Continuing our one mark questions for HSC Mathematics, questions from 25-32 are provided in this post. Check the following links for questions
25) The value of $ \int_{0}^{\frac{\pi}{2}}\frac{cos^\frac{5}{3}x}{cos^\frac{5}{3}x+sin^\frac{5}{3}x}\;dx $ is
a) $ \frac{\pi}{4} $
b) $ \frac{\pi}{2} $
c) $ \frac{\pi}{3} $
d) $ \frac{2\pi}{3} $
26) The volume generated when the region bounded by y=x, y=1, x=0 is rotated about x-axis is:
a) $ \frac{\pi}{3} $
b) $ \frac{2\pi}{3} $
c) $ \frac{\pi}{2} $
d) $ \frac{\pi}{4} $
27) The C.S.A of a sphere of radius 5 intercepted between two parallel planes of distance 2 and 4 from the centre is:
a) $ 20\pi $
b) $ 40\pi $
c) $ 10\pi $
d) $ 30\pi $
28) The area between the ellipse $ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $ and its auxiliary circle is:
a) $ \pi b(a-b) $
b) $ 2\pi a(a-b) $
c) $ \pi a(a-b) $
d) $ 2\pi b(a-b) $
29) The differential equation of all circles with centre at the origin is:
a) xdy + ydx = 0
b) xdy – ydx = 0
c) xdx + ydy = 0
d) xdx – ydy = 0
30) If $ \frac{\mathrm{dy} }{\mathrm{d} x}=\frac{x-y}{x+y} $ then
a) $ 2xy+y^2-x^2=c $
b) $ x^2+y^2-x+y=c $
c) $ x^2-y^2-2xy=c $
d) $ x^2+y^2-2xy=c $
31) The P.I. of $ (D^2+D)y=5e^{-x} $
a) $ 5e^{-x} $
b) $ -5e^{-x} $
c) $ 5xe^{-x} $
d) $ -5xe^{-x} $
32) The differential equation obtained by eliminating a and b from $ y=ae^{3x}+be^{-3x} $ is:
a) $ \frac{\mathrm{d^2}y }{\mathrm{d}x^2}+9y=0 $
b) $ \frac{\mathrm{d^2}y }{\mathrm{d}x^2}-9y=0 $
c) $ \frac{\mathrm{d^2}y }{\mathrm{d}x^2}-9\frac{\mathrm{d}y }{\mathrm{d}x}=0 $
d) $ \frac{\mathrm{d^2}y }{\mathrm{d}x^2}+9x=0 $
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