91Ó°ÊÓ

If in an A.P., \(S_{n}=p \cdot n^{2}\) and \(S_{m}=p . m^{2}\) where \(S_{r}\) denotes the sum of \(r\) terms of the A.P., then \(S_{p}\) is equal to (A) \(\frac{1}{2} p^{3}\) (B) \(m n p\) (C) \(p^{3}\) (D) \((m+n) p^{2}\)

If \(b_{1}, b_{2}\) and \(b_{3}\left(b_{1}>0\right)\) are three successive terms of a G.P. with common ratio \(r\), the value of \(r\) for which the inequality \(b_{3}>4 b_{2}-3 b_{1}\) holds, is given by (A) \(r>3\) (B) \(r<1\) (C) \(r=2.5\) (D) \(r=1.7\)

If \(p, q, r\) are positive and are in A.P., the roots of quadratic equation \(p x^{2}+q x+r=0\) are all real for (A) \(\left|\frac{r}{p}-7\right| \geq 4 \sqrt{3}\) (B) \(\left|\frac{p}{r}-7\right| \geq 4 \sqrt{3}\) (C) all \(p\) and \(r\) (D) no \(p\) and \(r\)

The sum to \(n\) terms of the series \(\frac{1}{3}+\frac{5}{9}+\frac{19}{27}+\frac{65}{81}+\ldots\) is (A) \(n-\frac{\left(3^{n}-2^{n}\right)}{2^{n}}\) (B) \(n-\frac{2\left(3^{n}-2^{n}\right)}{3^{n}}\) (C) \(2^{n}-1\) (D) \(3^{n}-1\)

Sum to \(n\) terms of the series \(\frac{1}{5 !}+\frac{1 !}{6 !}+\frac{2 !}{7 !}+\frac{3 !}{8 !}+\ldots\) is(A) \(\frac{2}{5 !}-\frac{1}{(n+1) !}\) (B) \(\frac{1}{4}\left(\frac{1}{4 !}-\frac{n !}{(n+4) !}\right)\) (C) \(\frac{1}{4}\left(\frac{1}{3 !}-\frac{3 !}{(n+2) !}\right)\) (D) None of these

If \(a, b, c, d\) and \(p\) are distinct real numbers such that \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2 p(a b+b c+c d)+\left(b^{2}+c^{2}+d^{2}\right)\) \(\leq 0\) then \(a, b, c, d\) are in (A) A.P. (B) G.P. (C) H.P. (D) \(a b=c d\)

If \(a+b+c=3\) and \(a>0, b>0, c>0\), then the greatest value of \(a^{2} b^{3} c^{2}\) is (A) \(\frac{3^{10} \cdot 2^{4}}{7^{7}}\) (B) \(\frac{3^{9} \cdot 2^{4}}{7^{7}}\) (C) \(\frac{3^{8} \cdot 2^{4}}{7^{7}}\) (D) None of these

If \(\left|\begin{array}{ccc}a & b & a \alpha-b \\ b & c & b \alpha-c \\ 2 & 1 & 0\end{array}\right|=0\) and \(\alpha \neq \frac{1}{2}\), then (A) \(a, b, c\) are in A.P. (B) \(a, b, c\) are in G.P. (C) \(a, b, c\) are in H.P. (D) None of these

Suppose \(a, b, c\) are in A.P. and \(a^{2}, b^{2}, c^{2}\) are in G.P. If \(a

If \(a_{1}, a_{2}, \ldots, a_{n}\) are in A.P. with common difference \(d \neq 0\), then sum of the series \(\sin d\left[\sec a_{1} \sec a_{2}+\sec \right.\) \(\left.a_{2} \sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right]\) is (A) \(\tan a_{n}-\tan a_{1}\) (B) \(\cot a_{n}-\cot a_{1}\) (C) \(\sec a_{n}-\sec a_{1}\) (D) \(\operatorname{cosec} a_{n}-\operatorname{cosec} a_{1}\)