Q. If the pair of lines represented by ax^2 + 2hxy + by^2 = 0 are perpendicular, then:
-
A.
a + b = 0
-
B.
a - b = 0
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C.
h = 0
-
D.
a = b
Solution
For the lines to be perpendicular, the condition a + b = 0 must hold.
Correct Answer: A — a + b = 0
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Q. If the pair of lines represented by the equation ax^2 + 2hxy + by^2 = 0 are perpendicular, then:
-
A.
a + b = 0
-
B.
a - b = 0
-
C.
h = 0
-
D.
a = b
Solution
For the lines to be perpendicular, the condition a*b + h^2 = 0 must hold.
Correct Answer: A — a + b = 0
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Q. If the parabola y = ax^2 + bx + c has its vertex at (1, -2), what is the value of a if it passes through the point (0, 0)?
Solution
Using the vertex form y = a(x - 1)^2 - 2 and substituting (0, 0), we get 0 = a(0 - 1)^2 - 2 => 2 = a => a = 2.
Correct Answer: B — 2
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Q. If the parabola y^2 = 16x opens to the right, what is the value of p?
Solution
In the equation y^2 = 4px, we have 4p = 16, thus p = 4.
Correct Answer: B — 4
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Q. If the parabola y^2 = 20x opens to the right, what is the value of p?
Solution
In the equation y^2 = 4px, we have 4p = 20, thus p = 20/4 = 5.
Correct Answer: A — 5
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Q. If the peak current in an AC circuit is 5 A, what is the average current over one complete cycle?
-
A.
5 A
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B.
2.5 A
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C.
0 A
-
D.
7.07 A
Solution
The average current over one complete cycle is I_avg = I_peak/√2 = 5/√2 = 2.5 A.
Correct Answer: B — 2.5 A
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Q. If the peak voltage of an AC source is 200 V, what is the RMS voltage?
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A.
100 V
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B.
141.42 V
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C.
200 V
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D.
282.84 V
Solution
The RMS voltage (V_rms) is given by V_rms = V_peak / √2. Therefore, V_rms = 200 V / √2 = 141.42 V.
Correct Answer: B — 141.42 V
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Q. If the peak voltage of an AC source is 220 V, what is the RMS voltage?
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A.
110 V
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B.
154 V
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C.
220 V
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D.
311 V
Solution
The RMS voltage (V_rms) is given by V_rms = V_peak / √2. Therefore, V_rms = 220 V / √2 = 110 V.
Correct Answer: A — 110 V
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Q. If the period of a satellite in a circular orbit is T, what is the relationship between T and the radius r of the orbit?
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A.
T ∝ r
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B.
T ∝ r²
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C.
T ∝ √r
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D.
T ∝ 1/√r
Solution
T = 2π√(r³/GM), thus T ∝ √r.
Correct Answer: C — T ∝ √r
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Q. If the photoelectric effect is observed, what can be inferred about the incident light?
-
A.
It has a frequency below the threshold
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B.
It has a frequency equal to the threshold
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C.
It has a frequency above the threshold
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D.
It has any frequency
Solution
The photoelectric effect occurs only when the frequency of the incident light is above the threshold frequency.
Correct Answer: C — It has a frequency above the threshold
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Q. If the points (1, 2), (3, 4), and (5, 6) are collinear, what is the slope of the line?
-
A.
1
-
B.
2
-
C.
0
-
D.
undefined
Solution
Slope = (4-2)/(3-1) = 1, and it remains the same for other points.
Correct Answer: A — 1
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Q. If the points A(1, 2), B(3, 4), and C(5, 6) are collinear, what is the area of triangle ABC?
Solution
Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 0.
Correct Answer: A — 0
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Q. If the polynomial P(x) = x^3 - 6x^2 + 11x - 6 has a root at x = 1, what is P(2)?
Solution
P(2) = 2^3 - 6(2^2) + 11(2) - 6 = 8 - 24 + 22 - 6 = 0.
Correct Answer: D — 3
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Q. If the position vector of a point is (5, 12), what is its distance from the origin?
Solution
Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Correct Answer: A — 13
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Q. If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?
-
A.
(2, 3, 4)
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B.
(4, 6, 8)
-
C.
