ECAT — Engineering College Admission Test (UET)
Take the ECAT Full-Length Mock Test 5 (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. The locus of the point z = x + iy in the Argand plane satisfying |z − 1| = |z + i| is the line
Answer: D. x + y = 0
Explanation: |z − 1|² = (x − 1)² + y² and |z + i|² = x² + (y + 1)². Equating gives −2x + 1 = 2y + 1, so x + y = 0.
Q2. The range of the function f(x) = 3 − 2cos x, x ∈ ℝ, is
Answer: A. [1, 5]
Explanation: Since −1 ≤ cos x ≤ 1, we get −2 ≤ −2cos x ≤ 2, so 1 ≤ f(x) ≤ 5.
Q3. In the group (ℤ₁₀, addition modulo 10), the order of the element 4 is
Answer: B. 5
Explanation: Multiples of 4 modulo 10: 4, 8, 2, 6, 0. The identity 0 is first reached after 5 additions, so the order is 5.
Q4. If A = [[0, 1], [−1, 0]], then A²⁰²³ equals
Answer: A. [[0, −1], [1, 0]]
Explanation: A² = −I, so A⁴ = I. Since 2023 = 4 × 505 + 3, A²⁰²³ = A³ = A²·A = −A = [[0, −1], [1, 0]].
Q5. The matrix [[1, 2, 3], [2, 4, k], [1, 1, 1]] is singular when k equals
Answer: B. 6
Explanation: Expanding along row 1: (4 − k) − 2(2 − k) + 3(2 − 4) = k − 6. It is zero when k = 6.
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Q6. If the roots of ax² + bx + c = 0 are in the ratio 2 : 3, then
Q7. If (x + 4)/((x + 1)(x + 2)²) = A/(x + 1) + B/(x + 2) + C/(x + 2)², then the value of C is
Q8. The value of Σ k(k + 1) for k = 1 to 10 is
Q9. Five persons sit around a circular table. The number of arrangements in which two particular persons P and Q are not seated next to each other is
Q10. A bag holds 4 red and 6 blue marbles. Two are drawn at random without replacement. The probability that they have the same colour is
Q11. The statement 2ⁿ > n² is false for n = 2, 3 and 4. Using mathematical induction, the smallest n₀ for which the statement is true for every integer n ≥ n₀ is
Q12. If sin θ + cos θ = 1/2, then sin 2θ equals
Q13. In a triangle two sides measure 5 and 6 units and the angle between them is 150°. Its area in square units is
Q14. The value of sin(sin⁻¹(3/5) + sin⁻¹(5/13)) is
Q15. The sum of all solutions of 2cos²θ + 3 sinθ = 0 in the interval [0, 2π) is
Q16. The value of lim(x→∞) (1 + 3/x)^(2x) is
Q17. The value of lim(x→0) (√(1 + x) − √(1 − x))/x is
Q18. If a > 0 is a constant, then d/dx (a^(x²)) equals
Q19. For the circle x² + y² = 25, the value of d²y/dx² at the point (3, 4) is
Q20. The derivative of f(x) = |x − 2| at x = 2
Q21. The radius of a circular oil spill grows at 2 cm/s. When the radius is 5 cm, the area is increasing at the rate of
Q22. An open box is made from a 12 cm × 12 cm square sheet by cutting equal squares from the corners and folding up the sides. The maximum possible volume is
Q23. ∫ dx/(√x (1 + √x)) equals
Q24. ∫ dx/(1 + cos x) is equal to
Q25. The value of ∫ from 0 to π/2 of sin⁵x dx is
Q26. A curve satisfies the differential equation x(dy/dx) = 2y for x > 0 and passes through the point (1, 3). The value of y at x = 2 is
Q27. Minimise z = 2x + 3y subject to x + y ≥ 4, x ≥ 1, y ≥ 0. The minimum value of z is
Q28. A parabola has its vertex at (1, 2) and its focus at (3, 2). Its equation is
Q29. The length of the latus rectum of the hyperbola x²/9 − y²/16 = 1 is
Q30. If |a| = 3, |b| = 4 and a · b = 6, then |a × b| is
Q31. A quantity has the dimensions M L² T⁻² A⁻¹ in terms of mass, length, time and current. Which quantity is it?
Q32. What is the magnitude of the vector product of A = 2i + j and B = j + 3k?
Q33. A 100 N sign is hung from the middle of a light cable whose two halves each make an angle of 30° with the horizontal. What is the tension in each half of the cable?
