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Mathematics important questions

From 4 NEB Class 12 past papers: the chapters that keep coming back and their most important questions, each with a solved model answer. No guarantees; study the whole syllabus.

Important questions with answers

From the most-tested chapters first, each with a solved model answer.

120818 marksNumericalCh 6: Calculus

Answer

(a) A curve $y^2=8x$ changes its abscissa and ordinate at the same rate; the ordinate is double the abscissa. Justify with calculation.

(b) $\int\frac{dx}{a+b\cos x}$ depends on the values of $a$ and $b$. Explain.

(a) Differentiate $y^2=8x$: $2y\dfrac{dy}{dt}=8\dfrac{dx}{dt}.$ Given $\dfrac{dy}{dt}=\dfrac{dx}{dt}$: $2y=8\Rightarrow y=4$, and then

$$ \begin{aligned} x &= \dfrac{y^2}{8} \ &= 2. \end{aligned} $$

$$ \begin{aligned} y &= 4 \ &= 2x \end{aligned} $$

(ordinate is double the abscissa) at the point $(2,4)$, as required.

(b) Substituting $t=\tan\tfrac x2$ gives $\int\dfrac{2,dt}{(a+b)+(a-b)t^2}.$ The form of the answer depends on the sign of $\dfrac{a-b}{a+b}$: if $a>b$ the integral is a $\tan^{-1}$ (inverse-tangent) form; if $a<b$ it is a logarithmic form; if $a=b$ it reduces to a simple rational integral. Hence the result depends on the relative values of $a$ and $b.$

(c) $M,dx+N,dy=0$ is exact iff $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}.$ Example: $2xy,dx+x^2,dy=0$ has $M=2xy,\ N=x^2$ with

$$ \begin{aligned} M_y &= 2x \ &= N_x, \end{aligned} $$

so it is exact and its solution is $x^2y=C.$

220728 marksCh 6: Calculus

Answer

(a) Evaluate $\int\frac{dx}{(x-1)^2(x-2)^3}$.

(b) Reduce $\frac{dy}{dx}+\frac{y}{x}=y^2$ to linear form and solve it. OR: Solve $\frac{dy}{dx}=\frac{y+1}{x+y+1}$.

(a) Partial fractions: $$\frac{1}{(x-1)^2(x-2)^3}=\frac{-3}{x-1}+\frac{-1}{(x-1)^2}+\frac{3}{x-2}+\frac{-2}{(x-2)^2}+\frac{1}{(x-2)^3}.$$ Integrating, $$\int=-3\ln|x-1|+\frac{1}{x-1}+3\ln|x-2|+\frac{2}{x-2}-\frac{1}{2(x-2)^2}+C.$$

(b) Bernoulli: divide by $y^2$ and put $v=y^{-1}$ ($v'=-y^{-2}y'$): $v'-\dfrac vx=-1.$ I.F. $=\tfrac1x$: $\left(\dfrac vx\right)'=-\dfrac1x\Rightarrow\dfrac vx=-\ln x+C.$ So $\dfrac1y=x(C-\ln|x|)$, i.e. $y=\dfrac{1}{x(C-\ln|x|)}.$

OR. Treat $x$ as function of $y$: $\dfrac{dx}{dy}-\dfrac{x}{y+1}=1.$ I.F. $=\dfrac1{y+1}$: $\dfrac{x}{y+1}=\ln|y+1|+C$, so $x=(y+1)(\ln|y+1|+C).$

320718 marksCh 6: Calculus

Answer

(a) Evaluate $\int\frac{dx}{1-3\sin x}$.

(b) Solve $xy\frac{dy}{dx}=x^2+y^2$. OR: Solve $\frac{dy}{dx}+\frac{2x}{1+x^2},y=\frac{1}{(1+x^2)^2}$.

(a) Put $t=\tan\tfrac x2$: $$ \begin{aligned} \int\frac{2,dt}{t^2-6t+1} &= \int\frac{2,dt}{(t-3)^2-8} \ &= \frac{1}{2\sqrt2}\ln\left\frac{t-3-2\sqrt2}{t-3+2\sqrt2}\right+C,\quad t=\tan\tfrac x2. \end{aligned} $$ (b) Homogeneous: $\dfr...

420708 marksCh 6: Calculus

Answer

(a) Evaluate $\int\frac{dx}{1+2\sin x}$.

(b) Solve $(1+x^2)\frac{dy}{dx}+2xy=4x^2$. OR: Solve $(x^2+y^2),dy=xy,dx$.

(a) Put $t=\tan\tfrac x2$, $\sin x=\dfrac{2t}{1+t^2}$, $dx=\dfrac{2,dt}{1+t^2}$: $$ \begin{aligned} \int\frac{2,dt}{t^2+4t+1} &= \int\frac{2,dt}{(t+2)^2-3} \ &= \frac{1}{\sqrt3}\ln\left\frac{t+2-\sqrt3}{t+2+\sqrt3}\right+C,\quad t=\t...

520726 marksNumericalCh 6: Calculus

Answer

(a) Compute $\int\frac{\coth x,dx}{\sinh x-9,\operatorname{cosech} x}$.

(b) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x-x-1}{x^2}$.

(a) Write $\coth x=\dfrac{\cosh x}{\sinh x}$ and $\operatorname{cosech}x=\dfrac1{\sinh x}$: $$\int\frac{\cosh x/\sinh x}{\sinh x-9/\sinh x},dx=\int\frac{\cosh x}{\sinh^2x-9},dx.$$ Let $u=\sinh x,\ du=\cosh x,dx$: $\displaystyle\int\fr...

620726 marksNumericalCh 6: Calculus

Answer

State the mean value theorem and verify it for $f(x)=\sqrt{x^2-4}$ on $[2,4]$. OR: Find from first principles the derivative of $\ln(\cos^{-1}x)$.

Statement. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then some $c\in(a,b)$ has $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Verification. $f(2)=0$, $$ \begin{aligned} f(4) &= \sqrt{12} \ &= 2\sqrt3, \end{aligned} $$ so $\dfrac{f...

720716 marksNumericalCh 6: Calculus

Answer

(a) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2 x}$.

(b) Evaluate $\int\frac{dx}{\sqrt{2ax+x^2}}$.

