1. Physical Chemistry

• Chemical Energetics: Constructing Born-Haber cycles to calculate lattice energies () and hydration enthalpies.
• Electrochemistry: Standard electrode potentials , the Nernst equation, and electrochemical cells.
• Equilibria & Buffers: Calculating the pH of buffer solutions using the Henderson-Hasselbalch equation:

[ = K_a + () ]

2. Inorganic Chemistry

• Periodicity & Group Trends: Group 2 sulfates and hydroxides solubility trends; halogens oxidizing ability.
• Transition Elements: Formation of complex ions, - orbital splitting, ligand exchange reactions, and variable oxidation states.

3. Organic Chemistry

• Reaction Mechanisms: Electrophilic addition of alkenes, electrophilic substitution of benzene, and nucleophilic substitution of acyl chlorides.
• Analytical Techniques: Interpreting and NMR spectra, Infra-Red (IR) spectroscopy, and Mass Spectrometry to elucidate unknown organic structures.

1. Cambridge Assessment International Education (CIE 9709)

• Pure Mathematics 1 (P1): Quadratics, functions, coordinate geometry, circular measure, trigonometry, series, differentiation, and integration.
• Pure Mathematics 3 (P3): Algebra, logarithmic/exponential functions, trigonometry, numerical solutions, vectors, differential equations, and complex numbers.
• Mechanics (Paper 4 / M1): Forces and equilibrium, kinematics of motion, Newton’s laws of motion, work, energy, and power.
• Probability & Statistics 1 (Paper 5 / S1): Representation of data, probability, discrete random variables, and normal distribution.
• Probability & Statistics 2 (Paper 6 / S2): Poisson distribution, linear combinations of random variables, continuous random variables, sampling, and hypothesis testing.

2. Pearson Edexcel International A-Level (IAL)

Modular qualification structure comprising P1, P2, P3, P4 (Pure Mathematics), M1, M2 (Mechanics), S1, S2 (Statistics), and FP1–FP3 (Further Pure Mathematics).

1. Integration by Parts & Partial Fractions (P3 / P4)

Decomposing complex rational expressions into partial fractions prior to integrating by parts:

[ u , dx = uv - v , dx ]

2. Kinematics & Newton’s Laws on Inclined Planes (Mechanics M1)

Resolving forces parallel and perpendicular to an inclined plane with coefficient of friction :

[ F_{} = m a m g - (m g ) = m a ]

3. Normal Distribution & Hypothesis Testing (Statistics S1 / S2)

Standardizing a continuous random variable into standard normal form :

[ Z = ]

1. Theme: Structure

• Structure 1: Models of the particulate nature of matter (Kinetic molecular theory, gas laws).
• Structure 2: Atomic structure (Mass spectrometry, electron configurations, emission spectra).
• Structure 3: Counting particles by mass - The mole concept (Stoichiometry, limiting reactants, molar volume).
• Structure 4: Periodicity & Bonding (VSEPR theory, intermolecular forces, metallic and ionic lattices).

2. Theme: Reactivity

• Reactivity 1: What drives chemical reactions? (Thermodynamics, enthalpy changes , entropy , Gibbs free energy ).
• Reactivity 2: How fast do chemical reactions go? (Kinetics, rate equations, reaction mechanisms, vs. ).
• Reactivity 3: How far do chemical reactions go? (Equilibrium constants , Le Chatelier’s principle, electrochemical cells ).

1. Gibbs Free Energy & Spontaneity ()

[ G^= H^- TS^ ]

A reaction is thermodynamically spontaneous when . Remember to convert from to .

2. The Arrhenius Equation & Activation Energy ()

[ k = A e^{-} k = - () + A ]

Plotting against yields a straight line with a gradient of .

3. Nucleophilic Substitution Mechanisms ( vs. )

• : Two-step mechanism involving a carbocation intermediate; favored by tertiary halogenoalkanes. Rate equation: .
• : One-step concerted mechanism involving a transition state; favored by primary halogenoalkanes. Rate equation: .

1. Proof by Mathematical Induction (IB Math AA HL)

Proof by induction requires a clear three-step structure: establishing the base step (), stating the inductive hypothesis (), and completing the inductive step ():

[ P(n): _{i=1}^{n} i^3 = ^2 n ^+ ]

2. Complex Numbers in Euler and Polar Form (IB Math AA HL)

Expressing complex numbers in exponential form simplifies multiplication, division, and applying De Moivre’s Theorem:

[ z = r(+ i) = r e^{i} z^n = r^n e^{in} ]

3. Differential Equations (IB Math AA/AI HL)

Solving first-order linear differential equations by integrating factors or separation of variables:

[ + P(x)y = Q(x) ]

4. Hypothesis Testing & Chi-Squared Tests (IB Math AI HL/SL)

Utilizing Graphic Display Calculators (GDC like Casio fx-CG50 or TI-Nspire CX II) to evaluate test statistics, degrees of freedom, and -values against significance levels ().

