ECAT — Engineering College Admission Test (UET)

ECAT Full-Length Mock Test 6 (100 MCQs)

Take the ECAT Full-Length Mock Test 6 (100 MCQs): 100 multiple-choice questions (110 minutes) with instant scoring and a full answer review. Covers 30…

What this test covers

30 ECAT Mathematics
30 ECAT Physics
30 ECAT Chemistry
10 ECAT English

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

Q1. If a complex number z satisfies |z − 3 − 4i| = 5, the greatest possible value of |z| is
- A. 7
- B. 15
- C. 5
- D. 10 ✓
Answer: D. 10
Explanation: z lies on a circle with centre 3 + 4i (at distance 5 from the origin) and radius 5. The farthest point from the origin is at distance 5 + 5 = 10.
Q2. If f(x) = x/(1 − x), then (f ∘ f)(x) equals
- A. x/(1 − 2x) ✓
- B. (1 − x)/x
- C. x/(1 − x)²
- D. x²/(1 − x)²
Answer: A. x/(1 − 2x)
Explanation: f(f(x)) = [x/(1 − x)] / [1 − x/(1 − x)] = [x/(1 − x)] / [(1 − 2x)/(1 − x)] = x/(1 − 2x).
Q3. On the set ℝ − {−1}, the operation a ∗ b = a + b + ab is defined (it forms a group with identity 0). The inverse of 3 is
- A. −3
- B. −1/3
- C. −3/4 ✓
- D. 1/4
Answer: C. −3/4
Explanation: We need 3 ∗ b = 0: 3 + b + 3b = 0, so 4b = −3 and b = −3/4.
Q4. If [[1, 2], [3, 5]] X = [[1], [2]], then X equals
- A. [[−1], [−1]]
- B. [[1], [1]]
- C. [[−1], [1]] ✓
- D. [[1], [−1]]
Answer: C. [[−1], [1]]
Explanation: det = 5 − 6 = −1 and the inverse is [[−5, 2], [3, −1]]. X = [[−5 + 4], [3 − 2]] = [[−1], [1]].
Q5. If A is a real 3 × 3 skew-symmetric matrix, then det A equals
- A. 0 ✓
- B. 3
- C. 1
- D. −1
Answer: A. 0
Explanation: For a skew-symmetric matrix, det A = det(Aᵀ) = det(−A) = (−1)³ det A = −det A, so det A = 0.

The other 95 questions

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Q6. The solution set of the equation √(x + 7) = x + 1 over the real numbers is
- A. {2}
- B. ∅
- C. {−3}
- D. {2, −3}
Q7. If (x² + 2)/(x(x + 1)²) = A/x + B/(x + 1) + C/(x + 1)², then B equals
- A. −1
- B. −3
- C. 1
- D. 2
Q8. The 7th term of the harmonic progression 6, 4, 3, … is
- A. 3/2
- B. 2
- C. 6/7
- D. 2/3
Q9. The number of four-digit even numbers that can be formed from the digits 1, 2, 3, 4, 5 without repeating any digit is
- A. 48
- B. 24
- C. 60
- D. 72
Q10. For events A and B, P(A) = 0.5, P(B) = 0.4 and P(A ∪ B) = 0.7. Then P(B | A) equals
- A. 3/5
- B. 1/5
- C. 1/2
- D. 2/5
Q11. The value of C(8,0) + C(8,2) + C(8,4) + C(8,6) + C(8,8) is
- A. 128
- B. 64
- C. 256
- D. 127
Q12. If sec θ − tan θ = 1/2 and θ is an acute angle, then sin θ equals
- A. 4/5
- B. 3/4
- C. 5/4
- D. 3/5
Q13. The sides of a triangle are 3, 5 and 7. Its largest angle is
- A. 135°
- B. 100°
- C. 120°
- D. 150°
Q14. The number of solutions of the equation sin 2θ = sin θ in the closed interval [0, π] is
- A. 3
- B. 4
- C. 1
- D. 2
Q15. The value of tan⁻¹(1/4) + tan⁻¹(3/5) is
- A. π/6