(2t, 3t, 4t)
-
D.
(0, 0, 0)
Solution
Velocity vector = dr/dt = (2, 3, 4)
Correct Answer: A — (2, 3, 4)
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Q. If the position vector of a point P is (2, 3, 4), what is the distance from the origin to point P?
Solution
Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.
Correct Answer: B — 6
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Q. If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
-
A.
(1, 2, 3)
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B.
(2, 4, 6)
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C.
(0, 0, 0)
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D.
(x, y, z)
Solution
Projection of P onto a = ((P · a) / |a|^2) * a.
Correct Answer: D — (x, y, z)
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Q. If the position vector of a point P is given by r = (2t, 3t, 4t), find the coordinates of P when t = 1.
-
A.
(2, 3, 4)
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B.
(1, 1, 1)
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C.
(0, 0, 0)
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D.
(2, 4, 6)
Solution
Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).
Correct Answer: A — (2, 3, 4)
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Q. If the position vector of point P is (3, -2) and Q is (1, 4), what is the vector PQ?
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A.
(-2, 6)
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B.
(2, -6)
-
C.
(4, -6)
-
D.
(6, 2)
Solution
Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).
Correct Answer: A — (-2, 6)
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Q. If the position vector of point P is (3, 4) and Q is (1, 2), what is the vector PQ?
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A.
(2, 2)
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B.
(4, 6)
-
C.
(2, 4)
-
D.
(1, 1)
Solution
Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).
Correct Answer: A — (2, 2)
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Q. If the principal quantum number n = 4 and the azimuthal quantum number l = 2, what is the maximum number of electrons in this subshell?
Solution
The maximum number of electrons in a subshell is given by 2(2l + 1). For l = 2, it is 2(2*2 + 1) = 10.
Correct Answer: C — 14
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Q. If the principal quantum number n = 4, what are the possible values of l?
-
A.
0, 1, 2, 3
-
B.
1, 2, 3, 4
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C.
0, 1, 2, 3, 4
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D.
0, 1, 2
Solution
For n=4, l can take values 0, 1, 2, or 3.
Correct Answer: A — 0, 1, 2, 3
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Q. If the principal quantum number n = 4, what are the possible values of the azimuthal quantum number l?
-
A.
0, 1, 2, 3
-
B.
1, 2, 3, 4
-
C.
0, 1, 2, 3, 4
-
D.
0, 1, 2
Solution
For n=4, l can take values from 0 to n-1, which are 0, 1, 2, and 3.
Correct Answer: A — 0, 1, 2, 3
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Q. If the probability of an event A is 0.3, what is the probability of the event not occurring?
-
A.
0.7
-
B.
0.3
-
C.
0.5
-
D.
0.2
Solution
Probability of not A = 1 - P(A) = 1 - 0.3 = 0.7.
Correct Answer: A — 0.7
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Q. If the probability of an event A is 0.7, what is the probability of the event not occurring?
-
A.
0.3
-
B.
0.7
-
C.
0.5
-
D.
0.1
Solution
Probability of not A = 1 - P(A) = 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
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Q. If the probability of an event is 0.7, what is the probability of its complement?
-
A.
0.3
-
B.
0.7
-
C.
1
-
D.
0.5
Solution
Probability of complement = 1 - P(event) = 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
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Q. If the probability of an event is 0.7, what is the probability of the event not occurring?
-
A.
0.3
-
B.
0.7
-
C.
0.5
-
D.
0.1
Solution
Probability of not occurring = 1 - P(event) = 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
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Q. If the probability of event A is 0.4 and event B is 0.5, what is the probability of both events occurring if they are independent?
-
A.
0.2
-
B.
0.4
-
C.
0.5
-
D.
0.6
Solution
P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.
Correct Answer: A — 0.2
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Q. If the quadratic equation ax^2 + bx + c = 0 has roots 3 and -2, what is the value of a?
Solution
Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.
Correct Answer: A — 1
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Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has real roots, what is the condition on p?
-
A.
p > 2
-
B.
p < 2
-
C.
p = 2
-
D.
p >= 2
Solution
The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 => 4p^2 - 4p^2 + 16 >= 0, which is always true. Thus, p can be any real number.
Correct Answer: D — p >= 2
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