Differentiation to Solve Modified Sum of Binomial Coefficients

Q.

3sinA + 4cosA = 5 then what is the value of 4sinA - 3cosA

Q.

An A.P. consists of $50$ terms of which $3rd$ term is $12$ and the last term is $106$. Find the ${29}^{th}$ term.

Q. If $(1+x{)}^{15}=C_0+C_{1}x+C_2{x}^2+......+C_{15}{x}^{15},$ then $C_2+2C_3+3C_4+......+14C_{15}=$

1. ${14.2}^{14}$
2. ${13.2}^{14}+1$
3. ${13.2}^{14}-1$
4. None of these

Q.

When ${32}^{33}$ is divided by 34, it leaves the remainder

1. 2

2. 4

3. 8

4. 32

Q. The value of $\sum _{r=1}^{19}\frac{{}^{20}{C}_{r+1}\cdot (-1{)}^r}{2^{2r+1}}$ is

1. $2({\left(\frac{3}{4}\right)}^{20}+4)$
2. $-2({\left(\frac{3}{4}\right)}^{20}+4)$
3. $2({\left(\frac{3}{4}\right)}^{20}-4)$
4. $-2({\left(\frac{3}{4}\right)}^{20}-4)$

Q. If $f\left(x\right)=2x^2,$ find $\frac{f\left(3.8\right)-f\left(4\right)}{3.8-4}$

1. $1.56$
2. $156$
3. $15.6$
4. $0.156$

Q. Statement-1: ${\sum }_{r=0}^n(r+1){}^n{C}_r=(n+2)2^{n-1}.$
Statement-2: ${\sum }_{r=0}^n(r+1){}^n{C}_r{x}^r=(1+x{)}^n+nx(1+x{)}^{n-1}.$

1. Statemet -1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
2. Statement -1 is true, Statement -2 is false
3. Statemet -1 is false, Statement - 2 is ture
4. Statement -1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1

Q. Find the value of 'p' for which the vectors $3\hat{i}+2\hat{j}+9\hat{k}$ and $\hat{i}-2p\hat{j}+3\hat{k}$ are parallel. [CBSE 2014]

Q.

An AP starts with a positive fraction and every alternate term is an integer. If the sum of the first 11 terms is 33, then the fourth term is

1. 2

2. 3

3. 5

4. 6

Q.

Let $a,b,c$ be in AP and $\left|a\right|<1,\left|b\right|<1$, $\left|c\right|<1$. If $x=1+a+a^2+...+\infty ,y=1+b+b^2+...+\infty ,z=1+c+c^2+...+\infty$, then $x,y\&z$ are in:

1. AP

2. GP

3. HP

4. None of these

Q. Classify the following functions as injection, surjection or bijection :
(i) f : N → N given by f(x) = x²
(ii) f : Z → Z given by f(x) = x²
(iii) f : N → N given by f(x) = x³
(iv) f : Z → Z given by f(x) = x³
(v) f : R → R, defined by f(x) = |x|
(vi) f : Z → Z, defined by f(x) = x² + x
(vii) f : Z → Z, defined by f(x) = x − 5
(viii) f : R → R, defined by f(x) = sinx
(ix) f : R → R, defined by f(x) = x³ + 1
(x) f : R → R, defined by f(x) = x³ − x
(xi) f : R → R, defined by f(x) = sin²x + cos²x
(xii) f : Q − {3} → Q, defined by $f\left(x\right)=\frac{2x+3}{x-3}$
(xiii) f : Q → Q, defined by f(x) = x³ + 1
(xiv) f : R → R, defined by f(x) = 5x³ + 4
(xv) f : R → R, defined by f(x) = 3 − 4x
(xvi) f : R → R, defined by f(x) = 1 + x²
(xvii) f : R → R, defined by f(x) = $\frac{x}{x^2+1}$ [NCERT EXEMPLAR]

Q. Let ${(1-x+x^4)}^{10}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{40}{x}^{40}.$ Then the correct option(s) is (are)

1. $a_0+a_2+a_4+.....+a_{40}=\frac{3^{10}+1}{2}$
2. $a_0=a_{40}=1$
3. $a_1=a_{39}$
4. $a_{39}=0$

Q. If $(1+x{)}^{15}=C_0+C_{1}x+C_2{x}^2+......+C_{15}{x}^{15},$ then $C_2+2C_3+3C_4+......+14C_{15}=$

1. 
2. 
3. 
4. None of these

Q. lim x–>1 (x+1)⁴–2⁴/(2x+1)⁵–3⁵

Q.

