Engineering Entrance
Q. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)
Solution
The determinant of G is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [1, 0, 1]]. (2023)
Solution
Det(H) = 1(1*1 - 3*0) - 2(0*1 - 3*1) + 1(0*0 - 1*1) = 1(1) - 2(-3) + 1(-1) = 1 + 6 - 1 = 6.
Correct Answer: A — -1
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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)
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 the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)
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 \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)
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 distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)
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 limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)
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.
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.
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 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 (x/2 - 3)^6.
-
A.
729
-
B.
729/64
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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 value of 12 × 3 - 4 × 2. (2023) 2023
Solution
12 × 3 = 36 and 4 × 2 = 8, so 36 - 8 = 28.
Correct Answer: A — 28
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Q. Calculate the value of 12 × 8 - 24. (2021)
Solution
12 × 8 = 96, then 96 - 24 = 72.
Correct Answer: A — 72
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Q. Calculate the value of 5! (5 factorial). (2020)
-
A.
120
-
B.
100
-
C.
60
-
D.
24
Solution
5! = 5 × 4 × 3 × 2 × 1 = 120.
Correct Answer: A — 120
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Q. Calculate the value of 6^2 - 4^2. (2023) 2023
Solution
6^2 = 36 and 4^2 = 16, so 36 - 16 = 20.
Correct Answer: A — 20
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Q. Determine the coefficient of x^4 in the expansion of (2x - 3)^6.
-
A.
540
-
B.
720
-
C.
810
-
D.
960
Solution
The coefficient of x^4 is given by 6C4 * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.
Correct Answer: B — 720
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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(6, 0, 0), and C(0, 8, 0). (2023)
-
A.
(2, 2, 0)
-
B.
(2, 3, 0)
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C.
(3, 2, 0)
-
D.
(0, 0, 0)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3, (0+0+0)/3) = (2, 2.67, 0).
Correct Answer: A — (2, 2, 0)
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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2023)
-
A.
(1, 1, 0)
-
B.
(2, 1, 0)
-
C.
(4/3, 1, 0)
-
D.
(0, 1, 0)
Solution
Centroid G = ((0+4+0)/3, (0+0+3)/3, (0+0+0)/3) = (4/3, 1, 0).
Correct Answer: B — (2, 1, 0)
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Q. Determine the critical points of f(x) = e^x - 2x. (2021)
Solution
f'(x) = e^x - 2. Setting f'(x) = 0 gives e^x = 2, so x = ln(2).
Correct Answer: B — 1
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Q. Determine the distance from the point (3, 4) to the line 2x + 3y - 12 = 0.
Solution
Using the formula for distance from a point to a line, d = |Ax1 + By1 + C| / sqrt(A^2 + B^2), we find d = |2(3) + 3(4) - 12| / sqrt(2^2 + 3^2) = 3.
Correct Answer: B — 3
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Q. Determine the intervals where f(x) = -x^2 + 4x is concave up. (2023)
-
A.
(-∞, 0)
-
B.
(0, 2)
-
C.
(2, ∞)
-
D.
(0, 4)
Solution
f''(x) = -2, which is always negative, indicating concave down everywhere.
Correct Answer: C — (2, ∞)
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has increasing behavior. (2023)
-
A.
(-∞, 0)
-
B.
(0, 2)
-
C.
(2, ∞)
-
D.
(0, 4)
Solution
f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3). f'(x) > 0 for x in (0, 3).
Correct Answer: B — (0, 2)
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has local minima. (2020)
-
A.
(0, 2)
-
B.
(1, 3)
-
C.
(2, 4)
-
D.
(0, 1)
Solution
f'(x) = 4x^3 - 12x^2. Setting f'(x) = 0 gives x = 0, 3. Testing intervals shows local minima at (0, 2).
Correct Answer: A — (0, 2)
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Q. Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)
-
A.
(0, 1)
-
B.
(1, 3)
-
C.
(2, 5)
-
D.
(3, 4)
Solution
f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x = 0 or x = 2. f(2) = 5 is a local maximum.
Correct Answer: B — (1, 3)
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Q. Determine the maximum area of a triangle with a base of 10 units and height as a function of x. (2020)
Solution
Area = 1/2 * base * height = 5h. Max area occurs when h is maximized, thus Area = 50 when h = 10.
Correct Answer: B — 50
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Q. Determine the maximum height of the function f(x) = -x^2 + 6x + 5. (2020) 2020
Solution
The vertex occurs at x = 3. f(3) = -3^2 + 6*3 + 5 = 8.
Correct Answer: A — 8
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