Major Competitive Exams

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Q. Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

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

Solution

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.

Correct Answer: D — 4

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Q. Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)

A. 0
B. 2
C. 4
D. Undefined

Solution

Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.

Correct Answer: D — Undefined

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Q. Calculate the limit: lim (x -> 2) (x^3 - 8)/(x - 2)

A. 4
B. 8
C. 6
D. 2

Solution

Factoring gives lim (x -> 2) ((x - 2)(x^2 + 2x + 4))/(x - 2) = lim (x -> 2) (x^2 + 2x + 4) = 12.

Correct Answer: A — 4

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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)

A. 3/5
B. 0
C. 1
D. ∞

Solution

Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.

Correct Answer: A — 3/5

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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)

A. 3/5
B. 5/3
C. 1
D. 0

Solution

Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.

Correct Answer: A — 3/5

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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)

A. 0
B. 5/2
C. 2/5
D. ∞

Solution

Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.

Correct Answer: B — 5/2

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Q. Calculate the mean absolute deviation for the data set: 1, 2, 3, 4, 5.

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

Solution

Mean = 3. Mean Absolute Deviation = (|1-3| + |2-3| + |3-3| + |4-3| + |5-3|)/5 = (2 + 1 + 0 + 1 + 2)/5 = 1.5.

Correct Answer: B — 1.5

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Q. Calculate the mean of the following data: 5, 10, 15, 20.

A. 10
B. 12.5
C. 15
D. 17.5

Solution

Mean = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.

Correct Answer: B — 12.5

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Q. Calculate the mean of the following numbers: 10, 20, 30, 40, 50.

A. 30
B. 35
C. 40
D. 45

Solution

Mean = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30.

Correct Answer: A — 30

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Q. Calculate the mean of the following numbers: 4, 8, 12, 16, 20.

A. 10
B. 12
C. 14
D. 16

Solution

Mean = (4 + 8 + 12 + 16 + 20) / 5 = 60 / 5 = 12.

Correct Answer: C — 14

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Q. Calculate the median of the following set: 1, 2, 3, 4, 5, 6, 7, 8. (2020)

A. 4
B. 4.5
C. 5
D. 6

Solution

Arranging the numbers: 1, 2, 3, 4, 5, 6, 7, 8. The median is the average of the 4th and 5th numbers: (4 + 5) / 2 = 4.5.

Correct Answer: B — 4.5

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Q. Calculate the median of the following set: 22, 19, 25, 30, 28, 24.

A. 24
B. 25
C. 26
D. 27

Solution

Arrange the numbers: 19, 22, 24, 25, 28, 30. The median is the average of the 3rd and 4th values: (24 + 25) / 2 = 24.5.

Correct Answer: A — 24

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Q. Calculate the molality of a solution if the boiling point elevation is 1.024 °C. (K_b for water = 0.512 °C kg/mol)

A. 1 mol/kg
B. 2 mol/kg
C. 0.5 mol/kg
D. 0.25 mol/kg

Solution

Molality = ΔT_b / (i * K_b) = 1.024 / (2 * 0.512) = 1 mol/kg

Correct Answer: B — 2 mol/kg

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Q. Calculate the moment of inertia of a hollow sphere of mass M and radius R about an axis through its center.

A. 2/5 MR^2
B. 3/5 MR^2
C. 2/3 MR^2
D. MR^2

Solution

The moment of inertia of a hollow sphere about an axis through its center is I = 2/5 MR^2.

Correct Answer: B — 3/5 MR^2

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Q. Calculate the perimeter of a square with side length 4 cm. (2015)

A. 16 cm
B. 12 cm
C. 8 cm
D. 20 cm

Solution

Perimeter = 4 × side = 4 × 4 cm = 16 cm.

Correct Answer: A — 16 cm

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Q. Calculate the perimeter of a square with side length 6 cm. (2015)

A. 24 cm
B. 20 cm
C. 18 cm
D. 30 cm

Solution

Perimeter = 4 × side = 4 × 6 = 24 cm.