(a) This is a $0/0$ form. Differentiating numerator and denominator twice (each intermediate form is again $0/0$): $$ \begin{aligned} \lim{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2x} &= \lim{x\to0}\frac{e^x+e^{-x}+2\cos x}{2\cos2x} \ &= \f...

820716 marksNumericalCh 6: Calculus

Answer

State Rolle's theorem, interpret it geometrically, and verify it for $f(x)=x(x-3)^2$ on $[0,3]$. OR: Find from first principles the derivative of $\ln!\left(\sin\frac{x}{a}\right)$.

Statement. If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f(a)=f(b)$, then there is $c\in(a,b)$ with $f'(c)=0.$ Geometrically, the tangent at $x=c$ is horizontal (parallel to the $x$-axis). Verification. $f(0)=0=f(3)$. $...

920706 marksNumericalCh 6: Calculus

Answer

(a) Find the equation of the tangent to the curve $y=2x^3-5x^2+8$ at $(2,4)$.

(b) Evaluate $\int\frac{dx}{\sqrt{(x-\alpha)(x-\beta)}},\ (\beta>\alpha)$.

(a) $\dfrac{dy}{dx}=6x^2-10x$; at $x=2$, $$ \begin{aligned} \text{slope} &= 24-20 \ &= 4. \end{aligned} $$ Tangent: $$ \begin{aligned} y-4 &= 4(x-2)\ \Rightarrow\ y \ &= 4x-4. \end{aligned} $$ (b) Complete the square: $(x-\alpha)(x-\be...

1020706 marksNumericalCh 6: Calculus

Answer

State the mean value theorem, interpret it geometrically, and verify it for $f(x)=x(x-1)^2$ on $[0,2]$. OR: Find from first principles the derivative of $x^x$.

Statement. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c\in(a,b)$ with $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Geometrically, the tangent at $x=c$ is parallel to the chord joining $(a,f(a))$ and $(b,f(b)).$ Verif...

1120815 marksCh 6: Calculus

Answer

(a) Write the derivative of $\sinh x$.

(b) Under what condition is $\frac{d}{dx}(\operatorname{sech}^{-1}x)=-\frac{1}{x\sqrt{1-x^2}}$?

(d) Write $\int\frac{1}{\sqrt{x^2-a^2}},dx$.

(e) Write the integrating factor of $\frac{dy}{dx}+Py=Q$.

(a) $\dfrac{d}{dx}\sinh x=\cosh x.$ (b) Valid for $0<x<1$ (the domain of $\operatorname{sech}^{-1}x$ taking the positive branch). (c) $\Delta y$ is the actual change in $y$ for a change $\Delta x$; $dy=f'(x),dx$ is the approximate (diff...

1220815 marksNumericalCh 6: Calculus

Answer

(a) Find the equation of the tangent to $y=2x^3+x^2-7x-2$ at $x=2$.

(b) Evaluate $\int\frac{2x^2-x+4}{x^3+4x},dx$.

(a) $\dfrac{dy}{dx}=6x^2+2x-7$; at $x=2$,

$$ \begin{aligned} \text{slope} &= 24+4-7 \ &= 21. \end{aligned} $$

The point on the curve is $\big(2,,2(8)+4-14-2\big)=(2,4).$ Tangent: $$ \begin{aligned} y-4 &= 21(x-2)\ \Rightarrow\ y \ &= 21x-38. \end{aligned} $$

(b) Partial fractions with $x^3+4x=x(x^2+4)$: $\dfrac{2x^2-x+4}{x(x^2+4)}=\dfrac1x+\dfrac{x-1}{x^2+4}.$ $$\int=\ln|x|+\tfrac12\ln(x^2+4)-\tfrac12\tan^{-1}\tfrac x2+C.$$

1320811 marksNumericalCh 6: Calculus

Answer

$f'(x)=3x^2$ and $g'(x)=2x$. Using L'Hospital's rule, find $\lim_{x\to1}\frac{f(x)}{g(x)}$. A) $\frac32$ B) $\frac13$ C) $\frac12$ D) $\frac16$

A) $\dfrac32$. By L'Hospital's rule $$ \begin{aligned} \lim{x\to1}\dfrac{f(x)}{g(x)} &= \lim{x\to1}\dfrac{f'(x)}{g'(x)} \ &= \lim{x\to1}\dfrac{3x^2}{2x} \ &= \dfrac{3(1)}{2} \ &= \dfrac32. \end{aligned} $$

1420811 marksCh 6: Calculus

Answer

Which is $\int\frac{1}{x^2+4},dx$? A) $\frac12\tan^{-1}\frac2x+C$ B) $\frac12\tan^{-1}\frac x2+C$ C) $2\tan^{-1}\frac x2+C$ D) $\tan^{-1}\frac x2+C$

B) $\dfrac12\tan^{-1}\dfrac x2+C$. Using $\int\dfrac{dx}{x^2+a^2}=\dfrac1a\tan^{-1}\dfrac xa$ with $a=2.$

1520811 marksCh 6: Calculus

Answer

What is the degree of the ODE $\frac{d^3s}{dt^3}=\left(4+\left(\frac{d^2s}{dt^2}\right)^2\right)^{1/2}$? A) 1 B) 2 C) 3 D) 4

B) 2. Squaring to clear the radical, $\left(\dfrac{d^3s}{dt^3}\right)^2=4+\left(\dfrac{d^2s}{dt^2}\right)^2.$ The highest-order derivative $\dfrac{d^3s}{dt^3}$ then appears to power $2$, so the degree is $2.$

1620811 marksCh 6: Calculus

Answer

Which is an example of a homogeneous differential equation of first order? A) $3x,dy+2y,dx=4$ B) $x,dy-y,dx+1=0$ C) $x,dx+y,dy=2$ D) $(x^2+xy),dy-(xy-y^2),dx=0$

D) $(x^2+xy),dy-(xy-y^2),dx=0$. Every term is of the same (second) degree in $x,y$, so it can be written as $\dfrac{dy}{dx}=F!\left(\dfrac yx\right)$; the others contain constant terms and are not homogeneous.

1720818 marksNumericalCh 1: Algebra

Answer

(a) Prove that the coefficient of $x^n$ in $(1+x)^{2n}$ is twice the coefficient of $x^n$ in $(1+x)^{2n-1}$.

(b) Use De Moivre's theorem to find the square root of $-(2+2\sqrt3,i)$.

