ESAT Engineering and Science Admissions Test — Questions and Answers

Question 1: What type of questions appear in the ESAT?

Four-option multiple choice only (Correct answer)
Diagram labelling
Open-ended written responses
True/false statements

Correct answer: Four-option multiple choice only

The ESAT consists entirely of four-option multiple-choice questions across all parts.

Question 2: What type of molecule forms the phospholipid bilayer of cell membranes?

Purely hydrophobic lipids
Amphipathic phospholipids with hydrophilic heads and hydrophobic tails (Correct answer)
Carbohydrate chains
Cholesterol alone

Correct answer: Amphipathic phospholipids with hydrophilic heads and hydrophobic tails

Phospholipids are amphipathic, with polar (hydrophilic) heads facing water and non-polar (hydrophobic) tails forming the interior.

Question 3: What is the dot product of vectors (1, 2, 3) and (4, -1, 2)?

6
10
8 (Correct answer)
12

Correct answer: 8

(1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8.

Question 4: Which of the following subjects is most likely to be covered in the ESAT?

Literature Analysis
Art History
World History
Organic Chemistry (Correct answer)

Correct answer: Organic Chemistry

The ESAT is designed to assess foundational knowledge in engineering and science subjects. Organic Chemistry is a fundamental branch of chemistry that is highly relevant to many engineering and science disciplines, making it a likely topic for inclusion. In contrast, World History, Literature Analysis, and Art History fall outside the typical scope of a specialized engineering and science admissions test.

Question 5: What is the recommended approach for pacing through ESAT questions?

Spend as long as needed on each question
Spend about 90 seconds per question and move on if stuck (Correct answer)
Read the whole paper before answering
Answer only the easy questions first and skip hard ones entirely

Correct answer: Spend about 90 seconds per question and move on if stuck

With 27 questions in 40 minutes, allocating roughly 90 seconds per question and moving on if stuck maximises overall score.

Question 6: What does the ESAT Biology section typically assess?

Human geography
Clinical medicine
Pharmacology only
Cell biology, genetics, and physiology (Correct answer)

Correct answer: Cell biology, genetics, and physiology

The ESAT Biology component covers cell biology, genetics, physiology, ecology, and evolution.

Question 7: What is the Hardy-Weinberg equilibrium used for in genetics?

Predicting allele frequencies in a non-evolving population (Correct answer)
Determining protein folding
Calculating mutation rates
Measuring gene expression levels

Correct answer: Predicting allele frequencies in a non-evolving population

Hardy-Weinberg equilibrium describes the allele and genotype frequencies in an ideally non-evolving population, serving as a baseline for detecting evolution.

Question 8: In an ESAT physics problem, a 2 kg object accelerates at 5 m/s². What net force acts on it?

7 N
10 N (Correct answer)
3 N
2.5 N

Correct answer: 10 N

F = ma = 2 kg × 5 m/s² = 10 N.

Question 9: Which of the following is tested in the ESAT Mathematics component?

Essay writing
Programming logic
Calculus and algebra (Correct answer)
Data interpretation only

Correct answer: Calculus and algebra

The ESAT Mathematics component assesses calculus, algebra, geometry, and other core mathematical skills.

Question 10: What is the standard electrode potential of the standard hydrogen electrode (SHE)?

0.00 V (Correct answer)
0.76 V
1.00 V
-1.00 V

Correct answer: 0.00 V

The standard hydrogen electrode is assigned a potential of exactly 0.00 V by convention.

Question 11: How should a student approach studying for the physics section of the ESAT?

By memorizing equations without understanding their context
By studying unrelated subjects like biology and history
By focusing on understanding fundamental principles such as mechanics and electromagnetism, and practicing related problems (Correct answer)
By ignoring the physics section altogether

Correct answer: By focusing on understanding fundamental principles such as mechanics and electromagnetism, and practicing related problems

To effectively study for the physics section of the ESAT, a student should prioritize understanding fundamental principles like mechanics, electricity and magnetism, thermodynamics, and waves. Rote memorization of equations is insufficient; instead, focus on grasping the underlying concepts and their applications. Consistent practice with problems that require applying these principles will solidify understanding and improve problem-solving speed.

