Q. Find the coefficient of x^5 in the expansion of (2x - 3)^8.
-
A.
-6720
-
B.
6720
-
C.
-3360
-
D.
3360
Solution
The coefficient of x^5 is C(8,5) * (2)^5 * (-3)^3 = 56 * 32 * (-27) = -6720.
Correct Answer: A — -6720
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^7.
Solution
The coefficient of x^5 is C(7,5) = 21.
Correct Answer: C — 35
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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 + 2)^7.
Solution
The coefficient of x^5 is C(7,5) * 2^2 = 21 * 4 = 84.
Correct Answer: C — 56
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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 coefficient of x^6 in the expansion of (x + 1)^10. (2023)
-
A.
10
-
B.
45
-
C.
120
-
D.
210
Solution
The coefficient of x^6 is given by 10C6 = 210.
Correct Answer: D — 210
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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 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 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 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 constant term in the expansion of (3x - 4/x)^5.
Solution
The constant term occurs when the power of x is zero. The term is given by 5C2 * (3x)^2 * (-4/x)^3 = 10 * 9 * (-64) = -5760.
Correct Answer: A — -64
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Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)
Solution
The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.
Correct Answer: A — -64
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Q. Find the coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (0, 8). (2022)
-
A.
(2, 2)
-
B.
(2, 3)
-
C.
(3, 2)
-
D.
(4, 4)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).
Correct Answer: A — (2, 2)
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Q. Find the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2021)
-
A.
(4/3, 1, 0)
-
B.
(2, 1, 0)
-
C.
(1, 1, 0)
-
D.
(0, 0, 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. 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 (0, 0), (6, 0), and (0, 8).
-
A.
(2, 2)
-
B.
(2, 3)
-
C.
(3, 2)
-
D.
(4, 4)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).
Correct Answer: A — (2, 2)
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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)
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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 midpoint of the line segment joining A(2, -1, 3) and B(4, 3, 5). (2022)
-
A.
(3, 1, 4)
-
B.
(2, 1, 4)
-
C.
(3, 2, 3)
-
D.
(4, 2, 4)
Solution
Midpoint M = ((2+4)/2, (-1+3)/2, (3+5)/2) = (3, 1, 4).
Correct Answer: A — (3, 1, 4)
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Q. Find the coordinates of the midpoint of the line segment joining A(2, 3, 4) and B(4, 5, 6). (2023)
-
A.
(3, 4, 5)
-
B.
(2, 3, 4)
-
C.
(4, 5, 6)
-
D.
(5, 6, 7)
Solution
Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).
Correct Answer: A — (3, 4, 5)
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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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Q. Find the coordinates of the point on the curve y = x^3 - 3x + 2 where the tangent is horizontal.
-
A.
(0, 2)
-
B.
(1, 0)
-
C.
(2, 0)
-
D.
(3, 2)
Solution
f'(x) = 3x^2 - 3. Setting f'(x) = 0 gives x = 1. The point is (1, 0).
Correct Answer: B — (1, 0)
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Q. Find the coordinates of the point where the function f(x) = 3x^2 - 12x + 9 has a local maximum.
-
A.
(2, 3)
-
B.
(3, 0)
-
C.
(1, 1)
-
D.
(0, 9)
Solution
f'(x) = 6x - 12. Setting f'(x) = 0 gives x = 2. f(2) = 3(2^2) - 12(2) + 9 = 3.
Correct Answer: A — (2, 3)
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Q. Find the critical points of f(x) = x^3 - 3x^2 + 4.
-
A.
(0, 4)
-
B.
(1, 2)
-
C.
(2, 0)
-
D.
(3, 1)
Solution
Setting f'(x) = 3x^2 - 6x = 0 gives x(x - 2) = 0, so critical points are x = 0 and x = 2. Evaluating f(1) = 2.
Correct Answer: B — (1, 2)
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Q. Find the critical points of f(x) = x^4 - 8x^2 + 16. (2021)
-
A.
(0, 16)
-
B.
(2, 0)
-
C.
(4, 0)
-
D.
(1, 15)
Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 0 is a critical point.
Correct Answer: B — (2, 0)
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Q. Find the critical points of the function f(x) = 3x^4 - 8x^3 + 6.
-
A.
(0, 6)
-
B.
(2, -2)
-
C.
(1, 1)
-
D.
(3, 0)
Solution
f'(x) = 12x^3 - 24x^2. Setting f'(x) = 0 gives x^2(12x - 24) = 0, so x = 0 or x = 2. f(2) = 3(2^4) - 8(2^3) + 6 = -2.
Correct Answer: B — (2, -2)
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Q. Find the critical points of the function f(x) = x^3 - 3x^2 + 4.
-
A.
x = 0, 2
-
B.
x = 1, 2
-
C.
x = 1, 3
-
D.
x = 0, 1
Solution
To find critical points, set f'(x) = 0. f'(x) = 3x^2 - 6x = 3x(x - 2). Critical points are x = 0 and x = 2.
Correct Answer: B — x = 1, 2
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