- B. 3π/4
- C. π/3
- D. π/4
Q16. The value of lim(x→0) (tan x − sin x)/x³ is
- A. 1/3
- B. 1
- C. 1/2
- D. 0
Q17. The value of lim(x→1) (x⁵ − 1)/(x³ − 1) is
- A. 1
- B. 3/5
- C. 15
- D. 5/3
Q18. For x > 0, the derivative of x^(ln x) with respect to x is
- A. 2 ln x · x^(ln x − 1)
- B. 2 x^(ln x)
- C. x^(ln x) · ln x
- D. ln x · x^(ln x − 1)
Q19. If y = ln(1 + x), the value of d³y/dx³ at x = 0 is
- A. −2
- B. 1
- C. 6
- D. 2
Q20. A curve is given by x = a(t − sin t), y = a(1 − cos t), where a > 0. The value of dy/dx at t = π/2 is
- A. −1
- B. 2
- C. 1
- D. 0
Q21. The function f(x) = x³ − 3x² is decreasing in the interval
- A. (−∞, 0)
- B. (2, ∞)
- C. (−2, 0)
- D. (0, 2)
Q22. The rectangle of greatest area inscribed in a circle of radius 5 is a square. Its side is
- A. 5√2
- B. 10
- C. 5
- D. 5√3
Q23. ∫ x/(x² + 1)² dx is equal to
- A. −1/(2(x² + 1)) + c
- B. ln(x² + 1)/2 + c
- C. −1/(x² + 1) + c
- D. 1/(2(x² + 1)) + c
Q24. ∫ dx/(x(x + 1)) for x > 0 is equal to
- A. ln((x + 1)/x) + c
- B. ln(x/(x + 1)) + c
- C. ln x · ln(x + 1) + c
- D. ln(x(x + 1)) + c
Q25. The value of ∫ from 0 to 1 of x(1 − x)³ dx is
- A. 1/30
- B. 1/4
- C. 1/12
- D. 1/20
Q26. The curve satisfying dy/dx = sec²x and passing through (π/4, 3) meets the y-axis where y equals
- A. 1
- B. 4
- C. 3
- D. 2
Q27. The distance of the point (1, −2) from the line 3x − 4y + 5 = 0 is
- A. 4
- B. 16/5
- C. 6/5
- D. 16/25
Q28. In an ellipse the length of the latus rectum equals half the length of the major axis. Its eccentricity is
- A. 1/√3
- B. 1/2
- C. 1/√2
- D. √3/2
Q29. The equation of the circle with centre (2, −1) that touches the y-axis is
- A. (x − 2)² + (y + 1)² = 4
- B. (x + 2)² + (y − 1)² = 4
- C. (x − 2)² + (y + 1)² = 2
- D. (x − 2)² + (y + 1)² = 1
Q30. The vectors 2i − j + k, i + 2j − 3k and 3i + λj + 5k are coplanar. The value of λ is
- A. −4
- B. −2
- C. 2
- D. 4
Q31. The mass of a cylinder is measured as 50.0 ± 0.5 g, its radius as 2.00 ± 0.02 cm and its height as 10.0 ± 0.1 cm. What is the maximum percentage uncertainty in its density?
- A. 4%
- B. 3%
- C. 5%
- D. 2%
Q32. What is the angle between the vectors A = i + j and B = j + k?
- A. 60°
- B. 30°
- C. 90°
- D. 45°
Q33. Two parallel forces, each of magnitude 8 N, act in opposite directions along lines 0.25 m apart. What is the magnitude of the torque of this couple?
- A. 4 N m
- B. 0 N m
- C. 2 N m
- D. 32 N m
Q34. A ball is thrown vertically upward at 10 m s⁻¹ from the edge of a 15 m high platform and eventually lands on the ground below the platform. How long is it in the air? (g = 10 m s⁻²)
- A. 3 s
- B. 4 s
- C. 2 s
- D. 1 s
Q35. A 2 kg block and a 3 kg block touch each other on a smooth horizontal table. A 10 N horizontal force pushes the 2 kg block towards the 3 kg block. What force does the 2 kg block exert on the 3 kg block?
