A box contains 5 red balls and 7 blue balls. If two balls are drawn at random, find the probability that both balls are red.
Define a first-order differential equation. Solve the differential equation dy over dx equal to 3x squared.
Explain the concept of indefinite integration and how it differs from definite integration. Integrate the function f of x equal to x squared.
Explain the concept of maxima and minima of a function. Find the maximum and minimum values of the function f of x equal to 3x squared minus 12x plus 7.
State the conditions under which a function is continuous at a point. Determine whether the function f of x equal to x plus 1 is continuous at x equal to 0.
Define an equivalence relation. Provide an example of an equivalence relation on the set of real numbers.
Define a square matrix. How do you find the determinant of a 3×3 matrix? Explain with an example.
State Hess’s law and explain its application in determining enthalpy changes. Provide an example of its use in calculating enthalpy of formation.
Explain Boyle’s law and its significance. A gas occupies a volume of 4 liters at a pressure of 2 atm. Calculate its volume when the pressure is reduced to 1 atm, assuming temperature remains constant.
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