Real numbers Practice Questions - NDA Mathematics Question Bank

Question 1 of 5 NDA MCQ

Let z = [y] and y = [x] – x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?

Detailed Explanation

Solution:

Given: z = [y] and y = [x] – x

As we know x = [x] + {x}

⇒ [x] – x = –{x}

z = [–{x}]

As we know 0 < {x} < 1

⇒ z = –1

Hint: Use x = [x] + {x} and 0 < {x} < 1.

Question 2 of 5 NDA MCQ

How many real numbers satisfy the equation|x − 4| + |x − 7| = 15 -

Detailed Explanation

Explanations: We have, |x – 4|+|x – 7| = 15

There are two cases arise.

Case I: When x < 4

–x + 4 – x + 7 = 15 ⇒ n = –2

Case II: When x > 7

So, only 2 Solution possible.

Question 3 of 5 NDA MCQ

Let A = {x Î R: –1 < x < 1}. Which of the following is/are bijective functions from A to itself?

1. f(x) = x|x|

2. g(x) = cos(πx)

Select the correct answer using the code given below:

Detailed Explanation

Question 4 of 5 NDA MCQ

Let z = [y] and y = [x] – x, where [] is the greatest integer function. If x is not an integer but positive, then what is the value of z?

Detailed Explanation

Question 5 of 5 NDA MCQ

For the next five (05) items that follow:

Let u be a positive integer and f be a real number lying between 0 and 1.

Further,

Consider the following statements:

I. (u + v + f) is an integer.

II. (f + v) is an integer.

Which of the statements given above is/are correct?

Detailed Explanation

Solution

Option (c) is correct.

Explanation:

Consider the values given: \[ u + f = \left( \sqrt{2} + 1 \right)^{10} \] \[ v = \left( \sqrt{2} - 1 \right)^{10} \] Since \( u \) is a positive integer and \( f \) is a real number between 0 and 1, the expressions can be analyzed as follows: 1. \( f = \left( \sqrt{2} + 1 \right)^{10} - u \) Thus, \( f \) being less than 1 means: \[ \left( \sqrt{2} + 1 \right)^{10} < u + 1 \] Hence, \( u + f \) is an integer. 2. \( v = \left( \sqrt{2} - 1 \right)^{10} \) Here, since \( \sqrt{2} - 1 \) is positive and less than 1, \( v \) is also a positive real number. Therefore, \( f + v \) can be expressed as: \[ f + v = \left( \sqrt{2} + 1 \right)^{10} - u + \left( \sqrt{2} - 1 \right)^{10} \] It follows that \( f + v \) is also an integer. Both statements I and II are correct.

📌 Hints / Properties Used:

Integers and Real Numbers: Basic arithmetic properties.
Rational number properties applied to \( u+f \) and \( f+v\).

📊 Visual Diagram Suggestion:

Consider a graph showing the exponential growth of \( (\sqrt{2} + 1)^{10} \) and the decay of \( (\sqrt{2} - 1)^{10} \) on the same axes to illustrate how both functions behave inside the defined limits.

📖 Factual Verification & Reference:

Verified against standard algebraic properties and integer definition rules.

Real numbers

Solution

Unlock 10,000+ Practice Questions