Previous Year Questions on Inequalities

Inequalities - Previous Year Questions

Previous Year Questions on Inequalities

Previous year questions on Inequalities provide an essential resource for mastering mathematical reasoning and problem-solving. With a rich history rooted in Classical Algebra, inequalities are foundational in various mathematical fields. These questions, categorized topicwise and yearwise, are indispensable for students preparing for university examinations. For more resources, visit Mathematics Notes and Mathematics Questions.

Vidyasagar University

2023-24 (CBCS)

If \(x,y,z \) are positive real numbers and \(x+y+z=1 \), prove that \(8xyz\leq (1-x)(1-y)(1-z)\leq \frac{8}{27}\). [5]

2021-22 (CBCS)

If \(a_{1},a_{2},…,a_{n} \) be all positive real numbers and \(s=a_{1}+a_{2}+…+a_{n} \) then prove that \( \left(\frac{s-a_{1}}{n-1} \right)\left(\frac{s-a_{2}}{n-1} \right)…\left(\frac{s-a_{n}}{n-1} \right)\gt a_{1}a_{2}…a_{n}\) unless \(a_{1}=a_{2}=…=a_{n} \). [4]
If \(s_{n}=1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}\), prove that \(s_{n}\gt \frac{2n}{n+1}\) if \(n\gt 1\). [3]
Find the greatest value of \(xyz\) where \(x,y,z\) are positive real numbers satisfying \(xy+yz+zx=27\). [2]

2020-21 (CBCS)

Find the maximum value of \((x+2)^{5}(7-x)^{4}\) when \(-2\lt x \lt 7\). [2]
Find the maximum value of \((x+2)^{5}(7-x)^{4}\) when \(-2\lt x \lt 7\). [2]
If \(n) be a positive integer greater than \(2\), then prove that \((n!)^{2}\gt n^{n}\).
If \(n) be a positive integer, prove that
\(\frac{1}{\sqrt{4n+1}}\lt \frac{3.7.11….(4n-1)}{5.9.13….(4n+1)}\lt \sqrt{\frac{3}{4n+3}} \). [6]
If \(a_{1},a_{2},…,a_{n} \), \(b_{1},b_{2},…,b_{n} \) be all real numbers, then show that \(\left( a_{1}^{2}+a_{2}^{2}+…+a_{n}^{2} \right)\left( b_{1}^{2}+b_{2}^{2}+…+b_{n}^{2} \right) \gt \left( a_{1}b_{1}+a_{2}b_{2}+…+a_{n}b_{n} \right)\), when \(a_{1},a_{2},…,a_{n} \) and \(b_{1},b_{2},…,b_{n} \) are not propotional. [6]

2019-20 (CBCS)

Prove that the minimum value of \(x^{2}+y^{2}+z^{2}\) is \(\left(\frac{e}{7}\right)^{2}\) where \(x,y,z\) are positive eal numbers subject to the condition \(2x+3y+6z=c\), \(c\) being a constant. Find the values of \(x,y,z\) for which the minimum value attained. [3]
If \(s_{n}=1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}\) prove that
(i) \(s_{n}\gt \frac{2n}{n+1}\) if \(n\gt 1\)
(ii) \(\left( \frac{n-s_{n}}{n-1} \right) \gt \frac{1}{n}\) if \(n\gt 2\). [5]

2018-19 (CBCS)

State and prove Cauchy Schwarz’s inequality. [5]
If \(x,y,z\) be positive and \(x+y+z=1\), then show that \(8xyz\leq (1-x)(1-y)(1-z)\leq \frac{8}{27}\).
Show that \(3x(3x+1)^{2}\gt 4\left[(3n)!\right]^{\frac{1}{n}} \) where \(n\) be a positive integer greater that \(1\). [5]

2017-18 (CBCS)

If \(a,b,c,x,y,z\) be all real numbers and \(a^{2}+b^{2}+c^{2}=1\), \(x^{2}+y^{2}+z^{2}=1\) then prove that \(-1\leq ax+by+cz\leq 1 \).
If \(a_{1},a_{2},…,a_{n} \) be \(n\) positive rational numbers and \(s=a_{1}+a_{2}+…+a_{n} \), prove that
\(\left(\frac{s}{a_{1}}-1 \right)^{a_{1}}\left(\frac{s}{a_{2}}-1 \right)^{a_{2}}…\left(\frac{s}{a_{n}}-1 \right)^{a_{n}}\leq \left(n-1\right)^{s} \). [5]
If \(a_{1},a_{2},…,a_{n} \) be \(n \) real positive quantities then prove that \(A.M.\leq G.M. \leq H.M. \). [5]

FAQs

What are mathematical inequalities?
Inequalities are mathematical expressions that compare two values using symbols like \(\lt, \gt, \leq, \geq \).
Why study previous year questions on inequalities?
They help in identifying recurring patterns and important topics in examinations.
How are inequalities applied in real life?
Inequalities are used in optimization problems, statistics, and economic modeling.
Where can I find topicwise questions on inequalities?
Visit Inequalities Questions for organized resources.
Are solved previous year questions available?
Yes, solved papers can be found in various textbooks and online educational platforms.
What is the importance of inequalities in Classical Algebra?
Inequalities form the basis of many advanced topics in mathematics and are integral to Classical Algebra.
What types of inequalities are commonly asked?
Linear, quadratic, polynomial, and absolute value inequalities are frequently included in exams.
How can I practice inequalities effectively?
Regular practice of previous year questions and understanding their solutions is recommended.
What are some reliable books for inequalities?
Refer to Suggested Books for comprehensive resources.
Can inequalities be linked to other topics?
Yes, they are closely related to Complex Numbers and Theory of Equations.