Mathematics Syllabus for JEE Main

Mathematics Syllabus (JEE Main)

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

Q. A person standing 50 meters away from a building observes the top of the building at an angle of elevation of 45 degrees. What is the height of the building?

A. 25 meters
B. 50 meters
C. 35 meters
D. 45 meters

Solution

Height = distance * tan(angle) = 50 * tan(45°) = 50 meters.

Correct Answer: B — 50 meters

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Q. A person standing on the ground observes the top of a tree at an angle of elevation of 45 degrees. If the person is 10 meters away from the tree, what is the height of the tree?

A. 5 m
B. 10 m
C. 15 m
D. 20 m

Solution

Using tan(45°) = height/10, we have 1 = height/10. Therefore, height = 10 m.

Correct Answer: B — 10 m

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Q. A student has a 70% chance of passing an exam. What is the probability that the student fails the exam?

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

Solution

The probability of failing the exam is 1 - 0.7 = 0.3.

Correct Answer: A — 0.3

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Q. A student is selected at random from a class of 40 students, where 25 are boys and 15 are girls. What is the probability that the student is a girl given that the student is not a boy?

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

Solution

The total number of students that are not boys is 15 (girls). The probability of selecting a girl given that the student is not a boy is 15/15 = 1.

Correct Answer: C — 2/3

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Q. A student is selected at random from a group of 40 students, where 25 are studying Mathematics and 15 are studying Physics. What is the probability that the student is studying Mathematics given that they are not studying Physics?

A. 5/8
B. 3/8
C. 1/2
D. 1/3

Solution

If the student is not studying Physics, they must be studying Mathematics. Therefore, P(Math | Not Physics) = 1.

Correct Answer: A — 5/8

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Q. A student is selected at random from a group of students who study Mathematics and Physics. If 70% study Mathematics and 40% study both subjects, what is the probability that a student studies Physics given that they study Mathematics?

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

Solution

Using the formula P(Physics|Mathematics) = P(Physics and Mathematics) / P(Mathematics) = 0.4 / 0.7 = 0.571.

Correct Answer: A — 0.4

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Q. A student is selected from a class of 40 students, where 25 are girls and 15 are boys. What is the probability that the student is a girl given that the student is not a boy?

A. 1
B. 0
C. 1/2
D. 3/4

Solution

If the student is not a boy, they must be a girl. Therefore, the probability is 1.

Correct Answer: A — 1

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Q. A tower is 100 meters high. From a point on the ground, the angle of elevation to the top of the tower is 30 degrees. How far is the point from the base of the tower?

A. 100√3 meters
B. 50√3 meters
C. 200 meters
D. 150 meters

Solution

Distance = height / tan(angle) = 100 / (1/√3) = 100√3 meters.

Correct Answer: A — 100√3 meters

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Q. A tower is 40 meters high. From a point on the ground, the angle of elevation to the top of the tower is 60 degrees. How far is the point from the base of the tower?

A. 20√3 meters
B. 40 meters
C. 20 meters
D. 30 meters

Solution

Distance = height / tan(angle) = 40 / tan(60°) = 40 / √3 = 20√3 meters.

Correct Answer: A — 20√3 meters

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Q. A tower is 50 meters high. From a point on the ground, the angle of elevation to the top of the tower is 30 degrees. What is the distance from the point to the base of the tower?

A. 25√3 m
B. 50 m
C. 25 m
D. 50√3 m

Solution

Using tan(30°) = height/distance, we have distance = height/tan(30°) = 50/√3 = 25√3 m.

Correct Answer: A — 25√3 m

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Q. A tower is standing on a horizontal ground. The angle of elevation of the top of the tower from a point on the ground is 30 degrees. If the height of the tower is 50 meters, how far is the point from the base of the tower?

A. 50√3 m
B. 100 m
C. 50 m
D. 100√3 m

Solution

Using tan(30°) = height/distance, we have distance = height/tan(30°) = 50/(1/√3) = 50√3 m.

Correct Answer: A — 50√3 m

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Q. A tower is standing on a horizontal ground. The angle of elevation of the top of the tower from a point on the ground is 30 degrees. If the height of the tower is 10√3 m, how far is the point from the base of the tower?

A. 10 m
B. 5 m
C. 15 m
D. 20 m

Solution

Using tan(30°) = height/distance, we have 1/√3 = 10√3/distance. Therefore, distance = 10√3 * √3 = 30 m.

Correct Answer: A — 10 m

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Q. A tower is standing on a horizontal ground. The angle of elevation of the top of the tower from a point on the ground is 30 degrees. If the height of the tower is 10√3 meters, how far is the point from the base of the tower?