5. The Binomial Theorem (IB Math AA SL/HL)

Expanding expressions of the form using combination notation:

[ (a + b)^n = _{k=0}^{n} a^{n-k} b^k ]

🧪 1. IB Chemistry Past Papers Bank

Hiện tại, trang IB Chemistry Overview đã lưu trữ tổng cộng 546 tệp đề thi (QP) và đáp án (MS) chính thức:

• Higher Level (HL): 270 tệp
• Standard Level (SL): 276 tệp
• Cập nhật mới nhất: Bổ sung 186 tệp QP và MS cho cả 2 hệ HL & SL trong đợt cập nhật vừa qua.

📐 2. IB Mathematics Past Papers Bank

Hệ thống tra cứu tại IB Mathematics Overview hiện lưu trữ tổng cộng 1,099 tệp đề thi và đáp án:

Question 1 [Maximum mark: 7]

A group of 52 students have their heights measured in cm.

1. Use the cumulative frequency data to find an estimate of the median height. [2 marks]

The heights of the students are shown in the following frequency table where .

Height ( cm)	Frequency
	3
	8
	14

2. 1. Find the value of and the value of .
   2. Use the frequency table to find an estimate of the mean height, giving your answer to four significant figures. [5 marks]

Question 2 [Maximum mark: 4]

Consider a function which has a first derivative given by , where .

Find the range of values of for which is decreasing. [4 marks]

Question 3 [Maximum mark: 6]

A Ferris wheel with radius 20 metres rotates at a constant speed. The centre of the wheel is 23 metres above the ground. P is a point on the wheel.

The wheel starts moving with P at the lowest point and completes one revolution in 4 minutes.

The height, metres, of P above the ground after minutes is given by , where .

1. Find the value of , the value of and the value of . [4 marks]

The height can also be given as , where .

2. Find the smallest positive value of . [2 marks]

Question 4 [Maximum mark: 7]

Consider the function defined as , where .

The graph of has a local maximum at point A and a local minimum at point B. The line (AB) passes through the origin O at (0,0).

The region R is enclosed by the graph of and the line segment [OB].

1. Find the coordinates of point A and the coordinates of point B. [2 marks]

2. Find the area of R. [5 marks]

Question 5 [Maximum mark: 7]

A large group of adults have their heights measured. The heights of the adults are normally distributed with mean cm and standard deviation cm.

The probability that one of the adults chosen at random has a height greater than 171 cm is 0.4.

If one of the adults with a height greater than 171 cm is chosen at random, the probability that their height is greater than 175 cm is 0.35.

Find the value of and the value of . [7 marks]

Question 6 [Maximum mark: 4]

The air inside an airtight syringe is compressed so that its volume, , decreases.

The air pressure, Newtons per , within the syringe is inversely proportional to the volume so that .

At a given time, the volume of air is and is decreasing at a rate of per minute.

Find the rate of increase in the air pressure at this time. [4 marks]

Question 7 [Maximum mark: 6]

The continuous random variable has a probability distribution defined as

where and .

1. Show that . [2 marks]

It is given that .

2. Find the value of . [4 marks]

Question 8 [Maximum mark: 8]

Consider the equation , where and is a positive constant.

One of the solutions of the equation is .

1. Find the value of . [2 marks]

2. Write down the other two solutions of giving your answers in the form , where . [2 marks]

3. Find the solutions of giving your answers in the form , where . [2 marks]

The solutions of can be represented on an Argand diagram by three points A, B and C.

4. Find the area of the triangle ABC. [2 marks]

Question 9 [Maximum mark: 8]

Consider the curve where and .

The region enclosed by the curve is rotated through radians around the y-axis to form a solid, S.

1. Show that the volume of S is given by . [6 marks]

It is given that the volume of S is .

2. Find the value of . [2 marks]

Question 10 [Maximum mark: 13]

At the start of 2026, Lavinia receives a gift of . She wants to buy a boat which costs so she decides to invest her money.

On 1 January 2026 Lavinia invests her money in a bank account which pays interest at a nominal annual rate of 4.8%, compounded quarterly. The interest is paid into her account on the last day of each quarter. She makes no further deposits to, or withdrawals from, the account.

1. Find the amount of money Lavinia will have in her bank account on 1 January 2031. Give your answer correct to the nearest dollar. [3 marks]

2. Show that Lavinia will first have more than in her bank account during the year 2039. [3 marks]

The cost of the boat at the start of 2026 was , and it depreciates at a constant annual rate of . After one year the cost of the boat is .

3. Find the value of . [2 marks]

Due to this depreciation, Lavinia will be able to buy the boat before 2039. She will buy the boat as soon as she has enough money in her bank account to pay for it.

4. Determine the year during which Lavinia will buy the boat. [5 marks]

Question 11 [Maximum mark: 19]

A child’s toy is made from a solid right cone and a solid hemisphere of equal radius. The circular faces of each solid are joined to form a single circle C. The point P lies on the circumference of C, and the point V is the vertex of the cone.

All units are given in cm.

The toy is positioned so that the circle C lies completely within the plane defined by .

The point P has coordinates and the point V has coordinates .