- A. 5 N
- B. 10 N
- C. 4 N
- D. 6 N
Q36. A 6 kg body at rest explodes into two pieces of masses 2 kg and 4 kg. The 2 kg piece moves off at 12 m s⁻¹. What total kinetic energy is released?
- A. 72 J
- B. 288 J
- C. 144 J
- D. 216 J
Q37. A uniform rod of mass 3 kg and length 2 m lies on the ground. How much work is needed to raise it until it stands vertically on one end? (g = 10 m s⁻²)
- A. 60 J
- B. 15 J
- C. 30 J
- D. 6 J
Q38. A 20 g bullet moving at 100 m s⁻¹ strikes a wooden plank and leaves it 10 cm further on with a speed of 60 m s⁻¹. What is the average resistive force exerted by the plank?
- A. 1280 N
- B. 640 N
- C. 200 N
- D. 400 N
Q39. The coefficient of static friction between a car's tyres and a flat road is 0.5. What is the maximum speed at which the car can turn along a curve of radius 80 m without skidding? (g = 10 m s⁻²)
- A. 20 m s⁻¹
- B. 28 m s⁻¹
- C. 10 m s⁻¹
- D. 40 m s⁻¹
Q40. A uniform disc of mass 2 kg and radius 0.5 m spins at 20 rad s⁻¹. A constant braking torque of 4 N m is applied. How long does it take to stop? (I = ½MR² for a disc)
- A. 1.25 s
- B. 2.5 s
- C. 5 s
- D. 0.8 s
Q41. Air flows over the top of an aircraft wing at 60 m s⁻¹ and under it at 40 m s⁻¹. The wing area is 10 m² and the air density is 1.2 kg m⁻³. What is the approximate lift force? (Use Bernoulli's equation; ignore height differences.)
- A. 2.4 × 10⁴ N
- B. 1.2 × 10³ N
- C. 6.0 × 10³ N
- D. 1.2 × 10⁴ N
Q42. A pendulum has a period of 1.0 s on the ground. It is taken into a lift that accelerates downward at 7.5 m s⁻². What is its period there? (g = 10 m s⁻²)
- A. 1.41 s
- B. 2.0 s
- C. 0.5 s
- D. 1.15 s
Q43. A travelling wave is described by y = 0.05 sin(100πt − 4πx), with x and y in metres and t in seconds. What is the speed of the wave?
- A. 400 m s⁻¹
- B. 25 m s⁻¹
- C. 12.5 m s⁻¹
- D. 50 m s⁻¹
Q44. In a stationary sound wave in air, successive nodes are 0.4 m apart. Taking the speed of sound as 340 m s⁻¹, what is the frequency?
- A. 850 Hz
- B. 425 Hz
- C. 136 Hz
- D. 212.5 Hz
Q45. Light reflected from the surface of glass of refractive index √3 is completely plane polarised. What is the angle of incidence?
- A. 30°
- B. 45°
- C. 60°
- D. 35°
Q46. An object is placed 30 cm in front of a concave mirror of focal length 20 cm. Using 1/p + 1/q = 1/f (real distances positive), where is the image formed?
- A. 60 cm behind the mirror
- B. 12 cm in front of the mirror
- C. 20 cm in front of the mirror
- D. 60 cm in front of the mirror
Q47. How much heat is needed to convert 0.1 kg of ice at 0 °C into water at 20 °C? (Latent heat of fusion 336 kJ kg⁻¹, specific heat of water 4200 J kg⁻¹ K⁻¹)
- A. 42 kJ
- B. 33.6 kJ
- C. 8.4 kJ
- D. 420 kJ
Q48. Charges +Q and +4Q are fixed 0.3 m apart in air. At what point on the line between them is the electric field zero?
- A. 0.15 m from +Q
- B. 0.05 m from +Q
- C. 0.1 m from +Q
- D. 0.2 m from +Q
Q49. Three 6 μF capacitors are arranged so that two are in series and this pair is connected in parallel with the third. What is the equivalent capacitance?
- A. 9 μF
- B. 2 μF
- C. 12 μF
- D. 18 μF
Q50. A resistor's colour bands are yellow, violet, red and gold in that order. Given yellow = 4, violet = 7, red = 2 (the multiplier is 10 raised to this number) and gold = ±5%, what is its value?
