Chapter 4

The minimum value of \(\frac{x^{4}+y^{4}+z^{2}}{x y z .}\) for positive real numbers \(x, y, z\) is a. \(\sqrt{2}\) b. \(2 \sqrt{2}\) c. \(4 \sqrt{2}\) d. \(8 \sqrt{2}\)

A rod of fixed length \(k\) slides along the coordinate axes. If it meets the axes at \(A(a, 0)\) and \(B(0, b)\), then the minimum value of \(\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}\) is a. 0 b. 8 c. \(k^{2}-4+\frac{4}{k^{2}}\) d. \(k^{2}+4+\frac{4}{k^{2}}\)

The least value of \(6 \tan ^{2} \phi+54 \cot ^{2} \phi+18\) is (I) 54 when A.M. \(\geq\) G.M. is applicable for \(6 \tan ^{2} \phi, 54 \cot ^{2} \phi, 18\) (II) 54 when A.M. \(\geq G . M\). is applicable for \(6 \tan ^{2} \phi, 54 \cot ^{2} \phi\) and 18 is added further \((\mathrm{III}) 78\) when \(\tan ^{2} \phi=\cot ^{2} \phi\) a. (I) is correct, II is false b. (T) and (II) are correct \(\mathrm{C}_{\mathrm{i}}\) (III) is correct d. none of the above are correct

If \(a b^{2} c^{3}, a^{2} b^{3} c^{4}, a^{3} b^{4} c^{5}\) are in A.P. \((a, b, c>0)\), then the minimum vatue of \(a+b+c\) is a. 1 b. 3 c. 5 d. 9

If \(a, b, c\) are three distinct positive real numbers in G.P., then prove that \(c^{2}+2 a b>3 a c\).

If \(y=3^{x-1}+3^{-x-1}\), then the least value of \(y\) is a. 2 b. 6 c. \(2 / 3\) d. \(3 / 2\)

Minimum value of \((b+c) / a+(c+a) / b+(a+b) / c\) (for real positive numbers \(a, b, c)\) is a. 1 b. 2 c. 4 d. 6

In how many parts an integer \(N \geq 5\) should be dissected so that the product of the parts is maximized.

If the product of \(n\) positive numbers is \(n^{\pi}\), then their sum is a. \(a\) positive integer b. divisible by \(n\) c. equal to \(n+l / n\) d. never less than \(n^{2}\)

If \(x+y+z=1\) and \(x, y, z\) are positive, then show that $$ \left(x+\frac{1}{x}\right)^{2}+\left(x+\frac{1}{y}\right)^{2}+\left(z+\frac{1}{z}\right)^{2}>\frac{100}{3} $$

The minimum value of \(P=b c x+c a y+a b z\), when \(x y z=a b c\), is a. \(3 a b c\) b. \(6 a b c\) c. \(a b c\) d. \(4 a b c\)

If \(l, m, n\) be the three positive roots of the equation \(x^{3}-a x^{2}\) \(+b x-48=0\), then the minimum value of \((1 / l)+(2 / m)\) \(+(3 / n)\) equals a. 1 b. 2 c. \(3 / 2\) d. \(5 / 2\)

If positive numbers \(a, b, c\) be in H.P., then equation \(x^{2}-k x\) \(+2 b^{10 i}-a^{1011}-c^{i a 1}=0(k \in R)\) has a. both roots positive b. both roots negative c. one positive and one negative root d. both roots imaginary

For \(x^{2}-(a+3)(x)+4=0\) to have real solutions, the range of \(a\) is a. \((-\infty,-7] \cup[1, \infty)\) b. \((-3, \infty)\) c. \((-\infty,-7]\) d. \([1, \infty)\)

If \(a, b, c\) are 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 equal to a. 3 b. 6 c. 9 d. 12

If \(a, b, c \in R^{+}\), then \(\frac{b c}{b+c}+\frac{a c}{a+c}+\frac{a b}{a+b}\) is always a. \(\leq \frac{1}{2}(a+b+c)\) b. \(\geq \frac{1}{3} \sqrt{a b c}\) c. \(\leq \frac{1}{3}(a+b+c)\) d. \(\geq \frac{1}{2} \sqrt{a b c}\)

If \(a, b, c \in R^{*}\), then \((a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\) is always a. \(\geq 12\) b. \(\geq 9\) c. \(\leq 12\) d. none of these

If \(a, b, c \in R^{+}\), then the minimum value of \(a\left(b^{2}+c^{2}\right)+b\left(c^{2}+a^{2}\right)\) \(+c\left(a^{2}+b^{2}\right)\) is equal to a. \(a b c\) b. \(2 a b c\) c. \(3 a b c\) d. \(6 a b c\)

If \(a, b, c, d \in R^{+}\)and \(a, b, c, d\) are in H.P., then a. \(a+d>b+c\) b. \(a+b>c+d\) c. \(a+c>b+d\) d. none of these

If \(a, b, c \in R^{+}\)such that \(a+b+c=18\), then the maximum value of \(a^{2} b^{3} c^{4}\) is equad to a. \(2^{18} \times 3^{2}\) b. \(2^{18} \times 3^{3}\) c. \(2^{19} \times 3^{2}\) d. \(2^{19} \times 3^{3}\)