MHT-CET
Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)
-
A.
54
-
B.
81
-
C.
108
-
D.
729
Solution
The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.
Correct Answer: C — 108
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Q. Calculate the derivative of f(x) = 5x^5. (2016)
-
A.
25x^4
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B.
5x^4
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C.
20x^4
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D.
10x^4
Solution
The derivative f'(x) = d/dx(5x^5) = 25x^4.
Correct Answer: A — 25x^4
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Q. Calculate the derivative of f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023
-
A.
e^x(2x + 1)
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B.
e^x(2x - 1)
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C.
2xe^x
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D.
x^2e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x(2x + 1).
Correct Answer: A — e^x(2x + 1)
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Q. Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)
Solution
Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = 2(0 - 2) - 3(0 - 8) + 1(1) = -4 + 24 + 1 = 21.
Correct Answer: A — -10
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.
Correct Answer: A — 1
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 2*2) + 1(1*1 - 0*2) = 3(0 - 2) - 2(3 - 4) + 1(1) = -6 + 2 + 1 = -3.
Correct Answer: A — 1
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Q. Calculate the determinant of D = [[4, 2], [1, 3]]. (2020)
Solution
Determinant of D = (4*3) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)
Solution
Using the determinant formula, det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = -24 + 40 - 15 = 1.
Correct Answer: A — -14
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Q. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)
Solution
The determinant of G is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [1, 0, 1]]. (2023)
Solution
Det(H) = 1(1*1 - 3*0) - 2(0*1 - 3*1) + 1(0*0 - 1*1) = 1(1) - 2(-3) + 1(-1) = 1 + 6 - 1 = 6.
Correct Answer: A — -1
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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [2, 1, 0]]. (2020)
Solution
Det(H) = 1(1*0 - 3*1) - 2(0*0 - 3*2) + 1(0*1 - 1*2) = 1(0 - 3) - 2(0 - 6) + 1(0 - 2) = -3 + 12 - 2 = 7.
Correct Answer: A — -5
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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)
Solution
Det(J) = 1(1*1 - 2*0) - 2(0*1 - 1*1) + 1(0*0 - 1*1) = 1(1) - 2(-1) + 1(-1) = 1 + 2 - 1 = 2.
Correct Answer: C — 2
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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)
Solution
The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).
Correct Answer: A — -2
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Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)
Solution
The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).
Correct Answer: A — 1
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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)
Solution
Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.
Correct Answer: B — √14
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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)
Solution
Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.
Correct Answer: B — 5/2
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Q. Calculate the perimeter of a square with side length 4 cm. (2015)
-
A.
16 cm
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B.
12 cm
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C.
8 cm
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D.
20 cm
Solution
Perimeter = 4 × side = 4 × 4 cm = 16 cm.
Correct Answer: A — 16 cm
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Q. Calculate the perimeter of a square with side length 6 cm. (2015)
-
A.
24 cm
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B.
20 cm
-
C.
18 cm
-
D.
30 cm
Solution
Perimeter = 4 × side = 4 × 6 = 24 cm.
Correct Answer: A — 24 cm
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Q. Calculate the pH of a 0.01 M solution of NaHCO3. (2023)
-
A.
8.3
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B.
9.0
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C.
7.5
-
D.
8.0
Solution
NaHCO3 is a weak base. The pH can be calculated using the formula pH = 7 + 0.5(pKa - log[C]). pKa of HCO3- is about 10.3, so pH ≈ 8.3.
Correct Answer: A — 8.3
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Q. Calculate the pH of a 0.05 M NH4Cl solution (Kb for NH3 = 1.8 x 10^-5).
-
A.
4.75
-
B.
5.25
-
C.
5.75
-
D.
6.25
Solution
Using the formula for weak bases, pH = 14 - 0.5(pKb - logC) = 14 - 0.5(4.74 - log(0.05)) = 5.25.
Correct Answer: B — 5.25
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Q. Calculate the pH of a 0.1 M NaOH solution.
Solution
pOH = -log[OH-] = -log(0.1) = 1, thus pH = 14 - pOH = 14 - 1 = 13.
Correct Answer: C — 14
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Q. Calculate the pH of a 0.2 M solution of KOH.
Solution
pOH = -log(0.2) = 0.7, thus pH = 14 - 0.7 = 13.3.
Correct Answer: B — 13
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Q. Calculate the term independent of x in the expansion of (2x - 3)^5.
-
A.
-243
-
B.
0
-
C.
243
-
D.
81
Solution
The term independent of x is C(5,5) * (2x)^0 * (-3)^5 = 1 * 1 * (-243) = -243.
Correct Answer: A — -243
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Q. Calculate the term independent of x in the expansion of (x/2 - 3)^6.
-
A.
729
-
B.
729/64
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C.
729/32
-
D.
729/16
Solution
The term independent of x occurs when k = 3, which gives C(6,3) * (x/2)^3 * (-3)^3 = 20 * (1/8) * (-27) = -67.5.
Correct Answer: B — 729/64
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Q. Calculate the value of 12 × 3 - 4 × 2. (2023) 2023
Solution
12 × 3 = 36 and 4 × 2 = 8, so 36 - 8 = 28.
Correct Answer: A — 28
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Q. Calculate the value of 12 × 3 - 4. (2021)
Solution
12 × 3 = 36, then 36 - 4 = 32.
Correct Answer: B — 28
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Q. Calculate the value of 12 × 8 - 10. (2021)
Solution
12 × 8 = 96, then 96 - 10 = 86.
Correct Answer: B — 82
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Q. Calculate the value of 12 × 8 - 24. (2021)
Solution
12 × 8 = 96, then 96 - 24 = 72.
Correct Answer: A — 72
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Q. Calculate the value of 5! (5 factorial). (2020)
-
A.
120
-
B.
100
-
C.
60
-
D.
24
Solution
5! = 5 × 4 × 3 × 2 × 1 = 120.
Correct Answer: A — 120
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