(a) The coefficients are $\binom{2n}{n}$ and $\binom{2n-1}{n}.$ Now

$$ \begin{aligned} \frac{\binom{2n}{n}}{\binom{2n-1}{n}} &= \frac{(2n)!/(n!,n!)}{(2n-1)!/(n!,(n-1)!)} \ &= \frac{(2n)!}{(2n-1)!}\cdot\frac{(n-1)!}{n!} \ &= 2n\cdot\frac1n \ &= 2, \end{aligned} $$

so $\binom{2n}{n}=2\binom{2n-1}{n}.\ \blacksquare$

(b) $-(2+2\sqrt3,i)=-2-2\sqrt3,i$ has modulus $4$ and argument $240^\circ$ (third quadrant): $4(\cos240^\circ+i\sin240^\circ).$ By De Moivre the square roots are $2\left(\cos\tfrac{240^\circ+360^\circ k}{2}+i\sin\cdots\right)$ for $k=0,1$: $2(\cos120^\circ+i\sin120^\circ)=-1+\sqrt3,i$ and $2(\cos300^\circ+i\sin300^\circ)=1-\sqrt3,i.$ So the square roots are $\pm(-1+\sqrt3,i).$ (Check $(-1+\sqrt3 i)^2=-2-2\sqrt3 i.$)

(c) Note: $n!<2^n$ actually holds only for $n=1,2,3$ ($1<2,\ 2<4,\ 6<8$) and reverses from $n=4$ ($24>16$). The result provable by induction is $2^n<n!$ for $n\ge4$: base $2^4=16<24=4!$; assuming $2^k<k!$, then $2^{k+1}=2\cdot2^k<2\cdot k!<(k+1)k!=(k+1)!$ since $k+1>2.$ Hence $2^n<n!$ for all $n\ge4.\ \blacksquare$

1820728 marksNumericalCh 1: Algebra

Answer

(a) A person has 12 acquaintances, of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them are relatives?

(b) Let $(G,*)$ be a group, $a,b\in G$; prove

(i) $(a*b)^{-1}=b^{-1}*a^{-1}$ and

(ii) $(a^{-1})^{-1}=a$. OR: Solve for $x$ in a group $G$ (with elements $a,b,c$): $x^2=a^2$ and $x^5=e$.

(a) Choose 5 relatives from 8 and $2$ non-relatives from $4$:

$$ \begin{aligned} \binom85\binom42 &= 56\times6 \ &= 336. \end{aligned} $$

(b) (i)

$$ \begin{aligned} (ab)(b^{-1}a^{-1}) &= a(bb^{-1})a^{-1} \ &= aea^{-1} \ &= e, \end{aligned} $$

so $b^{-1}a^{-1}$ is the inverse of $ab.$ (ii) $a*a^{-1}=e$ shows $a$ is the inverse of $a^{-1}$, i.e. $(a^{-1})^{-1}=a.$

OR. From $x^5=e$ and $x^2=a^2$:

$$ \begin{aligned} x &= x^5\cdot x^{-4} \ &= (x^2)^{-2}\cdot x^{5} \end{aligned} $$

... more directly, since $\gcd(5,2)=1$ there exist integers with $2m+5n=1$ ($m=3,n=-1$):

$$ \begin{aligned} x &= x^{2\cdot3+5\cdot(-1)} \ &= (x^2)^3(x^5)^{-1} \ &= (a^2)^3\cdot e \ &= a^{6}. \end{aligned} $$

Hence $x=a^6.$

1920718 marksNumericalCh 1: Algebra

Answer

(a) In how many ways can the letters of "COMPUTER" be arranged so that

(i) all vowels are always together

(ii) the relative positions of vowels and consonants are unchanged?

(b) A binary operation $$ on $S={a,b,c}$ is given by a Cayley table (row/col order $a,b,c$: $a!:!a,b,c;\ b!:!b,c,a;\ c!:!c,a,b$). Show that $(S,)$ is a group. OR: Let $a,b,c$ be elements of a group $(G,*)$:

(i) if $a*b=b$ prove $a=e$;

(ii) if $a*b=e$ prove $b=a^{-1}$.

(a) COMPUTER has 8 distinct letters; vowels $O,U,E$ (3), consonants $C,M,P,T,R$ (5). (i) Treat the vowels as one block: $6!$ arrangements of block+consonants, times $3!$ within the block: $$ \begin{aligned} 6!\times3! &= 720\times6 \ &=...

2020726 marksNumericalCh 1: Algebra

Answer

(a) In how many ways can the letters of "ELEMENT" be arranged so that the vowels are always together?

(b) Prove that $\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+\cdots=e$.

(a) ELEMENT has vowels $E,E,E$ (identical) and consonants $L,M,N,T$. Treating the three E's as one block, we arrange the block with $L,M,N,T$: $5!=120$ ways (the block has only one internal arrangement since the E's are identical). (b) T...

2120726 marksCh 1: Algebra

Answer

State the binomial theorem. In the expansion of $(1+x)^n$, prove that the sum of the coefficients of the odd terms equals the sum of the coefficients of the even terms, and each equals $2^{n-1}$.

Binomial theorem. $$ \begin{aligned} (1+x)^n &= \displaystyle\sum{r=0}^{n}\binom nr x^r \ &= C0+C1x+C2x^2+\cdots+Cnx^n. \end{aligned} $$ Proof. Put $x=1$: $C0+C1+C2+\cdots+Cn=2^n.$ Put $x=-1$: $C0-C1+C2-\cdots=0$, i.e. $$(C0+C2+C4+\cdot...

2220716 marksNumericalCh 1: Algebra

Answer

(a) A man has 5 friends. In how many ways can he invite one or more of them to dinner?

(b) Find the coefficient of $x$ in the expansion of $\left(x^2+\frac{a^2}{x}\right)^5$.

(a) Each friend is either invited or not; excluding the case of none, the number of ways is $$2^5-1=31.$$ (b) General term $$ \begin{aligned} T{r+1} &= \binom5r(x^2)^{5-r}\left(\tfrac{a^2}{x}\right)^r \ &= \binom5r a^{2r}x^{10-3r}. \end...

2320716 marksCh 1: Algebra

Answer

Show that $1+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\frac{1+2+3+4}{4!}+\cdots=\frac{3e}{2}$.