- A. 4.7 kΩ ± 5%
- B. 47 Ω ± 5%
- C. 47 kΩ ± 5%
- D. 470 Ω ± 5%
Q51. A potentiometer wire has a potential gradient of 0.25 V m⁻¹. A cell is balanced against 5.6 m of the wire. What is its emf?
- A. 22.4 V
- B. 1.4 V
- C. 0.045 V
- D. 2.8 V
Q52. A long solenoid has 1000 turns per metre and carries 2 A. What is the magnetic field inside it near its centre? (μ₀ = 4π × 10⁻⁷ T m A⁻¹)
- A. 4π × 10⁻⁷ T
- B. 8π × 10⁻² T
- C. 4π × 10⁻⁴ T
- D. 8π × 10⁻⁴ T
Q53. A 0.5 m straight wire carrying 5 A is placed in a uniform 0.4 T field at an angle of 30° to the field direction. What is the force on the wire?
- A. 2.0 N
- B. 0.87 N
- C. 1.0 N
- D. 0.5 N
Q54. The current through a 2 H inductor falls uniformly from 3 A to 1 A in 0.1 s. What is the magnitude of the average induced emf?
- A. 4 V
- B. 60 V
- C. 40 V
- D. 20 V
Q55. The north pole of a bar magnet is pushed towards a coil along its axis. What polarity does the face of the coil nearer the magnet acquire due to the induced current?
- A. No polarity, because the current is zero
- B. North, attracting the approaching magnet
- C. North, repelling the approaching magnet
- D. South, attracting the approaching magnet
Q56. A series circuit with R = 12 Ω and capacitive reactance 16 Ω is connected to a 100 V (r.m.s.) AC supply. What average power does it dissipate?
- A. 500 W
- B. 375 W
- C. 300 W
- D. 833 W
Q57. Why does the resistance of a pure semiconductor decrease when it is heated?
- A. The atoms vibrate more and stop scattering carriers
- B. More electron–hole pairs are generated
- C. The energy gap becomes wider
- D. Fewer free carriers collide with the lattice
Q58. A 12 V supply feeds a 6 V Zener diode through a 300 Ω series resistor, and there is no load. What current flows through the Zener diode?
- A. 60 mA
- B. 20 mA
- C. 40 mA
- D. 30 mA
Q59. A γ-ray photon of energy 2.04 MeV creates an electron–positron pair near a nucleus. The rest energy of an electron is 0.51 MeV. How much kinetic energy is shared by the pair?
- A. 1.02 MeV
- B. 2.04 MeV
- C. 1.53 MeV
- D. 0.51 MeV
Q60. A radioactive sample has an activity of 640 Bq and a half-life of 6 hours. What is its activity after 1 day?
- A. 20 Bq
- B. 160 Bq
- C. 80 Bq
- D. 40 Bq
Q61. What volume of 0.10 M NaOH is needed to neutralize completely 25 cm³ of 0.10 M H₂SO₄?
- A. 50 cm³
- B. 25 cm³
- C. 12.5 cm³
- D. 100 cm³
Q62. In the hydrogen atom spectrum, the series of lines that lies in the visible region arises from electronic transitions to the level:
- A. n = 4
- B. n = 2
- C. n = 3
- D. n = 1
Q63. What are the total number of orbitals and the maximum number of electrons in the shell with n = 3?
- A. 6 orbitals and 12 electrons
- B. 9 orbitals and 18 electrons
- C. 9 orbitals and 9 electrons
- D. 3 orbitals and 6 electrons
Q64. A gas occupies 300 cm³ at 27 °C and constant pressure. Its volume at 127 °C is:
- A. 600 cm³
- B. 150 cm³
- C. 900 cm³
- D. 400 cm³
Q65. By what factor does the root mean square speed of gas molecules change when the temperature is raised from 27 °C to 927 °C?
