Question

Choose the correct option for the following statements. Statement 1 : If non-zero number a, b, c, d are in A.P, then the numbers abc, abd, acd, bcd are in H.P. Statement 2 : There exists an A.P. whose three terms are $\sqrt{2},\sqrt{3},\sqrt{5}.$ Statement 3 : If the sides of a right angled triangle are in G.P. then the common ratio of the G.P. can take two and only two values.

• A. T F F
• B. T F T
• C. F T T
• D. F T F

Answer 1

The correct option is B T F T

(i) a, b, c, d are in A.P.
Divide by abcd
$\frac{1}{bcd},\frac{1}{acd},\frac{1}{abd},\frac{1}{abc}$ will be in A.P.
Hence, statement 1 is True.
(ii) $Let{p}^{th},q^{th}and{r}^{th}termsbe\sqrt{2},\sqrt{3}and\sqrt{5}$
$A+(p-1)d=\sqrt{2}$ ....(i)
$A+(q-1)d=\sqrt{3}$ ....(ii)
$A+(r-1)d=\sqrt{5}$ ....(iii)
by (i), (ii) and (iii)
$(p-q)d=\sqrt{2}-\sqrt{3}(r-q)d=\sqrt{3}-\sqrt{5}$
$\frac{p-q}{r-q}=\frac{\sqrt{2}-\sqrt{3}}{\sqrt{3}-\sqrt{5}}$
Rational $\ne$ Irrational
Hence statement is false.
(iii)Let sides be a, ar, $a{r}^2$
$\Rightarrow (a{r}^2{)}^2=a^2+(ar{)}^2{r}^4-r^2-1=0r^2=\frac{1\pm \sqrt{5}}{2}r=\pm \sqrt{\frac{1+\sqrt{5}}{2}}$
Hence only two values
Statement is true.