The $n$th term is $$ \begin{aligned} \dfrac{1+2+\cdots+n}{n!} &= \dfrac{n(n+1)/2}{n!} \ &= \dfrac{n^2+n}{2,n!}. \end{aligned} $$ So the sum is $$ \begin{aligned} S &= \frac12\sum{n=1}^{\infty}\frac{n^2+n}{n!} \ &= \frac12\left(\sum{n=...

2420706 marksNumericalCh 1: Algebra

Answer

(a) In how many ways can the letters of the word "ELEMENT" be arranged?

(b) Show that $\log_e 2 = \frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+\cdots$

(a) ELEMENT has 7 letters with E repeated 3 times. $$ \begin{aligned} N &= \frac{7!}{3!} \ &= \frac{5040}{6} \ &= 840. \end{aligned} $$ (b) Using $\loge(1+x)=x-\tfrac{x^2}{2}+\tfrac{x^3}{3}-\cdots$ at $x=1$: $\loge 2=1-\tfrac12+\tfrac1...

2520706 marksCh 1: Algebra

Answer

If $(1+x)^n=C_0+C_1x+C_2x^2+\cdots+C_nx^n$, prove that $C_0C_n+C_1C_{n-1}+\cdots+C_nC_0=\dfrac{2n!}{(n!)^2}$.

Consider the identity $(1+x)^n(1+x)^n=(1+x)^{2n}.$ Left side: $\left(\sum{r=0}^n Cr x^r\right)\left(\sum{s=0}^n Cs x^s\right)$. The coefficient of $x^n$ collects all $r+s=n$: $$C0Cn+C1C{n-1}+C2C{n-2}+\cdots+CnC0.$$ Right side: coefficien...

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120818 marksNumericalCh 6: Calculus

Answer

(a) A curve $y^2=8x$ changes its abscissa and ordinate at the same rate; the ordinate is double the abscissa. Justify with calculation.

(b) $\int\frac{dx}{a+b\cos x}$ depends on the values of $a$ and $b$ . Explain.

(c) An exact differential equation has the form $M(x,y)\,dx+N(x,y)\,dy=0$ . Give a simple example and its characteristic.

(a) A curve $y^2=8x$ changes its abscissa and ordinate at the same rate; the ordinate is double the abscissa. Justify with calculation.

(b) $\int\frac{dx}{a+b\cos x}$ depends on the values of $a$ and $b$. Explain.

(a) Differentiate $y^2=8x$ : $2y\dfrac{dy}{dt}=8\dfrac{dx}{dt}.$ Given $\dfrac{dy}{dt}=\dfrac{dx}{dt}$ : $2y=8\Rightarrow y=4$ , and then

\begin{aligned} x &= \dfrac{y^2}{8} \\ &= 2. \end{aligned}

\begin{aligned} y &= 4 \\ &= 2x \end{aligned}

(ordinate is double the abscissa) at the point $(2,4)$ , as required.

(b) Substituting $t=\tan\tfrac x2$ gives $\int\dfrac{2\,dt}{(a+b)+(a-b)t^2}.$ The form of the answer depends on the sign of $\dfrac{a-b}{a+b}$ : if $a>b$ the integral is a $\tan^{-1}$ (inverse-tangent) form; if $a<b$ it is a logarithmic form; if $a=b$ it reduces to a simple rational integral. Hence the result depends on the relative values of $a$ and $b.$

(c) $M\,dx+N\,dy=0$ is exact iff $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}.$ Example: $2xy\,dx+x^2\,dy=0$ has $M=2xy,\ N=x^2$ with

\begin{aligned} M_y &= 2x \\ &= N_x, \end{aligned}

so it is exact and its solution is $x^2y=C.$

(a) Differentiate $y^2=8x$: $2y\dfrac{dy}{dt}=8\dfrac{dx}{dt}.$ Given $\dfrac{dy}{dt}=\dfrac{dx}{dt}$: $2y=8\Rightarrow y=4$, and then

$$ \begin{aligned} x &= \dfrac{y^2}{8} \ &= 2. \end{aligned} $$

$$ \begin{aligned} y &= 4 \ &= 2x \end{aligned} $$

(ordinate is double the abscissa) at the point $(2,4)$, as required.

(c) $M,dx+N,dy=0$ is exact iff $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}.$ Example: $2xy,dx+x^2,dy=0$ has $M=2xy,\ N=x^2$ with

$$ \begin{aligned} M_y &= 2x \ &= N_x, \end{aligned} $$

so it is exact and its solution is $x^2y=C.$

220728 marksCh 6: Calculus

Answer

(a) Evaluate $\int\frac{dx}{(x-1)^2(x-2)^3}$ .

(b) Reduce $\frac{dy}{dx}+\frac{y}{x}=y^2$ to linear form and solve it. OR: Solve $\frac{dy}{dx}=\frac{y+1}{x+y+1}$ .

(a) Evaluate $\int\frac{dx}{(x-1)^2(x-2)^3}$.

(b) Reduce $\frac{dy}{dx}+\frac{y}{x}=y^2$ to linear form and solve it. OR: Solve $\frac{dy}{dx}=\frac{y+1}{x+y+1}$.

(a) Partial fractions: $\frac{1}{(x-1)^2(x-2)^3}=\frac{-3}{x-1}+\frac{-1}{(x-1)^2}+\frac{3}{x-2}+\frac{-2}{(x-2)^2}+\frac{1}{(x-2)^3}.$ Integrating, $\int=-3\ln|x-1|+\frac{1}{x-1}+3\ln|x-2|+\frac{2}{x-2}-\frac{1}{2(x-2)^2}+C.$

(b) Bernoulli: divide by $y^2$ and put $v=y^{-1}$ ( $v'=-y^{-2}y'$ ): $v'-\dfrac vx=-1.$ I.F. $=\tfrac1x$ : $\left(\dfrac vx\right)'=-\dfrac1x\Rightarrow\dfrac vx=-\ln x+C.$ So $\dfrac1y=x(C-\ln|x|)$ , i.e. $y=\dfrac{1}{x(C-\ln|x|)}.$

OR. Treat $x$ as function of $y$ : $\dfrac{dx}{dy}-\dfrac{x}{y+1}=1.$ I.F. $=\dfrac1{y+1}$ : $\dfrac{x}{y+1}=\ln|y+1|+C$ , so $x=(y+1)(\ln|y+1|+C).$

OR. Treat $x$ as function of $y$: $\dfrac{dx}{dy}-\dfrac{x}{y+1}=1.$ I.F. $=\dfrac1{y+1}$: $\dfrac{x}{y+1}=\ln|y+1|+C$, so $x=(y+1)(\ln|y+1|+C).$

320718 marksCh 6: Calculus

Answer

(a) Evaluate $\int\frac{dx}{1-3\sin x}$ .

