Mathematics Syllabus (JEE Main)
Q. Calculate the scalar product of the vectors A = (1, 2, 3) and B = (4, 5, 6).
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Solution
A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
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Q. Calculate the scalar product of the vectors A = (4, -1, 2) and B = (2, 3, 1).
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Solution
A · B = 4*2 + (-1)*3 + 2*1 = 8 - 3 + 2 = 7.
Correct Answer: A — 10
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Q. Calculate the scalar product of the vectors K = (0, 1, 2) and L = (3, 4, 5).
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Solution
K · L = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.
Correct Answer: A — 10
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Q. Calculate the variance of the data set {2, 4, 4, 4, 5, 5, 7, 9}.
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Solution
Mean = 5, Variance = [(2-5)² + (4-5)² + (4-5)² + (4-5)² + (5-5)² + (5-5)² + (7-5)² + (9-5)²] / 8 = 4.
Correct Answer: B — 6
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Q. Calculate the variance of the data set {4, 8, 6, 5, 3}.
A.
2.5
B.
3.2
C.
1.5
D.
4.0
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Solution
Mean = (4+8+6+5+3)/5 = 5.2. Variance = [(4-5.2)² + (8-5.2)² + (6-5.2)² + (5-5.2)² + (3-5.2)²]/5 = 2.5.
Correct Answer: A — 2.5
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Q. Calculate the vector product of A = (3, 2, 1) and B = (1, 0, 2).
A.
(4, 5, -2)
B.
(2, 5, -3)
C.
(2, -5, 3)
D.
(5, -2, 3)
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Solution
A × B = |i j k|\n|3 2 1|\n|1 0 2| = (4, 5, -2)
Correct Answer: A — (4, 5, -2)
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Q. Calculate the weighted mean of the following data: (10, 2), (20, 3), (30, 5).
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Solution
Weighted Mean = (10*2 + 20*3 + 30*5) / (2 + 3 + 5) = 220 / 10 = 22.
Correct Answer: B — 25
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Q. Calculate the weighted mean of the following data: (2, 3), (4, 5), (6, 7) where the first number is the weight.
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Solution
Weighted mean = (2*3 + 4*5 + 6*7) / (2 + 4 + 6) = (6 + 20 + 42) / 12 = 68 / 12 = 5.67.
Correct Answer: C — 5.5
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Q. Calculate the weighted mean of the numbers 10, 20, and 30 with weights 1, 2, and 3 respectively.
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Solution
Weighted mean = (10*1 + 20*2 + 30*3) / (1 + 2 + 3) = 140 / 6 = 23.33.
Correct Answer: B — 25
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Q. Calculate the weighted mean of the scores 70, 80, and 90 with weights 1, 2, and 3 respectively.
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Solution
Weighted mean = (70*1 + 80*2 + 90*3) / (1 + 2 + 3) = 85.
Correct Answer: B — 85
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Q. Calculate ∫ from 0 to 1 of (1 - x^2) dx.
A.
1/3
B.
1/2
C.
2/3
D.
1
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Solution
The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.
Correct Answer: B — 1/2
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Q. Calculate ∫ from 0 to 1 of (1/x) dx.
A.
0
B.
1
C.
ln(1)
D.
ln(2)
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Solution
The integral evaluates to [ln(x)] from 0 to 1 = ln(1) - ln(0) which diverges.
Correct Answer: C — ln(1)
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Q. Calculate ∫ from 0 to 1 of (2x^2 + 3x + 1) dx.
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Solution
The integral evaluates to [2x^3/3 + (3/2)x^2 + x] from 0 to 1 = (2/3 + 3/2 + 1) = 3.
Correct Answer: C — 3
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Q. Calculate ∫ from 0 to 1 of (4x^3 - 2x^2 + x) dx.
A.
1/4
B.
1/3
C.
1/2
D.
1
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Solution
The integral evaluates to [x^4 - (2/3)x^3 + (1/2)x^2] from 0 to 1 = (1 - 2/3 + 1/2) = 1/6.
Correct Answer: C — 1/2
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Q. Calculate ∫ from 0 to 1 of (4x^3 - 3x^2 + 2x - 1) dx.