Question 12: Which of the following is an example of an oxidation reaction?

Cl₂ + 2e⁻ → 2Cl⁻
Cu²⁺ + 2e⁻ → Cu
Mg → Mg²⁺ + 2e⁻ (Correct answer)
Fe³⁺ + e⁻ → Fe²⁺

Correct answer: Mg → Mg²⁺ + 2e⁻

Oxidation is loss of electrons (OIL RIG); Mg losing 2 electrons to form Mg²⁺ is oxidation.

Question 13: Which university program is most likely to require a high score on the ESAT for admission?

Bachelor of Business Administration
Bachelor of Arts in History
Bachelor of Science in Mechanical Engineering (Correct answer)
Bachelor of Fine Arts in Painting

Correct answer: Bachelor of Science in Mechanical Engineering

Given that the ESAT is the Engineering & Science Admissions Test, a Bachelor of Science in Mechanical Engineering is the program most aligned with its assessment focus. Engineering degrees require a strong aptitude in mathematics and scientific principles, which the ESAT is designed to measure. Programs in humanities, fine arts, or business administration would not typically require this specialized test.

Question 14: What type of bonding involves the sharing of electron pairs between atoms?

Covalent bonding (Correct answer)
Hydrogen bonding
Metallic bonding
Ionic bonding

Correct answer: Covalent bonding

Covalent bonds are formed by the sharing of one or more electron pairs between atoms.

Question 15: What is the relationship between pressure, force, and area?

P = FA
P = F/A (Correct answer)
F = P²A
P = A/F

Correct answer: P = F/A

Pressure is defined as force per unit area: P = F/A.

Question 16: How many moles of NaCl are in 117 g? (Molar mass NaCl = 58.5 g/mol)

58.5 mol
0.5 mol
1 mol
2 mol (Correct answer)

Correct answer: 2 mol

n = mass / molar mass = 117 / 58.5 = 2 mol.

Question 17: What does the area under a velocity-time graph represent?

Displacement (Correct answer)
Acceleration
Power
Force

Correct answer: Displacement

The area under a v-t graph gives displacement, since displacement = velocity × time.

Question 18: Find the equation of the line perpendicular to y = 2x + 5 passing through (4, 1).

y = -x/2 - 3
y = x/2 + 3
y = -x/2 + 3 (Correct answer)
y = 2x - 7

Correct answer: y = -x/2 + 3

Perpendicular slope is -1/2; using point-slope form: y - 1 = -1/2(x - 4) → y = -x/2 + 3.

Question 19: In ESAT Chemistry, what is the first ionisation energy?

Energy released when one electron is gained
Energy to break one mole of bonds
Energy to remove one electron from a gaseous atom (Correct answer)
Energy required to melt one mole of a substance

Correct answer: Energy to remove one electron from a gaseous atom

First ionisation energy is the energy needed to remove one mole of electrons from one mole of gaseous atoms.

Question 20: What is Avogadro's constant?

6.022 × 10²³ mol⁻¹ (Correct answer)
9.81 × 10²³ mol⁻¹
1.38 × 10⁻²³ mol⁻¹
6.022 × 10²² mol⁻¹

Correct answer: 6.022 × 10²³ mol⁻¹

Avogadro's constant is 6.022 × 10²³ mol⁻¹, the number of particles in one mole.

Question 21: Which law states that the total momentum of a closed system remains constant?

Hooke's law
Conservation of energy
Newton's third law
Conservation of momentum (Correct answer)

Correct answer: Conservation of momentum

The law of conservation of momentum states that total momentum is constant in the absence of external forces.

Question 22: Which type of electromagnetic radiation has the highest frequency?

Gamma rays (Correct answer)
Ultraviolet
X-rays
Microwaves

Correct answer: Gamma rays

Gamma rays occupy the highest-frequency (and highest-energy) end of the electromagnetic spectrum.

Question 23: In ESAT Chemistry, what does a higher activation energy indicate?

