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

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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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Important Questions on Algebra

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IMPORTANT

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}}

.

HARD

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IMPORTANT

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

?

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 33

Let

a, b

and

c

be three distinct one-digit numbers. The maximum value of the sum of the roots of the equation

(x - a) (x - b) + (x - b) (x - c) = 0

is

k .

What is the value of

2 k

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 34

There are exactly

N

distinct rational numbers

k

such that

| k | < 200

and

5 x^{2} + k x + 12 = 0

has at least one integer solution for

x .

What is

N

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 35

Let

a

and

b

be the roots of

x^{2} + 2000 x + 1 = 0

and let

c

and

d

be the roots of

x^{2} - 2008 x + 1 = 0 .

Now,

P

is defined as

P = (a + c) (b + c) (a - d) (b - d) .

Then, find sum of all the digits of

P

.

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IOQM - PRMO and RMO

IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 36

Find the value of the smallest positive integer

m

such that the equation

x^{2} + 2 (m + 5) x + (100 m + 9) = 0

has only integer solutions.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 37

Given that

(m - 2)

is a positive integer and it is also a factor of

3 m^{2} - 3 m + 10

. Find the sum of all such values of

m .

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.3>Q 38

Let

a

and

b

be the roots of the equation

x^{2} - m x + 2 = 0

. Suppose that

(a + \frac{1}{b})

and

(b + \frac{1}{a})

are the roots of the equation

x^{2} - p x + q = 0 .

What is

2 q

?