Engineering Entrance Exam Preparation Resources

Engineering Entrance

MHT-CET

Q. Calculate the determinant of I = [[2, 3, 1], [1, 0, 2], [0, 1, 3]]. (2015)

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

Solution

Det(I) = 2(0*3 - 2*1) - 3(1*3 - 2*0) + 1(1*1 - 0*0) = 2(0 - 2) - 3(3) + 1(1) = -4 - 9 + 1 = -12.

Correct Answer: C — 3

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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)

A. 0
B. 1
C. 2
D. 3

Solution

Det(J) = 1(1*1 - 2*0) - 2(0*1 - 1*1) + 1(0*0 - 1*1) = 1(1) - 2(-1) + 1(-1) = 1 + 2 - 1 = 2.

Correct Answer: C — 2

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Q. Calculate the determinant of J = [[5, 6], [7, 8]]. (2014)

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

Solution

Det(J) = (5*8) - (6*7) = 40 - 42 = -2.

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix F = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2022)

A. 0
B. 1
C. 2
D. 3

Solution

The determinant of F is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)

A. -2
B. 2
C. 0
D. 1

Solution

The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \). (2020)

A. 5
B. 10
C. 1
D. 8

Solution

The determinant is \( 2*4 - 1*3 = 8 - 3 = 5 \).

Correct Answer: A — 5

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Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)

A. 1
B. 0
C. 2
D. 3

Solution

The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).

Correct Answer: A — 1

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Q. Calculate the determinant of \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \). (2021)

A. 1
B. 0
C. -1
D. 2

Solution

The determinant is \( 0*0 - 1*1 = 0 - 1 = -1 \).

Correct Answer: A — 1

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Q. Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \). (2021)

A. 0
B. 1
C. 2
D. 3

Solution

The determinant of this matrix is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. Calculate the determinant of \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2022)

A. 10
B. 8
C. 6
D. 12

Solution

The determinant is calculated as \( 3*4 - 2*1 = 12 - 2 = 10 \).

Correct Answer: B — 8

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Q. Calculate the distance from the point (1, 2, 3) to the origin (0, 0, 0). (2021)

A. √14
B. √6
C. √9
D. √12

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: A — √14

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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)

A. 3
B. √14
C. √6
D. √9

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: B — √14

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Q. Calculate the gravitational potential energy of a 2 kg mass at a height of 5 m. (g = 9.8 m/s²)

A. 98 J
B. 19.6 J
C. 39.2 J
D. 49 J

Solution

Potential Energy (PE) = m * g * h = 2 kg * 9.8 m/s² * 5 m = 98 J

Correct Answer: C — 39.2 J

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Q. Calculate the limit: lim (x -> 0) (tan(5x)/x) (2022)

A. 0
B. 1
C. 5
D. Undefined

Solution

Using the limit property lim (x -> 0) (tan(kx)/x) = k, we have lim (x -> 0) (tan(5x)/x) = 5.

Correct Answer: C — 5

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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 -> ∞) (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 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 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 term containing x^3 in the expansion of (2x + 5)^6. (2000)

A. 1500
B. 1800
C. 2000
D. 2500

Solution

The term containing x^3 is C(6,3) * (2)^3 * (5)^(6-3) = 20 * 8 * 125 = 20000.

Correct Answer: B — 1800

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Q. Calculate the term containing x^3 in the expansion of (x + 2)^7.

A. 56
B. 84
C. 112
D. 128

Solution

The term containing x^3 is C(7,3) * (2)^4 = 35 * 16 = 560.

Correct Answer: B — 84

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Q. Calculate the term independent of x in the expansion of (2x - 3)^5.

A. -243
B. 0
C. 243
D. 81

Solution

The term independent of x is C(5,5) * (2x)^0 * (-3)^5 = 1 * 1 * (-243) = -243.

Correct Answer: A — -243

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^3.

A. 12
B. 24
C. 36
D. 48

Solution

The term independent of x occurs when the powers of x cancel out. The term is C(3,0)(4)^3 + C(3,1)(-3x)(4)^2 + C(3,2)(-3x)^2(4) + C(3,3)(-3x)^3 = 64 - 36 + 0 + 0 = 28.

Correct Answer: B — 24

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x + 4)^5.

A. 80
B. 120
C. 160
D. 200

Solution

The term independent of x occurs when the powers of x cancel out. This can be calculated using the binomial expansion.

Correct Answer: C — 160

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Q. Calculate the term independent of x in the expansion of (2x^2 - 3x)^4.

A. -81
B. 108
C. -108
D. 81

Solution

The term independent of x occurs when 2k = 4, k = 2. The term is C(4,2) * (2x^2)^2 * (-3x)^2 = 6 * 4 * 9 = -216.

Correct Answer: C — -108

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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^6.

A. 729
B. 729/64
C. 729/32
D. 729/16

Solution

The term independent of x occurs when k = 3, which gives C(6,3) * (x/2)^3 * (-3)^3 = 20 * (1/8) * (-27) = -67.5.

Correct Answer: B — 729/64

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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^8.

A. -3
B. -8
C. 0
D. 256

Solution

The term independent of x occurs when k = 4. Thus, the term is C(8,4) * (x/2)^4 * (-3)^4 = 70 * (1/16) * 81 = 256.

Correct Answer: D — 256

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Engineering Entrance

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