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

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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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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 |

HARD

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IMPORTANT

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

HARD

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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 - 1.4>Q 41

On a

50

-question multiple choice math contest, students receive

4

points for a correct answer,

0

points for an answer left blank, and

- 1

point for an incorrect answer. Jasvinder's total score on the contest was

99

. What is the maximum number of questions that Jasvinder could have answered correctly?