Summation by Sigma Method

Q.

Question 30

If on increasing a, b decreases in such a manner that___remains ___and positive, then a and b are said to vary inversely with each other.

Q.

We can split the middle term of a⁴ + 5a² + 6 as :

1. a² and 4a²

2. 7a² and -2a²

3. 3a² and 2a²

4. 6a² and –a²

Q. If $\frac{1}{4}\times \frac{2}{6}\times \frac{3}{8}\times \frac{4}{10}\times \frac{5}{12}\times .....\times \frac{31}{64}=\frac{1}{2^x}$, then what is the value of $x$?

1. $6$
2. $36$
3. $42$
4. $48$

Q. IF $S_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ and $t_n={\sum }_{r=0}^n\frac{r}{{}^n{C}_r},$ then $\frac{t_n}{S_n}$ is equal to

1. $\frac{2n-1}{2}$
2. $\frac{1}{2}n-1$
3. $\frac{1}{2}n$
4. $n-1$

Q. complete the following series :-3, 6, 8, 16, 18, ?, ?

Q. ${\left(\frac{x^a}{x^b}\right)}^c\times {\left(\frac{x^b}{x^c}\right)}^a\times {\left(\frac{x^c}{x^a}\right)}^b$=_______.

1. $0$
2. $1$
3. $-1$
4. $2$

Q.

If we subtract $-4x^{2}y$ from $2x^{2}y$, we get ______.

1. 6$x^{2}y$

2. 6$x^2$$y^2$

3. -6$x^{2}y$

4. $-6x{y}^2$

Q. $(\frac{{27}^{-3}}{9^{-3}}{)}^{\frac{1}{5}}$, then answer is $\frac{1}{3^{\frac{3}{5}}}$

1. True
2. False

Q.

Which of the following series forms an AP?

$\left(i\right)a–2d,a–d,a,a+d,a+2d$

$\left(ii\right)a,a+d,a+2d,a+3d,a+4d$

$\left(iii\right)a–3d,a–d,a+d,a+3d$

1. $\left(ii\right)and\left(iii\right)$

2. $\left(i\right)and\left(ii\right)$

3. $\left(i\right),\left(ii\right)and\left(iii\right)$

4. $\left(i\right)and\left(iii\right)$

Q. Show that $n{(n+1)}^3<8(1^3+2^3+3^3+....+n^3)$.

Q. Subtract $7x-3x^2$ from $4x+8x^2$

1. $11x^2-3x$

2. $11x^2$

3. $-3x$

4. $11x^2-5x$

Q.

The sum of the product of the integers 1, 2, 3, ...., n taken two at a time is

1. $\frac{n(n+1)}{24}(3n^2-n-2)$

2. $\frac{n^2(n+1{)}^2}{2}(n^2+1)$

3. $\frac{n(n+1)}{18}(n^2-n-2)$

4. $\frac{n(n+1)(n+2)(n+3)}{24}$

Q. Divide :

1. $\frac{x^2{z}^{4}a}{y}$
2. $\frac{x^{2}za}{3y}$
3. $\frac{x{z}^{4}a}{3y}$
4. None of these

Q. Find the value of $\frac{3^{12+n}\times 9^{2n-7}}{3^{5n}}$

1. $\frac{1}{9}$
2. $\frac{4}{9}$
3. $\frac{3}{9}$
4. $\frac{2}{9}$

Q. If $a\ne 0$ and $a-\frac{1}{a}=4$, find $a^2+\frac{1}{a^2}$.

1. $11$
2. $21$
3. $18$
4. $14$

Q. The mean of 100 observations is $50$ and their standard deviation is $5$. The sum of the squares of all the observations is

1. $252500$
2. $250000$
3. $50000$
4. $255000$

Q. The value of $(5^{-1}\times 2^{-1})\times 6^{-1}$ is $\frac{1}{m}$, then $m$ is equal to

Q. If $a_1,a_2,a_3$ are in H.P and $f_k={\sum }_{r=1}^n{a}_r-a_k$; then $2^{{\alpha }_1},2^{{\alpha }_2},2^{{\alpha }_3},2^{{\alpha }_4},....$ are in
{where ${\alpha }_1=\frac{a_1}{f_1},{\alpha }_2=\frac{a_2}{f_2},{\alpha }_3=\frac{a_3}{f_3}....$}

1. G.P
2. H.P
3. A.P
4. none of these

Q. All equations are identities but all identities may not be equations.

1. False
2. True

Q. The value of ${\int }_{-1}^2\frac{|x|}{x}dx$ is?

1. $1$
2. $0$
3. $2$
4. $3$

Q. The product of $4a^2,-6b^2$ and $3a^2{b}^2$ is

1. $a^2{b}^2$
2. $13a^4{b}^4$
3. $-72a^4{b}^4$
4. $a^4{b}^4$

Q. The sum of first 9 terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$ is

1. 142
2. 192
3. 71
4. 96

Q.

Which expression matches the statement
"Subtract the sum of $5x$ and $3x$ from $12y$”?

1. $12y-5x+3x$

2. $12y-8x$

3. $5x+3x+12y$

4. $5x+3x-12y$