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علمِ احصا
10 سوالات: نیٹ ریاضی، جوابات اور وضاحت کے ساتھ۔
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What is d/dx of x^2 ln x?
- 1 2x ln x + x
- 2 2x / x
- 3 x ln x
- 4 2x ln x
جواب دیکھیں
درست جواب: 2x ln x + x
By the product rule, the derivative is 2x ln x + x^2 x (1/x) = 2x ln x + x.
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What is the integral of 1/x dx for x > 0?
- 1 ln x + C
- 2 -1/x^2 + C
- 3 x ln x + C
- 4 1/x^2 + C
جواب دیکھیں
درست جواب: ln x + C
The derivative of ln x is 1/x, so the integral of 1/x is ln x + C.
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What is the value of the integral of sin x from 0 to pi?
- 1 0
- 2 1
- 3 2
- 4 pi
جواب دیکھیں
درست جواب: 2
The antiderivative is -cos x. So the value is -cos(pi) + cos(0) = 1 + 1 = 2.
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What is the limit of (3x^2 + 1) / (x^2 - 5) as x tends to infinity?
- 1 0
- 2 3
- 3 -1/5
- 4 Infinity
جواب دیکھیں
درست جواب: 3
Divide top and bottom by x^2. The terms with x in the denominator vanish, leaving 3 / 1 = 3.
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What is d/dx of tan x?
- 1 cot x
- 2 sec x tan x
- 3 sec^2 x
- 4 -cosec^2 x
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What is the maximum value of f(x) = 3 + 4x - x^2?
- 1 3
- 2 7
- 3 4
- 4 2
جواب دیکھیں
درست جواب: 7
f'(x) = 4 - 2x = 0 gives x = 2, and f(2) = 3 + 8 - 4 = 7. The parabola opens downward, so this is a maximum.
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What is the area under the line y = x from x = 0 to x = 4?
- 1 16
- 2 4
- 3 8
- 4 12
جواب دیکھیں
درست جواب: 8
The area is the integral of x from 0 to 4, which is 16/2 = 8. It is also the area of a triangle with base 4 and height 4.
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What is d/dx of sin(x^2)?
- 1 cos(x^2)
- 2 2x cos(x^2)
- 3 2 sin x cos x
- 4 -2x cos(x^2)
جواب دیکھیں
درست جواب: 2x cos(x^2)
By the chain rule, the derivative is cos(x^2) multiplied by the derivative of x^2, which is 2x.
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What is the integral of x e^x dx?
- 1 e^x (x + 1) + C
- 2 x e^x + C
- 3 e^x (x - 1) + C
- 4 x^2 e^x / 2 + C
جواب دیکھیں
درست جواب: e^x (x - 1) + C
Integration by parts gives x e^x - e^x + C, which equals e^x (x - 1) + C.
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What is d/dx of sqrt(x^2 + 1)?
- 1 1 / (2 sqrt(x^2 + 1))
- 2 x / sqrt(x^2 + 1)
- 3 2x sqrt(x^2 + 1)
- 4 sqrt(x^2 + 1) / x
جواب دیکھیں
درست جواب: x / sqrt(x^2 + 1)
By the chain rule, the derivative is 1/2 (x^2 + 1)^(-1/2) x 2x = x / sqrt(x^2 + 1).