Q-2) If \(a,b,c\) be the sides of a triangle, then the minimum value of \(\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}\) is
a) \(3\) b) \(6\) c) \(9\) d) \(12\)
Q-3) For positive reals \(a,b,c\), \(\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b}\) is always
a) \(\le \frac{a+b+c}{2}\) b) \(\ge \frac{\sqrt{abc}}{3}\) c) \(\le \frac{a+b+c}{3}\) d) \(\ge \frac{\sqrt{abc}}{2}\)
Q-4) For positive reals \(a,b,c,d\) in H.P which of the follwing is/are true:
a) \(a+d>b+c\) b) \(a+b>c+d\) c) \(a+c>b+d\) d) None of these
Q-5) A rod of integral length \(N\) has to be broken into rods each of integral length. How many parts should we dissect this rod into such that the product of the lengths of these disected parts is maximised?
Q-6) In \(\triangle ABC\), internal angle bisectors \(AI,BI,CI\) meet the opposite sides in \(A’,B’,C’\) respectively. Show that \(\frac{AI\times BI\times CI}{AA’ \times BB’\times CC’}\le \frac{8}{27}\).
Q-7) If \(x,y,z \in \mathbb{R}^+\) be in A.P, then
a) \(y^2 \ge xz\) b) \(xy+yz \ge 2xz \) c) \(\frac{x+y}{2y-x}+\frac{y+z}{2y-z}\ge 4\) d) None of these
Q-8) An A.P & G.P has the same first and last term and the same number of terms. Which progression has the greater sum of its terms?
Q-9) Prove that
- \(5<\sqrt{5}+\sqrt[3]{5}+\sqrt[4]{5}\)
- \(8>\sqrt{8}+\sqrt[3]{8}+\sqrt[4]{8}\)
- \(n>\sqrt{n}+\sqrt[3]{n}+\sqrt[4]{n}\) for all integers \(n\ge 9\)
Q-10) For positve reals \(a,b,c\) show that
(Hint – Use AM-GM+Nesbitt’s Inequality)
Q-11) Let \(x,y\) be positve reals such that \(y^3+y \le x-x^3\). Show that
- \(y<x<1\)
- \(x^2+y^2<1\)
Q-12) Let \(a,b,c \in \mathbb{R}^+\) such that \(|a-b| \ge c\), \(|b-c| \ge a\), \(|c-a| \ge b\). Show that one of \(a,b,c\) is equal to the sum of the other 2.
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Hi, I’d appreciate a solution of the 3rd question. Any where from where I can get them?
How to solve the last qsn? Any hints?
Solutions?