(b) Solve $xy\frac{dy}{dx}=x^2+y^2$ . OR: Solve $\frac{dy}{dx}+\frac{2x}{1+x^2}\,y=\frac{1}{(1+x^2)^2}$ .

(a) Evaluate $\int\frac{dx}{1-3\sin x}$.

(b) Solve $xy\frac{dy}{dx}=x^2+y^2$. OR: Solve $\frac{dy}{dx}+\frac{2x}{1+x^2},y=\frac{1}{(1+x^2)^2}$.

(a) Put $t=\tan\tfrac x2$ : $\begin{aligned} \int\frac{2\,dt}{t^2-6t+1} &= \int\frac{2\,dt}{(t-3)^2-8} \\ &= \frac{1}{2\sqrt2}\ln\left\frac{t-3-2\sqrt2}{t-3+2\sqrt2}\right+C,\quad t=\tan\tfrac x2. \end{aligned}$ (b) Homogeneous: $\dfr...

420708 marksCh 6: Calculus

Answer

(a) Evaluate $\int\frac{dx}{1+2\sin x}$ .

(b) Solve $(1+x^2)\frac{dy}{dx}+2xy=4x^2$ . OR: Solve $(x^2+y^2)\,dy=xy\,dx$ .

(a) Evaluate $\int\frac{dx}{1+2\sin x}$.

(b) Solve $(1+x^2)\frac{dy}{dx}+2xy=4x^2$. OR: Solve $(x^2+y^2),dy=xy,dx$.

(a) Put $t=\tan\tfrac x2$ , $\sin x=\dfrac{2t}{1+t^2}$ , $dx=\dfrac{2\,dt}{1+t^2}$ : $$ \begin{aligned} \int\frac{2,dt}{t^2+4t+1} &= \int\frac{2,dt}{(t+2)^2-3} \ &= \frac{1}{\sqrt3}\ln\left\frac{t+2-\sqrt3}{t+2+\sqrt3}\right+C,\quad t=\t...

520726 marksNumericalCh 6: Calculus

Answer

(a) Compute $\int\frac{\coth x\,dx}{\sinh x-9\,\operatorname{cosech} x}$ .

(b) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x-x-1}{x^2}$ .

(a) Compute $\int\frac{\coth x,dx}{\sinh x-9,\operatorname{cosech} x}$.

(b) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x-x-1}{x^2}$.

(a) Write $\coth x=\dfrac{\cosh x}{\sinh x}$ and $\operatorname{cosech}x=\dfrac1{\sinh x}$ : $\int\frac{\cosh x/\sinh x}{\sinh x-9/\sinh x}\,dx=\int\frac{\cosh x}{\sinh^2x-9}\,dx.$ Let $u=\sinh x,\ du=\cosh x\,dx$ : $\displaystyle\int\fr...

620726 marksNumericalCh 6: Calculus

Answer

State the mean value theorem and verify it for $f(x)=\sqrt{x^2-4}$ on $[2,4]$ . OR: Find from first principles the derivative of $\ln(\cos^{-1}x)$ .

State the mean value theorem and verify it for $f(x)=\sqrt{x^2-4}$ on $[2,4]$. OR: Find from first principles the derivative of $\ln(\cos^{-1}x)$.

Statement. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then some $c\in(a,b)$ has $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Verification. $f(2)=0$ , $\begin{aligned} f(4) &= \sqrt{12} \\ &= 2\sqrt3, \end{aligned}$ so $\dfrac{f...

720716 marksNumericalCh 6: Calculus

Answer

(a) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2 x}$ .

(b) Evaluate $\int\frac{dx}{\sqrt{2ax+x^2}}$ .

(a) Using L'Hospital's rule, evaluate $\lim_{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2 x}$.

(b) Evaluate $\int\frac{dx}{\sqrt{2ax+x^2}}$.

(a) This is a $0/0$ form. Differentiating numerator and denominator twice (each intermediate form is again $0/0$ ): $$ \begin{aligned} \lim{x\to0}\frac{e^x+e^{-x}-2\cos x}{\sin^2x} &= \lim{x\to0}\frac{e^x+e^{-x}+2\cos x}{2\cos2x} \ &= \f...

820716 marksNumericalCh 6: Calculus

Answer

State Rolle's theorem, interpret it geometrically, and verify it for $f(x)=x(x-3)^2$ on $[0,3]$ . OR: Find from first principles the derivative of $\ln\!\left(\sin\frac{x}{a}\right)$ .

State Rolle's theorem, interpret it geometrically, and verify it for $f(x)=x(x-3)^2$ on $[0,3]$. OR: Find from first principles the derivative of $\ln!\left(\sin\frac{x}{a}\right)$.

Statement. If $f$ is continuous on $[a,b]$ , differentiable on $(a,b)$ and $f(a)=f(b)$ , then there is $c\in(a,b)$ with $f'(c)=0.$ Geometrically, the tangent at $x=c$ is horizontal (parallel to the $x$ -axis). Verification. $f(0)=0=f(3)$ . $...

920706 marksNumericalCh 6: Calculus

Answer

(a) Find the equation of the tangent to the curve $y=2x^3-5x^2+8$ at $(2,4)$ .

(b) Evaluate $\int\frac{dx}{\sqrt{(x-\alpha)(x-\beta)}},\ (\beta>\alpha)$ .

(a) Find the equation of the tangent to the curve $y=2x^3-5x^2+8$ at $(2,4)$.

(b) Evaluate $\int\frac{dx}{\sqrt{(x-\alpha)(x-\beta)}},\ (\beta>\alpha)$.

(a) $\dfrac{dy}{dx}=6x^2-10x$ ; at $x=2$ , $\begin{aligned} \text{slope} &= 24-20 \\ &= 4. \end{aligned}$ Tangent: $\begin{aligned} y-4 &= 4(x-2)\ \Rightarrow\ y \\ &= 4x-4. \end{aligned}$ (b) Complete the square: $(x-\alpha)(x-\be...

