ECAT — Engineering College Admission Test (UET)
Take the ECAT Full-Length Mock Test 1 (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. In how many ways can the letters A, B, C, D, E be arranged in a row so that A and B are NOT next to each other?
Answer: D. 72
Explanation: Total 5! = 120; arrangements with A and B together = 4! × 2 = 48; so 120 − 48 = 72.
Q2. The number of solutions of tan²x = 1 in the interval [0, 2π) is:
Answer: D. 4
Explanation: tan x = ±1 gives x = π/4, 3π/4, 5π/4 and 7π/4: four solutions.
Q3. The homogeneous system kx + y = 0, x + ky = 0 has a non-trivial solution when:
Answer: A. k = ±1
Explanation: A non-trivial solution needs the coefficient determinant to vanish: k² − 1 = 0, so k = 1 or k = −1.
Q4. The value of the determinant |1 2 3; 2 3 1; 3 1 2| is:
Answer: D. −18
Explanation: Expanding along row 1: 1(6 − 1) − 2(4 − 3) + 3(2 − 9) = 5 − 2 − 21 = −18.
Q5. The sum of the infinite geometric series 0.9 + 0.09 + 0.009 + … is:
Answer: A. 1
Explanation: a = 0.9, r = 0.1, so S∞ = 0.9/(1 − 0.1) = 1.
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Q6. The sum of the series 1 + 4 + 7 + … + 100 is:
Q7. In triangle ABC, ∠A = 45°, ∠B = 60° and a = 2√2. The side b is:
Q8. The modulus of the product (1 + i)(2 + 2i) is:
Q9. The coefficient of x³ in the expansion of (1 − x)⁶ is:
Q10. The term independent of x in the expansion of (x + 1/(2x))⁴ is:
Q11. The period of the function f(x) = tan 3x is:
Q12. If 2, b, 6 are in harmonic progression, then b equals:
Q13. The value of 2 tan⁻¹(1/3) is equal to:
Q14. The range of the function f(x) = 2 + 3 sin x is:
Q15. The value of sin 105° − sin 15° is:
Q16. lim (x→0) sin(πx)/x equals:
Q17. The domain of f(x) = ln(x − 1) is:
Q18. If f(x) = x² + 1 and g(x) = 2x, then (f∘g)(1) equals:
Q19. lim (x→∞) (1 + 4/x)ˣ equals:
Q20. lim (x→9) (√x − 3)/(x − 9) equals:
Q21. If f(x) = x³eˣ, then f′(1) equals:
Q22. d/dx (cot x) equals:
Q23. d/dx [ln(cos x)] equals:
Q24. If y = ln x, then d²y/dx² equals:
Q25. d/dx √(sin x) equals:
Q26. If x² − y² = 16, then dy/dx equals:
Q27. The slope of the tangent to y = ln x at x = 2 is:
Q28. For x > 0, the maximum value of f(x) = 12x − x³ is:
Q29. The function f(x) = eˣ − x is increasing for:
Q30. A stone dropped in a pond makes a circular ripple whose radius grows at 4 cm/s. How fast is the enclosed area growing when r = 10 cm?
Q31. Pressure is force per unit area. Which dimensional formula represents it?
Q32. Suppose velocity [V], time [T] and force [F] were chosen as the fundamental quantities instead of length, mass and time. What would be the dimensions of mass?
Q33. The sides of a rectangular card are measured as 5.0 ± 0.1 cm and 2.0 ± 0.1 cm. What is the percentage uncertainty in its area?
Q34. How many significant figures does the value 6.020 × 10²³ contain?
Q35. For two non-zero vectors the magnitude of their cross product equals their dot product. What is the angle between them?
Q36. Which vector has magnitude 10 units and points in the same direction as 3i + 4j?
Q37. A force of 5j N acts at the point with position vector 2i m. What is its torque about the origin?
Q38. Forces of 6 N due east and 8 N due north act on a particle. What single extra force will keep it in equilibrium?
Q39. A train moving at 30 m/s brakes with a uniform retardation of 1.5 m/s². How long does it take to stop?
Q40. An apple falls freely from rest from a tall tree. What distance does it cover between t = 3 s and t = 4 s? (g = 10 m/s²)
Q41. A ball is projected at 20 m/s over level ground and lands 20√3 m away. At which angles of projection is this possible? (g = 10 m/s²)
Q42. At the same instant, one ball is dropped from a height and another is thrown horizontally from the same height. Neglecting air resistance:
Q43. A 50 g ball travelling at 20 m/s is caught and brought to rest in 0.1 s. What average force acts on the catcher's hands?
Q44. A pump lifts 3600 kg of water per hour to a tank 20 m above it. What is its useful power output? (g = 10 m/s²)
Q45. The kinetic energy of a body increases by 44% while its mass stays the same. By what percentage does its momentum increase?
Q46. Compressing a spring by 5 cm stores 0.5 J of energy in it. What is its spring constant?
Q47. A small block slides from rest down a smooth curved ramp whose top is 5 m above its bottom. What is its speed at the bottom? (g = 10 m/s²)
Q48. What is the angular speed of the second hand of a clock?
Q49. A pilot flies a loop in a vertical circle of radius 400 m. At the lowest point his speed is 80 m/s. How many times his normal weight does he feel there? (g = 10 m/s²)
Q50. A uniform rod of mass 3 kg and length 1 m swings about a horizontal axis through one end (I = ML²/3). What is its rotational kinetic energy when its angular speed is 2 rad/s?
