Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 24

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Let $a, b$ and $c$ be the lengths of the three sides of a triangle. Suppose $a$ and $b$ are the roots of the equation $x^{2} + 4 (c + 2) = (c + 4) x$ and the largest angle of the triangle is $x °$ . Find the value of $x$ .

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Important Questions on Algebra

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IOQM - PRMO and RMO

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 25

How many ordered pairs of integers

(x, y)

satisfy the equation

x^{2} + y^{2} = 2 (x + y) + x y

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 26

Let

n

be a positive integer such that one of the roots of the quadratic equation

4 x^{2} - (4 \sqrt{3} + 4) x + \sqrt{3} n - 24 = 0

is an integer. Find the value of

n

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 27

Suppose that the two roots of the equation

\frac{1}{x^{2} - 10 x - 29} + \frac{1}{x^{2} - 10 x - 45} - \frac{2}{x^{2} - 10 x - 69} = 0

are

α

and

β .

Find the value of

(- α β) .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 28

Find the total number of pairs of

(x, y)

, where

x, y

belongs to integer that satisfy the equation

x + y = x^{2} - x y + y^{2}

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 29

Let

a

and

b

be two integers. Suppose that

\sqrt{7 - 4 \sqrt{3}}

is a root of the equation

x^{2} + a x + b = 0

. Find the value of

b - a

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 30

Let

p

be an integer such that both roots of the equation

5 x^{2} - 5 p x + (66 p - 1) = 0

are positive integers. Find the value of

p

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 31

Suppose

a

and

b

are the roots of

x^{2} + x \sin α + 1 = 0

while

c

and

d

are the roots of the equation

x^{2} + x \cos α - 1 = 0

. Find the value of

\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} + \frac{1}{d^{2}}

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 32

The zeros of the function

f (x) = x^{2} - a x + 2 a

are integers. What is the sum of all possible values of

a

?