Mathematics Syllabus (JEE Main)
Q. Calculate the determinant \( \begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix} \)
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
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 |.
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 |.
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 |.
Solution
The determinant evaluates to 0 as the rows are linearly dependent.
Correct Answer: A — -1
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Q. Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.
Solution
Using the determinant formula, we find that the determinant evaluates to 0.
Correct Answer: A — -1
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Q. Calculate the integral ∫ (x^2 + 2x + 1) dx.
-
A.
(1/3)x^3 + x^2 + x + C
-
B.
(1/3)x^3 + x^2 + C
-
C.
(1/3)x^3 + 2x^2 + C
-
D.
(1/3)x^3 + x^2 + x
Solution
The integral of x^2 is (1/3)x^3, the integral of 2x is x^2, and the integral of 1 is x. Thus, ∫ (x^2 + 2x + 1) dx = (1/3)x^3 + x^2 + x + C.
Correct Answer: A — (1/3)x^3 + x^2 + x + C
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Q. Calculate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
-
A.
(1/3)x^3 + x^2 + C
-
B.
x^2 + 2x + C
-
C.
x^2 + x + C
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D.
(1/3)x^3 + (1/2)x^2 + C
Solution
The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.
Correct Answer: A — (1/3)x^3 + x^2 + C
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Q. Calculate the integral ∫ (x^3 - 4x) dx.
-
A.
(1/4)x^4 - 2x^2 + C
-
B.
(1/4)x^4 - 2x^2
-
C.
(1/4)x^4 - 4x^2 + C
-
D.
(1/4)x^4 - 2x^2 + 1
Solution
The integral of x^3 is (1/4)x^4 and the integral of -4x is -2x^2. Therefore, ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
Correct Answer: A — (1/4)x^4 - 2x^2 + C
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Q. Calculate the integral ∫ cos^2(x) dx.
-
A.
(1/2)x + (1/4)sin(2x) + C
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B.
(1/2)x + C
-
C.
(1/2)x - (1/4)sin(2x) + C
-
D.
(1/2)x + (1/2)sin(2x) + C
Solution
Using the identity cos^2(x) = (1 + cos(2x))/2, we find that ∫ cos^2(x) dx = (1/2)x + (1/4)sin(2x) + C.
Correct Answer: A — (1/2)x + (1/4)sin(2x) + C
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Q. Calculate the integral ∫ from 0 to π of sin(x) dx.
Solution
The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.
Correct Answer: C — 2
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Q. Calculate the interquartile range (IQR) for the data set: 1, 3, 7, 8, 9, 10.
Solution
Q1 = 3, Q3 = 9; IQR = Q3 - Q1 = 9 - 3 = 6.
Correct Answer: A — 4
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Q. Calculate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
-
A.
0
-
B.
1/2
-
C.
1
-
D.
Infinity
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1.
Correct Answer: B — 1/2
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Q. Calculate the limit: lim (x -> 0) (e^x - 1)/x
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
Using the definition of the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = e^0 = 1.
Correct Answer: B — 1
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Q. Calculate the limit: lim (x -> 0) (tan(3x)/x)
-
A.
3
-
B.
1
-
C.
0
-
D.
Infinity
Solution
Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 3, so the limit is 3.
Correct Answer: A — 3
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer: C — 2
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
-
A.
0
-
B.
1
-
C.
2
-
D.
Undefined
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2 = (x + 1)/(x - 1). Thus, lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer: C — 2
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Q. Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
-
A.
0
-
B.
1
-
C.
3
-
D.
Undefined
Solution
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim (x -> 1) (x^2 + x + 1) = 3.
Correct Answer: C — 3
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Q. Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)
-
A.
0
-
B.
2
-
C.
4
-
D.
Undefined
Solution
Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.
Correct Answer: D — Undefined
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Q. Calculate the mean absolute deviation for the data set: 1, 2, 3, 4, 5.
Solution
Mean = 3. Mean Absolute Deviation = (|1-3| + |2-3| + |3-3| + |4-3| + |5-3|)/5 = (2 + 1 + 0 + 1 + 2)/5 = 1.5.
Correct Answer: B — 1.5
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Q. Calculate the mean of the following data: 5, 10, 15, 20.
-
A.
10
-
B.
12.5
-
C.
15
-
D.
17.5
Solution
Mean = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.
Correct Answer: B — 12.5
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Q. Calculate the mean of the following numbers: 4, 8, 12, 16, 20.
Solution
Mean = (4 + 8 + 12 + 16 + 20) / 5 = 60 / 5 = 12.
Correct Answer: C — 14
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Q. Calculate the range of the data set: 12, 15, 22, 30, 5.
Solution
Range = Maximum - Minimum = 30 - 5 = 25.
Correct Answer: A — 25
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Q. Calculate the range of the data set: 4, 8, 15, 16, 23, 42.
Solution
Range = Maximum - Minimum = 42 - 4 = 38.
Correct Answer: A — 38
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Q. Calculate the range of the data set: 8, 12, 15, 20, 22.
Solution
Range = Max - Min = 22 - 8 = 14.
Correct Answer: A — 10
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Q. Calculate the range of the following data set: 12, 15, 20, 22, 30.
Solution
Range = Maximum - Minimum = 30 - 12 = 18.
Correct Answer: C — 18
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Q. Calculate the range of the following data set: 15, 22, 8, 19, 30.
Solution
Range = max - min = 30 - 8 = 22.
Correct Answer: D — 30
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Q. Calculate the range of the following data set: 4, 8, 15, 16, 23, 42.
Solution
Range = Maximum - Minimum = 42 - 4 = 38.
Correct Answer: A — 38
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Q. Calculate the range of the following data set: 8, 12, 15, 7, 10.
Solution
Range = Maximum - Minimum = 15 - 7 = 8.
Correct Answer: A — 5
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Q. Calculate the scalar product of A = (1, 1, 1) and B = (2, 2, 2).
Solution
A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: D — 6
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