Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 31

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Let $m, n$ be integers such that $1 < m \leq n$ . Define $f (m, n) = (1 - \frac{1}{m}) \times (1 - \frac{1}{m + 1}) \times (1 - \frac{1}{m + 2}) \times \dots \times (1 - \frac{1}{n}) .$

If $S = f (2, 2049) + f (3, 2049) + f (4, 2049) + \dots + f (2049, 2049)$ , find the value of $\frac{S}{64}$

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Important Questions on Algebra

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 32

The number of positive integral solutions

(a, b, c, d)

satisfying

\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = 1

with the condition that

a < b < c < d

is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 33

For integers

a_{1}, \dots, a_{n} \in \{1, 2, 3, \dots, 9\}

, we use the notation

\bar{a_{1} a_{2} \dots a_{n}}

to denote the number

10^{n - 1} a_{1} + 10^{n - 2} a_{2} + \dots + 10 a_{n - 1} + a_{n} .

For example, when

a = 2

and

b = 0, \bar{a b}

denotes the number

20

. Given that

\bar{a b} = b^{2}

and

\bar{a c b c} = (\bar{b a})^{2}

. Find the value of

a + b + c

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 34

It is known that there is only one pair of positive integers

a

and

b

such that

a \leq b

and

a^{2} + b^{2} + 8 a b = 2010

. Find the value of

a + b

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 35

Let

f (x) = \frac{x^{2010}}{x^{2010} + (1 - x)^{2010}}

. Find the value of

\frac{f (\frac{1}{2011}) + f (\frac{2}{2011}) + f (\frac{3}{2011}) + \dots + f (\frac{2010}{2011})}{15}

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 36

If

x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20

, then if

x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} = k

, find

[k]

, where

[.]

denotes greatest integer function.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 37

If

x

and

y

non-zero real numbers such that

| x | + y = 3

and

| x | y + x^{3} = 0

, then find

| x - y |

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 38

How many distinct ordered triplets

(x, y, z)

satisfy the equations

x + 2 y + 4 z = 12

,

x y + 4 y z + 2 x z = 22

,

x y z = 6

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 39

How many ordered pairs of real numbers $(x, y)$ satisfy the following system of equations?

$x + 3 y = 3$

$| | x | - | y | | = 1$