ECAT — Engineering College Admission Test (UET)
Take the ECAT Full-Length Mock Test 4 (100 MCQs): 100 multiple-choice questions (110 minutes) with instant scoring and a full answer review. Covers 30…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. If z = 1 + i, then z¹⁰ equals
Answer: B. 32i
Explanation: (1 + i)² = 2i, so (1 + i)¹⁰ = (2i)⁵ = 32 i⁵ = 32i.
Q2. The domain of the real function f(x) = √(x² − 5x + 6) is
Answer: A. (−∞, 2] ∪ [3, ∞)
Explanation: We need x² − 5x + 6 = (x − 2)(x − 3) ≥ 0, which holds for x ≤ 2 or x ≥ 3, endpoints included since √0 = 0 is defined.
Q3. In the multiplicative group of non-zero residues modulo 7, the inverse of the element 5 is
Answer: C. 3
Explanation: We need b with 5b ≡ 1 (mod 7). Since 5 × 3 = 15 = 2×7 + 1, the inverse is 3.
Q4. If A = [[1, 2], [3, 4]], the value of det(A² − 2A) is
Answer: D. 16
Explanation: A² = [[7, 10], [15, 22]] and 2A = [[2, 4], [6, 8]], so A² − 2A = [[5, 6], [9, 14]] with determinant 70 − 54 = 16.
Q5. Let A be a 3 × 3 matrix with det A = 4. Then det(adj(2A)) equals
Answer: B. 1024
Explanation: For an n × n matrix, det(adj M) = (det M)ⁿ⁻¹. Here det(2A) = 2³ × 4 = 32, so det(adj(2A)) = 32² = 1024.
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Q6. One root of x² + px + 12 = 0 is 4. If the equation x² + px + q = 0 has equal roots, then q is
Q7. The partial fraction decomposition of (x² + 1)/(x² − 1) is
Q8. Three positive numbers are in G.P. with sum 21 and product 216. The largest of them is
Q9. The number of distinct arrangements of the letters of MISSISSIPPI in which all four S's occur together is
Q10. Two fair dice are thrown. Given that at least one die shows a 6, the probability that the sum is 8 is
Q11. In the expansion of (1 − 4x)^(−1/2) in ascending powers of x, the coefficient of x² is
Q12. The value of cos(2 tan⁻¹(1/3)) is
Q13. The exact value of sin 105° + sin 15° is
Q14. A triangle has sides 13, 14 and 15 (its area is 84 square units). Its circumradius is
Q15. The number of solutions of the equation cos 2θ = sin θ in the interval [0, 2π) is
Q16. The value of lim(x→∞) (√(x² + 4x) − x) is
Q17. The value of lim(x→0) (1 − cos 6x)/(x sin 3x) is
Q18. The derivative of tan⁻¹((1 + x)/(1 − x)) with respect to x is
Q19. If x = t² + 1 and y = t³ − t, then dy/dx at t = 2 equals
Q20. If y = sin 3x, the value of d³y/dx³ at x = 0 is
Q21. The local maximum value of f(x) = 2x³ − 9x² + 12x exceeds its local minimum value by
Q22. The tangent to the curve y = x³ − 3x + 2 at the point where x = 2 meets the y-axis at the point whose y-coordinate is
Q23. ∫ cos³x dx is equal to
Q24. ∫ x ln x dx is equal to
Q25. The value of ∫ from −1 to 2 of |x| dx is
Q26. The solution of the differential equation dy/dx = 2xy with y(0) = 1 has the value, at x = 1, equal to
Q27. The line through the point of intersection of x + y = 4 and 2x − y = 2 and perpendicular to x − 2y = 1 has equation
Q28. A circle passes through the origin and cuts off intercepts of 4 and 6 on the positive x-axis and y-axis respectively. Its radius is
Q29. For the ellipse 9x² + 25y² = 225, the distance between the two directrices is
Q30. A force F = 2i + 3j − k newtons moves a particle from the point (1, 0, 2) to the point (3, 1, 1) (distances in metres). The work done in joules is
Q31. A vernier caliper has 20 vernier divisions equal to 19 main-scale divisions, each main-scale division being 1 mm. It shows a positive zero error of 0.05 mm. A rod gives a main-scale reading of 12 mm with the 7th vernier division coinciding. What is the correct length of the rod?
Q32. What is the scalar component of A = 3i + 4j + k along B = 2i + j + 2k?
Q33. A uniform plank of mass 20 kg and length 4 m rests horizontally on supports at its two ends. A 50 kg person stands 1 m from the left end. What force does the left support exert? (g = 10 m s⁻²)
Q34. A car starts from rest and accelerates uniformly to 20 m s⁻¹ in 10 s, travels at that speed for 20 s, then decelerates uniformly to rest in 5 s. What total distance does it cover?
