## Some questions on progressions for IITJEE Mains

(1) If $\log_{2}(5.2^{x}+1)$, $_{4}(2^{1-x}+1)$ and 1 are in AP, then x is equal to

(a) $\frac{\log (5)}{\log (2)}$

(b) $\log_{2}(2/5)$

(c) $1-\frac{\log (5)}{\log (2)}$

(d) $\frac{\log (2)}{\log (5)}$

(2) Values of the positive integer m for which $n^{m}+1$ divides $1+n+n^{2}+ \ldots+n^{127}$ are

(a) 8 (b) 16 (c) 32 (d) 64

(3) If p, q, r are positive and are in AP, the roots of the quadratic equation $px^{2}+qx+r=0$ are all real for

(a) $|\frac{r}{p}-7| \geq 4\sqrt{3}$ (b) $|\frac{q}{r}-4| \geq 2\sqrt{3}$

(c) $|\frac{p}{r}-7| \geq 4\sqrt{3}$ (d) $|\frac{q}{p}-4| \geq 2\sqrt{3}$

(4) If sum of the GP p,1, $\frac{1}{p^{2}}$, $\frac{1}{p^{3}}$, … is 9/2, the value of p is

(a) 3

(b) 2/3

(c) 3/2

(d) 1/3

(5) The roots of $x^{3}+bx^{2}+cx+d=0$ are

(a) in AP if $2b^{3}-9bc+27d=0$

(b) in GP if $b^{3}d=c^{3}$

(c) in GP if $27d^{3}=9bcd^{2}-2c^{3}d$

(d) equal if $c^{3}=b^{3}+3bc$