Algebraic Expression

Q. If ${}^{20}{C}_r$ is the co-efficient of $x^r$ in the expansion of $(1+x{)}^{20},$ then the value $\sum _{r=0}^{20}{r}^2{}^{20}{C}_r$ is equal to:

1. $420\times 2^{19}$
2. $420\times 2^{18}$
3. $380\times 2^{19}$
4. $380\times 2^{18}$

Q.

Let $P=\left[\begin{array}{ccc}-30& 20& 56\\ 90& 140& 112\\ 120& 16& 14\end{array}\right]$ and $A=\left[\begin{array}{ccc}2& 7& {\omega }^2\\ -1& -\omega & 1\\ 0& -\omega & -\omega +1\end{array}\right]$

where $𝛚=\frac{(-1+i\sqrt{3})}{2}$, and $I_3$ be the identity matrix of order $3.$

If the determinant of the matrix $\begin{array}{rcl}& & {\left|P^{1}AP-1\right|}^2\end{array}$ is $\begin{array}{rcl}& & \alpha {\omega }^2\end{array}$, then the value of $\alpha$ is equal to

Q. Integrate (tanx.tan2x)dx

Q.

If $x,yandz$ are all different and

$\left[\begin{array}{ccc}x& x^2& 1+x^3\\ y& y^2& 1+y^3\\ z& z^2& 1+z^3\end{array}\right]=0$

then:

1. $xyz=-1$

2. $xyz=1$

3. $xyz=-2$

4. $xyz=2$

Q. The total number of two digit numbers $n$, such that $3^n+7^n$ is a multiple of $10$, is

Q. why adjA.A ia equal to determinantOf A.I

Q. Combine the simple sentences to form a meaningful compound sentence:
He must hurry. He will not catch the train. [Intermediate]

1. He must hurry to catch the train.
2. He must run fast or he will not catch the train.
3. He must run fast if he wants to catch the train.
4. If he does not hurry, he will not catch the train.

Q. If the third term in the expansion of ${[x+x^{{\log }_{10}x}]}^5$ is ${10}^6$, then $x$ can be

1. ${10}^{-1/3}$
2. $10$
3. ${10}^{-5/2}$
4. ${10}^2$

Q. If $T_0,T_1,\cdots ,T_n$ represent the terms in the expansion of $(x+a{)}^n$, then the value of $(T_0-T_2+T_4-\cdots {)}^2+(T_1-T_3+T_5-\cdots {)}^2$ is

1. $a^{2n}$
2. $(x+a{)}^{2n}$
3. $(x^2+a^2{)}^n$
4. $a^{2n}$

Q. If $T_0,T_1,\cdots ,T_n$ represent the terms in the expansion of $(x+a{)}^n$, then the value of $(T_0-T_2+T_4-\cdots {)}^2+(T_1-T_3+T_5-\cdots {)}^2$ is

1. $(x+a{)}^{2n}$
2. $(x^2+a^2{)}^n$
3. $a^{2n}$
4. $a^{2n}$

Q. $\underset{x\to 0}{lim}\frac{e^x+e^{-x}+2cosx-4}{x^4}$ is equal to

1. $0$
2. $\frac{1}{6}$
3. $-\frac{1}{6}$
4. $1$

Q. In an A.P, $5$ times the $5^{th}$ terms is equal to $8$ times the $8^{th}$ term, then its ${13}^{th}$ terms is

Q. Sum of the series $4+6+9+13+18+...$ to $n$ terms be $=\frac{n}{k}(n^2+3n+m)$.Find $m-k$ ?

Q. Combine the following to form a simple sentence:
He is very lazy. He will not succeed.

1. He is too lazy to succeed.
2. He will not succeed because he is very lazy.
3. If he is very lazy, he will not succeed.
4. He is very lazy and will not succeed.

Q. Combine the two simple sentences below to form a simple sentence:
He was courageous. He was rewarded.

1. He was courageous and he was rewarded.
2. He was rewarded for his courage.
3. He was rewarded because he was courageous.
4. Though he was courageous, he was not rewarded.

Q. ${\int }_{-\sqrt{2}}^{\sqrt{2}}\frac{2x^7+3x^6-10x^5-7x^3-12x^2+x+1}{x^2+2}dx$ equals

1. $\frac{\pi }{\sqrt{2}}-16\sqrt{2}$
2. $\frac{\pi }{2\sqrt{2}}-16\sqrt{2}$
3. $\frac{\pi }{2\sqrt{2}}-\frac{16}{5}\sqrt{2}$
4. None of these

Q. What is a Gamma Function?

Q. How many terms of the series $2+6+18+$...must be be taken to make the sum equal to $728$?

Q. Evaluate:
$(-5)-(-48)÷(-16)+(-2)\times 6$

Q. Find the $8th$ term of the geometric progression $2,\frac{2}{3},\frac{2}{9},\frac{2}{27},...$

Q. $(2m^2-3m+10)÷(m-5)$.

Q. How do you find the Maclaurin series for $e^x$?

Q. Combine the two simple sentences below to form a simple sentence:
They must practice hard. They will win.

1. If they practice hard, they will win.
2. Unless they practice hard, they cannot win.
3. They must practice hard and they will win.
4. They must practice hard to win.

Q. Write the first five term of the sequence whose$n^{th}$ term is $a_n=n\frac{n^2+5}{4}$.

Q. Find the sum of n terms of the series $1.5+2.15+3.31+4.53+5.81+....$