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Q. Find the coefficient of x^4 in the expansion of (x + 2)^6.

A. 15
B. 30
C. 45
D. 60

Solution

The coefficient of x^4 is C(6,4) * (2)^2 = 15 * 4 = 60.

Correct Answer: B — 30

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Q. Find the coefficient of x^4 in the expansion of (x + 3)^6.

A. 81
B. 162
C. 243
D. 729

Solution

The coefficient of x^4 is C(6,4) * (3)^2 = 15 * 9 = 135.

Correct Answer: C — 243

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Q. Find the coefficient of x^4 in the expansion of (x + 5)^7.

A. 210
B. 1260
C. 1750
D. 2450

Solution

The coefficient of x^4 is given by C(7,4) * 5^3 = 35 * 125 = 4375.

Correct Answer: B — 1260

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Q. Find the coefficient of x^4 in the expansion of (x - 1)^5.

A. -5
B. 10
C. -10
D. 5

Solution

The coefficient of x^4 is C(5,4) * (-1)^1 = 5 * (-1) = -5.

Correct Answer: C — -10

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Q. Find the coefficient of x^4 in the expansion of (x - 5)^6.

A. 150
B. 200
C. 250
D. 300

Solution

The coefficient of x^4 is given by C(6,4) * (-5)^2 = 15 * 25 = 375.

Correct Answer: B — 200

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)

A. -540
B. 540
C. -720
D. 720

Solution

The coefficient of x^5 is C(6,5) * (2)^5 * (-3)^1 = 6 * 32 * (-3) = -576.

Correct Answer: A — -540

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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^7. (2023)

A. 168
B. 252
C. 336
D. 504

Solution

The coefficient of x^5 is C(7,5) * (2)^5 * (-3)^2 = 21 * 32 * 9 = 6048.

Correct Answer: B — 252

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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 (3x + 2)^6.

A. 486
B. 729
C. 729
D. 486

Solution

The coefficient of x^5 in (3x + 2)^6 is C(6, 5)(3)^5(2)^1 = 6 * 243 * 2 = 2916.

Correct Answer: A — 486

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Q. Find the coefficient of x^5 in the expansion of (x + 1)^7.

A. 7
B. 21
C. 35
D. 42

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.

A. 56
B. 70
C. 80
D. 90

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.

A. 21
B. 42
C. 56
D. 70

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 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 = 2 - 5i.

A. 2 + 5i
B. 2 - 5i
C. -2 + 5i
D. -2 - 5i

Solution

The conjugate of z = 2 - 5i is z* = 2 + 5i.

Correct Answer: A — 2 + 5i

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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.

A. -64
B. 0
C. 32
D. 48

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)

A. -64
B. 32
C. 0
D. -32

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)
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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Major Competitive Exams

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