Mathematics Syllabus for JEE Main

Mathematics Syllabus (JEE Main)

Algebra Calculus Coordinate Geometry Sets, Relations & Functions Statistics & Probability Trigonometry Vector & 3D Geometry

Q. If the odds in favor of an event are 3:2, what is the probability of the event occurring?

A. 0.6
B. 0.4
C. 0.5
D. 0.3

Solution

Probability = Odds in favor / (Total odds) = 3 / (3 + 2) = 3/5 = 0.6.

Correct Answer: A — 0.6

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Q. If the pair of lines represented by ax^2 + 2hxy + by^2 = 0 are perpendicular, then:

A. a + b = 0
B. a - b = 0
C. h = 0
D. a = b

Solution

For the lines to be perpendicular, the condition a + b = 0 must hold.

Correct Answer: A — a + b = 0

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Q. If the pair of lines represented by the equation ax^2 + 2hxy + by^2 = 0 are perpendicular, then:

A. a + b = 0
B. a - b = 0
C. h = 0
D. a = b

Solution

For the lines to be perpendicular, the condition a*b + h^2 = 0 must hold.

Correct Answer: A — a + b = 0

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Q. If the parabola y = ax^2 + bx + c has its vertex at (1, -2), what is the value of a if it passes through the point (0, 0)?

A. 1
B. 2
C. 3
D. 4

Solution

Using the vertex form y = a(x - 1)^2 - 2 and substituting (0, 0), we get 0 = a(0 - 1)^2 - 2 => 2 = a => a = 2.

Correct Answer: B — 2

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Q. If the parabola y^2 = 16x opens to the right, what is the value of p?

A. 2
B. 4
C. 8
D. 16

Solution

In the equation y^2 = 4px, we have 4p = 16, thus p = 4.

Correct Answer: B — 4

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Q. If the parabola y^2 = 20x opens to the right, what is the value of p?

A. 5
B. 10
C. 20
D. 2

Solution

In the equation y^2 = 4px, we have 4p = 20, thus p = 20/4 = 5.

Correct Answer: A — 5

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Q. If the points (1, 2), (3, 4), and (5, 6) are collinear, what is the slope of the line?

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

Solution

Slope = (4-2)/(3-1) = 1, and it remains the same for other points.

Correct Answer: A — 1

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Q. If the points A(1, 2), B(3, 4), and C(5, 6) are collinear, what is the area of triangle ABC?

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

Solution

Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 0.

Correct Answer: A — 0

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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 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 probability of an event A is 0.3, what is the probability of the event not occurring?

A. 0.7
B. 0.3
C. 0.5
D. 0.2

Solution

Probability of not A = 1 - P(A) = 1 - 0.3 = 0.7.

Correct Answer: A — 0.7

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Q. If the probability of an event A is 0.7, what is the probability of the event not occurring?

A. 0.3
B. 0.7
C. 0.5
D. 0.1

Solution

Probability of not A = 1 - P(A) = 1 - 0.7 = 0.3.

Correct Answer: A — 0.3

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Q. If the probability of an event is 0.7, what is the probability of its complement?

A. 0.3
B. 0.7
C. 1
D. 0.5

Solution

Probability of complement = 1 - P(event) = 1 - 0.7 = 0.3.

Correct Answer: A — 0.3

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Q. If the probability of an event is 0.7, what is the probability of the event not occurring?

A. 0.3
B. 0.7
C. 0.5
D. 0.1

Solution

Probability of not occurring = 1 - P(event) = 1 - 0.7 = 0.3.

Correct Answer: A — 0.3

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Q. If the probability of event A is 0.4 and event B is 0.5, what is the probability of both events occurring if they are independent?

A. 0.2
B. 0.4
C. 0.5
D. 0.6

Solution

P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.

Correct Answer: A — 0.2

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Q. If the quadratic equation ax^2 + bx + c = 0 has roots 3 and -2, what is the value of a?

A. 1
B. 2
C. 3
D. 4

Solution

Using the fact that the product of the roots is c/a and the sum is -b/a, we can set a = 1, b = -1, c = -6.

Correct Answer: A — 1

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Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has real roots, what is the condition on p?

A. p > 2
B. p < 2
C. p = 2
D. p >= 2

Solution

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 => 4p^2 - 4p^2 + 16 >= 0, which is always true. Thus, p can be any real number.

Correct Answer: D — p >= 2

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Q. If the quadratic equation x^2 + 2px + p^2 - 4 = 0 has roots that are equal, what is the value of p?

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

Solution

Setting the discriminant to zero: (2p)^2 - 4(1)(p^2 - 4) = 0 leads to p = ±2.

Correct Answer: C — -2

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Q. If the quadratic equation x^2 + 2x + k = 0 has equal roots, what is the value of k?

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

Solution

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0, leading to k = 1.

Correct Answer: C — -1

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Q. If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition for k?

A. k < 0
B. k > 0
C. k >= 0
D. k <= 0

Solution

For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0 => 4 - 4k < 0 => k > 1.

Correct Answer: A — k < 0

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Q. If the quadratic equation x^2 + 2x + k = 0 has no real roots, what is the condition on k?

A. k < 0
B. k > 0
C. k >= 0
D. k <= 0

Solution

For no real roots, the discriminant must be less than zero: 2^2 - 4*1*k < 0, hence k > 1.

Correct Answer: A — k < 0

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are equal, what is the value of k?

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

Solution

For equal roots, the discriminant must be zero: 2^2 - 4*1*k = 0 leads to k = -1.

Correct Answer: D — -2

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Q. If the quadratic equation x^2 + 4x + c = 0 has one root equal to -2, what is the value of c?

A. 0
B. 2
C. 4
D. 6

Solution

If one root is -2, then substituting x = -2 gives: (-2)^2 + 4(-2) + c = 0 => 4 - 8 + c = 0 => c = 4.

Correct Answer: A — 0

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Q. If the quadratic equation x^2 + 4x + k = 0 has roots -2 and -2, what is the value of k?

A. 0
B. 4
C. 8
D. 16

Solution

Using the formula for roots, k = (-2)^2 - 4*(-2) = 4 + 8 = 12.

Correct Answer: B — 4

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Mathematics Syllabus (JEE Main)

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