ECAT — Engineering College Admission Test (UET)

ECAT Full-Length Mock Test 2 (100 MCQs)

Take the ECAT Full-Length Mock Test 2 (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.

1. Q1. The value of sin 15° cos 15° is:

   • A. 1/2
   • B. √3/4
   • C. 1/4 ✓
   • D. 1/8

   Answer: C. 1/4

   Explanation: sin 15° cos 15° = ½ sin 30° = ½ × ½ = 1/4.

2. Q2. The inverse of the conditional p → q is:

   • A. p → ~q
   • B. q → p
   • C. ~p → ~q ✓
   • D. ~q → ~p

   Answer: C. ~p → ~q

   Explanation: The inverse negates both parts without swapping them: ~p → ~q. (q → p is the converse; ~q → ~p is the contrapositive.)

3. Q3. If f(x) = 3x − 4, then f(f(2)) equals:

   • A. −4
   • B. 2 ✓
   • C. 6
   • D. 14

   Answer: B. 2

   Explanation: f(2) = 6 − 4 = 2, so f(f(2)) = f(2) = 2.

4. Q4. The in-radius of a triangle with sides 3, 4 and 5 is:

   • A. 1 ✓
   • B. 1.5
   • C. 2.5
   • D. 2

   Answer: A. 1

   Explanation: Area Δ = ½ × 3 × 4 = 6 (right triangle) and s = 6, so r = Δ/s = 1.

5. Q5. The roots of 4x² − 12x + 9 = 0 are:

   • A. 3/2 and 3/2 (real and equal) ✓
   • B. 3/2 and −3/2
   • C. 3 and 3/4
   • D. non-real complex numbers

   Answer: A. 3/2 and 3/2 (real and equal)

   Explanation: Discriminant = 144 − 144 = 0, and 4x² − 12x + 9 = (2x − 3)², so both roots equal 3/2.

The other 95 questions

Take the timed test (free, no sign-up) to see your score and the answer and explanation for every question.

6. Q6. If A = {1, 2} and B = {2, 3}, then the number of elements in (A × B) ∩ (B × A) is:

   • A. 4
   • B. 0
   • C. 2
   • D. 1

7. Q7. For every positive integer n, 3²ⁿ − 1 is divisible by:

   • A. 8
   • B. 9
   • C. 5
   • D. 6

8. Q8. The principal value of tan⁻¹(−√3) is:

   • A. 2π/3
   • B. π/3
   • C. −π/6
   • D. −π/3

9. Q9. The value of sin⁻¹(sin 5π/6) is:

   • A. π/6
   • B. 5π/6
   • C. −π/6
   • D. π/3

10. Q10. From the top of a building 20 m high, the angle of depression of a car on level ground is 30°. The distance of the car from the foot of the building is:

    • A. 10√3 m
    • B. 40 m
    • C. 20√3 m
    • D. 20/√3 m

11. Q11. The value of i⁵⁷ + i⁵⁸ is:

    • A. −1 − i
    • B. −1 + i
    • C. 1 + i
    • D. 1 − i

12. Q12. The partial fraction decomposition of (2x + 3)/(x² + 3x + 2) is:

    • A. 1/(x + 1) + 1/(x + 2)
    • B. 1/(x + 1) − 1/(x + 2)
    • C. 3/(x + 1) − 1/(x + 2)
    • D. 2/(x + 1) + 1/(x + 2)

13. Q13. The expression sin⁴θ − cos⁴θ is equal to:

    • A. −cos 2θ
    • B. cos 2θ
    • C. sin 2θ
    • D. 1

14. Q14. Given that α and β (α ≠ β) both satisfy x² + x + 1 = 0, what is the value of α² + β²?

    • A. 1
    • B. −1
    • C. −3
    • D. 3

15. Q15. The value of sin 120° + cos 150° is:

    • A. √3
    • B. −√3
    • C. 1
    • D. 0

16. Q16. In the Maclaurin series of cos x, the coefficient of x⁴ is:

    • A. 1/24
    • B. 1/4
    • C. 1/12
    • D. −1/24

17. Q17. The slope of the normal to the curve y = 2x² at x = 1 is:

    • A. −4
    • B. 1/4
    • C. 4
    • D. −1/4

18. Q18. ∫ (1/x + eˣ) dx equals:

    • A. 1/x + eˣ + c
    • B. ln|x| + eˣ + c
    • C. −1/x² + eˣ + c
    • D. ln|x| + xeˣ + c

19. Q19. ∫ sin 2x dx equals:

    • A. −sin 2x/2 + c
    • B. cos 2x/2 + c
    • C. −2 cos 2x + c
    • D. −cos 2x/2 + c

