1.The point of intersection of the tangents drawn to
the curve \[x^{2}y=1-y\] at the points where it is met by
the curve \[xy=1-y\] is given by
a) (0, - 1)
b) (1, 1)
c) (0, 1)
d) (1, –1)
Explanation:
2. The equation of the tangent to the curve \[y=\left(2x-1\right)e^{2\left(1-x\right)}\] at the point of its maximum is
a) y = 1
b) x = 1
c) x + y = 1
d) x - y = - 1
Explanation:
3. The distance of the point on \[y=x^{4}+3x^{2}+2x\] which
is nearest to the line y = 2x - 1 is
a) \[4/\sqrt{5}\]
b) \[3/\sqrt{5}\]
c) \[2/\sqrt{5}\]
d) \[1/\sqrt{5}\]
Explanation:
4. If the function \[f\left(x\right)=x^{2}+a/x\] has a local minimum
at x = 2, then the value of a is
a) 8
b) 16
c) 18
d) 12
Explanation:
5. The coordinates of the point on the curve
\[\left(x^{2}+1\right)\left(y-3\right)=x\] where a tangent to the curve has
the greatest slope are given by
a) \[\left(\sqrt{3},3+\sqrt{3}/4\right)\]
b) \[\left(-\sqrt{3},3-\sqrt{3}/4\right)\]
c) (0, 3)
d) Both a and c
Explanation:
6. The critical points of the function \[f\left(x\right)=\left(x-2\right)^{2/3}\left(2x+1\right)\] are
a) - 1 and 2
b) 1
c) 1 and - 1/2
d) 1 and 2
Explanation:
7. The function \[f\left(x\right)=\sin x\cos^{2}x\] has extremum at
a) \[x=\pi/2\]
b) \[x=\pi\]
c) \[x=\cos^{-1}\left(-\sqrt{2/3}\right)\]
d) Both a and c
Explanation:
8. The function \[f\left(x\right)=2\log \left(x-2\right)-x^{2}+4x+1\] increases in the interval
a) (1, 2)
b) (2, 3)
c) (5/2, 3)
d) Both b and c
Explanation:
9. The critical points of the function f'(x), where \[f\left(x\right)=\frac{\mid x-2\mid}{x^{3}}\] are
a) 0
b) 1
c) 3
d) -1
Explanation:
10. y = log x satisfies for x > 1, the equality
a) x - 1 > y
b) \[x^{2}-1>y\]
c) (x - 1)/x < y
d) Both a and c
Explanation:
