1. We are given the function $$f(x) = \int_0^x (\cos^{23} y)(2 + \sin^{23} y) \, dy$$ and asked on which intervals $$f$$ is increasing. 2. Since $$f(x)$$ is defined as an integral from 0 to $$x$$ of the function $$g(y) = (\cos^{23} y)(2 + \sin^{23} y)$$, by the Fundamental Theorem of Calculus, the derivative $$f'(x) = g(x) = (\cos^{23} x)(2 + \sin^{23} x)$$. 3. To determine where $$f$$ is increasing, we need where $$f'(x) > 0$$, i.e., where $$g(x) > 0$$. 4. Note that: - $$\cos^{23} x$$ has the same sign as $$\cos x$$ because 23 is odd. - $$2 + \sin^{23} x$$ is always positive because $$\sin^{23} x$$ ranges between $$-1$$ and $$1$$, so $$2 + \sin^{23} x \geq 1$$. 5. Therefore, the sign of $$g(x)$$ depends solely on the sign of $$\cos x$$. 6. $$\cos x > 0$$ on intervals where $$x \in (-\pi/2, \pi/2)$$. 7. Checking the multiple-choice options, the only interval fully contained within $$(-\pi/2, \pi/2)$$ is option (B), which is $$( -\pi/2, \pi/2 )$$. 8. For question 30, the curve is given by: $$c(t) = (\cos^{3} t, \sin^{3} t)$$ for $$t \in [0, 2\pi]$$. 9. This curve is a special case known as an astroid, which looks like a diamond-shaped figure with vertices at $$(\pm 1, 0)$$ and $$(0, \pm 1)$$. 10. Among the options, (A) describes a diamond shape with vertices at exactly these points. 11. For question 31, we roll 4 dice. 12. Total outcomes: $$6^4 = 1296$$. 13. The probability that all four are distinct is: $$\frac{6}{6} \times \frac{5}{6} \times \frac{4}{6} \times \frac{3}{6} = \frac{6 \times 5 \times 4 \times 3}{6^4} = \frac{360}{1296} = \frac{5}{18}$$. 14. Probability at least two are the same = $$1 - \text{probability all distinct} = 1 - \frac{5}{18} = \frac{13}{18}$$. 15. Answer (B) $$\frac{13}{18}$$. 16. Question 32: Determine which set is NOT a group under multiplication. 17. Key points for groups of complex numbers under multiplication: - The set must be closed, contain identity (1), and every element must have an inverse. 18. (A) $$\{a+bi: a,b \text{ positive rationals}\}$$: This set is not closed under multiplication because product of positives is positive, but zero not included because $$a,b$$ are positive, so zero not included. But the identity $$1+0i$$ is not included since $$b=0$$ is not positive. So no identity element. Hence, (A) is NOT a group. 19. (B) All nonzero complex numbers: group. 20. (C) Integers $$a,b$$ with $$a^2+b^2=1$$: possible values $$(\pm 1,0), (0, \pm 1)$$ - four elements, form a group (unit complex integers). 21. (D) Rational $$a,b$$ with $$a^2 + b^2 =1$$: group of rational points on unit circle, closed under multiplication. 22. (E) Real $$a,b$$ with $$a^2+b^2=1$$: unit circle, known group. 23. Answer: (A) is NOT a group under multiplication. 24. Question 33: Spherical tank radius $$r=5$$ meters, water depth $$h=2$$ meters, depth decreasing at $$\frac{dh}{dt} = -\frac{1}{3}$$ m/s. 25. Volume $$V$$ of spherical cap of water depth $$h$$ is: $$V = \pi h^2 \left(r - \frac{h}{3}\right)$$. 26. Differentiate $$V$$ with respect to $$t$$: $$\frac{dV}{dt} = \pi \left(2h \frac{dh}{dt} (r - \frac{h}{3}) + h^{2} \left(- \frac{1}{3} \right) \frac{dh}{dt} \right)$$. 27. Substitute $$r=5$$, $$h=2$$, $$\frac{dh}{dt} = -\frac{1}{3}$$: $$\frac{dV}{dt} = \pi \left[ 2 \times 2 \times \left(5 - \frac{2}{3} \right) \times \left(-\frac{1}{3}\right) + 2^2 \times \left(-\frac{1}{3}\right) \times \left(- \frac{1}{3} \right) \right]$$ Calculate inside: $$5 - \frac{2}{3} = \frac{15}{3} - \frac{2}{3} = \frac{13}{3}$$ So, $$\frac{dV}{dt} = \pi \left[4 \times \frac{13}{3} \times \left(-\frac{1}{3}\right) + 4 \times \frac{1}{9} \right] = \pi \left[-\frac{52}{9} + \frac{4}{9} \right] = \pi \times \left(- \frac{48}{9}\right) = -\frac{16}{3} \pi$$ Since it's decreasing, rate of decrease of volume is $$\frac{16}{3} \pi$$. Answer: (A) $$\frac{16}{3} \pi$$.