20. Q20. ∫ x²·e^(x³) dx equals:

    • A. e^(x³)/3 + c
    • B. x³e^(x³)/3 + c
    • C. 3e^(x³) + c
    • D. e^(x³) + c

21. Q21. ∫ dx/(x² + 4x + 5) equals:

    • A. tan⁻¹(x + 4) + c
    • B. tan⁻¹(x + 2) + c
    • C. ½ tan⁻¹((x + 2)/2) + c
    • D. ln(x² + 4x + 5) + c

22. Q22. ∫ x cos 2x dx equals:

    • A. −(x sin 2x)/2 + (cos 2x)/4 + c
    • B. (x sin 2x)/2 − (cos 2x)/4 + c
    • C. (x sin 2x)/2 + (cos 2x)/4 + c
    • D. x sin 2x + cos 2x + c

23. Q23. ∫ dx/√(x + 1) equals:

    • A. 2√(x + 1) + c
    • B. ½√(x + 1) + c
    • C. √(x + 1) + c
    • D. −2/√(x + 1) + c

24. Q24. ∫₀^(π/3) sec²x dx equals:

    • A. 1
    • B. 1/√3
    • C. 2
    • D. √3

25. Q25. ∫₀³ x² dx equals:

    • A. 27
    • B. 9
    • C. 6
    • D. 3

26. Q26. ∫₁² (2x − 1) dx equals:

    • A. 2
    • B. 3
    • C. 1
    • D. 4

27. Q27. The area under the curve y = 1/x between x = 1 and x = e² is:

    • A. 1
    • B. e²
    • C. 2
    • D. e² − 1

28. Q28. The solution of dy/dx = cos x with y(0) = 1 is:

    • A. y = 1 − sin x
    • B. y = cos x
    • C. y = sin x
    • D. y = sin x + 1

29. Q29. ∫₀^(ln 3) e^(2x) dx equals:

    • A. 9/2
    • B. 8
    • C. 2
    • D. 4

30. Q30. The slope of a line perpendicular to 3x − 4y = 7 is:

    • A. −3/4
    • B. 3/4
    • C. −4/3
    • D. 4/3

31. Q31. A string 60 cm long, fixed at both ends, vibrates in three loops. What is the wavelength of the waves on it?

    • A. 180 cm
    • B. 40 cm
    • C. 20 cm
    • D. 60 cm

32. Q32. An ambulance siren of frequency 660 Hz approaches a stationary observer at 30 m/s. What frequency does the observer hear? (speed of sound 330 m/s)

    • A. 726 Hz
    • B. 720 Hz
    • C. 600 Hz
    • D. 605 Hz

33. Q33. Astronauts on the Moon cannot hear each other speak directly because:

    • A. sound waves are transverse
    • B. the Moon's gravity is too weak
    • C. sound needs a material medium to travel through
    • D. sound travels too fast in a vacuum

34. Q34. A pipe open at both ends has a fundamental frequency of 400 Hz. If one end is now closed, its fundamental frequency becomes:

    • A. 1200 Hz
    • B. 200 Hz
    • C. 800 Hz
    • D. 400 Hz

35. Q35. With violet light of wavelength 400 nm, a double-slit arrangement gives fringes 2 mm apart. What is the fringe spacing with orange light of 600 nm, everything else unchanged?

    • A. 3 mm
    • B. 4.5 mm
    • C. 2 mm
    • D. 1.33 mm

36. Q36. Light of wavelength 600 nm produces its first-order maximum at 30° with a certain grating. What is the grating spacing d?

    • A. 0.6 μm
    • B. 2.4 μm
    • C. 0.3 μm
    • D. 1.2 μm

37. Q37. Which of these waves cannot be polarized?

    • A. Light waves
    • B. Transverse waves on a string
    • C. Sound waves in air
    • D. Radio waves

38. Q38. No steady interference pattern is seen when light from two separate electric bulbs overlaps, because the two bulbs:

    • A. are not coherent sources
    • B. emit only longitudinal waves
    • C. are too far apart
    • D. emit light of too low intensity

39. Q39. A convex lens of focal length 5 cm is used as a magnifying glass with the image at the near point (25 cm). What is its magnifying power?

    • A. 30
    • B. 4
    • C. 5
    • D. 6

40. Q40. An astronomical telescope in normal adjustment has an objective of focal length 100 cm and an eyepiece of focal length 5 cm. What is the length of the telescope?