Q34. A boat heads straight across a river 100 m wide at 5 m s⁻¹ relative to the water. The river flows at 3 m s⁻¹. How far downstream does the boat land from the point directly opposite its start?
Q35. A 10 kg block on a rough horizontal floor is pushed by a horizontal force of 50 N. The coefficient of kinetic friction is 0.3. What is the block's acceleration? (g = 10 m s⁻²)
Q36. A 2 kg ball moving at 6 m s⁻¹ makes a head-on perfectly elastic collision with a stationary 4 kg ball. What is the speed of the 4 kg ball just after the collision?
Q37. A 0.5 kg ball is thrown vertically upward at 20 m s⁻¹ and returns to the thrower's hand at 16 m s⁻¹ because of air resistance. How much mechanical energy did it lose?
Q38. A constant force of 20 N acts on a 5 kg body starting from rest. What power does the force develop at the end of 5 s?
Q39. A skater spinning at 3 rad s⁻¹ with moment of inertia 4 kg m² pulls in her arms, reducing it to 1.5 kg m². What is her new angular speed?
Q40. A satellite orbits at a radius 4 times that of another satellite around the same planet. How does its orbital period compare with that of the nearer one?
Q41. A large open tank of water has a small hole in its side 1.8 m below the water surface. At what speed does water leave the hole? (g = 10 m s⁻²)
Q42. A body executes simple harmonic motion with amplitude A. At a displacement of A/2 from the mean position, what fraction of its total energy is kinetic?
Q43. A 1.2 m string of mass 12 g is fixed at both ends and held at a tension of 36 N. What is the frequency of its third harmonic?
Q44. A stationary source emits a 500 Hz note. An observer moves directly towards it at 34 m s⁻¹. Taking the speed of sound as 340 m s⁻¹, what frequency does the observer hear?
Q45. In a double-slit experiment the fringe spacing in air is 1.2 mm. If the whole apparatus is immersed in water (refractive index 4/3), what is the new fringe spacing?
Q46. A thin glass prism of refracting angle 5° and refractive index 1.6 is used with small angles of incidence. What is the angle of deviation?
Q47. A heat engine absorbs 900 J from the hot reservoir and rejects 600 J to the cold reservoir in each cycle. Each cycle takes 0.4 s. What is its power output?
Q48. A +2 nC charge and a −1 nC charge are 10 cm apart in air. What is the electric potential at the midpoint of the line joining them? (k = 9 × 10⁹ N m² C⁻²)
Q49. A 5 μF parallel-plate capacitor stays connected to a 10 V battery while a dielectric of constant 4 completely fills the gap. What energy does it then store?
Q50. A 6 Ω and a 3 Ω resistor are connected in parallel, and this pair is in series with a 4 Ω resistor across a 12 V battery of negligible internal resistance. What current flows through the 3 Ω resistor?
Q51. A 2 kW immersion heater warms 4 kg of water by 20 K. Taking c = 4200 J kg⁻¹ K⁻¹ and ignoring losses, how long does this take?
Q52. Two long parallel wires 5 cm apart carry currents of 2 A and 3 A in the same direction. What is the force per unit length between them? (μ₀ = 4π × 10⁻⁷ T m A⁻¹)
Q53. A proton (mass 1.6 × 10⁻²⁷ kg, charge 1.6 × 10⁻¹⁹ C) moves at 10⁶ m s⁻¹ perpendicular to a 0.1 T uniform magnetic field. What is the radius of its circular path?
Q54. A 0.5 m rod slides at constant speed 4 m s⁻¹ along rails in a uniform 0.25 T field perpendicular to the rails. The total circuit resistance is 0.5 Ω. What force must be applied to keep the rod moving uniformly?
Q55. A square loop of side 0.2 m is placed in a uniform 0.5 T magnetic field so that the normal to the loop makes an angle of 60° with the field. What is the magnetic flux through the loop?
Q56. An AC voltage V = 200 sin(100πt) volts is applied across a 50 Ω resistor. What is the average power dissipated?
Q57. Two wires of the same material and length are stretched by the same load. The radius of wire B is twice that of wire A. What is the ratio of the extension of A to that of B?
Q58. When both inputs of a NAND gate are connected together to one input line, the gate behaves as which single-input gate?
Q59. An X-ray tube is operated at 31 kV. What is the shortest wavelength of the X-rays it emits? (hc = 1240 eV nm)
Q60. ²³⁸₉₂U emits one α-particle and then the daughter undergoes two β⁻ decays. What is the final nucleus?
