Q. Find the area under the curve y = x^2 from x = 0 to x = 3.
Solution
Area = ∫ from 0 to 3 of x^2 dx = [1/3 * x^3] from 0 to 3 = 9.
Correct Answer: A — 9
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Q. Find the area under the curve y = x^2 from x = 1 to x = 3.
-
A.
8/3
-
B.
10/3
-
C.
9/3
-
D.
7/3
Solution
The area is given by the integral ∫ (x^2) dx from 1 to 3. This evaluates to [x^3/3] from 1 to 3 = (27/3 - 1/3) = 26/3.
Correct Answer: B — 10/3
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Q. Find the area under the curve y = x^4 from x = 0 to x = 1.
-
A.
1/5
-
B.
1/4
-
C.
1/3
-
D.
1/2
Solution
The area under the curve y = x^4 from 0 to 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.
Correct Answer: A — 1/5
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Q. Find the area under the curve y = x^4 from x = 0 to x = 2.
Solution
The area is given by the integral from 0 to 2 of x^4 dx. This evaluates to [x^5/5] from 0 to 2 = (32/5) = 16.
Correct Answer: C — 16
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Q. Find the argument of the complex number z = -1 - i.
-
A.
-3π/4
-
B.
3π/4
-
C.
π/4
-
D.
-π/4
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.
Correct Answer: A — -3π/4
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Q. Find the arithmetic mean of the first five prime numbers.
Solution
First five primes: 2, 3, 5, 7, 11. Mean = (2 + 3 + 5 + 7 + 11) / 5 = 28 / 5 = 5.6.
Correct Answer: C — 7
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Q. Find the arithmetic mean of the numbers 12, 15, 18, 21, and 24.
Solution
Mean = (12 + 15 + 18 + 21 + 24) / 5 = 90 / 5 = 18.
Correct Answer: A — 18
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Q. Find the coefficient of x^0 in the expansion of (2x + 3)^4.
Solution
The coefficient of x^0 is C(4, 0) * (2x)^0 * 3^4 = 1 * 81 = 81.
Correct Answer: A — 81
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Q. Find the coefficient of x^1 in the expansion of (x + 2)^5.
Solution
The coefficient of x^1 is C(5,1) * 2^4 = 5 * 16 = 80.
Correct Answer: B — 20
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Q. Find the coefficient of x^2 in the expansion of (3x - 4)^6.
-
A.
540
-
B.
720
-
C.
480
-
D.
360
Solution
The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.
Correct Answer: B — 720
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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^6.
-
A.
-540
-
B.
-720
-
C.
540
-
D.
720
Solution
The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.
Correct Answer: A — -540
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Q. Find the coefficient of x^3 in the expansion of (x + 1/2)^6.
Solution
The coefficient of x^3 is C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Correct Answer: B — 15
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Q. Find the coefficient of x^3 in the expansion of (x + 2)^6.
-
A.
80
-
B.
120
-
C.
160
-
D.
240
Solution
The coefficient of x^3 is C(6,3) * (2)^3 = 20 * 8 = 160.
Correct Answer: B — 120
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^5.
Solution
The coefficient of x^3 is C(5,3) * (-1)^2 = 10.
Correct Answer: A — -10
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^6.
-
A.
-20
-
B.
-15
-
C.
-10
-
D.
-6
Solution
The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.
Correct Answer: A — -20
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Q. Find the coefficient of x^3 in the expansion of (x - 3)^5.
-
A.
-135
-
B.
-90
-
C.
-60
-
D.
-45
Solution
The coefficient of x^3 is C(5,3) * (-3)^2 = 10 * 9 = -90.
Correct Answer: A — -135
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Q. Find the coefficient of x^4 in the expansion of (3x - 2)^6.
-
A.
540
-
B.
720
-
C.
810
-
D.
960
Solution
Using the binomial theorem, the coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: C — 810
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^8.