The ratio of the A.M and G.M. of two positive numbers a and b, is m: n. Show that .

Q. Remainder when $(2{)}^{1004}$ is divided by 17 is

1. 14
2. 16
3. 7
4. 10

Q. Evaluate the following integrals:

$\int \frac{x}{\left(x^2+1\right)\left(x-1\right)}\mathrm{d}x$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=\frac{e^{\left|x\right|}-e^{-x}}{e^x+e^{-x}}.\mathrm{Then},$
(a) f is a bijection
(b) f is an injection only
(c) f is surjection on only
(d) f is neither an injection nor a surjection

Q. If $y=\frac{e^x-e^{-x}}{e^x+e^{-x}}$, prove that $\frac{dy}{dx}=1-y^2$

Q. Express the vector $\overrightarrow{a}=5\hat{i}-2\hat{j}+5\hat{k}$ as the sum of two vectors such that one is parallel to the vector $\overrightarrow{b}=3\hat{i}+\hat{k}$ and other is perpendicular to $\overrightarrow{b}$.

Q. The value of $\sum _{r=2}^{10}{}^r{C}_2\cdot {}^{10}{C}_r$ is not divisible by

1. $10$
2. $5$
3. $7$
4. $3$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are vectors of lengths $2,3,4$. If $\overrightarrow{a}$ is orthogonal to $\overrightarrow{b}+\overrightarrow{c},\overrightarrow{b}$ is orthogonal to $\overrightarrow{c}+\overrightarrow{a}$ and $\overrightarrow{c}$ is orthogonal to $\overrightarrow{a}+\overrightarrow{b}$, then the modulus of $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ is

1. $\sqrt{29}$
2. $\sqrt{39}$
3. $\sqrt{45}$
4. $2\sqrt{19}$

Q. $7^{2n}+3^{n-1}\cdot 2^{3n-3}$ is divisible by

1. $25$
2. $24$
3. $13$
4. $9$

Q. If $\sum \sum _{0\le i<j\le n}j{}^n{C}_i=320$.
Then the value of $n$ is

Q. A series whose nth term is $\left(\frac{n}{x}\right)+y$ then sum of r terms will be

1. 
2. 
3. 
4. 

Q. The sum of the series $a-(a+d)+(a+2d)-(a+3d)+\cdots$upto$(2n+1)$ terms is

1. $a+2nd$
2. $2nd$
3. $a+nd$
4. $-nd$

Q. If $\int \frac{4e^x+6e^{-x}}{9e^x-4e^{-x}}dx=ax+b\log (9e^{2x}+c)+d$, then descending order of $a,b,c$, is

1. $a,b,c$
2. $b,c,a$
3. $b,a,c$
4. $c,a,b$

Q. In the expansion of ${(x^3+{3.2}^{-{\log }_2{x}^3})}^{11}$

1. there appears a term with the power $x^2$
2. there does not appear a term with the power $x^2$
3. there appear a term with the power $x^{-3}$
4. the ratio of the co-efficient of $x^3$ to that of $x^{-3}$ is $\frac{1}{3}$

Q. $\underset{r=1}{\overset{n}{\sum r\cdot }}nCr=$

1. $n.2^{n-1}$
2. $n$
3. $2n$
4. $2^n$

Q. The sum upto $(2n+1)$ terms of the series $a^2-(a+d{)}^2+(a+2d{)}^2-(a+3d{)}^2+....$ is

1. $a^2+2nad+n(n-1)d^2$
2. $a^2+3n{d}^2$
3. $a^2+3nad+n(n-1)d^2$
4. $a^2+2nad+n(2n+1)d^2$