    • A. 20 cm
    • B. 500 cm
    • C. 105 cm
    • D. 95 cm

41. Q41. Light is guided along an optical fibre by:

    • A. refraction at the outer surface into air
    • B. diffraction
    • C. total internal reflection
    • D. polarization

42. Q42. What is the critical angle for light going from glass of refractive index √2 into air?

    • A. 45°
    • B. 30°
    • C. 90°
    • D. 60°

43. Q43. What is the efficiency of a Carnot engine working between 600 K and 300 K?

    • A. 100%
    • B. 50%
    • C. 200%
    • D. 25%

44. Q44. A gas in a sealed rigid container absorbs 400 J of heat. What is the change in its internal energy?

    • A. +200 J
    • B. zero
    • C. +400 J
    • D. −400 J

45. Q45. At what temperature will the r.m.s. speed of the molecules of a gas be twice its value at 27 °C?

    • A. 1200 °C
    • B. 54 °C
    • C. 927 °C
    • D. 108 °C

46. Q46. When an ideal gas is compressed isothermally:

    • A. heat flows out of the gas equal to the work done on it
    • B. its internal energy increases
    • C. its temperature rises
    • D. no heat is exchanged

47. Q47. A Carnot engine has an efficiency of 1/6. When the sink temperature is lowered by 65 K (source unchanged), its efficiency doubles. What are the source and sink temperatures?

    • A. 600 K and 500 K
    • B. 390 K and 325 K
    • C. 390 K and 65 K
    • D. 325 K and 260 K

48. Q48. Two identical point charges, each +q, repel each other with force F. Half the charge of one is transferred to the other without changing their separation. The force becomes:

    • A. F
    • B. F/2
    • C. 3F/4
    • D. 3F/2

49. Q49. Two equal positive point charges are fixed a certain distance apart. The electric field at the midpoint between them is:

    • A. directed towards one of the charges
    • B. perpendicular to the line joining them
    • C. twice the field of one charge
    • D. zero

50. Q50. A 2 μF capacitor is charged to 50 V. How much charge does it store?

    • A. 25 μC
    • B. 2500 μC
    • C. 100 μC
    • D. 0.04 μC

51. Q51. Two 4 μF capacitors are connected in parallel across a 10 V supply. What total energy is stored?

    • A. 4 × 10⁻⁴ J
    • B. 8 × 10⁻⁴ J
    • C. 2 × 10⁻⁴ J
    • D. 1 × 10⁻⁴ J

52. Q52. An electron starts from rest and is accelerated through a potential difference of 500 V. What speed does it reach? (e = 1.6 × 10⁻¹⁹ C, mₑ = 9.1 × 10⁻³¹ kg)

    • A. 1.33 × 10⁹ m/s
    • B. 1.76 × 10¹⁴ m/s
    • C. 1.33 × 10⁷ m/s
    • D. 4.19 × 10⁵ m/s

53. Q53. A point charge q sits at the exact centre of a closed cube. What is the electric flux through any one face of the cube?

    • A. q/6ε₀
    • B. q/ε₀
    • C. q/4ε₀
    • D. zero

54. Q54. Resistors of 2 Ω, 3 Ω and 6 Ω are connected in parallel. What is their equivalent resistance?

    • A. 0.5 Ω
    • B. 11 Ω
    • C. 3.67 Ω
    • D. 1 Ω

55. Q55. A current of 2 A flows through a 10 Ω resistor. At what rate is heat produced in it?

    • A. 40 W
    • B. 5 W
    • C. 200 W
    • D. 20 W

56. Q56. A cell of emf 1.5 V has an internal resistance of 0.5 Ω. What current flows if its terminals are short-circuited by a thick copper wire?

    • A. 3 A
    • B. 0.75 A
    • C. 1.5 A
    • D. 0.33 A

57. Q57. A uniform wire of resistance 10 Ω is folded in half so that the two halves lie side by side and act as a single conductor. What is its new resistance?

    • A. 20 Ω
    • B. 2.5 Ω
    • C. 10 Ω
    • D. 5 Ω

58. Q58. A bulb rated 40 W, 220 V is operated on a 110 V supply. Assuming its resistance stays constant, what power does it consume?

    • A. 80 W
    • B. 40 W
    • C. 20 W
    • D. 10 W

59. Q59. Cells of emf 6 V and 2 V (negligible internal resistance) are connected in series, opposing each other, with a 4 Ω resistor. What current flows?