Q51. Taking g = 10 m/s² and R = 6.4 × 10⁶ m, roughly how long does a satellite orbiting just above the Earth's surface take for one revolution?
Q52. Two raindrops are made of the same water, but one has three times the radius of the other. Using Stokes' law, the ratio of their terminal velocities (large : small) is:
Q53. A tap fills a 10 litre bucket in 20 s. The tap opening has a cross-sectional area of 1 cm². With what speed does water leave the tap?
Q54. Water flows steadily through a horizontal pipe of varying width. Where is the water pressure lowest?
Q55. Water flows through a horizontal pipe. At a wide section the pressure is 1.5 × 10⁵ Pa and the speed 2 m/s; at a narrow section the speed is 6 m/s. What is the pressure at the narrow section? (density of water 1000 kg/m³)
Q56. A particle moves so that its acceleration is a = −16x (SI units). What is its period of oscillation?
Q57. A spring is cut into two equal halves and the same mass is hung from one half. Compared with the period on the whole spring, the new period is:
Q58. At the mean position of a body executing SHM:
Q59. The acceleration due to gravity on the Moon is one-sixth of that on the Earth. A pendulum clock taken to the Moon will have a period:
Q60. A stretched string fixed at both ends has two successive resonant frequencies of 300 Hz and 400 Hz. What is its fundamental frequency?
Q61. How many moles of oxygen atoms are present in 0.5 mol of Ca(NO₃)₂?
Q62. The mass of a single water molecule is about (Nₐ = 6.022 × 10²³ mol⁻¹):
Q63. 0.2 mol of a gas has a mass of 8.8 g. The gas could be:
Q64. In the thermite reaction 2Al + Fe₂O₃ → Al₂O₃ + 2Fe, 5.4 g of Al reacts with excess Fe₂O₃. The mass of iron formed is (Al = 27, Fe = 56):
Q65. In paper chromatography a spot travels 3 cm while the solvent front travels 12 cm. The Rf value of the substance is:
Q66. When extracting a solute from water into ether, it is more efficient to:
Q67. Which substance can be purified by sublimation?
Q68. A gas in a sealed rigid container at 27°C and 1 atm is heated to 327°C. Its pressure becomes:
Q69. Equal masses of CH₄ and O₂ are mixed in an empty container. The fraction of the total pressure exerted by O₂ is:
Q70. For a fixed mass of gas at constant pressure, a graph of volume against temperature in °C is a straight line that, when extended, cuts the temperature axis at:
Q71. The kinetic gas equation is:
Q72. The attraction between an HCl molecule and an argon atom is an example of:
Q73. The boiling points of the noble gases increase from He to Xe because:
Q74. A metal crystallizes in a body-centred cubic lattice with an edge length of 300 pm. The distance between nearest-neighbour atoms is about:
Q75. How many NaCl formula units are present in one unit cell of the sodium chloride (rock salt) structure?
Q76. How many electrons in a ground-state chlorine atom (Z = 17) have l = 1?
Q77. The wavelength of a photon with an energy of 2.0 eV is about (hc = 1240 eV nm):
Q78. The energy of a photon of wavelength 200 nm compared with one of wavelength 400 nm is:
Q79. How many radial nodes does a 3s orbital have?
Q80. The splitting of spectral lines when atoms are placed in a magnetic field (Zeeman effect) is explained by the:
Q81. The shape of the nitrate ion, NO₃⁻, is:
Q82. According to MO theory, the bond order of nitric oxide (NO, 15 electrons) is:
Q83. MgO has a far higher lattice energy than NaCl mainly because:
Q84. Which molecule has the smallest bond angle?
Q85. In which species is the central atom sp² hybridized?
Q86. For NaCl, the lattice energy is −788 kJ mol⁻¹ and the total hydration energy of its ions is −784 kJ mol⁻¹. The enthalpy of solution of NaCl is:
Q87. How much heat is needed to warm 36 g of water from 25°C to 75°C (c = 4.18 J g⁻¹ K⁻¹)?
Q88. For CO(g) + ½O₂(g) → CO₂(g) at 298 K, the value of ΔH − ΔE is about (R = 8.314 J K⁻¹ mol⁻¹):
Q89. In the energy profile of an exothermic reaction:
Q90. For N₂O₄(g) ⇌ 2NO₂(g), the equilibrium concentrations are [N₂O₄] = 0.04 M and [NO₂] = 0.02 M. Kc is:
Q91. Choose the correct option to fill in the blank: The factory workers ___ on strike since last Monday.
Q92. Choose the correct option to fill in the blank: Neither the students nor the teacher ___ aware of the change in the schedule.
Q93. Choose the correct option to fill in the blank: She would have caught the bus if she ___ faster.
Q94. Choose the correct option to fill in the blank: My grandfather ___ in this village all his life; he still lives in the same house.
Q95. Choose the correct option to fill in the blank: The new hospital is equipped ___ solar panels on every roof.
Q96. Choose the correct option to fill in the blank: ___ Arabian Sea lies to the south of Pakistan.
Q97. In which sentence is the article 'a' or 'an' used correctly?
Q98. Choose the correct option to fill in the blank: We waited under the shelter ___ the rain stopped.
Q99. Choose the correct option to fill in the blank: The jewellery in this box ___ to my grandmother.
Q100. Which option correctly completes the sentence? 'Walking along the beach, ___.'