Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 16

EASY

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IMPORTANT

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The value of ${(625^{\log_{5} 2015})}^{\frac{1}{4}}$ is a four-digit number $A$ . What is $\frac{A}{31}$ ?

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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 - 1.6>Q 17

When

p = \sum_{k = 1}^{6} k \ln k

, the number

e^{p}

is an integer. The largest power of

2

that is a factor of

e^{p}

is

2^{R} .

What is

R

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 18

There are

q

values of

x

which satisfy

\log_{10} (x - 40) + \log_{10} (60 - x) < 2

. What is the value of

q

?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 19

Let

m > 1

and

n > 1

be integers. Suppose that the product of the solutions for

x

of the equation

8 (\log_{n} x) (\log_{m} x) - 7 \log_{n} x - 6 \log_{m} x - 2013 = 0

is the smallest possible integer. What is

m + n

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 20

The solution of the equation

7^{x + 7} = 8^{x}

can be expressed in the form

x = \log_{b} 7^{7}

. What is

21 b

?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 21

x = a b c

(

3

-digit number) satisfy the equation

\log_{\sqrt{2}} \sqrt{x} + \log_{2} x + \log_{4} (x^{2}) + \log_{8} (x^{3}) + \log_{16} (x^{4}) = 40

. What is

a b

(

2

digit-number)?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 22

There are

t

distinct four-tuples

(a, b, c, d)

of rational numbers with

a \log_{10} 2 + b \log_{10} 3 + c \log_{10} 5 + d \log_{10} 7 = 2005

. What is

25 \times t

?

HARD

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IMPORTANT

Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 23

Let

S

be the set of ordered triples

(x, y, z)

of real numbers for which

\log_{10} (x + y) = z

and

\log_{10} (x^{2} + y^{2}) = z + 1 .

There are real numbers

a

and

b

such that for all ordered triples

(x, y, z)

in

S

, we have

x^{3} + y^{3} = a 10^{3 z} + b 10^{2 z}

. What is the value of

4 (a + b) ?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.6>Q 24

Find the number of positive integers

x

that satisfy the inequality

|3 + \log_{x} \frac{1}{3}| < \frac{8}{3}

.