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پی ڈی ایف میں ڈنلوڈ کرنے کے لیےاس لنک پر کلک کریں۔
Basic Math PDF File
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چار آپشن میں سے کسی ایک پر کلک کرنے سے جواب سرخ ہو جائے گا۔
مندرجہ ذیل میں سے کون سا صوابدیدی استحکام کے خاتمے کے ذریعہ تفریق مساوات کی تشکیل کے سلسلے میں سچ ہے؟
If 'n' arbitrary constant is present, the given equation should be differentiated 'n' number of times
Elimination of the arbitrary constant by replacing it using derivative
The given equation should be differentiated with respect to independent variable
None of these
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Explanation
To form a differential equation from a given function containing arbitrary constants:
If the function has 'n' arbitrary constants , differentiate the equation 'n' times with respect to the independent variable. This process removes the arbitrary constants , leaving a differential equation.
f(x) = 3x^2 + 4x + 1
f(x) = x^3 + 2x^2 + 6x
f(x) = x^3 + x^2y + 4y^2
None of these
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Explanation
Implicit functions involve both dependent (y) and independent (x) variables in the same equation without explicitly solving for y. Explicit functions directly express y in terms of x (e.g., f(x) = 3x² + 4x + 1).
13/2
7/2
3/2
None of these
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Explanation
To find the value of c that satisfies the Mean Value Theorem, we need to find the derivative of the function f(x) = x^2 + 2x + 1.
f'(x) = 2x + 2
The Mean Value Theorem states that there exists a point c in the interval [1, 2] such that:
f'(c) = (f(2) - f(1)) / (2 - 1)
First, let's calculate f(2) and f(1):
f(2) = 2^2 + 2(2) + 1 = 9
f(1) = 1^2 + 2(1) + 1 = 4
Now, we can calculate the slope:
(f(2) - f(1)) / (2 - 1) = (9 - 4) / 1 = 5
Now, we set f'(c) equal to 5:
2c + 2 = 5
Solving for c, we get:
2c = 3
c = 3/2
Since c = 3/2 is between 1 and 2, it satisfies the Mean Value Theorem.
Homogeneous equation
Orthogonal method
Variable separable
None of these
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Explanation
Homogeneous equations and Variable separable methods are standard techniques for solving differential equations. Orthogonal method is not a commonly recognized standard method for solving differential equations.
Does not exist
1
0
None of these
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Explanation
The given limit is lim (x,y) → (0,0) sin(y)/x .As x x approaches 0 , the denominator becomes 0 , causing an undefined form . Along different paths (e.g., setting y = 0 y = 0 or x = 0 x = 0 ), the limit gives different results, meaning the limit does not exist .
Bisection method
Taylor's method
Newton-Raphson method
None of these
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Explanation
The half-interval method is a root-finding technique that repeatedly divides an interval in half to locate a root. It is commonly known as the Bisection Method in numerical analysis.
0
2
1/2
None of these
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Explanation
The limit lim x→∞ 2/x is equal to 0 .
As x approaches infinity , the denominator x becomes very large, and the numerator 2 remains constant .
Therefore, the ratio 2/x approaches 0 .
Michel Rolle
Augustin Louis Cauchy
Parameshvara
None of these
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Explanation
Mean Value Theorem (MVT) was stated and rigorously proved by Augustin Louis Cauchy in the 19th century. It generalizes Rolle's Theorem . It is fundamental in calculus for understanding differentiable functions.
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) .