Q35. Masses of 3 kg and 1 kg hang over a light frictionless pulley (Atwood machine) and are released from rest. What is the tension in the string? (g = 10 m s⁻²)
Q36. A 0.2 kg ball is dropped from a height of 20 m onto a hard floor and rebounds to a height of 5 m. What is the magnitude of the impulse given to the ball by the floor? (g = 10 m s⁻², ignore air resistance)
Q37. A force acting along the x-axis is 10 N from x = 0 to x = 4 m, and then falls uniformly to zero at x = 8 m. What work does it do between x = 0 and x = 8 m?
Q38. A motor with an efficiency of 80% raises a 500 kg load at a constant speed of 0.4 m s⁻¹. What power must be supplied to the motor? (g = 10 m s⁻²)
Q39. A car travels at 20 m s⁻¹ round a frictionless banked curve of radius 80 m. At what angle must the road be banked? (g = 10 m s⁻²)
Q40. A planet has 8 times the mass and twice the radius of the Earth. A person weighs 600 N on Earth. What would be her weight on the surface of this planet?
Q41. Water (density 1000 kg m⁻³) flows through a horizontal pipe at 2 m s⁻¹ in a wide section. The pipe narrows to one quarter of the area. What is the pressure drop between the wide and narrow sections?
Q42. A 0.5 kg mass on a spring of constant 200 N m⁻¹ oscillates with an amplitude of 10 cm. What is its speed when it is 6 cm from the mean position?
Q43. A tuning fork of unknown frequency gives 4 beats per second with a 256 Hz fork. When the unknown fork is loaded with a little wax, the beat frequency rises to 6 per second. What was its original frequency?
Q44. A pipe closed at one end is 0.85 m long. Taking the speed of sound as 340 m s⁻¹, what is the frequency of its fifth harmonic?
Q45. A fish swims 1.2 m below the surface of still water (refractive index 4/3). How deep does it appear to be to an observer looking vertically down from the air?
Q46. A convex lens of power +4 D is placed in contact with a concave lens of power −1.5 D. What is the focal length of the combination?
Q47. A gas at a constant pressure of 2 × 10⁵ Pa expands from 3 litres to 8 litres while absorbing 1500 J of heat. What is the change in its internal energy? (1 litre = 10⁻³ m³)
Q48. Charges of +4 μC and −4 μC are placed 0.6 m apart in air. What is the magnitude of the electric field at the midpoint of the line joining them? (k = 9 × 10⁹ N m² C⁻²)
Q49. Capacitors of 6 μF and 3 μF are connected in series across a 12 V battery. What is the potential difference across the 3 μF capacitor?
Q50. A copper coil has a resistance of 50 Ω at 0 °C. Taking the temperature coefficient of resistance to be 4 × 10⁻³ K⁻¹, what is its resistance at 50 °C?
Q51. A 12 V battery of internal resistance 1 Ω is connected to a 5 Ω and a 20 Ω resistor in parallel. What is the terminal voltage of the battery?
Q52. A rectangular coil of 50 turns and area 0.02 m² carries 0.5 A in a uniform field of 0.4 T, with the plane of the coil parallel to the field. What torque acts on the coil?
Q53. In a velocity selector, crossed uniform fields E = 2000 V m⁻¹ and B = 0.01 T act on a charged particle beam. What speed do the particles that pass undeflected have?
Q54. Two coils have a mutual inductance of 0.2 H. The current in the primary changes by 3 A in 0.02 s. What average emf is induced in the secondary?
Q55. A series circuit has R = 30 Ω, inductive reactance 60 Ω and capacitive reactance 20 Ω and is connected to a 100 V (r.m.s.) AC supply. What r.m.s. current flows?
Q56. A steel wire of Young's modulus 2 × 10¹¹ Pa is stretched to a strain of 10⁻³ (within the elastic limit). How much elastic energy is stored per unit volume?
Q57. An ideal operational amplifier is connected as an inverting amplifier with R_in = 10 kΩ and a feedback resistor R_f = 100 kΩ. What is the output voltage for an input of +0.3 V?
Q58. Light of wavelength 310 nm falls on a metal of work function 2.5 eV. What stopping potential is needed to stop the fastest photoelectrons? (hc = 1240 eV nm)
Q59. A nucleus of 4 nucleons has a mass defect of 0.03 u. Taking 1 u = 931 MeV/c², what is its binding energy per nucleon?
Q60. A stone is thrown horizontally from the top of a 45 m high cliff at 12 m s⁻¹. How far from the foot of the cliff does it land? (g = 10 m s⁻², ignore air resistance)
Q61. A mixture of 4 g of hydrogen and 16 g of oxygen is ignited and reacts completely according to 2H₂ + O₂ → 2H₂O. What mass of water is formed? (H = 1, O = 16)
Q62. How many unpaired electrons are present in the Mn²⁺ ion (Z = 25)?
