Sum of Binomial Coefficients of Even Numbered Terms

Q.

Let ${(1+x+2x^2)}^{20}=a_0+a_{1}x+a_2{x}^2+\dots \dots +a_{40}{x}^{40}$. Then, $a_1+a_3+a_5+\dots .+a_{37}$ is equal to

1. $2^{20}(2^{20}+21)$

2. $2^{19}(2^{20}+21)$

3. $2^{20}(2^{20}–21)$

4. $2^{19}(2^{20}–21)$

Q. Let $(x+10{)}^{50}+(x-10{)}^{50}$
$=a_0+a_{1}x+a_2{x}^2+...+a_{50}{x}^{50}$, for all
$x\in R$; then $\frac{a_2}{a_0}$ is equal to :

1. $12.75$
2. $12.50$
3. $12.25$
4. $12.00$

Q. The value of $C_0^2+3C_1^2+5C_2^2+\cdots$ up to $51$ terms is equal to
( where $C_r={}^{50}{C}_r$)

1. $51\cdot 2^{50}$
2. $102\cdot {}^{100}{C}_{50}$
3. $51\cdot {}^{100}{C}_{50}$
4. $99\cdot {}^{100}{C}_{50}$

Q.

Show that subtraction and division are not binary operations on $N$.

Q. If $n$ is the degree of the polynomial, ${\left[\frac{2}{\sqrt{5x^3+1}-\sqrt{5x^3-1}}\right]}^8+{\left[\frac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]}^8$ and $m$ is the coefficient of $x^n$ in it, then the ordered pair $(n,m)$ is equal to :

1. $(8,5(10{)}^4)$
2. $(12,(20{)}^4)$
3. $(24,(10{)}^8)$
4. $(12,8(10{)}^4)$

Q.

The sum of the coefficients of all the integral powers of x in $(1+3\sqrt{x}{)}^{100}$ is

1. 2⁹⁹ [2¹⁰⁰ - 1]

2. 2⁹⁹ [2¹⁰⁰ + 1]

3. 4¹⁰⁰ + 2¹⁰⁰

4. None of these

Q. If the sum of the series ${}^{40}{C}_0+{}^{40}{C}_4+{}^{40}{C}_8+\cdots +{}^{40}{C}_{40}$ is $2^a(2^a+1)$, then the value of $a$ is

Q. The value of $C_0^2+3C_1^2+5C_2^2+\cdots$ up to $51$ terms is equal to
( where $C_r={}^{50}{C}_r$)

1. $51\cdot {}^{100}{C}_{50}$
2. $99\cdot {}^{100}{C}_{50}$
3. $51\cdot 2^{50}$
4. $102\cdot {}^{100}{C}_{50}$

Q. If ${(1+x-2x^2)}^6=1+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots +a_{12}{x}^{12},$ then the value of
$a_2+a_4+a_6+\cdots +a_{12}$ will be

1. $31$
2. $64$
3. $32$
4. $1024$

Q. ${\left(\frac{1}{x^{a-b}}\right)}^{\frac{1}{(a-c)}}.{\left(\frac{1}{x^{b-c}}\right)}^{\frac{1}{(b-a)}}.{\left(\frac{1}{x^{c-a}}\right)}^{\frac{1}{(c-b)}}=$

1. $0$
2. $1$
3. $a+b+c$
4. $(a-b+c{)}^2$

Q.

If $(1-x+x^2{)}^n=a_0+a_{1}x+a_2{x}^2+...a_{2n}{x}^{2n}$, find the value of $a_0+a_2+a_4+.....+a_{2n}.$

Q. For any positive integer $m,n$ with $n\ge m,$ let $\left(\begin{array}{c}n\\ m\end{array}\right)={}^n{C}_m.$ Then
$\left(\begin{array}{c}n\\ m\end{array}\right)+\left(\begin{array}{c}n-1\\ m\end{array}\right)+\left(\begin{array}{c}n-2\\ m\end{array}\right)+\cdots +\left(\begin{array}{c}m\\ m\end{array}\right)$ is equal to

1. ${}^n{C}_{m+1}$
2. ${}^n{C}_m$
3. ${}^{n+1}{C}_{m+1}$
4. ${}^{n+1}{C}_m$

Q. For each $n\in N,3^{2n}-1$ is divisible by

1. $8$
2. $9$
3. $16$
4. $32$

Q. Let $(x+10{)}^{50}+(x-10{)}^{50}=a_0+a_{1}x+a_2{x}^2+...+a_{50}{x}^{50}$, for all $x\in R$, then $\frac{a_2}{a_0}$ is equal to:

1. $12.75$
2. $12.50$
3. $12.25$
4. $12.00$

Q. If the sum of the coefficients of all even powers of $x$ in the product $(1+x+x^2+x^3+....+x^{2n})(1-x+x^2-x^3....+x^{2n})$ is $61$, then $n$ is equal to

Q. If $\left|a\right|=2$ $,\left|b\right|=3$ , $\left|c\right|=4$ and $a+b+c=0$ then the value of $b\cdot c+c\cdot a+a\cdot b$ is equal to

1. $19/2$
2. $-19/2$
3. $-29/2$
4. $29/2$

Q. There will be no term containing $x^{2r}$ in the expansion of ${(x+x^{-2})}^{n-3}$ if $(n-2r)$ is positive but not a multiple of

1. $11$
2. $5$
3. $3$
4. $2$

Q. Assertion :$f:R\to R$ is a function defined by $f\left(x\right)=\frac{5x-8}{3}$ then $f^{-1}\left(x\right)=\frac{3x+8}{5}$ Reason: $f\left(x\right)$ is not a bijection.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. If $C_0,C_1,C_2,\dots C_n$ are coefficients of expansion $(1+x{)}^n$ then prove that:
$C_0+3.C_1+5.C_2+\dots +(2n+1).C_n=(n+1)2^n$.

Q. Find the sum $2C_o+\frac{2^2}{2}{C}_1+\frac{2^3}{3}{C}_2+\frac{2^4}{4}{C}_3+...+\frac{2^{11}}{11}{C}_{10}$

1. $\frac{3^{11}}{11}$
2. $\frac{3^{10}+1}{11}$
3. $\frac{3^{11}-1}{11}$
4. $\frac{3^{10}-1}{11}$

Q. If $A\ne A^2=I,$ then $|I+A|=$

Q. Consider the piecewise defined function $\{\begin{array}{cc}\sqrt{-x},& \text{if}x<0\\ 0,& \text{if}0\le x\le 4\\ x-4,& \text{if}x>4\end{array}$.Choose the answer which best describes the continuity of this function-

1. the function is right continuous at $x=0$
2. the function is unbounded and therefore cannot be continuous
3. the function has a removable discontinuity at 0 and 4, but is continuous on the rest of the real line
4. the function is continuous on the entire real line

Q.

Find the sum of 10 terms of the series whose $n^{th}$ term is 3.$2^n$.

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Q. Evaluate the following integrals:
$\int \frac{\sqrt{16+{\left(\log x\right)}^2}}{x}dx$

Q. Evaluate the following integrals:
$\int \frac{x}{\sqrt{x^4+a^4}}dx$