explanation

Split the fraction: (2x + 1)/x = 2 + 1/x. The integral is 2x + ln x + C.

Question 37Calculus

Find the integral of x^2 ln x dx using integration by parts.

  1. (x^3/3) ln x + C
  2. (x^3/3) ln x - x^3/9 + C Correct answer.
  3. (x^3/3) ln x - x^3/3 + C
  4. x^3 ln x - x^3 + C

Explanation

By parts with u = ln x, dv = x^2 dx: v = x^3/3. Integral = (x^3/3)ln x - integral of (x^3/3)(1/x) dx = (x^3/3)ln x - (1/3)(x^3/3) = (x^3/3)ln x - x^3/9 + C.

Question 38Calculus

The function f(x) = 2x^3 - 9x^2 + 12x is increasing on which interval?

  1. 1 < x < 2
  2. x > 0 only
  3. x < 1 or x > 2 Correct answer.
  4. all real x

Explanation

f'(x) = 6x^2 - 18x + 12 = 6(x - 1)(x - 2). This is positive when x < 1 or x > 2, so f is increasing on those intervals.

Question 39Calculus

Evaluate the definite integral from 1 to 2 of (1/x^2) dx.

  1. 1
  2. 1/2 Correct answer.
  3. -1/2
  4. ln 2

Explanation

Integral of x^(-2) is -x^(-1) = -1/x. Evaluating from 1 to 2: (-1/2) - (-1) = -1/2 + 1 = 1/2.

Question 40Calculus

Find dy/dx by implicit differentiation if x^2 + xy + y^2 = 7, evaluated in general form.

  1. -(2x + y)/(x + 2y) Correct answer.
  2. (2x + y)/(x + 2y)
  3. -(2x + y)/(2x + y)
  4. (x + 2y)/(2x + y)

Explanation

Differentiate: 2x + y + x(dy/dx) + 2y(dy/dx) = 0. Collect: (x + 2y)(dy/dx) = -(2x + y), so dy/dx = -(2x + y)/(x + 2y).

Question 41Calculus

The total revenue is R(x) = 100x - 0.5x^2 dollars for x units. What output x maximises revenue?

  1. 50
  2. 100 Correct answer.
  3. 200
  4. 0

Explanation

R'(x) = 100 - x = 0 gives x = 100. Since R''(x) = -1 < 0, this is a maximum. Revenue is maximised at 100 units.

Question 42Calculus

Differentiate y = e^(x^2) with respect to x.

  1. e^(x^2)
  2. 2x e^(2x)
  3. 2x e^(x^2) Correct answer.
  4. x^2 e^(x^2 - 1)

Explanation

Chain rule: d/dx of e^(u) = e^(u) du/dx with u = x^2, du/dx = 2x. So dy/dx = 2x e^(x^2).

Question 43Calculus

The amount of a drug in the body is A = 50 e^(-0.2t) mg, where t is in hours. At what rate is the amount changing when t = 0?

  1. -50 mg/hour
  2. 10 mg/hour
  3. -0.2 mg/hour
  4. -10 mg/hour Correct answer.

Explanation

dA/dt = 50 * (-0.2) e^(-0.2t) = -10 e^(-0.2t). At t = 0, e^0 = 1, so the rate is -10 mg/hour (decreasing).

Question 44Probability Statistics

Two events A and B are independent with P(A) = 0.4 and P(B) = 0.5. What is P(A and B)?

  1. 0.2 Correct answer.
  2. 0.9
  3. 0.1
  4. 0.45

Explanation

For independent events, P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.

Question 45Probability Statistics

Given P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.3, find P(A | B).

  1. 0.5
  2. 0.6 Correct answer.
  3. 0.3
  4. 0.18

Explanation

P(A | B) = P(A and B)/P(B) = 0.3/0.5 = 0.6 by the definition of conditional probability.

Question 46Probability Statistics

A factory has two machines. Machine A produces 60% of items with a 2% defect rate; Machine B produces 40% with a 5% defect rate. What is the overall probability that a randomly chosen item is defective?

  1. 0.035
  2. 0.070
  3. 0.032 Correct answer.
  4. 0.020

Explanation

By the law of total probability: P(defective) = 0.6(0.02) + 0.4(0.05) = 0.012 + 0.020 = 0.032.

Question 47Probability Statistics

A discrete random variable X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3 respectively. Find E(X).

  1. 2.1 Correct answer.
  2. 2.0
  3. 1.8
  4. 2.5

Explanation

E(X) = sum of x * P(x) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2 + 1.0 + 0.9 = 2.1.

Question 48Probability Statistics

For the random variable X in a probability distribution, if E(X) = 2 and E(X^2) = 6, what is Var(X)?

  1. 4
  2. 2 Correct answer.
  3. 6
  4. 8

Explanation

Var(X) = E(X^2) - [E(X)]^2 = 6 - 2^2 = 6 - 4 = 2.

Question 49Probability Statistics

If X follows a binomial distribution B(10, 0.3), what is the mean of X?

  1. 2.1
  2. 7
  3. 3 Correct answer.
  4. 0.3

Explanation

For X ~ B(n, p), the mean is np = 10 * 0.3 = 3.

Question 50Probability Statistics

If X ~ B(10, 0.3), what is the variance of X?

  1. 3
  2. 0.7
  3. 7
  4. 2.1 Correct answer.

Explanation

For a binomial distribution, Var(X) = np(1 - p) = 10 * 0.3 * 0.7 = 2.1.