Q61. When 4.8 g of magnesium burns completely in oxygen to form MgO, 6.0 g of MgO is actually collected. The percentage yield is (Mg = 24, O = 16):
Q62. What is the maximum number of electrons in an atom that can have the quantum numbers n = 3 and mₗ = +1?
Q63. The number of neutrons in a nucleus of ²³⁸₉₂U is:
Q64. A mixture of 2 g of H₂ and 14 g of N₂ exerts a total pressure of 3.0 atm. The partial pressure of hydrogen is (H = 1, N = 14):
Q65. At very high pressures the compression factor Z = PV/nRT of real gases becomes greater than 1 mainly because:
Q66. A pressure cooker cooks food faster mainly because:
Q67. Which of the following ions is planar, with all O–X–O bond angles equal to 120° (central atom sp² hybridized)?
Q68. In the carbonate ion CO₃²⁻, represented by three equivalent resonance structures, the average C–O bond order is:
Q69. Given bond enthalpies H–H = 436, Cl–Cl = 243 and H–Cl = 432 kJ mol⁻¹, the enthalpy change for H₂ + Cl₂ → 2HCl is:
Q70. The solubility of Ag₂CrO₄ in water is 2.0 × 10⁻⁴ mol dm⁻³. Its solubility product Ksp is:
Q71. The pH of a 0.10 M aqueous solution of ammonia (Kb = 1.0 × 10⁻⁵ at 25 °C) is:
Q72. The osmotic pressure of a 0.10 M glucose solution at 300 K is (R = 0.082 dm³ atm K⁻¹ mol⁻¹):
Q73. In the electrolysis of concentrated aqueous sodium chloride (brine) with inert electrodes, the product at the cathode is:
Q74. During the discharge of a lead–acid storage battery, the density of the electrolyte:
Q75. A first-order reaction has a rate constant k = 0.0693 min⁻¹ (ln 2 = 0.693). The time needed for 87.5% of the reactant to be consumed is:
Q76. Chlorine has a higher electron affinity than fluorine mainly because:
Q77. Which ion imparts an apple-green colour to a flame?
Q78. Aluminium oxide, Al₂O₃, is described as amphoteric because it:
Q79. Bleaching powder, CaOCl₂, is manufactured by passing chlorine gas over:
Q80. In acidic medium, KMnO₄ (M = 158 g mol⁻¹) is reduced to Mn²⁺. Its equivalent mass is:
Q81. In the blast furnace, limestone is added to the charge in order to:
Q82. How many structural isomers of molecular formula C₄H₁₀O are alcohols?
Q83. The Wurtz reaction of 2-bromopropane with sodium metal in dry ether gives mainly:
Q84. Dehydrohalogenation of 2-bromobutane with alcoholic KOH gives mainly:
Q85. When ethyne is passed through a red-hot iron tube, the main product is:
Q86. Hot alkaline KMnO₄ oxidizes ethylbenzene, followed by acidification, to:
Q87. Chlorobenzene is much less reactive than chloroethane toward nucleophilic substitution mainly because:
Q88. An excess of bromine water is added to 9.4 g of phenol (C₆H₅OH = 94 g mol⁻¹). The mass of 2,4,6-tribromophenol (M = 331 g mol⁻¹) precipitated, assuming 100% yield, is:
Q89. Which compound undergoes the aldol condensation in dilute NaOH?
Q90. Glycine, alanine and valine are joined to form a tripeptide. The number of water molecules eliminated is:
Q91. Choose the correct option to fill in the blank: The tailor ___ the suit for three days when the customer cancelled the order.
Q92. Choose the correct option to fill in the blank: The child is allergic ___ penicillin.
Q93. Choose the correct option to fill in the blank: ___ European delegation arrived in ___ university hall at noon.
Q94. Identify the part of the sentence that contains an error. (A) Hardly had he (B) entered the room (C) than the lights (D) went out.
Q95. Identify the part of the sentence that contains an error. (A) If he would have (B) listened to the advice, (C) he would have avoided (D) the accident.
Q96. Choose the word closest in meaning to ABATE.
Q97. Choose the word most nearly opposite in meaning to CORDIAL.
Q98. Choose the word that best completes the sentence: Because the evidence was so ___, the judge refused to rely on it.
Q99. Choose the correct passive form: Why did the manager reject the proposal?
Q100. Choose the correct indirect (reported) speech: She said to me, "Where did you buy this book?"