1020706 marksNumericalCh 6: Calculus

Answer

State the mean value theorem, interpret it geometrically, and verify it for $f(x)=x(x-1)^2$ on $[0,2]$ . OR: Find from first principles the derivative of $x^x$ .

State the mean value theorem, interpret it geometrically, and verify it for $f(x)=x(x-1)^2$ on $[0,2]$. OR: Find from first principles the derivative of $x^x$.

Statement. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , there exists $c\in(a,b)$ with $f'(c)=\dfrac{f(b)-f(a)}{b-a}.$ Geometrically, the tangent at $x=c$ is parallel to the chord joining $(a,f(a))$ and $(b,f(b)).$ Verif...

1120815 marksCh 6: Calculus

Answer

(a) Write the derivative of $\sinh x$ .

(b) Under what condition is $\frac{d}{dx}(\operatorname{sech}^{-1}x)=-\frac{1}{x\sqrt{1-x^2}}$ ?

(d) Write $\int\frac{1}{\sqrt{x^2-a^2}}\,dx$ .

(e) Write the integrating factor of $\frac{dy}{dx}+Py=Q$ .

(a) Write the derivative of $\sinh x$.

(b) Under what condition is $\frac{d}{dx}(\operatorname{sech}^{-1}x)=-\frac{1}{x\sqrt{1-x^2}}$?

(d) Write $\int\frac{1}{\sqrt{x^2-a^2}},dx$.

(e) Write the integrating factor of $\frac{dy}{dx}+Py=Q$.

1220815 marksNumericalCh 6: Calculus

Answer

(a) Find the equation of the tangent to $y=2x^3+x^2-7x-2$ at $x=2$ .

(b) Evaluate $\int\frac{2x^2-x+4}{x^3+4x}\,dx$ .

(a) Find the equation of the tangent to $y=2x^3+x^2-7x-2$ at $x=2$.

(b) Evaluate $\int\frac{2x^2-x+4}{x^3+4x},dx$.

(a) $\dfrac{dy}{dx}=6x^2+2x-7$ ; at $x=2$ ,

\begin{aligned} \text{slope} &= 24+4-7 \\ &= 21. \end{aligned}

The point on the curve is $\big(2,\,2(8)+4-14-2\big)=(2,4).$ Tangent:

\begin{aligned} y-4 &= 21(x-2)\ \Rightarrow\ y \\ &= 21x-38. \end{aligned}

(b) Partial fractions with $x^3+4x=x(x^2+4)$ : $\dfrac{2x^2-x+4}{x(x^2+4)}=\dfrac1x+\dfrac{x-1}{x^2+4}.$ $\int=\ln|x|+\tfrac12\ln(x^2+4)-\tfrac12\tan^{-1}\tfrac x2+C.$

(a) $\dfrac{dy}{dx}=6x^2+2x-7$; at $x=2$,

$$ \begin{aligned} \text{slope} &= 24+4-7 \ &= 21. \end{aligned} $$

The point on the curve is $\big(2,,2(8)+4-14-2\big)=(2,4).$ Tangent: $$ \begin{aligned} y-4 &= 21(x-2)\ \Rightarrow\ y \ &= 21x-38. \end{aligned} $$

(b) Partial fractions with $x^3+4x=x(x^2+4)$: $\dfrac{2x^2-x+4}{x(x^2+4)}=\dfrac1x+\dfrac{x-1}{x^2+4}.$ $$\int=\ln|x|+\tfrac12\ln(x^2+4)-\tfrac12\tan^{-1}\tfrac x2+C.$$

1320811 marksNumericalCh 6: Calculus

Answer

$f'(x)=3x^2$ and $g'(x)=2x$ . Using L'Hospital's rule, find $\lim_{x\to1}\frac{f(x)}{g(x)}$ . A) $\frac32$ B) $\frac13$ C) $\frac12$ D) $\frac16$

$f'(x)=3x^2$ and $g'(x)=2x$. Using L'Hospital's rule, find $\lim_{x\to1}\frac{f(x)}{g(x)}$. A) $\frac32$ B) $\frac13$ C) $\frac12$ D) $\frac16$

A) $\dfrac32$ . By L'Hospital's rule $\begin{aligned} \lim{x\to1}\dfrac{f(x)}{g(x)} &= \lim{x\to1}\dfrac{f'(x)}{g'(x)} \\ &= \lim{x\to1}\dfrac{3x^2}{2x} \\ &= \dfrac{3(1)}{2} \\ &= \dfrac32. \end{aligned}$

1420811 marksCh 6: Calculus

Answer

Which is $\int\frac{1}{x^2+4}\,dx$ ? A) $\frac12\tan^{-1}\frac2x+C$ B) $\frac12\tan^{-1}\frac x2+C$ C) $2\tan^{-1}\frac x2+C$ D) $\tan^{-1}\frac x2+C$

Which is $\int\frac{1}{x^2+4},dx$? A) $\frac12\tan^{-1}\frac2x+C$ B) $\frac12\tan^{-1}\frac x2+C$ C) $2\tan^{-1}\frac x2+C$ D) $\tan^{-1}\frac x2+C$

B) $\dfrac12\tan^{-1}\dfrac x2+C$ . Using $\int\dfrac{dx}{x^2+a^2}=\dfrac1a\tan^{-1}\dfrac xa$ with $a=2.$

B) $\dfrac12\tan^{-1}\dfrac x2+C$. Using $\int\dfrac{dx}{x^2+a^2}=\dfrac1a\tan^{-1}\dfrac xa$ with $a=2.$

1520811 marksCh 6: Calculus

Answer

What is the degree of the ODE $\frac{d^3s}{dt^3}=\left(4+\left(\frac{d^2s}{dt^2}\right)^2\right)^{1/2}$ ? A) 1 B) 2 C) 3 D) 4

What is the degree of the ODE $\frac{d^3s}{dt^3}=\left(4+\left(\frac{d^2s}{dt^2}\right)^2\right)^{1/2}$? A) 1 B) 2 C) 3 D) 4

1620811 marksCh 6: Calculus

Answer

Which is an example of a homogeneous differential equation of first order? A) $3x\,dy+2y\,dx=4$ B) $x\,dy-y\,dx+1=0$ C) $x\,dx+y\,dy=2$ D) $(x^2+xy)\,dy-(xy-y^2)\,dx=0$

Which is an example of a homogeneous differential equation of first order? A) $3x,dy+2y,dx=4$ B) $x,dy-y,dx+1=0$ C) $x,dx+y,dy=2$ D) $(x^2+xy),dy-(xy-y^2),dx=0$

D) $(x^2+xy)\,dy-(xy-y^2)\,dx=0$ . Every term is of the same (second) degree in $x,y$ , so it can be written as $\dfrac{dy}{dx}=F\!\left(\dfrac yx\right)$ ; the others contain constant terms and are not homogeneous.