A. 10 m
B. 20 m
C. 30 m
D. 40 m

Solution

Using tan(30°) = height/distance, we have 1/√3 = 10√3/distance. Therefore, distance = 10√3 * √3 = 30 m.

Correct Answer: B — 20 m

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Q. A tree casts a shadow of 20 m when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

A. 10 m
B. 15 m
C. 20 m
D. 25 m

Solution

Using tan(30°) = height/20, we have 1/√3 = height/20. Therefore, height = 20/√3 ≈ 11.55 m.

Correct Answer: A — 10 m

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Q. A tree casts a shadow of 20 m when the angle of elevation of the sun is 45 degrees. What is the height of the tree?

A. 10 m
B. 20 m
C. 30 m
D. 40 m

Solution

Using tan(45°) = height/shadow, we have 1 = height/20. Therefore, height = 20 m.

Correct Answer: B — 20 m

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Q. A tree casts a shadow of 20 meters when the angle of elevation of the sun is 30 degrees. What is the height of the tree?

A. 20√3 meters
B. 10√3 meters
C. 30 meters
D. 40 meters

Solution

Height = shadow * tan(angle) = 20 * tan(30°) = 20 * (1/√3) = 20√3 meters.

Correct Answer: A — 20√3 meters

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Q. A tree is 15 meters tall. From a point on the ground, the angle of elevation to the top of the tree is 30 degrees. How far is the point from the base of the tree?

A. 15√3 meters
B. 30 meters
C. 45 meters
D. 10 meters

Solution

Distance = height / tan(angle) = 15 / tan(30°) = 15√3 meters.

Correct Answer: A — 15√3 meters

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Q. Calculate the area between the curves y = x and y = x^2 from x = 0 to x = 1.

A. 0.25
B. 0.5
C. 0.75
D. 1

Solution

The area is given by the integral from 0 to 1 of (x - x^2) dx. This evaluates to [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.

Correct Answer: B — 0.5

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Q. Calculate the area between the curves y = x^2 and y = 2x from x = 0 to x = 2.

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

Solution

The area is given by the integral from 0 to 2 of (2x - x^2) dx. This evaluates to [x^2 - x^3/3] from 0 to 2 = (4 - 8/3) = 4/3.

Correct Answer: A — 2

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Q. Calculate the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.

A. 4
B. 8
C. 6
D. 2

Solution

The area is given by the integral from 0 to 2 of (4 - x^2) dx. This evaluates to [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 16/3.

Correct Answer: A — 4

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Q. Calculate the area under the curve y = cos(x) from x = 0 to x = π/2.

A. 1
B. 0
C. π/2
D. 2

Solution

The area under the curve y = cos(x) from x = 0 to x = π/2 is given by ∫(from 0 to π/2) cos(x) dx = [sin(x)] from 0 to π/2 = 1 - 0 = 1.

Correct Answer: A — 1

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Q. Calculate the area under the curve y = x^2 + 2x from x = 0 to x = 2.

A. 4
B. 6
C. 8
D. 10

Solution

The area under the curve is given by ∫(from 0 to 2) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 2 = (8/3 + 4) = 20/3.

Correct Answer: B — 6

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Q. Calculate the area under the curve y = x^4 from x = 0 to x = 2.

A. 4
B. 8
C. 16
D. 32

Solution

The area under the curve y = x^4 from x = 0 to x = 2 is given by ∫(from 0 to 2) x^4 dx = [x^5/5] from 0 to 2 = (32/5) - 0 = 32/5.

Correct Answer: B — 8

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Q. Calculate 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) = d/dx(e^(2x)) = 2e^(2x).

Correct Answer: A — 2e^(2x)

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Q. Calculate the derivative of f(x) = x^2 * e^x.

A. (2x + x^2)e^x
B. 2xe^x
C. x^2e^x
D. (x^2 + 2x)e^x

Solution

Using the product rule, f'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.

Correct Answer: D — (x^2 + 2x)e^x

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Q. Calculate the determinant of the matrix [[1, 2], [3, 4]].

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

Solution

Determinant = (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).

A. 5
B. 10
C. 7
D. 8

Solution

The determinant is calculated as \( 2*4 - 3*1 = 8 - 3 = 5 \).

Correct Answer: A — 5

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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).

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

Solution

The determinant is calculated as (2*7) - (3*5) = 14 - 15 = -1.

Correct Answer: A — 1

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Q. Calculate the determinant of the matrix: | 1 1 1 | | 2 2 2 | | 3 3 3 |

A. 0
B. 3
C. 6
D. 9

Solution

The rows are linearly dependent, hence the determinant is 0.

Correct Answer: A — 0

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Q. Calculate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \).

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

Solution

The determinant is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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