Explanation

Frugal means کفایت شعار یا کم خرچ

Rich means امیر

Miserable means غمگین

Happy means خوش

Extravagant means فضول خرچ

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Explanation

Elated means پرجوش

Thrilled means پرجوش

Overjoyed بے حد خوشی ہوئی۔

Explanation

Meticulous means محتاط

Conscientious means باضمیر

Precise means عین مطابق

Explanation

Fragile means کمزور

Robust means مضبوط

Explanation

The integrating factor for a first-order linear differential equation dy/dx + P(x)y = Q(x) is exp (∫P(x)dx).

This is a well-known result in differential equations.

Explanation

The notation 7 ∈ Sr (5) indicates that point 7 belongs to the circle Sr (5), which means it is on the circle.

Since point 7 is on the circle, the distance between the center (point 5) and point 7 is equal to the radius r.

The correct answer is d(5,7) < r.

Explanation

In a group (G, *), the binary operation * is defined in such a way that the result of combining any two elements is always an element in the group. Therefore, if a, b ∈ G, then a * b ∈ G.

Explanation

The Differential Equation ydx-2xdy=0 represents the family of 'Parabola'.

Explanation

• In a group (G, *), the binary operation * is defined such that for all a, b ∈ G, a * b ∈ G.
• This means that the result of combining any two elements of the group using the group operation is always an element of the group.

Explanation

• A cycle of length 2 is called a transposition.
• A transposition is a permutation that swaps two elements, leaving the others unchanged.
• For example, the permutation (1 2) is a transposition that swaps the elements 1 and 2.