1720818 marksNumericalCh 1: Algebra

Answer

(a) Prove that the coefficient of $x^n$ in $(1+x)^{2n}$ is twice the coefficient of $x^n$ in $(1+x)^{2n-1}$ .

(b) Use De Moivre's theorem to find the square root of $-(2+2\sqrt3\,i)$ .

(a) Prove that the coefficient of $x^n$ in $(1+x)^{2n}$ is twice the coefficient of $x^n$ in $(1+x)^{2n-1}$.

(b) Use De Moivre's theorem to find the square root of $-(2+2\sqrt3,i)$.

(a) The coefficients are $\binom{2n}{n}$ and $\binom{2n-1}{n}.$ Now

\begin{aligned} \frac{\binom{2n}{n}}{\binom{2n-1}{n}} &= \frac{(2n)!/(n!\,n!)}{(2n-1)!/(n!\,(n-1)!)} \\ &= \frac{(2n)!}{(2n-1)!}\cdot\frac{(n-1)!}{n!} \\ &= 2n\cdot\frac1n \\ &= 2, \end{aligned}

so $\binom{2n}{n}=2\binom{2n-1}{n}.\ \blacksquare$

(b) $-(2+2\sqrt3\,i)=-2-2\sqrt3\,i$ has modulus $4$ and argument $240^\circ$ (third quadrant): $4(\cos240^\circ+i\sin240^\circ).$ By De Moivre the square roots are $2\left(\cos\tfrac{240^\circ+360^\circ k}{2}+i\sin\cdots\right)$ for $k=0,1$ : $2(\cos120^\circ+i\sin120^\circ)=-1+\sqrt3\,i$ and $2(\cos300^\circ+i\sin300^\circ)=1-\sqrt3\,i.$ So the square roots are $\pm(-1+\sqrt3\,i).$ (Check $(-1+\sqrt3 i)^2=-2-2\sqrt3 i.$ )

(c) Note: $n!<2^n$ actually holds only for $n=1,2,3$ ( $1<2,\ 2<4,\ 6<8$ ) and reverses from $n=4$ ( $24>16$ ). The result provable by induction is $2^n<n!$ for $n\ge4$ : base $2^4=16<24=4!$ ; assuming $2^k<k!$ , then $2^{k+1}=2\cdot2^k<2\cdot k!<(k+1)k!=(k+1)!$ since $k+1>2.$ Hence $2^n<n!$ for all $n\ge4.\ \blacksquare$

(a) The coefficients are $\binom{2n}{n}$ and $\binom{2n-1}{n}.$ Now

$$ \begin{aligned} \frac{\binom{2n}{n}}{\binom{2n-1}{n}} &= \frac{(2n)!/(n!,n!)}{(2n-1)!/(n!,(n-1)!)} \ &= \frac{(2n)!}{(2n-1)!}\cdot\frac{(n-1)!}{n!} \ &= 2n\cdot\frac1n \ &= 2, \end{aligned} $$

so $\binom{2n}{n}=2\binom{2n-1}{n}.\ \blacksquare$

1820728 marksNumericalCh 1: Algebra

Answer

(a) A person has 12 acquaintances, of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them are relatives?

(b) Let $(G,*)$ be a group, $a,b\in G$ ; prove

(i) $(a*b)^{-1}=b^{-1}*a^{-1}$ and

(ii) $(a^{-1})^{-1}=a$ . OR: Solve for $x$ in a group $G$ (with elements $a,b,c$ ): $x^2=a^2$ and $x^5=e$ .

(a) A person has 12 acquaintances, of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them are relatives?

(b) Let $(G,*)$ be a group, $a,b\in G$; prove

(i) $(a*b)^{-1}=b^{-1}*a^{-1}$ and

(ii) $(a^{-1})^{-1}=a$. OR: Solve for $x$ in a group $G$ (with elements $a,b,c$): $x^2=a^2$ and $x^5=e$.

(a) Choose 5 relatives from 8 and $2$ non-relatives from $4$ :

\begin{aligned} \binom85\binom42 &= 56\times6 \\ &= 336. \end{aligned}

(b) (i)

\begin{aligned} (a*b)*(b^{-1}*a^{-1}) &= a*(b*b^{-1})*a^{-1} \\ &= a*e*a^{-1} \\ &= e, \end{aligned}

so $b^{-1}*a^{-1}$ is the inverse of $a*b.$ (ii) $a*a^{-1}=e$ shows $a$ is the inverse of $a^{-1}$ , i.e. $(a^{-1})^{-1}=a.$

OR. From $x^5=e$ and $x^2=a^2$ :

\begin{aligned} x &= x^5\cdot x^{-4} \\ &= (x^2)^{-2}\cdot x^{5} \end{aligned}

... more directly, since $\gcd(5,2)=1$ there exist integers with $2m+5n=1$ ( $m=3,n=-1$ ):

\begin{aligned} x &= x^{2\cdot3+5\cdot(-1)} \\ &= (x^2)^3(x^5)^{-1} \\ &= (a^2)^3\cdot e \\ &= a^{6}. \end{aligned}

Hence $x=a^6.$

(a) Choose 5 relatives from 8 and $2$ non-relatives from $4$:

$$ \begin{aligned} \binom85\binom42 &= 56\times6 \ &= 336. \end{aligned} $$

(b) (i)

$$ \begin{aligned} (ab)(b^{-1}a^{-1}) &= a(bb^{-1})a^{-1} \ &= aea^{-1} \ &= e, \end{aligned} $$

so $b^{-1}a^{-1}$ is the inverse of $ab.$ (ii) $a*a^{-1}=e$ shows $a$ is the inverse of $a^{-1}$, i.e. $(a^{-1})^{-1}=a.$

