daad
- In a H.P., $T_{p}=q(p+q)$ and $T_{q}=p(p+q)$, then $p$ and $q$ are roots of the equation
(A) $x^{2}-T_{p+q} x+T_{p q}=0$
(B) $x^{2}-T_{p q} x+T_{p+q}=0$
(C) $x^{2}-2 T_{p+q} x+T_{p q}=0$
(D) $x^{2}-T_{p q} x+2 T_{p+q}=0$
- The sum of the infinite series $1+(1+a) x+\left(1+a+a^{2}\right) x^{2}+\left(1+a+a^{2}+a^{3}\right) x^{3}+\ldots$ where $a>0, x<1$ is
(A) $\frac{1}{(1-x)(1-a)}$
(B) $\frac{1}{(1-x)(1-a x)}$
(C) $\frac{1}{(1-a)(1-a x)}$
(D) $\frac{1}{(1-x)(1+a)}$
- $\frac{1}{1!(n-1)!}-\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}-\ldots \ldots .$. is equal to
(A) $\frac{2^{n-1}}{n!}$ for all $n \in N$
(B) $\frac{2^{n-1}}{n!}$ for odd values of $n$ only
(C) $\frac{2^{n-1}}{n!}$ for even values of $n$ only
(D) none of these
- If $\mathrm{x}>0$, then the minimum value of $\mathrm{x}^{1000}+\mathrm{x}^{900}+\mathrm{x}^{90}+\mathrm{x}^{6}+\frac{1996}{\mathrm{x}}$ is
(A) 1000
(B) 2000
(C) 1996
(D) 3000
Solution
- A sequence of real numbers $a_{1}, a_{2}, a_{3} \ldots$ is such that $a_{1}=0,\left|a_{2}\right|=\left|a_{1}+1\right|,\left|a_{3}\right|=\left|a_{2}+1\right| \ldots$ and
$\left|a_{n}\right|=\left|a_{n-1}+1\right|$, the arithmetic mean $\frac{a_{1}+a_{2}+a_{3}+\ldots .+a_{n}}{n}$ is
(A) $\geq-\frac{1}{3}$
(B) $\leq-\frac{1}{2}$
(C) $\geq 1$
(D) none of these
- If $a, b, c$ are in H.P., then the value of $\frac{a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}}{a^{2} c^{2}}$ is
(A) $9 \mathrm{ac}-6 \mathrm{~b}^{2}$
(B) $3 a c-2 b^{2}$
(C) $9 a c-4 b^{2}$
(D) $9 a c-2 b^{2}$
- Consider straight line $a x+b y=c$ where $a b c \in R^{+}$and $a, b, c$ are distinct. This line meets the coordinate axes at $P$ and $Q$ respectively. If area of $\triangle O P Q$, ’ $O$ ’ being origin does not depend upon $a, b$ and $c$ then
(A) a, b, c are in GP
(B) $a, b, c$ are in HP
(C) a, b, c are in AP
(D) none of these
- For the series $21,22,23, \ldots, k-1, k$; the A.M. and G.M. of the first and last numbers exist in the given series. If ’ $k$ ’ is a three digits number, then ’ $k$ ’ can attain
(A) 5 values
(B) 6 values
(C) 2 values
(D) 4 values