The integral $$\int_1^3 {\frac{1}{{\text{x}}}{\text{dx,}}} $$ when evaluated by using Simpson's 1/3 rule on two equal subintervals each of length 1, equals

With respect to the numerical evaluation of the definite integral $${\text{K}} = \int_{\text{a}}^{\text{b}} {{{\text{x}}^2}{\text{dx,}}} $$ where a and b are given, which of the following statements is/are TRUE?
I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.

A

I only

B

II only

C

Both I and II

D

Neither I nor II

The integral $$\int\limits_{{{\text{x}}_1}}^{{{\text{x}}_2}} {{{\text{x}}^2}{\text{dx}}} $$ with x₂ > x₁ > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If $$I$$ is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

A

J > $$I$$

B

J

C

J = $$I$$

D

Insufficient data to determine the relationship

A common convention is: • is evaluated first • and are evaluated next • Quantifiers are evaluated next • is evaluated last.

A

True

B

False

Statement : It has been reported by one of the TV channels that the answer papers of Board examination of one State are evaluated by students studying in the same standard with the help of model answers as instructed by the examiners.

Courses of Action :
I. All such examiners should be immediately suspended from their official positions.
II. All such papers evaluated by the students should be immediately confiscated and got evaluated by qualified teachers.
III. The Board should explore possibilities even though they are remote, of getting the answer papers of this examination evaluated by computerised machines.

A

Only I follows

B

Only II follows

C

Only III follows

D

Only I and II follow

E

All follow

A common convention is:
• is evaluated first
• and are evaluated next
• Quantifiers are evaluated next
• is evaluated last.

A

True

B

False

The minimum number of equal length subintervals needed to approximate $$\int\limits_1^2 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx}}} $$ to an accuracy of at least $$\frac{1}{3} \times {10^{ - 6}}$$ using the trapezoidal rule is

A

1000e

B

1000

C

100e

D

100

The value of the definite integral $$\int_1^{\text{e}} {\sqrt {\text{x}} } \ln \left( {\text{x}} \right){\text{dx}}$$ is

A

$$\frac{4}{9}\sqrt {{{\text{e}}^3}} + \frac{2}{9}$$

B

$$\frac{2}{9}\sqrt {{{\text{e}}^3}} - \frac{4}{9}$$

C

$$\frac{2}{9}\sqrt {{{\text{e}}^3}} + \frac{4}{9}$$

D

$$\frac{4}{9}\sqrt {{{\text{e}}^3}} - \frac{2}{9}$$

Choose the set of figures which follows the given rule.
Rule: As the circle decreases in size, its sectors increase in number.
Rule Detection mcq question image

A

1

B

2

C

3

D

4

Choose the set of figures which follows the given rule.
Rule: Closed figures become more and more open and open figures become more and more closed.
Rule Detection mcq question image

A

1

B

2

C

3

D

4

Choose the set of figures which follows the given rule.
Rule: The series becomes simpler as it proceeds.
Rule Detection mcq question image

A

1

B

2

C

3

D

4

The integral $$\int_1^3 {\frac{1}{{\text{x}}}{\text{dx,}}} $$ when evaluated by using Simpson's 1/3 rule on two equal subintervals each of length 1, equals

Related Questions