List of top Mathematics Questions on Operations on Real Numbers

Let α and β be real numbers. Consider a \(3 \times 3\) matrix A such that \(A^2 = 3A + \alpha I\). If \(A^4 = 21A + \beta I\), then
Let \(λ ≠ 0\) be a real number. Let α, β be the roots of the equation \(14x^2 – 31x + 3λ = 0\) and α, γ be the roots of the equation \(35x^2 – 53x + 4λ = 0\). Then \(\frac{3\alpha}{\beta}\) and \(\frac{4\alpha}{\lambda}\) are the roots of the equation
Let for some real numbers \(α\) and \(β\), \(a = α – iβ\). If the system of equations \(4ix + (1 + i) y = 0\) and \(8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0 \)has more than one solution, then \(\frac{α}{β}\) is equal to

For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and 
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)2 is equal to _____.

Let A = [1, 2, 3, 4, 5], m be the number of relations such as 4x ≤ 5y XRY and n be the minimum number of elements to be added from A × A to make a symmetric relation. Then the value of n + m.

Classify the following numbers as rational or irrational:

(i) \(2 - \sqrt5\)  

(ii) \((3 + \sqrt23) - \sqrt23\) 

(iii) \(\frac{2 \sqrt{7}} { 7 \sqrt7}\)

(iv) \(\frac{1}{\sqrt{2}}\)

(v) 2π 

Simplify each of the following expressions: 

(i) (3 + √3)(2 + √2) 

(ii) (3 + √3)(3 - √3) 

(iii) (√5 + √2 )2 

(iv) (√5 - √2)(√5 + √2)

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π =\(\frac{ c}{d}\). This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Represent √9.3 on the number line

Rationalise the denominators of the following: 

(i) \(\frac{1 }{ \sqrt{7 }}\)

(ii) \(\frac{1 }{ \sqrt{7 }-\sqrt6}\)

(iii) \(\frac{1 }{ \sqrt{5}+\sqrt2}\)

(iv) \(\frac{1 }{ \sqrt{7}-2}\)