Mathematics Syllabus (JEE Main)
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
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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 60 meters high. From a point on the ground, the angle of elevation to the top of the tower is 45 degrees. How far is the point from the base of the tower?
A.
30 meters
B.
60 meters
C.
45 meters
D.
75 meters
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Solution
Distance = height / tan(angle) = 60 / 1 = 60 meters.
Correct Answer: B — 60 meters
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Q. A tower is 80 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.
40 m
B.
80 m
C.
20 m
D.
60 m
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Solution
Distance = height / tan(angle) = 80 / √3 = 40 m.
Correct Answer: A — 40 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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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.
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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)
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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
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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]].
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Solution
Determinant = (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. Calculate the determinant of the matrix \( B = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).
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Solution
The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).
Correct Answer: D — 10
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Q. Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{pmatrix} \).
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Solution
The determinant is calculated as \( 2(0*2 - 1*1) - 1(1*2 - 3*1) + 3(1*1 - 3*0) = 0 \).
Correct Answer: A — -1
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Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
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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 \( \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).
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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 |
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Solution
The rows are linearly dependent, hence the determinant is 0.
Correct Answer: A — 0
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Q. Calculate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \).
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Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Calculate the determinant \( \begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix} \)
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Solution
The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).
Correct Answer: A — 1
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Q. Calculate the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)
A.
ad - bc
B.
ab + cd
C.
ac - bd
D.
bc - ad
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Solution
The determinant is calculated as \( ad - bc \).
Correct Answer: A — ad - bc
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Q. Calculate the determinant | 1 0 0 | | 0 1 0 | | 0 0 1 |.
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Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. Calculate the determinant | 2 3 | | 4 5 | + | 1 1 | | 1 1 |.
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Solution
The first determinant is -2 and the second is 0, so the total is -2 + 0 = -2.
Correct Answer: B — 1
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Q. Calculate the determinant: | 2 3 1 | | 1 0 2 | | 0 1 3 |.
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Solution
The determinant evaluates to 0 as the rows are linearly dependent.
Correct Answer: A — -1
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