Engineering Entrance
Q. What is the value of k if the equation x² - 4x + k = 0 has no real roots? (2021)
Solution
For no real roots, the discriminant must be less than zero: (-4)² - 4*1*k < 0, hence k > 4.
Correct Answer: B — 6
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Q. What is the value of k if the equation x² - kx + 16 = 0 has roots 4 and 4? (2021)
Solution
Since the roots are equal, k = 4 + 4 = 8.
Correct Answer: A — 8
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Q. What is the value of sec(0°)?
Solution
sec(0°) = 1/cos(0°) = 1/1 = 1
Correct Answer: B — 1
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Q. What is the value of sec(30°)? (2015)
-
A.
√3/2
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B.
2/√3
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C.
2
-
D.
√3
Solution
sec(30°) = 1/cos(30°) = 1/(√3/2) = 2/√3 = √3
Correct Answer: D — √3
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Q. What is the value of sec(45°)? (2019)
Solution
sec(45°) = 1/cos(45°) = 1/(√2/2) = √2.
Correct Answer: B — √2
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Q. What is the value of sin(45°)? (2016)
-
A.
1/√2
-
B.
1/2
-
C.
√3/2
-
D.
1
Solution
sin(45°) = 1/√2
Correct Answer: A — 1/√2
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Q. What is the value of tan(0°)? (2018)
-
A.
0
-
B.
1
-
C.
undefined
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D.
∞
Q. What is the value of tan(90°)? (2022)
-
A.
0
-
B.
1
-
C.
undefined
-
D.
∞
Solution
tan(90°) is undefined.
Correct Answer: C — undefined
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Q. What is the value of the 5th term in the expansion of (3x - 2)^6? (2023)
-
A.
-540
-
B.
540
-
C.
720
-
D.
360
Solution
The 5th term is given by C(6,4) * (3x)^4 * (-2)^2 = 15 * 81 * 4 = -4860.
Correct Answer: A — -540
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Q. What is the value of the azimuthal quantum number (l) for a d orbital? (2021) 2021
Solution
The azimuthal quantum number l for d orbitals is 2.
Correct Answer: C — 2
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Q. What is the value of the azimuthal quantum number (l) for a p-orbital? (2015)
Solution
The azimuthal quantum number (l) for a p-orbital is 1.
Correct Answer: B — 1
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Q. What is the value of the coefficient of x^5 in the expansion of (x + 2)^7?
Solution
The coefficient of x^5 is given by 7C5 * (2)^2 = 21 * 4 = 84.
Correct Answer: C — 56
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Q. What is the value of the determinant of the matrix B = [[2, 3], [5, 7]]? (2020)
Solution
Determinant of B = (2*7) - (3*5) = 14 - 15 = -1.
Correct Answer: C — 11
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Q. What is the value of the determinant of the matrix B = [[5, 1], [2, 3]]? (2022)
Solution
Determinant of B = (5*3) - (1*2) = 15 - 2 = 13.
Correct Answer: A — 10
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Q. What is the value of the term containing x^5 in the expansion of (x + 1/2)^8? (2020)
Solution
The term containing x^5 is C(8,5)(1/2)^3 = 56 * 1/8 = 7.
Correct Answer: B — 56
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Q. What is the value of the term containing x^5 in the expansion of (x + 2)^8? (2020)
-
A.
112
-
B.
128
-
C.
256
-
D.
64
Solution
The term containing x^5 is C(8,5)(2)^3 = 56 * 8 = 448.
Correct Answer: B — 128
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Q. What is the value of x where f(x) = x^3 - 3x has a local maximum? (2022)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = ±1. f(1) = -2.
Correct Answer: C — 1
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Q. What is the value of \( |D| \) for the matrix \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)? (2023)
Solution
The determinant is \( 0*0 - 1*1 = 0 - 1 = -1 \).
Correct Answer: A — 1
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Q. What is the vertex of the parabola represented by the equation y = x² - 4x + 3? (2022)
-
A.
(2, -1)
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B.
(2, 1)
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C.
(1, 2)
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D.
(3, 0)
Solution
The vertex can be found using the formula x = -b/2a. Here, x = 2, and substituting back gives y = -1.
Correct Answer: A — (2, -1)
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Q. What is the vertex of the parabola represented by the equation y = x² - 6x + 8? (2023)
-
A.
(3, -1)
-
B.
(3, -5)
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C.
(2, -4)
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D.
(2, -2)
Solution
The vertex can be found using the formula x = -b/2a = 6/2 = 3. Substituting x = 3 into the equation gives y = -1.
Correct Answer: A — (3, -1)
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Q. What is the voltage across a 12Ω resistor carrying a current of 1.5A? (2023)
-
A.
18V
-
B.
12V
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C.
6V
-
D.
24V
Solution
Using Ohm's law, V = IR = 1.5A * 12Ω = 18V.
Correct Answer: A — 18V
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Q. What is the voltage across a 12Ω resistor carrying a current of 3A? (2023)
-
A.
36V
-
B.
24V
-
C.
12V
-
D.
18V
Solution
Using Ohm's law, V = IR = 3A * 12Ω = 36V.
Correct Answer: A — 36V
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Q. What is the voltage across a 5Ω resistor carrying a current of 3A? (2023)
-
A.
15V
-
B.
10V
-
C.
5V
-
D.
20V
Solution
Using Ohm's law, V = IR = 3A * 5Ω = 15V.
Correct Answer: A — 15V
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Q. What is the volume of 0.2 M NaOH required to neutralize 0.1 M HCl in 250 mL? (2020)
-
A.
125 mL
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B.
250 mL
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C.
500 mL
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D.
100 mL
Solution
Using the formula M1V1 = M2V2, (0.1 M)(0.25 L) = (0.2 M)(V2). V2 = 0.125 L = 125 mL.
Correct Answer: A — 125 mL
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Q. What is the volume of 1 M HCl solution needed to obtain 0.5 moles of HCl? (2022)
-
A.
0.5 L
-
B.
1 L
-
C.
2 L
-
D.
0.25 L
Solution
Volume = moles / molarity = 0.5 moles / 1 M = 0.5 L.
Correct Answer: A — 0.5 L
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Q. What is the volume of 1 mole of an ideal gas at STP? (2019) 2019
-
A.
22.4 L
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B.
24 L
-
C.
1 L
-
D.
10 L
Solution
At STP, 1 mole of an ideal gas occupies 22.4 liters.
Correct Answer: A — 22.4 L
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Q. What is the volume of a 2 M NaOH solution needed to obtain 4 moles of NaOH? (2021) 2021
-
A.
2 L
-
B.
1 L
-
C.
0.5 L
-
D.
4 L
Solution
Volume (L) = moles / molarity = 4 moles / 2 M = 2 L.
Correct Answer: A — 2 L
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Q. What is the volume of a cube with side length 3 units? (2023)
Solution
Volume = side³ = 3³ = 27 cubic units.
Correct Answer: B — 27
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Q. What is the volume of a cylinder with radius 2 and height 5? (2023)
-
A.
20π
-
B.
10π
-
C.
15π
-
D.
25π
Solution
Volume = πr²h = π(2²)(5) = 20π cubic units.
Correct Answer: A — 20π
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Q. What is the volume of a sphere with radius 5 units? (2021)
-
A.
500/3π
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B.
125/3π
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C.
100/3π
-
D.
200/3π
Solution
Volume = (4/3)πr³ = (4/3)π(5)³ = (4/3)π(125) = 500/3π cubic units.
Correct Answer: A — 500/3π
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