    • A. 0.5 A
    • B. 2 A
    • C. 4 A
    • D. 1 A

60. Q60. An electron moves parallel to a uniform magnetic field. The magnetic force on it is:

    • A. maximum
    • B. directed along the field
    • C. evB
    • D. zero

61. Q61. 50 cm³ of 0.2 M HCl is mixed with 50 cm³ of 0.1 M NaOH. The pH of the resulting solution is about:

    • A. 7.0
    • B. 1.0
    • C. 2.0
    • D. 1.3

62. Q62. The conjugate base of HSO₄⁻ is:

    • A. SO₃²⁻
    • B. SO₄²⁻
    • C. H₂SO₄
    • D. H₃O⁺

63. Q63. In qualitative analysis, NH₄Cl is added before NH₄OH to precipitate group III hydroxides. Its role is to:

    • A. Suppress the ionization of NH₄OH by the common-ion effect, keeping [OH⁻] low
    • B. Act as a catalyst
    • C. Oxidize the metal ions
    • D. Increase the concentration of OH⁻

64. Q64. A weak acid HA has Ka = 1 × 10⁻⁵ at 25°C. The Kb of its conjugate base A⁻ is:

    • A. 1 × 10⁻⁹
    • B. 1 × 10⁻¹⁴
    • C. 1 × 10⁻⁵
    • D. 1 × 10⁵

65. Q65. What volume of 18 M H₂SO₄ is needed to prepare 1 dm³ of 0.9 M H₂SO₄?

    • A. 20 cm³
    • B. 500 cm³
    • C. 162 cm³
    • D. 50 cm³

66. Q66. 1 mol of a non-volatile solute is dissolved in 99 mol of a solvent whose vapour pressure is 30 torr. The lowering of vapour pressure is:

    • A. 29.7 torr
    • B. 0.3 torr
    • C. 3 torr
    • D. 0.03 torr

67. Q67. Fish may suffocate in water heated by the effluent of a power station because:

    • A. Warm water contains more dissolved oxygen
    • B. Heat converts dissolved O₂ into O₃
    • C. The solubility of oxygen in water decreases as temperature rises
    • D. Water becomes acidic when warmed

68. Q68. The oxidation number of phosphorus in pyrophosphoric acid, H₄P₂O₇, is:

    • A. +3
    • B. +5
    • C. +7
    • D. +4

69. Q69. How many faradays are needed to reduce 1 mol of Cr₂O₇²⁻ completely to Cr³⁺?

    • A. 2 F
    • B. 3 F
    • C. 6 F
    • D. 12 F

70. Q70. Which metal cannot liberate hydrogen from dilute HCl?

    • A. Fe
    • B. Cu
    • C. Mg
    • D. Zn

71. Q71. In a Daniell cell (Zn | Zn²⁺ || Cu²⁺ | Cu), electrons in the external circuit flow:

    • A. From copper to zinc
    • B. Through the salt bridge
    • C. In both directions alternately
    • D. From zinc to copper

72. Q72. A reaction occurs in two steps: A + B → X (slow) and X + B → C (fast). The rate law is:

    • A. Rate = k[X][B]
    • B. Rate = k[A][B]²
    • C. Rate = k[C]
    • D. Rate = k[A][B]

73. Q73. A first-order reaction has a half-life of 10 min. The percentage of reactant remaining after 30 min is:

    • A. 6.25%
    • B. 33.3%
    • C. 12.5%
    • D. 25%

74. Q74. For the Arrhenius equation, a graph of ln k against 1/T is a straight line whose slope equals:

    • A. Ea/R
    • B. ln A
    • C. −Ea/R
    • D. −RT/Ea

75. Q75. The rate of a reaction doubles for every 10°C rise in temperature. When the temperature is raised from 20°C to 50°C, the rate increases by a factor of:

    • A. 3
    • B. 9
    • C. 8
    • D. 6

76. Q76. The element with atomic number 37 belongs to:

    • A. Period 5, group 1
    • B. Period 4, group 17
    • C. Period 5, group 2
    • D. Period 4, group 1

77. Q77. Across period 3 (Na to Ar), the melting point is highest for:

    • A. Na
    • B. P
    • C. Al
    • D. Si

78. Q78. Washing soda crystals, Na₂CO₃·10H₂O, lose water when exposed to air and become powdery. This is called:

    • A. Hydrolysis
    • B. Sublimation
    • C. Efflorescence
    • D. Deliquescence

79. Q79. BeCl₂ is largely covalent while the other group 2 chlorides are ionic because:

    • A. Be has a very low ionization energy
    • B. Be is the largest group 2 atom
    • C. Cl⁻ is very small
    • D. Be²⁺ is very small and highly polarizing