Faster reaction rate
Slower reaction rate (Correct answer)
Higher product yield
Exothermic reaction

Correct answer: Slower reaction rate

Higher activation energy means fewer molecules have sufficient energy to react, resulting in a slower rate.

Question 24: How long is the ESAT in total (all parts combined)?

30 minutes
4 hours
Approximately 2 hours (Correct answer)
1 hour

Correct answer: Approximately 2 hours

The ESAT consists of two parts each lasting approximately 40 minutes, totalling around 80 minutes of test time.

Question 25: Why is it important to take practice tests when preparing for the ESAT?

To avoid studying altogether
To memorize answers without understanding concepts
To learn new material not covered in textbooks
To become familiar with the test format and improve time management skills (Correct answer)

Correct answer: To become familiar with the test format and improve time management skills

Taking practice tests is crucial for ESAT preparation because it allows students to become thoroughly familiar with the test format, question types, and overall structure. This familiarity reduces anxiety on test day and, more importantly, helps students develop and refine their time management skills. By simulating the actual testing environment, practice tests enable students to pace themselves effectively and complete all sections within the allotted time.

Question 26: Which organelle is the site of protein synthesis in eukaryotic cells?

Lysosome
Golgi apparatus
Ribosome (Correct answer)
Vacuole

Correct answer: Ribosome

Ribosomes are the organelles that carry out translation, assembling amino acids into proteins.

Question 27: What is the product of the complex numbers (2 + i) and (1 - 3i)?

5 - 5i (Correct answer)
5 + 5i
2 - 6i
-1 - 5i

Correct answer: 5 - 5i

(2+i)(1-3i) = 2 - 6i + i - 3i² = 2 - 5i + 3 = 5 - 5i, since i² = -1.

Question 28: Why do universities use the ESAT as part of their admissions process?

To identify students with strong potential for success in rigorous engineering and science programs (Correct answer)
To assess candidates' skills in arts and literature
To provide scholarships based on performance
To increase the number of applicants

Correct answer: To identify students with strong potential for success in rigorous engineering and science programs

Universities utilize the ESAT as part of their admissions process to identify students who possess strong potential for success in their rigorous engineering and science programs. The test provides an objective measure of a candidate's aptitude and preparedness in key technical subjects. This helps institutions select applicants who are most likely to thrive academically and contribute to these challenging fields.

Question 29: What does Faraday's law of electromagnetic induction state?

Current is proportional to voltage
EMF is proportional to the rate of change of magnetic flux (Correct answer)
Force on a charge equals qvB
Magnetic field lines form closed loops

Correct answer: EMF is proportional to the rate of change of magnetic flux

Faraday's law states that induced EMF = -dΦ/dt, where Φ is the magnetic flux.

Question 30: What is the process by which plants make glucose using light energy?

Respiration
Photosynthesis (Correct answer)
Osmosis
Transpiration

Correct answer: Photosynthesis

Photosynthesis converts light energy, CO₂, and water into glucose and oxygen in chloroplasts.

Question 31: What is the role of enzymes in biological reactions?

They change the overall free energy of the reaction
They provide energy for reactions
They act as biological catalysts, lowering activation energy (Correct answer)
They are consumed during reactions

Correct answer: They act as biological catalysts, lowering activation energy

Enzymes are biological catalysts that speed up reactions by lowering the activation energy without being consumed.

Question 32: What is the marking scheme for the ESAT?

Two marks per correct, minus one for wrong
One mark per correct answer, no penalty for wrong answers (Correct answer)
Fraction marks for partially correct answers
Marks deducted for all incorrect answers

Correct answer: One mark per correct answer, no penalty for wrong answers

The ESAT uses positive marking only: one mark for each correct answer with no penalty for incorrect responses.

Question 33: What is a key difference between the ESAT and typical A-level exams?

ESAT is purely essay-based
ESAT covers the same breadth at the same depth as A-level
ESAT allows more time per question than A-level exams
ESAT questions are shorter but require faster application of knowledge (Correct answer)

Correct answer: ESAT questions are shorter but require faster application of knowledge

The ESAT demands rapid application of knowledge across concise multiple-choice questions, unlike the longer structured answers of A-level exams.

Question 34: What is the molar mass of water (H₂O)?

34 g/mol
18 g/mol (Correct answer)
20 g/mol
16 g/mol

Correct answer: 18 g/mol

H₂O: 2(1) + 16 = 18 g/mol, using standard atomic masses.

Question 35: What is one of the key outcomes for universities that use the ESAT in their admissions process?

They can determine students' interest in sports and activities.
They can decrease the number of applicants.
They can better align their incoming students with the demands of their engineering and science programs. (Correct answer)
They can assess students' skills in foreign languages.

Correct answer: They can better align their incoming students with the demands of their engineering and science programs.

A key outcome for universities using the ESAT is the ability to better align their incoming student body with the rigorous demands of their engineering and science programs. By assessing foundational knowledge and problem-solving skills, the ESAT helps universities admit students who are academically prepared for the curriculum. This leads to higher student retention and success rates within these challenging fields.

Question 36: What is lim(x→0) of sin(x)/x?

undefined
1 (Correct answer)
0
∞

Correct answer: 1

lim(x→0) sin(x)/x = 1 is a fundamental limit, provable via L'Hôpital's rule or the squeeze theorem.

Question 37: What is the half-life concept in radioactive decay as tested in ESAT?

Time for all nuclei to decay
Time for the activity to double
Time for half the mass to be lost as energy
Time for half the nuclei to decay (Correct answer)

Correct answer: Time for half the nuclei to decay

Half-life is the time taken for half the radioactive nuclei in a sample to undergo decay.

Question 38: According to De Moivre's theorem, (cos θ + i sin θ)^5 equals which of the following?

cos(5θ) + i sin(5θ) (Correct answer)
cos(θ/5) + i sin(θ/5)
5cos(θ) + 5i sin(θ)
cos²(5θ) - sin²(5θ)

Correct answer: cos(5θ) + i sin(5θ)

De Moivre's theorem states (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), so with n = 5 the result is cos(5θ) + i sin(5θ).

Question 39: In ESAT Physics, what does Coulomb's law describe?

The force on a current-carrying wire
The electric field inside a conductor
The force between two point charges (Correct answer)
The energy stored in a capacitor

Correct answer: The force between two point charges

Coulomb's law gives the electrostatic force F = kq₁q₂/r² between two point charges.

Question 40: What does the term 'natural selection' mean in evolutionary biology?

All individuals in a population survive equally
Traits are inherited from only one parent
Organisms choose their own traits
Individuals with advantageous traits survive and reproduce more (Correct answer)

Correct answer: Individuals with advantageous traits survive and reproduce more

Natural selection is the process by which heritable traits that improve survival and reproduction become more common over generations.

Question 41: What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + ...?

3
3/2 (Correct answer)
2/3
1/2

Correct answer: 3/2

For |r| < 1, the sum is a/(1-r) = 1/(1 - 1/3) = 3/2.

Question 42: Which resource is officially recommended for ESAT preparation?

Online social media study groups exclusively
Undergraduate textbooks
Past NSAA/ENGAA papers and specimen materials published by Cambridge/Imperial (Correct answer)
General GCSE revision guides only

Correct answer: Past NSAA/ENGAA papers and specimen materials published by Cambridge/Imperial

Official specimen papers and the previous NSAA/ENGAA past papers (the ESAT's predecessors) are the most authentic preparation resources available.

Question 43: What is the ideal gas law?

PV² = nRT
PV = nRT (Correct answer)
P = nRT/V²
PV = nT

Correct answer: PV = nRT

The ideal gas law PV = nRT relates pressure, volume, moles, the gas constant R, and temperature.

Question 44: The mathematics section of the ESAT typically includes questions on which of the following topics?

