SPSC 25 YEARS PAST PAPERS 2001 TO 2026
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چار آپشن میں سے کسی ایک پر کلک کرنے سے جواب سرخ ہو جائے گا۔
The derivation of g(x) be equal to 0
The functions, f(x) and g(x) be continous in [a,b]
The functions f(x) and g(x) be derivable in (a, b)
None of these
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Explanation
Cauchy's Mean Value Theorem states that if two functions f(x) and g(x) are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and g'(x) ≠ 0 for any x in (a, b), then there exists a point c in (a, b) such that: (f'(c) / g'(c)) = (f(b) - f(a)) / (g(b) - g(a))
The necessary conditions for Cauchy's Mean Value Theorem are :
1. The functions f(x) and g(x) are continuous on [a, b]. 2. The functions f(x) and g(x) are differentiable on (a, b). 3. g'(x) ≠ 0 for any x in (a, b).
Taylor Series of a function
Leibniz theorem
Mean value theorem
None of these
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Explanation
Rolle's Theorem is a special case of the Mean Value Theorem (MVT) where the function has equal values at two endpoints. It ensuring the derivative is zero at some point in the interval. It is derived from MVT by adding the condition f ( a ) = f ( b ) f(a) = f(b)
Positive Integer
Negative Integer
Rational Number
None of these
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Explanation
Leibniz theorem is also known as the Leibniz rule . It is used for finding the n n derivative of a product of two functions . It is defined only when n n is a positive integer .
1/2
Undefined
4
None of these
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Explanation
To find the third derivative of f(x) = ln(1-x)/e^x, we'll first find the first, second, and third derivatives.
Using the quotient rule and chain rule, we can find the derivatives. After calculating, we get:
f'''(0) = 1/2
So, the third derivative at x = 0 is 1/2.
0
1
1/2
None of these
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Explanation
Let the quadratic polynomial be f(x) = ax^2 + bx + c.
Given that f'(1) = 1 and f''(1) = 2.
f'(x) = 2ax + b
f''(x) = 2a
f'(1) = 1 => 2a + b = 1
f''(1) = 2 => 2a = 2 => a = 1
Substituting a = 1 in the first equation, we get:
2 + b = 1 => b = -1
Now, f(x) = x^2 - x + c
f(1) = 1^2 - 1 + c = c
f(0) = 0^2 - 0 + c = c
f(1) - f(0) = c - c = 0
Polite
Delight
Hatred
None of these
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Explanation
ANGUISH means تکلیف Delight means خوشی
Merciful
Grief
Pity
None of these
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Explanation
Ruthless means بے رحم
Merciful means مہربان
Whim
Passion
Disinclination
Both B and C
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Explanation
Velleity means خواہش کی کمی
Passion شوق
Passion is a strong and intense emotion, desire, or enthusiasm for something,
The opposite of “Velleity,” which implies a weak desire.
In rage
From nowhere
Campout fast
None of these
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Explanation
" Out of the blue " means something happening unexpectedly or suddenly . Example : "The news of his promotion came out of the blue ."
Will not cost
Will not coasting
Had not cost
None of these
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Explanation
The correct sentence is: " This watch will not cost very much ." اس گھڑی پر بہت زیادہ لاگت نہیں آئے گی۔ " Will not cost " is correct because it refers to a future event .
