Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 18

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Given that the real numbers $s, t$ satisfy $19 s^{2} + 99 s + 1 = 0, t^{2} + 99 t + 19 = 0$ and st $\neq 1 .$ Find the value of $|\frac{s t + 4 s + 1}{t}| .$

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Important Questions on Algebra

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 19

Given that

a = 8 - b

and

c^{2} = a b - 16,

prove that

a = b .

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 20

Given that

α, β

are roots of the equation

x^{2} - 7 x + 8 = 0

, where

α > β

. Find the value of

\frac{2}{α} + 3 β^{2}

without solving the equation.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 21

Both roots of the quadratic equation

x^{2} - 63 x + k = 0

are prime numbers. The number of possible values of

k

is

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 22

Find the sum of all possible values of $a$ such that the following equation has real root in $x$ :

${(x - a)}^{2} + {(x^{2} - 3 x + 2)}^{2} = 0$

EASY

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 23

Let

p

be a real number such that the equation

2 y^{2} - 8 y = p

has only one solution. Then find the value of

p + 64 .

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

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

.

HARD

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IMPORTANT

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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IMPORTANT

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

.