Solution
The coefficient of x^5 is C(8,5) = 56.
Correct Answer: B — 70
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Q. Find the coefficient of x^5 in the expansion of (x + 3)^8.
-
A.
56
-
B.
168
-
C.
336
-
D.
672
Solution
The coefficient of x^5 is C(8,5) * (3)^3 = 56 * 27 = 1512.
Correct Answer: B — 168
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Q. Find the coefficient of x^5 in the expansion of (x - 3)^7.
-
A.
-1890
-
B.
-2187
-
C.
-2401
-
D.
-2430
Solution
The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -1890.
Correct Answer: A — -1890
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Q. Find the condition for the lines represented by the equation 2x^2 + 3xy + y^2 = 0 to be parallel.
-
A.
D = 0
-
B.
D > 0
-
C.
D < 0
-
D.
D = 1
Solution
For the lines to be parallel, the discriminant D must be equal to 0.
Correct Answer: A — D = 0
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Q. Find the condition for the lines represented by the equation ax^2 + 2hxy + by^2 = 0 to be parallel.
-
A.
h^2 = ab
-
B.
h^2 > ab
-
C.
h^2 < ab
-
D.
h^2 = 0
Solution
The condition for the lines to be parallel is given by h^2 = ab.
Correct Answer: A — h^2 = ab
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Q. Find the condition for the lines represented by the equation ax^2 + 2hxy + by^2 = 0 to be perpendicular.
-
A.
ab + h^2 = 0
-
B.
ab - h^2 = 0
-
C.
a + b = 0
-
D.
a - b = 0
Solution
The condition for the lines to be perpendicular is given by the relation ab + h^2 = 0.
Correct Answer: A — ab + h^2 = 0
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Q. Find the conjugate of the complex number z = 5 - 6i.
-
A.
5 + 6i
-
B.
5 - 6i
-
C.
-5 + 6i
-
D.
-5 - 6i
Solution
The conjugate of z = 5 - 6i is z̅ = 5 + 6i.
Correct Answer: A — 5 + 6i
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Q. Find the coordinates of the centroid of the triangle with vertices at (0, 0), (6, 0), and (3, 6).
-
A.
(3, 2)
-
B.
(3, 3)
-
C.
(2, 3)
-
D.
(0, 0)
Solution
Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3) = (9/3, 6/3) = (3, 2).
Correct Answer: B — (3, 3)
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Q. Find the coordinates of the centroid of the triangle with vertices at (1, 2), (3, 4), and (5, 6).
-
A.
(3, 4)
-
B.
(2, 3)
-
C.
(4, 5)
-
D.
(5, 6)
Solution
Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).
Correct Answer: B — (2, 3)
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Q. Find the coordinates of the focus of the parabola y^2 = -12x.
-
A.
(-3, 0)
-
B.
(-2, 0)
-
C.
(3, 0)
-
D.
(2, 0)
Solution
The equation y^2 = -12x can be rewritten as (y - 0)^2 = 4p(x - 0) with p = -3, so the focus is at (-3, 0).
Correct Answer: A — (-3, 0)
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Q. Find the coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x - 3y + 6 = 0.
-
A.
(0, 2)
-
B.
(1, 1)
-
C.
(2, 0)
-
D.
(3, -1)
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (1, 1).
Correct Answer: B — (1, 1)
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Q. Find the coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0.
-
A.
(2, 0)
-
B.
(1, 1)
-
C.
(0, 2)
-
D.
(3, 2)
Solution
Using the formula for foot of perpendicular, we find the coordinates to be (3, 2).
Correct Answer: D — (3, 2)
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Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the slope of the tangent is 0.
-
A.
(1, 0)
-
B.
(0, 2)
-
C.
(2, 0)
-
D.
(3, 2)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x^2 = 1, so x = 1 or x = -1. f(1) = 0, f(-1) = 4. The point is (1, 0).
Correct Answer: A — (1, 0)
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