1. Find a vector equation for the line passing through V and the centre of C. [3 marks]

2. Find the coordinates of the centre of C. [3 marks]

3. 1. Show that the radius of C is cm.
   2. Find the total surface area of the toy. [6 marks]

The toy is rotated so that the position of the vertex is fixed at V and the image of the point P has coordinates .

4. Find angle giving your answer in degrees. [3 marks]

5. Find an equation of the plane containing the points P, V and , giving your answer in the form , where . [4 marks]

Question 12 [Maximum mark: 21]

The function is defined as , where and .

First consider the case where is finite.

1. 1. Show that .
   2. Find in terms of . [4 marks]

Now consider the case where the graph of has a vertical asymptote, an oblique asymptote with equation , and at least one point of zero gradient.

2. Write down the equation of the vertical asymptote. [1 mark]

3. Show that . [3 marks]

4. Show that . [6 marks]

5. In the case where and

   1. sketch the graph of , showing any asymptotes with their equations, the value of the intercept with the y-axis, and the coordinates of any local maximum or minimum points;
   2. solve . [7 marks]

Question 1 [Maximum mark: 5]

Consider the function , where and is a real constant.

The axis of symmetry of the graph of has equation .

1. Show that . [2 marks]

2. Find the coordinates of the vertex of the graph of . [2 marks]

3. Hence, write down the range of . [1 mark]

Question 2 [Maximum mark: 5]

A particle P is travelling along a straight line, with displacement measured relative to a point O on the line. The velocity, , of the particle at time seconds is given by

where .

1. Find . [2 marks]

Initially the particle is at O.

2. Find an expression for the displacement, metres, in terms of . [3 marks]

Question 3 [Maximum mark: 6]

Consider a geometric sequence, , with common ratio , where each term in the sequence is positive.

It is given that and .

1. Find the value of . [3 marks]

2. Find the value of , giving your answer in the form , where and . [3 marks]

Question 4 [Maximum mark: 7]

1. Show that for . [4 marks]

2. Hence, solve the equation where . [3 marks]

Question 5 [Maximum mark: 9]

Consider two events, A and B.

It is given that , and , where .

1. Show that . [3 marks]

It is given that A and B are independent.

2. Find the possible values of . [6 marks]

Question 6 [Maximum mark: 4]

Consider a piece of paper in the shape of a circle with centre O and radius cm, where . A second circle with centre O and radius 6 cm is drawn inside the first and shaded.

The points A and B are on the circumference of the larger circle such that the acute angle . The paper is cut along the lines AO and BO and the sector AOB with a central angle of is removed.

After removing the sector, it is now given that

Find the value of , giving your answer in the form , where .

Question 7 [Maximum mark: 6]

Reyn is investigating the fertility of a group of adult female foxes. He has developed a clinical test that aims to identify whether or not a given fox is pregnant, and he wants to evaluate the reliability of this test.

Reyn knows that:

• 40% of the foxes in the group are pregnant
• if a fox is pregnant, the probability that it will test positive is 90%
• if a fox is not pregnant, the probability it will test positive is 20%.

1. Find the probability that a randomly selected fox from the group will test positive. [3 marks]

2. Find the probability that a randomly selected fox from the group is pregnant, given that it tested positive. [3 marks]

Question 8 [Maximum mark: 8]

Consider the polynomial where and .

Three of the roots of are , , , where , .

1. By considering the sum and the product of the roots of ,
   1. show that ;
   2. find the value of . [5 marks]
2. Show that , where are integers to be determined. [3 marks]

Question 9 [Maximum mark: 8]

Consider non-zero vectors , and in three-dimensional space.

The non-zero vector is defined as .

1. Show that . [2 marks]

It is given that and are non-zero vectors.

2. 1. Justify why .
   2. Show that . [4 marks]
3. Hence, state the geometrical relationship between and , briefly justifying your answer. [2 marks]

Question 10 [Maximum mark: 15]

Consider the function , where and where and are real constants.

The graph of has a point of inflexion at .

1. Show that and . [6 marks]

It is given that the tangents to the graph of at and are horizontal.

2. 1. Show that .
   2. Find the value of .
   3. State whether has a local maximum or a local minimum at , justifying your answer. [7 marks]

The graph of intersects the y-axis at the point P.

3. Show that the tangent to the graph of at passes through P. [2 marks]

Question 11 [Maximum mark: 19]

1. Prove the identity . [3 marks]

2. Hence, solve the equation for where . [6 marks]

3. By differentiating both sides of the identity in part (a), show that . [5 marks]

Consider the right-angled triangle , where , and , where and all lengths are given in metres.

QR has length metres and has area square metres.

4. Calculate the ratio in the form where . [5 marks]

Question 12 [Maximum mark: 18]

Rowan is investigating the spread of an infection through a population of rabbits. He models , the number (in thousands) of infected rabbits in the population, at time days.

Rowan models the spread of the infection by the differential equation

where and .

Initially, 1500 rabbits are infected.

1. Express in the form , where . [3 marks]

2. Hence, show that , where is a constant to be determined. [8 marks]

3. Find an expression for in the form where . [5 marks]

According to Rowan’s model, the number of infected rabbits approaches a limit, , over the long term.

4. Find the value of . [2 marks]