80. Q80. The formula of potash alum is:

    • A. KAlO₂
    • B. K₂SO₄·Al₂(SO₄)₃
    • C. KAl(SO₄)₂·6H₂O
    • D. K₂SO₄·Al₂(SO₄)₃·24H₂O

81. Q81. Which group 14 oxide is amphoteric?

    • A. SiO₂
    • B. CO
    • C. CO₂
    • D. PbO

82. Q82. Silicones are polymers that contain:

    • A. A –Si–O–Si– backbone with organic groups attached to silicon
    • B. Si–H bonds only
    • C. A –C–C– backbone with silicon side groups
    • D. Only silicon and carbon

83. Q83. The oxidation number of nitrogen in nitrous acid, HNO₂, is:

    • A. +3
    • B. −3
    • C. +1
    • D. +5

84. Q84. SO₂ bleaches coloured materials temporarily, whereas chlorine bleaches them permanently. This is because:

    • A. SO₂ bleaches by reduction, and air later re-oxidizes the colour
    • B. Chlorine bleaches by reduction
    • C. SO₂ only works in dry conditions
    • D. SO₂ bleaches by oxidation

85. Q85. At room temperature the stable allotrope of sulphur is rhombic sulphur. Above 95.5°C it changes into:

    • A. Plastic sulphur
    • B. Colloidal sulphur
    • C. S₂ gas
    • D. Monoclinic sulphur

86. Q86. Which reaction does NOT take place?

    • A. I₂ + 2KCl → 2KI + Cl₂
    • B. F₂ + 2KCl → 2KF + Cl₂
    • C. Br₂ + 2KI → 2KBr + I₂
    • D. Cl₂ + 2KI → 2KCl + I₂

87. Q87. The oxidation number of chlorine in chloric acid, HClO₃, is:

    • A. +3
    • B. +7
    • C. +5
    • D. +1

88. Q88. Helium is used instead of hydrogen to fill airships mainly because helium:

    • A. Is more reactive
    • B. Is lighter than hydrogen
    • C. Is non-flammable
    • D. Is cheaper

89. Q89. Ti³⁺ salts are violet in solution while Ti⁴⁺ salts are colourless. This is because:

    • A. Ti⁴⁺ is paramagnetic
    • B. Ti³⁺ has a completely filled d subshell
    • C. Ti³⁺ has one 3d electron that can undergo d–d transitions, while Ti⁴⁺ has none
    • D. Ti⁴⁺ is a larger ion

90. Q90. The oxidation state of vanadium in V₂O₅, the catalyst of the contact process, is:

    • A. +5
    • B. +3
    • C. +2
    • D. +4

91. Q91. Identify the part of the sentence that contains an error. (A) When it begun to rain, (B) the children (C) were playing (D) in the park.

    • A. (A) When it begun to rain,
    • B. (B) the children
    • C. (C) were playing
    • D. (D) in the park.

92. Q92. Identify the part of the sentence that contains an error. (A) She (B) has (C) a lot of (D) knowledges about plants.

    • A. (A) She
    • B. (B) has
    • C. (C) a lot of
    • D. (D) knowledges about plants.

93. Q93. Identify the part of the sentence that contains an error. (A) He (B) did not (C) attend the class (D) because of he was ill.

    • A. (A) He
    • B. (B) did not
    • C. (C) attend the class
    • D. (D) because of he was ill.

94. Q94. Identify the part of the sentence that contains an error. (A) From the earth, (B) the moon looks (C) much bigger then the stars (D) because it is nearer.

    • A. (A) From the earth,
    • B. (B) the moon looks
    • C. (C) much bigger then the stars
    • D. (D) because it is nearer.

95. Q95. Identify the part of the sentence that contains an error. (A) My mother (B) bought (C) two dozens (D) eggs from the market.

    • A. (A) My mother
    • B. (B) bought
    • C. (C) two dozens
    • D. (D) eggs from the market.

96. Q96. Identify the part of the sentence that contains an error. (A) Can you (B) tell me (C) where (D) does he live?

    • A. (A) Can you
    • B. (B) tell me
    • C. (C) where
    • D. (D) does he live?

97. Q97. Choose the word closest in meaning to PROFOUND.

    • A. shallow
    • B. quick
    • C. deep
    • D. loud

98. Q98. Choose the word closest in meaning to THWART.

    • A. prevent
    • B. reward
    • C. encourage
    • D. imitate

99. Q99. Choose the word closest in meaning to CONSPICUOUS.

    • A. rarely used
    • B. very costly
    • C. easily noticed
    • D. well hidden

100. Q100. Choose the word most nearly opposite in meaning to TRIVIAL.

     • A. petty
     • B. trifling
     • C. significant
     • D. minor