Q63. Using rₙ ∝ n²/Z for hydrogen-like species, the ratio of the radius of the n = 3 orbit of hydrogen to that of the n = 2 orbit of He⁺ is:
Q64. A gas at 2.46 atm occupies 10.0 dm³ at 300 K. Taking R = 0.082 dm³ atm K⁻¹ mol⁻¹, the amount of gas is:
Q65. Equal volumes of CH₄ and SO₂ effuse through the same small hole under the same conditions. If CH₄ takes 20 s, SO₂ will take (C = 12, H = 1, S = 32, O = 16):
Q66. A metal crystallizes in a face-centred cubic lattice with unit cell edge a = 4√2 Å. Its atomic radius, using the contact along the face diagonal, is:
Q67. How many σ bonds and π bonds are present in the molecule CH₂=CH–C≡CH (but-1-en-3-yne)?
Q68. According to molecular orbital theory, the bond order of the C₂ molecule (12 electrons, with the σ2p orbital above the π2p orbitals) is:
Q69. The standard enthalpies of combustion of graphite and carbon monoxide are −394 and −284 kJ mol⁻¹ respectively. The enthalpy of formation of CO(g) from graphite and oxygen is:
Q70. 1.0 mol each of A and B are mixed in a 1.0 dm³ vessel and react as A + B ⇌ C. At equilibrium 0.5 mol of C is present. The value of Kc is:
Q71. A buffer is made from 0.10 M ethanoic acid (pKa = 4.74) and 1.0 M sodium ethanoate. Its pH is:
Q72. 18 g of glucose (M = 180 g mol⁻¹) is dissolved in 500 g of water (Kf = 1.86 °C kg mol⁻¹). The freezing point of the solution is:
Q73. The standard reduction potentials are E°(Zn²⁺/Zn) = −0.76 V and E°(Ag⁺/Ag) = +0.80 V. The standard EMF of the cell Zn | Zn²⁺ || Ag⁺ | Ag is:
Q74. A charge of 9650 C is passed through aqueous CuSO₄ using inert electrodes. The mass of copper deposited is (Cu = 63.5; F = 96500 C mol⁻¹):
Q75. Doubling [A] quadruples the rate of a reaction, while doubling [B] only doubles it. The units of the rate constant k, with concentration in mol dm⁻³ and time in s, are:
Q76. Aluminium has a lower first ionization energy than magnesium mainly because:
Q77. Plaster of Paris is formed by heating gypsum at about 120 °C. The number of water molecules per CaSO₄ unit in plaster of Paris is:
Q78. In the Ostwald process, ammonia is first oxidized by air over a platinum–rhodium catalyst at about 800 °C. The main nitrogen-containing product is:
Q79. Among the hydrogen halides, HI is the strongest acid in aqueous solution mainly because:
Q80. How many moles of silver chloride are precipitated when excess AgNO₃ is added to 1 mol of [Co(NH₃)₅Cl]Cl₂?
Q81. Normal, unpolluted rain water has a pH of about 5.6 rather than 7.0 because it contains:
Q82. The IUPAC name of (CH₃)₂CH–CH(OH)–CH₃ is:
Q83. Monochlorination of 2-methylpropane under ultraviolet light gives how many structurally different products?
Q84. The major product of the addition of HBr to 2-methylpropene is:
Q85. How many sigma (σ) bonds are present in one molecule of propene, CH₃–CH=CH₂?
Q86. Which of the following undergoes electrophilic nitration most readily?
Q87. Alkaline hydrolysis of (CH₃)₃CBr occurs by the SN1 mechanism. The rate law is:
Q88. 4.6 g of ethanol (M = 46 g mol⁻¹) reacts completely with excess sodium metal. The volume of hydrogen liberated at STP (22.4 dm³ mol⁻¹) is:
Q89. Which compound gives a yellow precipitate of iodoform with I₂ and NaOH?
Q90. Which reagent gives effervescence with benzoic acid but no effervescence with phenol?
Q91. Choose the correct option to fill in the blank: By the time the surveyors submitted their report, the river ___ its course twice.
Q92. Choose the correct option to fill in the blank: The council has succeeded ___ reducing the waiting time for permits.
Q93. Which sentence uses the articles correctly?
Q94. Identify the part of the sentence that contains an error. (A) Despite of the heavy rain, (B) the match (C) was played (D) as scheduled.
Q95. Identify the part of the sentence that contains an error. (A) One of the most significant (B) discovery of the century (C) was made (D) by a young chemist.
Q96. Choose the word closest in meaning to VERBOSE.
Q97. Choose the word most nearly opposite in meaning to CANDID.
Q98. Choose the word that best completes the sentence: The minister's reply was so ___ that reporters could not tell whether he agreed or disagreed.
Q99. Choose the correct passive form: The board will appoint a new director next month.
Q100. Choose the correct indirect (reported) speech: He said, "I will submit the form tomorrow."