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
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B.
D > 0
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C.
D < 0
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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
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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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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 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 - 6x^2 + 9x.
-
A.
(0, 0)
-
B.
(3, 0)
-
C.
(2, 0)
-
D.
(1, 0)
Solution
f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Critical points are (1, f(1)) and (3, f(3)).
Correct Answer: B — (3, 0)
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Q. Find the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6).
-
A.
(-3, 6, -3)
-
B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, -2, 1)
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer: A — (-3, 6, -3)
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Q. Find the derivative of f(x) = 1/x.
-
A.
-1/x^2
-
B.
1/x^2
-
C.
-2/x^2
-
D.
1/x
Solution
Using the power rule, f'(x) = -1/x^2.
Correct Answer: A — -1/x^2
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Q. Find the derivative of f(x) = 3x^2 + 5x - 7.
-
A.
6x + 5
-
B.
3x + 5
-
C.
6x - 5
-
D.
3x^2 + 5
Solution
Using the power rule, f'(x) = d/dx(3x^2) + d/dx(5x) - d/dx(7) = 6x + 5.
Correct Answer: A — 6x + 5
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Q. Find the derivative of f(x) = 5x^4 - 3x + 2.
-
A.
20x^3 - 3
-
B.
15x^3 - 3
-
C.
20x^4 - 3
-
D.
5x^3 - 3
Solution
Using the power rule, f'(x) = 20x^3 - 3.
Correct Answer: A — 20x^3 - 3
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Q. Find the derivative of f(x) = e^(2x) at x = 0.
Solution
f'(x) = 2e^(2x). At x = 0, f'(0) = 2e^0 = 2.
Correct Answer: B — 2
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Q. Find the derivative of f(x) = e^(2x).
-
A.
2e^(2x)
-
B.
e^(2x)
-
C.
2xe^(2x)
-
D.
e^(x)
Solution
Using the chain rule, f'(x) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. Find the derivative of f(x) = e^(x^2).
-
A.
2xe^(x^2)
-
B.
e^(x^2)
-
C.
x e^(x^2)
-
D.
2e^(x^2)
Solution
Using the chain rule, f'(x) = e^(x^2) * 2x = 2x e^(x^2).
Correct Answer: A — 2xe^(x^2)
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Q. Find the derivative of f(x) = e^x * ln(x) at x = 1.
Solution
Using the product rule, f'(x) = e^x * ln(x) + e^x/x. At x = 1, this simplifies to 0.
Correct Answer: A — 1
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Q. Find the derivative of f(x) = e^x * sin(x) at x = 0.
Solution
Using the product rule, f'(0) = e^0 * sin(0) + e^0 * cos(0) = 0 + 1 = 1.
Correct Answer: A — 1
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Q. Find the derivative of f(x) = ln(x^2 + 1) at x = 1.
Solution
f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = (2*1)/(1^2 + 1) = 2/2 = 1.
Correct Answer: B — 1
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Q. Find the derivative of f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
-
C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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