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Q. If the polynomial f(x) = x^3 - 6x^2 + 11x - 6 is factored, which of the following is one of its factors?

A. x - 1
B. x + 2
C. x - 3
D. x + 1

Solution

The polynomial can be factored as (x - 1)(x - 2)(x - 3), so x - 1 is one of its factors.

Correct Answer: A — x - 1

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Q. If the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 is evaluated at x = 1, what is the result?

A. 1
B. 0
C. 2
D. 3

Solution

Substituting x = 1 into the polynomial gives f(1) = 1 - 4 + 6 - 4 + 1 = 0.

Correct Answer: A — 1

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Q. If the polynomial f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1, what can be inferred about its symmetry?

A. It is symmetric about the y-axis.
B. It is symmetric about the x-axis.
C. It is symmetric about the origin.
D. It is symmetric about the line x = 1.

Solution

The polynomial can be rewritten in a form that shows symmetry about the line x = 1.

Correct Answer: D — It is symmetric about the line x = 1.

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Q. If the polynomial g(x) = x^2 + bx + c has a double root, what can be inferred about its discriminant?

A. It is greater than zero.
B. It is less than zero.
C. It is equal to zero.
D. It can be any value.

Solution

A polynomial has a double root when its discriminant is equal to zero.

Correct Answer: C — It is equal to zero.

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Q. If the polynomial g(x) = x^2 + bx + c has roots -2 and 3, what is the value of b?

A. 1
B. -1
C. 5
D. -5

Solution

Using Vieta's formulas, the sum of the roots (-2 + 3) = 1, which means b = -1.

Correct Answer: C — 5

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Q. If the polynomial P(x) = x^2 + bx + c has roots 3 and -2, what is the value of b?

A. 1
B. 5
C. -1
D. -5

Solution

Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.

Correct Answer: B — 5

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Q. If the polynomial P(x) = x^2 - 5x + 6 has roots r1 and r2, what is the value of r1 + r2?

A. 5
B. -5
C. 6
D. -6

Solution

According to Vieta's formulas, the sum of the roots r1 + r2 of the polynomial x^2 - 5x + 6 is equal to the coefficient of x (which is -(-5)) = 5.

Correct Answer: A — 5

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Q. If the polynomial P(x) = x^2 - 5x + 6 is factored, what are the roots of the polynomial?

A. 2 and 3
B. 1 and 6
C. 3 and 2
D. 0 and 5

Solution

Factoring the polynomial gives (x - 2)(x - 3), so the roots are 2 and 3.

Correct Answer: A — 2 and 3

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Q. If the polynomial P(x) = x^2 - 5x + 6 is factored, what are the roots?

A. 2 and 3
B. 1 and 6
C. 3 and 2
D. 0 and 6

Solution

Factoring the polynomial P(x) gives (x - 2)(x - 3), so the roots are 2 and 3.

Correct Answer: A — 2 and 3

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Q. If the polynomial P(x) = x^3 - 3x^2 + 4 has a local maximum at x = 1, what is the value of P(1)?

A. 2
B. 0
C. 1
D. 4

Solution

Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

Correct Answer: A — 2

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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)?

A. 0
B. 1
C. 2
D. 3

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 population of a city increases by 10% every year, what will be the population after 2 years if the current population is 1000? (2021)

A. 1210
B. 1100
C. 1020
D. 1200

Solution

Population after 1 year = 1000 * 1.10 = 1100. Population after 2 years = 1100 * 1.10 = 1210.

Correct Answer: A — 1210

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Q. If the population of a city is 1 million and it grows at a rate of 2% per year, what will be the population after one year? (2023)

A. 1.02 million
B. 1.05 million
C. 1.1 million
D. 1.1 million

Solution

Population after one year = Current Population * (1 + Growth Rate) = 1 million * (1 + 0.02) = 1 million * 1.02 = 1.02 million.

Correct Answer: A — 1.02 million

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Q. If the population of a country increases by 5% every year, what will be the population after 2 years if the current population is 100,000?

A. 110,250
B. 105,000
C. 120,000
D. 102,500

Solution

Population after 2 years = 100,000 * (1 + 0.05)^2 = 100,000 * 1.1025 = 110,250.

Correct Answer: A — 110,250

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Q. If the population of a town increases by 10% every year, what will be the population after 2 years if the current population is 1000? (2021)

A. 1210
B. 1100
C. 1020
D. 1200

Solution

Population after 1 year = 1000 * 1.10 = 1100. Population after 2 years = 1100 * 1.10 = 1210.

Correct Answer: A — 1210

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Q. If the position vector of a point is (5, 12), what is its distance from the origin?

A. 13
B. 12
C. 11
D. 10

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)
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?

A. 5
B. 6
C. 7
D. 8

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)
B. (2, 4, 6)
C. (0, 0, 0)
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)
B. (1, 1, 1)
C. (0, 0, 0)
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?

A. (-2, 6)
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?

A. (2, 2)
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 position vector of point P is given by r = 2i + 3j + 4k, what is the distance from the origin to point P?

A. 5
B. 6
C. 7
D. 8

Solution

Distance = |r| = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29.

Correct Answer: B — 6

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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-coordinate of point P? (2020)

A. 2
B. 3
C. 4
D. 5

Solution

The x-coordinate of point P is the coefficient of i in the position vector, which is 2.

Correct Answer: A — 2

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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-component of r?

A. 2
B. 3
C. 4
D. 5

Solution

The x-component of r is 2.

Correct Answer: A — 2

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Q. If the position vector of point P is given by r = 3i + 4j, what is the distance of point P from the origin?

A. 5
B. 7
C. 4
D. 3

Solution

Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. If the potential at a point is 5 V and the electric field is 2 V/m, what is the distance from the reference point? (2023)

A. 2.5 m
B. 10 m
C. 7.5 m
D. 5 m

Solution

Distance (d) = Potential (V) / Electric field (E) = 5 V / 2 V/m = 2.5 m.

Correct Answer: B — 10 m

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Q. If the potential difference across a capacitor is halved, what happens to the energy stored in the capacitor? (2019)

A. Halved
B. Doubled
C. Remains the same
D. Quadrupled

Solution

The energy stored in a capacitor is given by U = 1/2 CV^2. If V is halved, the energy becomes U' = 1/2 C(V/2)^2 = 1/8 CV^2, which is halved.

Correct Answer: A — Halved

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Q. If the potential difference across a capacitor is increased, what happens to the charge stored in it? (2020)

A. It decreases
B. It remains the same
C. It increases
D. It becomes zero

Solution

The charge (Q) stored in a capacitor is directly proportional to the potential difference (V) across it, given by Q = CV.

Correct Answer: C — It increases

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Q. If the potential difference between two points is 10 V and the work done to move a charge of 2 C between these points is: (2020)

A. 5 J
B. 10 J
C. 20 J
D. 40 J

Solution

Work done (W) = Charge (Q) × Potential difference (V) = 2 C × 10 V = 20 J.

Correct Answer: C — 20 J

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Major Competitive Exams

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