1.The points of contact of the vertical
tangents to \[x=2 -3 \sin\theta, y=3+2 \cos\theta\]
are
a) (2, 5), (2, 1)
b) (– 1, 3), (5, 3)
c) (2, 5), (5, 3)
d) (– 1, 3), (2, 1)
Explanation:
2. If \[f\left(x\right)=\frac{x}{\sin x}\] and \[g\left(x\right)=\frac{x}{\tan x}\] where
\[0 < x\leq 1\] , then in this interval
a) both f(x) and g(x) are increasing functions
b) both f(x) and g(x) are decreasing functions
c) f(x) is an increasing function
d) g(x) is an increasing function
Explanation:
3. The set of all values of a for which the
function
\[f\left(x\right)=\left(\frac{\sqrt{a+4}}{1-a}-1\right)x^{5}-3x+\log 5\]
decreases for all real x is
a)\[\left(-3 ,\frac{5-\sqrt{27}}{2}\right)\cup \left(2,\infty\right)\]
b) \[\left(-4 ,\frac{3-\sqrt{21}}{2}\right]\cup \left(1,\infty\right)\]
c) \[\left(-\infty ,\infty\right)\]
d) \[\left[-1 ,\infty\right)\]
Explanation: Differentiating, we get
4. The function \[g\left(x\right)=e^{ax^{2}}\frac{\log\left(\pi+x\right)}{\log\left(e+x\right)}\left(x\geq 0,a>0\right)\]
is
a) increasing on \[\left[0 ,\infty\right)\]
b) decreasing on \[\left[0 ,\infty\right)\]
c) increasing on \[\left[0,\pi/e\right)\] and decreasing on \[\left[\pi/e ,\infty\right)\]
d) decreasing on \[\left[0,\pi/e\right)\] and increasing on
\[\left[\pi/e ,\infty\right)\]
Explanation:
5. The function \[f\left(x\right)=\left(3x^{4}+40x^{3}-0.06x^{2}-1.2x\right)\]
a) decreases on \[\left(-\infty ,0\right)\]
b) increases on \[\left(0 ,\infty\right)\]
c) x = –10 is a point of minimum
d) x = 0.1 is a point of maximum
Explanation:
6. Let \[f\left(x\right)=xe^{x\left(1-x\right)}\] , then f (x) is
a) increasing on [– 1/2, 1]
b) decreasing on R
c) increasing on R
d) decreasing on [– 1/2, 1]
Explanation:
7. The equation \[e^{x-1}+x-2=0\] as
a) one real root
b) two real root
c) three real root
d) four real root
Explanation: Clearly, x = 1 satisfies the given equation
8. The function f satisfying \[\frac{f\left(b\right)-f\left(a\right)}{b-a}\neq f'\left(x\right)\]
for any \[x\epsilon\] (a, b) is
a) \[f\left(x\right)=x^{1/3},a=-1 , b=1\]
b) \[f\left(x\right)=\begin{cases}2 & x=1\\x^{2}&\ 1< x< 2 ,a=1 , b=2 \\1 &\ x=2 \end{cases}\]
c) \[f\left(x\right)=x \mid x\mid ; a=-1 , b=1\]
d) \[f\left(x\right)=1/x ; a=1 , b=4\]
Explanation: The functions in (a), (c), (d) satisfying hypothesis of Lagrange’s Mean Value theorem so there is
9. Suppose f is differentiable on R and \[a\leq f'\left(x\right) \leq b\] for all \[x\epsilon R \] where a, b > 0. If f (0) = 0, then
a) \[f\left(x\right)\leq\min\left(ax ,bx\right)\]
b) \[f\left(x\right)\geq\max\left(ax ,bx\right)\]
c) \[a\leq f\left(x\right)\leq b\]
d) \[ax\leq f\left(x\right)\leq bx\]
Explanation: For x > 0. Applying Lagrange’s theorem on
10. The minimum value of f(x) = |3 – x| +
|2 + x| + |5 – x| is
a) 0
b) 7
c) 8
d) 10
Explanation: f can be written as