Algebra, Calculus, and Trigonometry (Correct answer)
Music Theory
Political Science and Economics
Language and Grammar

Correct answer: Algebra, Calculus, and Trigonometry

The mathematics section of the ESAT is designed to evaluate a candidate's readiness for advanced STEM studies. Therefore, it typically includes questions on core higher-level mathematics topics such as Algebra, Calculus, and Trigonometry. These subjects form the foundational mathematical language and problem-solving tools essential for success in engineering and science programs.

Question 45: What is produced when an acid reacts with a base?

Salt and water (Correct answer)
Water and hydrogen gas
Oxide and water
Salt and carbon dioxide

Correct answer: Salt and water

An acid-base neutralisation reaction produces a salt and water as products.

Question 46: Which molecule carries genetic information in most organisms?

RNA
ATP
Protein
DNA (Correct answer)

Correct answer: DNA

Deoxyribonucleic acid (DNA) stores genetic information in the sequence of its nucleotide bases.

Question 47: In ESAT Chemistry, what does the rate equation r = k[A]²[B] indicate?

Zero order overall
Second order in B and first order in A
First order in both A and B
Second order in A and first order in B, third order overall (Correct answer)

Correct answer: Second order in A and first order in B, third order overall

The rate equation shows the reaction is second order in A, first order in B, and third order overall.

Question 48: In the ESAT Physics component, Ohm's Law is expressed as:

R = IV
P = IV²
I = V²/R
V = IR (Correct answer)

Correct answer: V = IR

Ohm's Law states that voltage (V) equals current (I) multiplied by resistance (R).

Question 49: What is Newton's second law of motion as typically tested in the ESAT Physics section?

F = mv
F = ma (Correct answer)
p = mv²
E = mc²

Correct answer: F = ma

Newton's second law states that the net force equals mass multiplied by acceleration (F = ma).

Question 50: What is the pH of a solution with [H⁺] = 1 × 10⁻³ mol/L?

7
-3
11
3 (Correct answer)

Correct answer: 3

pH = -log[H⁺] = -log(10⁻³) = 3.

Question 51: What is the role of the ribosome in a cell?

Lipid digestion
Protein synthesis (translation) (Correct answer)
DNA replication
Energy production

Correct answer: Protein synthesis (translation)

Ribosomes read mRNA codons and assemble amino acids into polypeptide chains during translation.

Question 52: What type of reaction converts alkenes to alkanes using hydrogen gas?

Halogenation
Hydrogenation (Correct answer)
Hydrolysis
Oxidation

Correct answer: Hydrogenation

Hydrogenation adds H₂ across the double bond of an alkene, converting it to an alkane.

Question 53: What should a candidate do if they encounter a question with an unfamiliar topic in the ESAT?

Change all previous answers
Leave it completely blank
Spend as long as possible trying to recall the answer
Apply logical reasoning and elimination to make the best guess, then move on (Correct answer)

Correct answer: Apply logical reasoning and elimination to make the best guess, then move on

With no negative marking, applying reasoning to eliminate options and making a best guess is always better than leaving a blank.

Question 54: What is the product of anaerobic respiration in animal cells?

Glucose and water
Ethanol and CO₂
Lactate and a small amount of ATP (Correct answer)
Oxygen and ATP

Correct answer: Lactate and a small amount of ATP

In animal cells, anaerobic respiration converts pyruvate to lactate, regenerating NAD⁺ for continued glycolysis.

Question 55: Solve for x: log₂(x) + log₂(4) = 5

8 (Correct answer)
4
16
32

Correct answer: 8

log₂(4x) = 5 means 4x = 32, so x = 8.

Question 56: A projectile is launched horizontally from a cliff. Which quantity remains constant throughout its flight (ignoring air resistance)?

Vertical velocity
Speed
Kinetic energy
Horizontal velocity (Correct answer)

Correct answer: Horizontal velocity

With no air resistance, the horizontal velocity is unchanged because no horizontal force acts on the projectile.

Question 57: How many complex roots does x⁴ + 1 = 0 have?

4 (Correct answer)
1
0
2

Correct answer: 4

A degree-4 polynomial always has exactly 4 roots (counting multiplicity) in the complex numbers.

Question 58: What is the value of sin(π/6)?