Correct Answer: A — 24 cm

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Q. Calculate the pH of a 0.01 M solution of NaHCO3. (2023)

A. 8.3
B. 9.0
C. 7.5
D. 8.0

Solution

NaHCO3 is a weak base. The pH can be calculated using the formula pH = 7 + 0.5(pKa - log[C]). pKa of HCO3- is about 10.3, so pH ≈ 8.3.

Correct Answer: A — 8.3

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Q. Calculate the pH of a 0.05 M NH4Cl solution (Kb for NH3 = 1.8 x 10^-5).

A. 4.75
B. 5.25
C. 5.75
D. 6.25

Solution

Using the formula for weak bases, pH = 14 - 0.5(pKb - logC) = 14 - 0.5(4.74 - log(0.05)) = 5.25.

Correct Answer: B — 5.25

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Q. Calculate the pH of a 0.1 M acetic acid solution (Ka = 1.8 x 10^-5).

A. 2.87
B. 3.87
C. 4.87
D. 5.87

Solution

Using the formula for weak acids, pH = 0.5(pKa - log[C]), where pKa = -log(1.8 x 10^-5) ≈ 4.74. Thus, pH = 0.5(4.74 - log(0.1)) = 3.87.

Correct Answer: B — 3.87

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Q. Calculate the pH of a 0.1 M NaOH solution.

A. 12
B. 13
C. 14
D. 11

Solution

pOH = -log[OH-] = -log(0.1) = 1, thus pH = 14 - pOH = 14 - 1 = 13.

Correct Answer: C — 14

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Q. Calculate the pH of a 0.2 M solution of KOH.

A. 12
B. 13
C. 14
D. 11

Solution

pOH = -log(0.2) = 0.7, thus pH = 14 - 0.7 = 13.3.

Correct Answer: B — 13

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Q. Calculate the pH of a buffer solution containing 0.1 M acetic acid and 0.1 M sodium acetate.

A. 4.76
B. 5.76
C. 6.76
D. 7.76

Solution

Using Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]); pKa of acetic acid = 4.76, so pH = 4.76 + log(1) = 4.76

Correct Answer: B — 5.76

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Q. Calculate the pH of a solution that is 0.1 M in acetic acid (Ka = 1.8 x 10^-5).

A. 2.87
B. 3.87
C. 4.87
D. 5.87

Solution

Using the formula for weak acids, pH = 0.5(pKa - logC) = 0.5(4.74 - log(0.1)) = 3.87.

Correct Answer: B — 3.87

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Q. Calculate the range of the data set: 12, 15, 22, 30, 5.

A. 25
B. 27
C. 30
D. 22

Solution

Range = Maximum - Minimum = 30 - 5 = 25.

Correct Answer: A — 25

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Q. Calculate the range of the data set: 4, 8, 15, 16, 23, 42.

A. 38
B. 36
C. 34
D. 32

Solution

Range = Maximum - Minimum = 42 - 4 = 38.

Correct Answer: A — 38

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Q. Calculate the range of the data set: 8, 12, 15, 20, 22.

A. 10
B. 12
C. 14
D. 15

Solution

Range = Max - Min = 22 - 8 = 14.

Correct Answer: A — 10

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Q. Calculate the range of the following data set: 12, 15, 20, 22, 30.

A. 10
B. 12
C. 18
D. 20

Solution

Range = Maximum - Minimum = 30 - 12 = 18.

Correct Answer: C — 18

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Q. Calculate the range of the following data set: 15, 22, 8, 19, 30.

A. 7
B. 15
C. 22
D. 30

Solution

Range = max - min = 30 - 8 = 22.

Correct Answer: D — 30

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Q. Calculate the range of the following data set: 4, 8, 15, 16, 23, 42.

A. 38
B. 36
C. 34
D. 32

Solution

Range = Maximum - Minimum = 42 - 4 = 38.

Correct Answer: A — 38

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Q. Calculate the range of the following data set: 8, 12, 15, 7, 10.

A. 5
B. 8
C. 10
D. 12

Solution

Range = Maximum - Minimum = 15 - 7 = 8.

Correct Answer: A — 5

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Major Competitive Exams

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