Relation between AM and GM — practice questions

Question 1 · difficulty L2 · understanding

The minimum value of $$f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$$, where a, $$x \in R$$ and a > 0, is equal to :

• A. $$a + {1 \over a}$$
• B. 2a
• C. a + 1
• D. $$2\sqrt a$$

Question 2 · difficulty L3 · understanding

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. and $\mathrm{a}^2, 2 \mathrm{~b}^2, \mathrm{c}^2$ are in G.P. If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, then $9\left(\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2\right)$ is equal to $\_\_\_\_$ .

Question 3 · difficulty L3 · understanding

If the arithmetic mean and geometric mean of the p th and q th terms of the sequence $$-$$16, 8, $$-$$4, 2, ...... satisfy the equation 4x 2 $$-$$ 9x + 5 = 0, then p + q is equal to __________.

Question 4 · difficulty L3 · understanding

Let m be the minimum possible value of $${\log _3}({3^{{y_1}}} + {3^{{y_2}}} + {3^{{y_3}}})$$, where $${y_1},{y_2},{y_3}$$ are real numbers for which $${{y_1} + {y_2} + {y_3}}$$ = 9. Let M be the maximum possible value of $$({\log _3}{x_1} + {\log _3}{x_2} + {\log _3}{x_3})$$, where $${x_1},{x_2},{x_3}$$ are positive real numbers for which $${{x_1} + {x_2} + {x_3}}$$ = 9. Then the value of $${\log _2}({m^3}) + {\log _3}({M^2})$$ is ...........

Question 5 · difficulty L3 · understanding

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}$, $a > 2$. If $\alpha$ is such that $a$, $4$, $\alpha$, $b$ are in A.P., then the equation $\alpha x^2 - a x + 2(\alpha - 2b) = 0$ has :

• A. one root in $(1, 4)$ and another in $(-2, 0)$
• B. one root in $(0, 2)$ and another in $(-4, -2)$
• C. both roots in the interval $(-2, 0)$
• D. complex roots of magnitude less than $2$

Question 6 · difficulty L3 · understanding

Let three real numbers $$a, b, c$$ be in arithmetic progression and $$a+1, b, c+3$$ be in geometric progression. If $$a>10$$ and the arithmetic mean of $$a, b$$ and $$c$$ is 8, then the cube of the geometric mean of $$a, b$$ and $$c$$ is

• A. 120
• B. 316
• C. 312
• D. 128

Question 7 · difficulty L3 · understanding

Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :

• A. -4
• B. -1
• C. 13
• D. 11

Question 8 · difficulty L3 · understanding

Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4}+G_{2}^{4}+G_{3}^{4}+G_{1}^{2} G_{3}^{2}$ is equal to :

• A. $\left(A_{1}+A_{2}\right)^{2} G_{1} G_{3}$
• B. $\left(A_{1}+A_{2}\right) G_{1}^{2} G_{3}^{2}$
• C. $2\left(A_{1}+A_{2}\right) G_{1}^{2} G_{3}^{2}$
• D. $2\left(A_{1}+A_{2}\right) G_{1} G_{3}$

Question 9 · difficulty L3 · understanding

Let $$a, b, c$$ and $$d$$ be positive real numbers such that $$a+b+c+d=11$$. If the maximum value of $$a^{5} b^{3} c^{2} d$$ is $$3750 \beta$$, then the value of $$\beta$$ is

• A. 110
• B. 108
• C. 90
• D. 55

Question 10 · difficulty L3 · understanding

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

• A. 21
• B. 22
• C. 23
• D. 24