OR. From $x^5=e$ and $x^2=a^2$:

$$ \begin{aligned} x &= x^5\cdot x^{-4} \ &= (x^2)^{-2}\cdot x^{5} \end{aligned} $$

... more directly, since $\gcd(5,2)=1$ there exist integers with $2m+5n=1$ ($m=3,n=-1$):

$$ \begin{aligned} x &= x^{2\cdot3+5\cdot(-1)} \ &= (x^2)^3(x^5)^{-1} \ &= (a^2)^3\cdot e \ &= a^{6}. \end{aligned} $$

Hence $x=a^6.$

1920718 marksNumericalCh 1: Algebra

Answer

(a) In how many ways can the letters of "COMPUTER" be arranged so that

(i) all vowels are always together

(ii) the relative positions of vowels and consonants are unchanged?

(b) A binary operation $*$ on $S=\{a,b,c\}$ is given by a Cayley table (row/col order $a,b,c$ : $a\!:\!a,b,c;\ b\!:\!b,c,a;\ c\!:\!c,a,b$ ). Show that $(S,*)$ is a group. OR: Let $a,b,c$ be elements of a group $(G,*)$ :

(i) if $a*b=b$ prove $a=e$ ;

(ii) if $a*b=e$ prove $b=a^{-1}$ .

(a) In how many ways can the letters of "COMPUTER" be arranged so that

(i) all vowels are always together

(ii) the relative positions of vowels and consonants are unchanged?

(i) if $a*b=b$ prove $a=e$;

(ii) if $a*b=e$ prove $b=a^{-1}$.

2020726 marksNumericalCh 1: Algebra

Answer

(a) In how many ways can the letters of "ELEMENT" be arranged so that the vowels are always together?

(b) Prove that $\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+\cdots=e$ .

(a) In how many ways can the letters of "ELEMENT" be arranged so that the vowels are always together?

(b) Prove that $\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+\cdots=e$.

(a) ELEMENT has vowels $E,E,E$ (identical) and consonants $L,M,N,T$ . Treating the three E's as one block, we arrange the block with $L,M,N,T$ : $5!=120$ ways (the block has only one internal arrangement since the E's are identical). (b) T...

2120726 marksCh 1: Algebra

Answer

State the binomial theorem. In the expansion of $(1+x)^n$ , prove that the sum of the coefficients of the odd terms equals the sum of the coefficients of the even terms, and each equals $2^{n-1}$ .

State the binomial theorem. In the expansion of $(1+x)^n$, prove that the sum of the coefficients of the odd terms equals the sum of the coefficients of the even terms, and each equals $2^{n-1}$.

Binomial theorem. $\begin{aligned} (1+x)^n &= \displaystyle\sum{r=0}^{n}\binom nr x^r \\ &= C0+C1x+C2x^2+\cdots+Cnx^n. \end{aligned}$ Proof. Put $x=1$ : $C0+C1+C2+\cdots+Cn=2^n.$ Put $x=-1$ : $C0-C1+C2-\cdots=0$ , i.e. $$(C0+C2+C4+\cdot...

2220716 marksNumericalCh 1: Algebra

Answer

(a) A man has 5 friends. In how many ways can he invite one or more of them to dinner?

(b) Find the coefficient of $x$ in the expansion of $\left(x^2+\frac{a^2}{x}\right)^5$ .

(a) A man has 5 friends. In how many ways can he invite one or more of them to dinner?

(b) Find the coefficient of $x$ in the expansion of $\left(x^2+\frac{a^2}{x}\right)^5$.

(a) Each friend is either invited or not; excluding the case of none, the number of ways is $2^5-1=31.$ (b) General term $$ \begin{aligned} T{r+1} &= \binom5r(x^2)^{5-r}\left(\tfrac{a^2}{x}\right)^r \ &= \binom5r a^{2r}x^{10-3r}. \end...

2320716 marksCh 1: Algebra

Answer

Show that $1+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\frac{1+2+3+4}{4!}+\cdots=\frac{3e}{2}$ .

Show that $1+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\frac{1+2+3+4}{4!}+\cdots=\frac{3e}{2}$.

The $n$ th term is $\begin{aligned} \dfrac{1+2+\cdots+n}{n!} &= \dfrac{n(n+1)/2}{n!} \\ &= \dfrac{n^2+n}{2\,n!}. \end{aligned}$ So the sum is $$ \begin{aligned} S &= \frac12\sum{n=1}^{\infty}\frac{n^2+n}{n!} \ &= \frac12\left(\sum{n=...

2420706 marksNumericalCh 1: Algebra

Answer

(a) In how many ways can the letters of the word "ELEMENT" be arranged?

(b) Show that $\log_e 2 = \frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+\cdots$

(a) In how many ways can the letters of the word "ELEMENT" be arranged?

(b) Show that $\log_e 2 = \frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+\cdots$

(a) ELEMENT has 7 letters with E repeated 3 times. $\begin{aligned} N &= \frac{7!}{3!} \\ &= \frac{5040}{6} \\ &= 840. \end{aligned}$ (b) Using $\loge(1+x)=x-\tfrac{x^2}{2}+\tfrac{x^3}{3}-\cdots$ at $x=1$ : $\loge 2=1-\tfrac12+\tfrac1...

2520706 marksCh 1: Algebra

Answer

If $(1+x)^n=C_0+C_1x+C_2x^2+\cdots+C_nx^n$ , prove that $C_0C_n+C_1C_{n-1}+\cdots+C_nC_0=\dfrac{2n!}{(n!)^2}$ .

If $(1+x)^n=C_0+C_1x+C_2x^2+\cdots+C_nx^n$, prove that $C_0C_n+C_1C_{n-1}+\cdots+C_nC_0=\dfrac{2n!}{(n!)^2}$.

Consider the identity $(1+x)^n(1+x)^n=(1+x)^{2n}.$ Left side: $\left(\sum{r=0}^n Cr x^r\right)\left(\sum{s=0}^n Cs x^s\right)$ . The coefficient of $x^n$ collects all $r+s=n$ : $C0Cn+C1C{n-1}+C2C{n-2}+\cdots+CnC0.$ Right side: coefficien...