- A. It becomes 3.5 times as great
- B. It becomes four times as great
- C. It becomes twice as great
- D. It becomes 5.9 times as great
Q66. Evaporation of a liquid produces a cooling effect because:
- A. the intermolecular forces become stronger as the liquid warms up
- B. the heat of vaporization is released into the surroundings of the liquid
- C. the vapour pressure of the liquid falls to zero at that temperature
- D. the fastest molecules escape, lowering the average kinetic energy of the rest
Q67. Which property is shown by crystalline solids but not by amorphous solids such as glass?
- A. Isotropic behaviour
- B. Absence of long-range order
- C. Softening over a range of temperature
- D. A sharp, definite melting point
Q68. The H–N–H bond angle in NH₄⁺ is larger than in NH₃ because:
- A. nitrogen is more electronegative in NH₄⁺ than in NH₃
- B. the N–H bonds in NH₄⁺ are all fully ionic bonds
- C. NH₄⁺ has no lone pair on nitrogen
- D. NH₄⁺ is sp² hybridized while NH₃ is sp³ hybridized
Q69. Which of the following molecules has zero dipole moment even though it contains polar bonds?
- A. 1,1-dichloroethene
- B. chloroethene
- C. cis-1,2-dichloroethene
- D. trans-1,2-dichloroethene
Q70. In a calorimeter, 250 g of water (specific heat 4.2 J g⁻¹ °C⁻¹) warms by 4.0 °C when 0.025 mol of a fuel is burnt completely. Ignoring the heat capacity of the calorimeter, the enthalpy of combustion of the fuel is:
- A. −16.8 kJ mol⁻¹
- B. −168 kJ mol⁻¹
- C. −4.2 kJ mol⁻¹
- D. +168 kJ mol⁻¹
Q71. For PCl₅(g) ⇌ PCl₃(g) + Cl₂(g), Kc = 0.040 mol dm⁻³ at 300 K. With R = 0.082 dm³ atm K⁻¹ mol⁻¹, Kp is:
- A. 1.64 atm
- B. 0.984 atm
- C. 0.0016 atm
- D. 0.040 atm
Q72. A weak monobasic acid has Ka = 1.0 × 10⁻⁵. Its degree of ionization in 0.10 M solution is:
- A. 10%
- B. 1.0%
- C. 0.010%
- D. 0.10%
Q73. What is the molarity of concentrated hydrochloric acid that is 36.5% HCl by mass and has a density of 1.20 g cm⁻³? (HCl = 36.5)
- A. 1.20 M
- B. 12.0 M
- C. 36.5 M
- D. 10.0 M
Q74. A zinc-coated (galvanized) iron sheet still protects iron from rusting after the coating is scratched because:
- A. zinc reacts with iron at the scratch to form a rust-proof Fe–Zn alloy
- B. zinc is a poorer conductor of electricity than iron is
- C. zinc is more easily oxidized than iron, so it corrodes first
- D. iron is a stronger reducing agent than zinc is, so it stays intact
Q75. A catalyst increases the rate of a reversible reaction but does not change its equilibrium constant because it:
- A. speeds up the forward reaction but not the reverse one
- B. increases the enthalpy change of the overall reaction
- C. raises the potential energy of the products that are formed
- D. lowers the activation energy of both directions equally
Q76. Lithium carbonate, unlike the other alkali metal carbonates, decomposes on heating to the oxide and CO₂. This is an example of:
- A. the common ion effect in solutions
- B. the diagonal relationship of Li with Mg
- C. the lanthanoid contraction in the f-block
- D. the inert pair effect shown by heavy elements
Q77. Which of the following sulphates has the lowest solubility in water?
- A. MgSO₄
- B. CaSO₄
- C. SrSO₄
- D. BaSO₄
Q78. The tendency of lead to be more stable as Pb²⁺ than as Pb⁴⁺ is due to:
- A. extended π bonding
- B. the inert pair effect
- C. the lanthanoid contraction
- D. hydrogen bonding
Q79. In ice, each water molecule is hydrogen-bonded to how many nearest neighbours?
- A. 6
- B. 4
- C. 2
- D. 3
Q80. Which of the following ions is paramagnetic?
