Download App

MATHEMATICAL REASONING — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSMATHEMATICAL REASONING3 Marks
Question
Check the validity of the statements given below by the method given against it. p: The sum of an irrational number and a rational number is irrational (by contradiction method).
✓

Answer

The given statement is as follows. p: the sum of an irrational number and a rational number is irrational. Let us assume that the given statement, p, is false. That is, we assume that the sum of an irrational number and a rational number is rational. Therefore, $\sqrt{\text{a}}+\frac{\text{b}}{\text{c}}=\frac{\text{d}}{\text{e}},$ where $\sqrt{\text{a}}$ is irrational and and b, c, d, e are integers. ⇒ de − bc = a But here, $\frac{\text{d}}{\text{e}}-\frac{\text{b}}{\text{c}}$ is a rational number and $\sqrt{\text{a}}$ is an irrational number. This is a contradiction. Therefore, our assumption is wrong. Therefore, the sum of an irrational number and a rational number is rational. Thus, the given statement is true.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 3 marks)" from MATHEMATICAL REASONING→MATHEMATICAL REASONING — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

How many numbers can be formed with the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd places?
→
If the sum of n terms of an A.P. is $\text{np}+\frac{1}{2}\text{n}(\text{n}-1)$ Q, where P and Q are constants, find the common difference.
→
Write the following relation as the sets of ordered pairs: A relation R on the set {0, 1, 2, ....., 10} defined by 2x + 3y = 12.
→
In the expansion of $(1+a)^{m+n}$, prove that the coefficients of $a^m$ and $a^n$ are equal.
→
Solve the following system of equations in R. $\Big|\text{x}+\frac{1}{3}\Big|>\frac{8}{3}$
→
Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\frac{\pi}{4}}}\frac{1-\tan\text{x}}{\text{x}-\frac{\pi}{4}}$
→
Find the values of the other five trigonometric functions in the following: $\cot\text{x}=\frac{12}{5},$ x in quadrant III.
→
Prove that : sin 3x + sin 2x - sin x $ = 4\sin \;x\cos \frac{x}{2}\cos \frac{{3x}}{2}$
→
The perpendicular from the origin to the line y = mx + c meets it at the point (-1, 2). Find the value of m and c.
→
Evaluate the following one sided limits: $\lim\limits_{\text{x}\rightarrow0^-}\frac{\text{x}^2-3\text{x}+2}{\text{x}^3-2\text{x}^2}$
→