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

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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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Important Questions on Algebra

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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 39

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

$x + 3 y = 3$

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

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.4>Q 40

What is the sum of all the solutions of

x = |2 x - |60 - 2 x||

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.5>Q 2

The two digits in Sunil's age are the same as the digits in Ramesh's age, but in reverse order. In five years Sunil will be twice as old as Ramesh will be, then what is the difference in their current ages?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.5>Q 3

Driving at a constant speed, Kavita usually takes

180

minutes to drive from her house to her mother's house.
One day Kavita begins the drive at her usual speed, but after driving

\frac{1}{3}

of the way, she hits a bad snowstorm and reduces her speed by

20 km

per hour. This time the trip takes her a total of

276

minutes. Kavita is now

H kms

far from her mother's house. What is

\frac{H}{5}

?