11
The function f(x) = 2x3 - 3x2 - 36x + 2 has its maxima at
A)
x = -2 only
B)
x = 0 only
C)
x = 3 only
D)
both x = -2 and x = 3
Answer & Solution
Answer: Option
A
12
The minimum value of the function $${\text{f}}\left( {\text{x}} \right) = \left( {\frac{{{{\text{x}}^3}}}{3}} \right) - {\text{x}}$$ occurs at
A)
x = 1
B)
x = -1
C)
x = 0
D)
x = $$\frac{1}{{\sqrt 3 }}$$
Answer & Solution
Answer: Option
A
13
The function f(x) = 2x - x2 + 3 has
A)
a maxima at x = 1 and a minima at x = 5
B)
a maxima at x = 1 and a minima at x = -5
C)
only a maxima at x = 1
D)
only a minima at x = 1
Answer & Solution
Answer: Option
C
14
The function y = |2 - 3x|
A)
is continuous $$\forall {\text{x}} \in {\text{R}}$$ and differentiable $$\forall {\text{x}} \in {\text{R}}$$
B)
is continuous $$\forall {\text{x}} \in {\text{R}}$$ and differentiable $$\forall {\text{x}} \in {\text{R}}$$ except at x = $$\frac{3}{2}$$
C)
is continuous $$\forall {\text{x}} \in {\text{R}}$$ and differentiable $$\forall {\text{x}} \in {\text{R}}$$ except at x = $$\frac{2}{3}$$
D)
is continuous $$\forall {\text{x}} \in {\text{R}}$$ except x = 3 and differentiable $$\forall {\text{x}} \in {\text{R}}$$
Answer & Solution
Answer: Option
C
15
Which of the following is correct?
A)
$$\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\sin \,4{\text{x}}}}{{\sin \,2{\text{x}}}}} \right) = 1{\text{ and }}\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\tan {\text{x}}}}{{\text{x}}}} \right) = 1$$
B)
$$\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\sin \,4{\text{x}}}}{{\sin \,2{\text{x}}}}} \right) = \infty {\text{ and }}\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\tan {\text{x}}}}{{\text{x}}}} \right) = 1$$
C)
$$\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\sin \,4{\text{x}}}}{{\sin \,2{\text{x}}}}} \right) = 2{\text{ and }}\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\tan {\text{x}}}}{{\text{x}}}} \right) = \infty $$
D)
$$\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\sin \,4{\text{x}}}}{{\sin \,2{\text{x}}}}} \right) = 2{\text{ and }}\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{\tan {\text{x}}}}{{\text{x}}}} \right) = 1$$
Answer & Solution
Answer: Option
D
16
The double integral $$\int\limits_0^{\text{a}} {\int\limits_0^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$ is equivalent to
A)
$$\int\limits_0^{\text{x}} {\int\limits_0^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$
B)
$$\int\limits_0^{\text{a}} {\int\limits_{\text{x}}^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$
C)
$$\int\limits_0^{\text{a}} {\int\limits_{\text{x}}^{\text{a}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dy dx}}} } $$
D)
$$\int\limits_0^{\text{a}} {\int\limits_0^{\text{a}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$
Answer & Solution
Answer: Option
C
17
The values of x for which the function $${\text{f}}\left( {\text{x}} \right) = \frac{{{{\text{x}}^2} - 3{\text{x}} - 4}}{{{{\text{x}}^2} + 3{\text{x}} - 4}}$$ is NOT continuous are
A)
4 and -1
B)
4 and 1
C)
-4 and 1
D)
-4 and -1
Answer & Solution
Answer: Option
C
18
The expression $${\text{V}} = \int_0^{\text{H}} {\pi {{\text{R}}^2}{{\left( {1 - \frac{{\text{h}}}{{\text{H}}}} \right)}^2}} {\text{dh}}$$ for the volume of a cone is equal to
A)
$$\int_0^{\text{R}} {\pi {{\text{R}}^2}{{\left( {1 - \frac{{\text{h}}}{{\text{H}}}} \right)}^2}} {\text{dr}}$$
B)
$$\int_0^{\text{R}} {\pi {{\text{R}}^2}{{\left( {1 - \frac{{\text{h}}}{{\text{H}}}} \right)}^2}} {\text{dh}}$$
C)
$$\int_0^{\text{H}} {2\pi {\text{rH}}{{\left( {1 - \frac{{\text{r}}}{{\text{R}}}} \right)}^2}} {\text{dh}}$$
D)
$$4\int_0^{\text{R}} {\pi {\text{rH}}{{\left( {1 - \frac{{\text{r}}}{{\text{R}}}} \right)}^2}} {\text{dr}}$$
Answer & Solution
Answer: Option
D
19
Consider the function f(x) = |x| in the interval -1 < x ≤ 1. At the point x = 0, f(x) is
A)
continuous and differentiable
B)
non continuous and differentiable
C)
continuous and non-differentiable
D)
neither continuous nor differentiable
Answer & Solution
Answer: Option
C
20
For a scalar function f(x, y, z) = x2 + 3y2 + 2z2 , the gradient at the point P(1, 2, -1) is
A)
$$2\overrightarrow {\text{i}} + 6\overrightarrow {\text{j}} + 4\overrightarrow {\text{k}} $$
B)
$$2\overrightarrow {\text{i}} + 12\overrightarrow {\text{j}} - 4\overrightarrow {\text{k}} $$
C)
$$2\overrightarrow {\text{i}} + 12\overrightarrow {\text{j}} + 4\overrightarrow {\text{k}} $$
D)
$$\sqrt {56} $$
Answer & Solution
Answer: Option
B