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Solution
The integral evaluates to [x^4 - x^3 + x^2 - x] from 0 to 1 = (1 - 1 + 1 - 1) = 0.
Correct Answer: B — 1
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Q. Calculate ∫ from 0 to 1 of (4x^3 - 4x^2 + 1) dx.
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Solution
The integral evaluates to [x^4 - (4/3)x^3 + x] from 0 to 1 = 1 - (4/3) + 1 = 2/3.
Correct Answer: A — 1
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Q. Calculate ∫ from 0 to 1 of (6x^2 - 4x + 1) dx.
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Solution
The integral evaluates to [2x^3 - 2x^2 + x] from 0 to 1 = (2 - 2 + 1) = 1.
Correct Answer: A — 1
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Q. Calculate ∫ from 0 to 1 of (x^2 * e^x) dx.
A.
1/e
B.
2/e
C.
3/e
D.
4/e
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Solution
Using integration by parts, the integral evaluates to (2/e - 1/e) = 1/e.
Correct Answer: B — 2/e
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Q. Calculate ∫ from 0 to 1 of (x^2 + 1/x^2) dx.
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Solution
The integral evaluates to [x^3/3 - 1/x] from 0 to 1 = (1/3 - 1) = -2/3.
Correct Answer: C — 3
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Q. Calculate ∫ from 0 to 1 of (x^2 + 2x + 1) dx.
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Solution
The integral evaluates to [x^3/3 + x^2 + x] from 0 to 1 = (1/3 + 1 + 1) - (0) = 7/3.
Correct Answer: C — 3
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Q. Calculate ∫ from 0 to 1 of (x^2 + 4x + 4) dx.
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Solution
The integral evaluates to [x^3/3 + 2x^2 + 4x] from 0 to 1 = (1/3 + 2 + 4) = 25/3.
Correct Answer: C — 5
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Q. Calculate ∫ from 0 to 1 of (x^4 - 2x^2 + 1) dx.
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Solution
The integral evaluates to [x^5/5 - 2x^3/3 + x] from 0 to 1 = (1/5 - 2/3 + 1) = (15/15 - 10/15 + 3/15) = 8/15.
Correct Answer: D — 2/3
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Q. Calculate ∫ from 0 to 1 of (x^4 - 2x^3 + x^2) dx.
A.
0
B.
1/5
C.
1/3
D.
1/2
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Solution
The integral evaluates to [x^5/5 - (2/4)x^4 + (1/3)x^3] from 0 to 1 = (1/5 - 1/2 + 1/3) = 1/30.
Correct Answer: B — 1/5
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Q. Calculate ∫ from 0 to 2 of (x^3 - 3x^2 + 4) dx.
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Solution
The integral evaluates to [x^4/4 - x^3 + 4x] from 0 to 2 = (4 - 8 + 8) - 0 = 4.
Correct Answer: C — 6
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Q. Calculate ∫ from 0 to π of sin(x) dx.
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Solution
The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.
Correct Answer: C — 2
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Q. Calculate ∫ from 0 to π/2 of sin(x) cos(x) dx.
A.
1/2
B.
1
C.
π/4
D.
π/2
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Solution
Using the identity sin(2x) = 2sin(x)cos(x), the integral becomes 1/2 ∫ from 0 to π/2 of sin(2x) dx = 1/2 [-1/2 cos(2x)] from 0 to π/2 = 1/2 [0 - (-1/2)] = 1/4.
Correct Answer: A — 1/2
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Q. Calculate ∫ from 0 to π/2 of sin^2(x) dx.
A.
π/4
B.
π/2
C.
π/3
D.
π/6
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Solution
Using the identity sin^2(x) = (1 - cos(2x))/2, the integral evaluates to π/4.
Correct Answer: A — π/4
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Q. Calculate ∫ from 1 to 3 of (2x + 1) dx.
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Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer: B — 6
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Q. Calculate ∫_0^1 (4x^3 - 3x^2 + 2) dx.
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Solution
∫_0^1 (4x^3 - 3x^2 + 2) dx = [x^4 - x^3 + 2x] from 0 to 1 = (1 - 1 + 2) - (0) = 2.
Correct Answer: B — 2
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Q. Calculate ∫_0^1 (e^x) dx.
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Solution
∫_0^1 e^x dx = [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
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