Evaluate the Given limit: lim r → 1 π r 2

We have been given lim r → 1 π r 2 Now, substituting `r=1`, we get lim r → 1 π r 2 = π 1 2 = π

Evaluate: lim x → c x - 22 7

Given that, lim x → c x - 22 7 The required limit is a limit of a polynomial function. Hence, the limit is the value of the function at the prescribed point. Therefore, lim x → c x - 22 7 = c - 22 7

Evaluate: lim x → 4 4 x + 3 x - 2

Given lim x → 4 4 x + 3 x - 2 The required limit is a limit of some rational function. Hence, the limit is the value of the function at the prescribed point. Therefore, lim x → 4 4 x + 3 x - 2 = 4 4 + 3 4 - 2 = 16 + 3 2 = 19 2

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Important Questions on Mechanical Properties of Fluids

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Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

Suppose that

20

pillars of the same height have been erected along the boundary of circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is:

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

There are

m

men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by

84

, then the value of

m

is :

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Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

A committee of

11

member is to be formed from

8

males and

5

females. If

m

is the number of ways the committee is formed with at least

6

males and

n

is the number of ways the committee is formed with at least

3

females, then

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

Consider a class of

5

girls and

7

boys. The number of different teams consisting of

2

girls and

3

boys that can be formed from this class, if there are two specific boys

A

and

B,

who refuse to be the members of the same team, is:

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

Let

f : R \to R

be a positive increasing function with

\lim_{x \to \infty} \frac{f (3 x)}{f (x)} = 1

. Then,

\lim_{x \to \infty} \frac{f (2 x)}{f (x)}

is equal to

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

The Number of ways of choosing

10

objects out of

31

objects of which

10

are identical and the remaining

21

are distinct, is:

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

The value of

\sum_{r = 1}^{10} r \cdot \frac{{}^{n}C_{r}}{{}^{n}C_{r - 1}}

is equal to

MEDIUM

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

\lim_{x \to 0} \frac{\sin ({πcos}^{2} x)}{x^{2}}

is equal to

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

The value of

n \in I

for which the function

f (x) = \frac{\sin n x}{\sin (\frac{x}{n})}

has

4 π

as its period is:

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

{\sum_{i = 1}^{20} (\frac{^{20} C_{i - 1}}{^{20} C_{i} +^{20} C_{i - 1}})}^{3} = \frac{k}{21},

then

k

equals

EASY

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

Let

f (x) = \begin{array}{l} x^{2} - 1, 0 < x < 2 \\ 2 x + 3, 2 \leq x < 3 \end{array}

, find the quadratic equation whose roots are

\lim_{x \to 2^{-}} f (x)

and

\lim_{x \to 2^{+}} f (x)

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

[\cdot]

denote the greatest integer function then

\lim_{n \to \infty} \frac{[x] + [2 x] + \dots . + [n x]}{n^{2}}

is -

HARD

Physics>Mechanical Properties of Matter>Mechanical Properties of Fluids>Fluid Mechanics

Let $f (x)$ be a polynomial of degree four and having its extreme values at $x = 1$ and $x = 2$ . If $\lim_{x \to 0} [1 + \frac{f (x)}{x^{2}}] = 3$ , then $f (2)$ is equal to