- A. Sc³⁺
- B. Cu²⁺
- C. Cu⁺
- D. Zn²⁺
Q81. In the stratosphere, ozone is destroyed in a catalytic chain reaction by:
- A. sulphur dioxide released from volcanic eruptions
- B. methane released from rice fields and landfills
- C. Cl atoms released from CFCs by UV light
- D. carbon dioxide produced by fuel combustion
Q82. Which compound exhibits geometrical (cis–trans) isomerism?
- A. But-2-ene
- B. But-1-ene
- C. 2-Methylpropene
- D. Propene
Q83. In the free-radical chlorination of methane, which equation represents a chain termination step?
- A. CH₃• + Cl₂ → CH₃Cl + Cl•
- B. CH₃• + Cl• → CH₃Cl
- C. Cl₂ → 2Cl•
- D. Cl• + CH₄ → HCl + CH₃•
Q84. Dehydration of butan-2-ol with concentrated H₂SO₄ at 170 °C gives mainly:
- A. but-1-ene
- B. but-2-ene
- C. butane
- D. 2-methylpropene
Q85. Hydration of propyne in the presence of HgSO₄ and dilute H₂SO₄ finally gives:
- A. propanone
- B. propan-2-ol
- C. propanoic acid
- D. propanal
Q86. How many different dibromobenzenes (C₆H₄Br₂) are possible?
- A. 6
- B. 2
- C. 3
- D. 4
Q87. Which of the following reacts fastest with ethanolic silver nitrate to give a precipitate?
- A. 1-iodopropane
- B. 1-bromopropane
- C. 1-fluoropropane
- D. 1-chloropropane
Q88. Which alcohol is not oxidized by hot acidified potassium dichromate under ordinary conditions?
- A. 2-Methylpropan-2-ol
- B. Ethanol
- C. Butan-1-ol
- D. Propan-2-ol
Q89. Reduction of butanone with LiAlH₄ followed by hydrolysis gives:
- A. butan-1-ol
- B. butanal
- C. butan-2-ol
- D. 2-methylpropan-2-ol
Q90. When egg white is heated and the protein coagulates (denaturation), which level of protein structure is NOT destroyed?
- A. The secondary structure
- B. The tertiary structure
- C. The hydrogen bonding between chains
- D. The primary structure (amino acid sequence)
Q91. Choose the correct option to fill in the blank: Since the new principal ___ office last year, attendance has improved.
- A. took
- B. takes
- C. had taken
- D. has taken
Q92. Choose the correct option to fill in the blank: The villagers are deeply grateful ___ the doctor for saving the child.
- A. at
- B. to
- C. on
- D. with
Q93. Choose the correct option to fill in the blank: ___ hour of silence was observed in memory of ___ victims.
- A. An ... a
- B. The ... an
- C. An ... the
- D. A ... the
Q94. Identify the part of the sentence that contains an error. (A) She prefers (B) tea than (C) coffee (D) in the morning.
- A. (A) She prefers
- B. (B) tea than
- C. (C) coffee
- D. (D) in the morning.
Q95. Identify the part of the sentence that contains an error. (A) I have been (B) living in this house (C) since five years (D) with my uncle.
- A. (A) I have been
- B. (B) living in this house
- C. (C) since five years
- D. (D) with my uncle.
Q96. Choose the word closest in meaning to BENIGN.
- A. severe
- B. contagious
- C. sudden
- D. harmless
Q97. Choose the word most nearly opposite in meaning to TRANSIENT.
- A. enduring
- B. fleeting
- C. temporary
- D. brief
Q98. Choose the word that best completes the sentence: Although the plan looked ___ on paper, it failed when put into practice.
- A. flawless
- B. reckless
- C. doubtful
- D. disastrous
Q99. Choose the correct passive form: People say that honesty is the best policy.
- A. It was said that honesty is the best policy.
- B. It has said that honesty is the best policy.
- C. Honesty is said that the best policy.
- D. It is said that honesty is the best policy.
Q100. Choose the correct indirect (reported) speech: They said, "Hurrah! We have won the match."
- A. They exclaimed with joy that they would win the match.
- B. They exclaimed with joy that they have won the match.
- C. They exclaimed with joy that they had won the match.
- D. They exclaimed with joy that we had won the match.