1/2 (Correct answer)
1
√3/2
√2/2

Correct answer: 1/2

sin(30°) = sin(π/6) = 1/2 is a standard trigonometric value.

Question 59: How far in advance should candidates begin preparing for the ESAT?

No preparation is needed
At least 3–6 months before the test date (Correct answer)
The night before
1 week before

Correct answer: At least 3–6 months before the test date

Given the ESAT's challenging A-level+ content, experts recommend 3–6 months of structured preparation to consolidate subject knowledge and exam technique.

Question 60: For vectors a = (1, 2, 3) and b = (4, 5, 6), what is the dot product a · b?

17
45
32 (Correct answer)
28

Correct answer: 32

a · b = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.

Question 61: What is the SI unit of electric charge?

Farad (F)
Ampere (A)
Coulomb (C) (Correct answer)
Volt (V)

Correct answer: Coulomb (C)

The SI unit of electric charge is the coulomb (C), equal to one ampere-second.

Question 62: What is the difference between mitosis and meiosis?

Mitosis produces haploid cells; meiosis produces diploid cells
Mitosis produces 2 identical diploid cells; meiosis produces 4 haploid cells (Correct answer)
Both produce identical haploid cells
Mitosis only occurs in gametes

Correct answer: Mitosis produces 2 identical diploid cells; meiosis produces 4 haploid cells

Mitosis produces two genetically identical diploid cells for growth/repair; meiosis produces four genetically diverse haploid gametes.

Question 63: What is the function of mitochondria in a cell as tested in ESAT Biology?

Site of photosynthesis
Controls cell division
Site of aerobic respiration and ATP production (Correct answer)
Site of protein synthesis

Correct answer: Site of aerobic respiration and ATP production

Mitochondria are the sites of aerobic respiration, producing the majority of a cell's ATP.

Question 64: In a series circuit with resistors R₁ = 3Ω and R₂ = 6Ω, what is the total resistance?

4.5 Ω
18 Ω
9 Ω (Correct answer)
2 Ω

Correct answer: 9 Ω

Resistors in series add directly: R_total = R₁ + R₂ = 3 + 6 = 9 Ω.

Question 65: How many distinct real roots does x⁴ - 5x² + 4 = 0 have?

2
4 (Correct answer)
3
0

Correct answer: 4

Substituting u = x² gives (u-1)(u-4) = 0, so u = 1 or u = 4, yielding x = ±1 and x = ±2: four distinct real roots.

Question 66: What is a buffer solution?

A solution that changes colour with pH
A solution of neutral pH
A solution that conducts electricity
A solution that resists pH change on addition of small amounts of acid or base (Correct answer)

Correct answer: A solution that resists pH change on addition of small amounts of acid or base

A buffer solution contains a weak acid and its conjugate base, resisting pH change when small amounts of acid or alkali are added.

Question 67: Which type of institution is most likely to use the ESAT for selecting candidates?

Music Conservatories
Liberal Arts Colleges
Technical Universities and Engineering Schools (Correct answer)
Culinary Institutes

Correct answer: Technical Universities and Engineering Schools

The ESAT is tailored for evaluating candidates for engineering and science fields. Consequently, Technical Universities and Engineering Schools are the institutions most likely to utilize this test. These specialized institutions focus heavily on STEM disciplines and require applicants to demonstrate strong foundational knowledge in these areas.

Question 68: In ESAT Physics, what is the principle of superposition for waves?

Waves speed up when they overlap
Waves cancel each other out completely
Wave frequency doubles when two waves meet
The resultant displacement is the sum of individual displacements (Correct answer)

Correct answer: The resultant displacement is the sum of individual displacements

Superposition states that when two or more waves overlap, the total displacement equals the algebraic sum of individual displacements.

Question 69: What is the empirical formula of a compound with 40% C, 6.67% H, and 53.33% O by mass?

CH₄O
C₂H₂O
CH₂O (Correct answer)
C₂H₄O₂

Correct answer: CH₂O

Dividing by atomic masses: C:H:O = 40/12 : 6.67/1 : 53.33/16 ≈ 3.33 : 6.67 : 3.33 = 1:2:1 → CH₂O.

