Institute of Engineering · official sample
IOE Model Questions 2080
The 19 sample questions IOE published with the 2080 entrance syllabus to show the question style. Section A questions carry 1 mark and Section B questions 2 marks. The worked answers are ours.
Section A · 1 mark each
1. The value of k for which one root of the equation 3x² + 7x + 6 − k = 0 is equal to zero is
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Put x = 0 in the equation.
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Answer: D. 6
If x = 0 is a root, substituting gives 6 − k = 0, so k = 6.
2. If the line x + y − 1 = 0 touches the parabola y² = kx, then the value of k is
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Substitute the line into the parabola and set the discriminant to zero.
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Answer: D. −4
Put x = 1 − y: y² = k(1 − y), so y² + ky − k = 0. Tangency needs a zero discriminant: k² + 4k = 0, so k = 0 or k = −4. k = 0 is not a parabola, so k = −4.
3. The derivative of sech⁻¹x is valid for
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Where is 1/(x√(1 − x²)) defined?
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Answer: B. x < 1
The official options don't include the exact condition 0 < x < 1; “x < 1” is the closest.
d/dx (sech⁻¹x) = −1/(x√(1 − x²)), which is defined only for 0 < x < 1. Of the options, x < 1 is the one that fits (strictly the condition is 0 < x < 1).
4. The integrating factor of the differential equation dy/dx + (x/(1 + x))y = 1 + x is
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Write x/(1 + x) as 1 − 1/(1 + x) before integrating.
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Answer: B. eˣ/(1 + x)
IF = e^(∫x/(1 + x) dx) = e^(∫(1 − 1/(1 + x)) dx) = e^(x − ln(1 + x)) = eˣ/(1 + x).
8. Two ice blocks when pressed together join to form one block, because
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Think of regelation.
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Answer: B. melting point falls with pressure
Pressure lowers the melting point of ice, so the ice at the contact melts; when the pressure is released the water refreezes and joins the blocks (regelation).
9. If you are given three equal resistors, how many different combinations of these resistors are possible?
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List series, parallel and the two mixed arrangements.
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Answer: C. 4
Using all three: all in series (3R), all in parallel (R/3), two in series with the third in parallel (2R/3), and two in parallel with the third in series (3R/2): four combinations.
12. Any p-orbital can accommodate up to
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One orbital, not the whole subshell.
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Answer: D. two electrons with anti-parallel spins
Each orbital (one of px, py, pz) holds at most two electrons, and by Pauli's exclusion principle they must have opposite spins. Six is the capacity of the whole p subshell.
14. In Williamson’s synthesis, a haloalkane is treated with
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It is a method of making ethers.
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Answer: C. sodium alkoxide
Williamson ether synthesis: R–X + R′–ONa → R–O–R′ + NaX, an SN2 reaction of a haloalkane with sodium alkoxide.
16. He is one person I count ………… in times of difficulty.
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Which preposition makes “count” mean “rely”?
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Answer: D. on
“Count on someone” means to rely on them.
17. Which of the following pairs have the same initial vowel sound?
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Look for the unstressed schwa.
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Answer: A. ago, alone
Both ago /əˈɡəʊ/ and alone /əˈləʊn/ begin with the schwa /ə/. inn /ɪ/ vs eel /iː/, owl /aʊ/ vs all /ɔː/, up /ʌ/ vs on /ɒ/ differ.
18. The word ‘procrastination’ has its stress on the ………… syllable.
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-tion words stress the syllable before -tion.
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Answer: C. 4th
pro-cras-ti-NA-tion /prəˌkræstɪˈneɪʃn/: words ending in -tion are stressed on the syllable just before the ending, which is the 4th here.
Section B · 2 marks each
5. If a⃗ = i + 2j + 3k and b⃗ = 2i + 3j − 2k, then the projection of b⃗ on a⃗ is
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Use (a·b)/|a|.
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Answer: A. 2/√14 or C. √(2/7)
2/√14 and √(2/7) are the same number, so options A and C are both correct as printed in the official model set.
Projection of b on a = (a·b)/|a| = (2 + 6 − 6)/√(1 + 4 + 9) = 2/√14.
6. If a lottery ticket is drawn at random from a lottery box containing 10 prizes and 25 blanks, what is the probability of getting a prize?
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Favourable over total tickets.
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Answer: C. 2/7
There are 10 + 25 = 35 tickets, 10 of them prizes: P = 10/35 = 2/7.
7. To what depth below the surface of a sea should a rubber ball be taken so that its volume decreases by 1%? Bulk modulus of rubber is 9 × 10⁸ N m⁻². (Take density of sea water 1000 kg/m³ and g = 10 m/s².)
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Pressure needed = B × (ΔV/V), then h = P/(ρg).
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Answer: D. 1 km
The exact depth is about 900 m; “1 km” is the nearest option.
ΔV/V = P/B, so P = 0.01 × 9 × 10⁸ = 9 × 10⁶ Pa. Depth h = P/(ρg) = 9 × 10⁶/(1000 × 10) = 900 m, which is about 1 km.
10. The ratio of the areas within the electron orbits for the first excited state to the ground state for the hydrogen atom is
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r ∝ n² and area ∝ r².
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Answer: A. 16 : 1
Bohr radius rₙ ∝ n², so area ∝ r² ∝ n⁴. For n = 2 vs n = 1: 2⁴ : 1⁴ = 16 : 1.
11. 21 g of magnesium carbonate was treated with 50 mL of 12 N hydrochloric acid. How many molecules of unreacted hydrochloric acid remain?
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Find the HCl used by 0.25 mol MgCO₃ and subtract.
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Answer: B. 6.025 × 10²²
MgCO₃ (M = 84): 21/84 = 0.25 mol, which needs 0.5 mol HCl (MgCO₃ + 2HCl → MgCl₂ + H₂O + CO₂). HCl taken = 0.050 L × 12 = 0.6 mol. Left over = 0.1 mol = 0.1 × 6.025 × 10²³ = 6.025 × 10²² molecules.
13. What mass of copper will be deposited by passing 2 faradays of electricity through a solution of a copper(II) salt?
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How many faradays does one mole of Cu²⁺ need?
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Answer: C. 63.5 g
Cu²⁺ + 2e⁻ → Cu: 2 F deposit 1 mol of copper = 63.5 g.
15. It …… two weeks since he left the city.
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The time runs from the past up to now.
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Answer: C. has been
A period of time from a past point up to now takes the present perfect: “It has been two weeks since…”.
Read the passage, then answer the questions below
19. In the English language, the animals are
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Look at the adjectives the passage lists.
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Answer: B. used for unpleasant comparisons
The passage says English ‘transfers negative human traits to animals’ and gives the language ‘as many unpleasant adjectives as it could possibly want’.
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