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- MCQs with Answers (FSc/ICS Part 1) @fsc:fsc_part_1_mcqs
- (B) formula * (C) rational fraction * (D) theorem </col> <col sm="6"> * An arrangement of the num
- Important Questions: HSSC-I @fsc-part1-ptb
- uestions:ch08-mathematical-induction-and-binomial-theorem]] * [[fsc-part1-ptb:important-questions:ch09-f
- MTH424: Convex Analysis (Spring 2024) @atiq
- vex hull and their properties, Best approximation theorem. Convex functions, Basic definitions, properties,
- Definitions: FSc Part 1 (Mathematics): PTB @fsc-part1-ptb
- \sqrt{x+2}+\sqrt{x-3}=7$ * **Remainder Theorem:** If a polynomial $f(x)$ of degree $n \geq 1$ is... as a polynomial function of $x$. * **Factor Theorem:** The polynomial $(x-a)$ is a factor of the poly... = Chapter 08: Mathematical induction and binomial theorem ===== * **Binomial Theorem:** An algebraic expression consisting of two terms is called binomial expre
- Topology and Functional Analysis Solved Paper by Noman Khalid @notes
- Component * First category * Baire's category theorem * Absolutely convergent series * Finite dimen
- Question 5 & 6 Review Exercise 7 @math-11-kpk:sol:unit07
- n write $$ (0.99)^5=(1-0.01)^5 $$ Using binomial theorem, we have $$ \begin{aligned} & (1-0.01)^5 \cong{ }
- Question 9 Exercise 7.3 @math-11-kpk:sol:unit07
- right)(1-x)^2 \end{aligned} $$ Applying binomial theorem $$ \begin{aligned} & =\left(x^2+2 x+1\right)[1+2
- Question 7 and 8 Exercise 7.3 @math-11-kpk:sol:unit07
- f the above given equation and apply the binomial theorem $$ \begin{aligned} & (1+x)^{\frac{1}{4}}+(1-x)^{\
- Question 5 and 6 Exercise 7.3 @math-11-kpk:sol:unit07
- }\right)^{-2} \end{aligned} $$ Applying binomial theorem and neglecting $\frac{1}{x^3}$ etc $$ \begin{alig
- Question 3 Exercise 7.3 @math-11-kpk:sol:unit07
- )^{-\frac{1}{2}} \text {. } $$ Applying binomial theorem, $$ \begin{aligned} & (1-x)^{\frac{1}{2}}(1+x)^{\
- Question 14 Exercise 7.3 @math-11-kpk:sol:unit07
- x^p-q x^q=p(1+h)^p-q(1+h)^q $$ Applying binomial theorem on the R.H.S of the above last equation, $$ \begi
- Question 13 Exercise 7.3 @math-11-kpk:sol:unit07
- x\right]^{-1} \end{aligned} $$ Applying binomial theorem now $$ \begin{aligned} & {\left[1+\left(\frac{n+1
- Question 1 Exercise 7.3 @math-11-kpk:sol:unit07
- i) (1 - x) $\frac{1}{2}$ Solution: Using binomial theorem to tind the four terms $$ \begin{aligned} & (1-x)... s \\ & \end{aligned} $$ Solution: Using binomial theorem $$ \begin{aligned} & (1-x)^{\frac{3}{2}}=1-\frac{
- Question 8 Exercise 7.2 @math-11-kpk:sol:unit07
- \end{aligned} Hence by perpe:ts of like binomial theorem, we hisu that: $p+1$ - 5.. 1 - 6 icrm is numerica
- Question 7 Exercise 7.2 @math-11-kpk:sol:unit07
- (iii) $(a+b)^5+(a-b)^5$ Solution: Using binomial theorem $$ \begin{aligned} (a+b)^5+(a-b)^5&=\left[\left(\