Question 70: What is the Maclaurin series for e^x up to and including the x³ term?

x + x²/2 + x³/6
1 + x + 2x² + 6x³
1 + x + x²/2 + x³/6 (Correct answer)
1 + x + x² + x³

Correct answer: 1 + x + x²/2 + x³/6

The Maclaurin series for e^x is Σ xⁿ/n!, so 1 + x + x²/2! + x³/3! = 1 + x + x²/2 + x³/6.

Question 71: In the human nervous system, what is the role of a synapse?

The junction where signals are transmitted between neurons via neurotransmitters (Correct answer)
The site where action potentials are generated
The structure that stores neurotransmitters permanently
The region that myelinates axons

Correct answer: The junction where signals are transmitted between neurons via neurotransmitters

A synapse is the gap between two neurons where neurotransmitters are released to transmit signals across the junction.

Question 72: Which technique is most appropriate for evaluating ∫x·e^x dx?

Integration by parts (Correct answer)
Substitution
Partial fractions
Trigonometric substitution

Correct answer: Integration by parts

Integration by parts with u = x and dv = e^x dx gives ∫x·e^x dx = x·e^x - e^x + C.

Question 73: Which integration technique would best solve ∫ x·eˣ dx in an ESAT question?

Substitution
Integration by parts (Correct answer)
Partial fractions
Trigonometric substitution

Correct answer: Integration by parts

Integration by parts (∫u dv = uv - ∫v du) is used for products of a polynomial and exponential.

Question 74: What is the volume of the solid formed when y = √x (0 ≤ x ≤ 4) is rotated 360° about the x-axis?

8π (Correct answer)
16π
4π
32π

Correct answer: 8π

V = π∫₀⁴ (√x)² dx = π∫₀⁴ x dx = π[x²/2]₀⁴ = 8π.

Question 75: What shape does a molecule with 4 bonding pairs and no lone pairs adopt?

Bent
Trigonal planar
Tetrahedral (Correct answer)
Linear

Correct answer: Tetrahedral

VSEPR theory predicts a tetrahedral geometry for 4 bonding pairs with no lone pairs (e.g., CH₄).

Question 76: Which of the following describes the work-energy theorem?

Power equals force times velocity squared
Net work done on an object equals its change in kinetic energy (Correct answer)
Energy is always conserved in all forms
Work equals force times time

Correct answer: Net work done on an object equals its change in kinetic energy

The work-energy theorem states W_net = ΔKE = ½mv² - ½mu².

Question 77: A wave has frequency 500 Hz and wavelength 0.68 m. What is its speed?

340 m/s (Correct answer)
735 m/s
0.68 m/s
500 m/s

Correct answer: 340 m/s

v = fλ = 500 × 0.68 = 340 m/s, which is the speed of sound in air.

Question 78: What is the gravitational potential energy of a 5 kg mass at a height of 10 m? (g = 10 m/s²)

250 J
50 J
500 J (Correct answer)
5000 J

Correct answer: 500 J

GPE = mgh = 5 × 10 × 10 = 500 J.

Question 79: Which periodic trend describes the decrease in atomic radius across a period (left to right)?

Decreasing atomic mass
Decreasing number of electron shells
Increasing nuclear charge attracting electrons more strongly (Correct answer)
Increasing electron shielding

Correct answer: Increasing nuclear charge attracting electrons more strongly

Across a period, protons increase while electrons are added to the same shell, so greater nuclear charge pulls electrons closer.

Question 80: Evaluate the definite integral ∫₀^π sin(x) dx.

2 (Correct answer)
π
0
1

Correct answer: 2

[-cos(x)]₀^π = -cos(π) + cos(0) = 1 + 1 = 2.

ESAT Engineering and Science Admissions Test

The ESAT (Engineering and Science Admissions Test) is used by Cambridge, Imperial College London, and other UK universities for engineering and science admission. It assesses mathematics, physics, chemistry, and biology across multiple timed paper sections.

Questions

120 min

Time Limit

60%

To Pass

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