Detailed Solutions: Algebra
Algebra is one of the most important chapters in Mathematics for SSC, Banking, Railway, NDA, and other competitive exams. Questions from algebra are frequently asked in equations, expressions, and formula-based problems.
Below are 4 important Algebra questions with solutions to help you understand the basics clearly.
What is Algebra in Mathematics?
Algebra is the branch of mathematics that uses letters and symbols (like x, y, a, b) to represent numbers and form equations.
Question: 1
यदि \( x + \frac{1}{x} = 5 \) , तो \( x^2 + \frac{1}{x^2} \) का मान ज्ञात करें।
If \( x + \frac{1}{x} = 5 \) , then find the value of \( x^2 + \frac{1}{x^2} \).
[A] 27 [B] 23
[C] 20 [D] 25
⚡Solution:
Answer (b) : दिया है : \( x + \frac{1}{x} = 5 \)
Step-1: दोनों तरफ वर्ग करें ।
\( (x + \frac{1}{x} )^2 = 5^2 \)
Step-2: \( x^2 + \frac{1}{x^2} + 2 × x × \frac{1}{x} = 25 \)
Step-3: \( x^2 + \frac{1}{x^2} + 2 = 25 \)
Step-4: \( x^2 + \frac{1}{x^2} = 25 – 2 = 23\)
Question: 2
यदि α और β, समीकरण \(x^2 – 5x + 6 = 0\) के मूल हैं, तो \(α^2 + β^2\) का मान ज्ञात कीजिए।
If α and β are the roots of \( x^2 – 5x + 6 = 0 \) , find the value of \( α ^2 + β ^2 \).
[A] 13 [B] 6
[C] 25 [D] 5
⚡Solution:
Answer (a) : 13
Step-1: समीकरण के मूलों का योगफल = α + β = \(\frac{-b}{a}\) = \(\frac{-(-5)}{1}\) = 5
Step-2: समीकरण के मूलों का गुणनफल = α × β = \(\frac{c}{a}\) = \(\frac{6}{1}\) = 6
Step-3: समीकरण α + β = 5 में दोनों तरफ वर्ग करें।
\( (α + β)^2 = 5^2 \)
\( α^2 + β^2 + 2αβ = 25\)
Step-4: α × β = 6 रखें।
\( α^2 + β^2 + 2 × 6 = 25\)
\( α^2 + β^2 = 25 – 12 = 13 \)
Question: 3
यदि \( x + y + z = 15\) और \( xy + yz + zx = 71 \) हैं, तो \(x^2 + y^2 + z^2 \) का मान ज्ञात कीजिए।
If \( x + y + z = 15\) and \( xy + yz + zx = 71 \) , find the value of \(x^2 + y^2 + z^2 \).
[A] 225 [B] 54
[C] 142 [D] 83
⚡Solution:
Answer (d) : 13
सर्वसमिका \( ( x + y + z )^2 = x^2 + y^2 + z^2 + 2( xy + yz + zx ) \) से
दिया है : \( x + y + z = 15\) और \( xy + yz + zx = 71 \)
सभी मान सर्वसमिका में रखें ।
\( 15^2 = x^2 + y^2 + z^2 + 2 ( 71 ) \)
\( 225 = x^2 + y^2 + z^2 + 142 \)
\( 225 – 142 = x^2 + y^2 + z^2 \)
\( 83 = x^2 + y^2 + z^2 \)
Question: 4
यदि x और y धनात्मक संख्याएँ हैं, जहाँ x – y = 5 और xy = 150 है, तो (x + y ) का मान क्या होगा ?
If x and y are positive numbers, where x – y = 5 and xy = 150, then what is the value of (x + y)?
[A] 625 [B] 25
[C] 150 [D] 600
⚡Solution:
Answer (b) : 25
दिया है : x – y = 5 और xy = 150
Formula : \( ( x + y ) = \sqrt{ ( x – y )^2 + 4xy }\) में x – y और xy का मान रखें
\( ( x + y ) = \sqrt{ ( 5 )^2 + 4 × 150}\)
\( ( x + y ) = \sqrt{ 25 + 600}\)
\( ( x + y ) = \sqrt{625}\)
\( ( x + y ) = 25\)
Why Algebra Questions Are Important for Exams
- Forms the base of advanced mathematics
- Used in equations, formulas, and problem solving
- Frequently asked in school and competitive exams
- High-scoring topic
Conclusion
These Algebra important questions are perfect for building a strong foundation. Practice them to improve your speed and accuracy in exams.