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Discussion Forums -> Organic Chemistry -> ethyl carbocation is more stable then n-propyl why? -> Go to message

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Ethyl carbocation has more no. of alpha hydrogens (3) than n propyl (2), hence due to hyper conjugation it is more stable.

Discussion Forums -> Algebra -> find the inverse of [2 0 -1 5 1 0 -> Go to message

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What's the mistake ? I've rechecked but haven't found any.

Discussion Forums -> Algebra -> How can I solve this: x4+x3+x2+x+1=0 (4,3,2-->exponents) ? thnx -> Go to message

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The solution involves two steps :

(i)Eliminating the term

(ii)Factorizing the biquadratic expression and equating it to the first degree term.

To eliminate the term :

Replace with and expand the equation.

The coefficient of will be

Sub

You get an equation of the form

If are the roots of the biquadratic equation

Discussion Forums -> Algebra -> find the inverse of [2 0 -1 5 1 0 -> Go to message

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C1 --> C1+C3

C3 --> C1+C3

C1 --> C1-C3

C3 --> -5(C2)+C3

C1--> C1+3(C3)

C2 --> C2-C3

Discussion Forums -> Analytical Geometry -> circle problem please solve -> Go to message

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PA.PB=(PT)^2

It is a well known and important result..you can google it for the derivation.

Discussion Forums -> Algebra -> coefficient of x^99 in the expansion of (x-1) (x-3) (x-5).......(x-199) is ? -> Go to message

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For a polynomial equation of the form $ax^n+bx^{n-1}+...k=0$

sum of roots taken one at a time = $\frac{-b}{a}$

The given polynomial on expansion will be of the form
$x^{100}+k_1x^{99}+...k_n = 0$

$\Rightarrow -k_1=(1+3+5+...199)$

$\Rightarrow$ coefficient of $x^{99}=(1+3+5+...199)$

Discussion Forums -> Integral Calculus -> help !!!! integration question dx /(x^4 + x^6) , how to start? -> Go to message

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Resolve into partial fractions

${\frac{1}{x^4+x^6}}$ $={\frac{-1}{x^2}+\frac{1}{x^4}+\frac{1}{1+x^2}}\$

Discussion Forums -> Organic Chemistry -> 1 litre gasflask contains 3 gases at a total pressure of 1500mm.Ratio of their partial pressure are -> Go to message

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$(2+0.5+0.5) \times x=1500mm$

$x=500mm$

Hence the pressure exerted by each gas :

$2 \times 500mm, 0.5 \times 500mm, 0.5 \times 500mm$

As the temperature remains constant throughout,

$2 \times 500mm \times 1l=760mm \times V_1$

$0.5 \times 500mm \times 1l=760mm \times V_2$

$0.5 \times 500mm \times 1l=760mm \times V_3$

Discussion Forums -> Differential Calculus -> please solve this and give the instructions too.... lim (2 - a/x)^tan(pi*x/2a) x->a -> Go to message

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Let $L=$ $Lt_{x \rightarrow a}\ {\left(2-\frac{a}{x}\right)^{tan\left(\frac{\pi x}{2a}\right)}$

$ln L=Lt_{x \rightarrow a}\ {tan\left(\frac{\pi x}{2a}\right)}ln{\left(2-\frac{a}{x}\right)}$
$=Lt_{x \rightarrow a}\ \frac{ln{\left(2-\frac{a}{x}\right)}}{cot\left(\frac{\pi x}{2a}\right)}$

Using L'Hospital's rule

$ln L=Lt_{x \rightarrow a}\ \frac{-\frac{1}{a\left(2-\frac{a}{x}\right)}}{\frac{- \pi}{2a}csc^2\left(\frac{\pi x}{2a}\right)}$

$\Rightarrow L=e^{\frac{2}{\pi}}$

Discussion Forums -> Trignometry -> The interior angles of a polygon are in AP. least angle is 120 and common difference is 5 . Find the -> Go to message

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The interior angles are 120,125,130,...

Sum of interior angles of a polygon = 180×(n-2) =(n/2)(240+(n-1)5)

Solving,we get n = 16 or n = 9

Discussion Forums -> Optics -> gud question -> Go to message

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Let the object distance be 'x' and image distance be 'y'

$\frac{1}{x}+\frac{1}{y}=-\frac{2}{R}$

$-\frac{dy}{dt}=\frac{dx}{dt}.\frac{y^2}{x^2}$

$At\ t < \frac{d}{v}$

$x=-(d-vt)$

$\frac{1}{y}=\frac{1}{d-vt}-\frac{2}{R}$

$\frac{dy}{dt}=-v\times \ \frac{R^2}{(R-2(d-vt))^2}$

$At\ t > \frac{d}{v}$

The block comes to rest and the mirror moves with the velocity v

$v_{b,m}=v_b-v_m=-v$

$x=-vt+d$

$v_{i,m}=-\frac{y^2}{x^2}\ v_{o,m}$

$=v \times \frac{R^2}{(2(vt-d)-R)^2}$

$v_i=v+v \times \frac{R^2}{(2(vt-d)-R)^2}$

Discussion Forums -> Mechanics -> three particles of mass 20g,30g & 40g are initially moving along a positive dirrection of the three -> Go to message

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As $F_{ext}=0$ , momentum can be conserved.

$20 \times \left(20 \vec{i}+30 \vec{j}+40 \vec{k}\right)=0+30(10 \vec{i}+20 \vec{k})+40\vec{v}$

Discussion Forums -> Mechanics -> during the paddling of cycle,the force of friction exerted by the ground on the two wheels is in whi -> Go to message

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This might be helpful

http://www.goiit.com/posts/list/mechanics-direction-of-friction-30397.htm

Discussion Forums -> Algebra -> If the quadratic equations x^2 - ax + b = 0 and x^2 + bx - a = 0 have a common root, show that a - b -> Go to message

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Let

$f(x)=x^2 - ax + b$
$g(x)=x^2 + bx - a$

Let $p$ be the common root

$\Rightarrow p^2-ap+b=p^2+bp-a=0$

Solving both, we get $p=1$

$\Rightarrow f(1)=g(1)=0$

$\Rightarrow 1-a+b=0$

Hence proved.

Discussion Forums -> Algebra -> . The number of solutions of √(3x2+6+7) + √(5x2+10x+14) = 4 - 2x - x2 is............... -> Go to message

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$\sqrt{3x^2+6x+7} + \sqrt{5x^2+10x+14} = 4 - 2x - x^2$

$\sqrt{3(x+1)^2+4}+\sqrt{5(x+1)^2+9}=-((x+1)^2-5)$

Let $\sqrt{3(x+1)^2+4}=p, \sqrt{5(x+1)^2+9}=q, -((x+1)^2-5)=r$

$p_{min} =2,\ q_{min} =3,\ r_{max} =5$

$p_{min} +q_{min} =r_{max}$

As the value of $(x+1)^2$ increases, value of $p+q$ increases while the value of $r$ decreases.

Hence there can be only one solution, $x=-1$

Discussion Forums -> Mechanics -> when abody is projected vertically up from ground,the ratio of its potential to kinetic energiesat a -> Go to message

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$\frac{KE}{PE}=\frac{4}{3}$

$\frac{KE+PE}{PE}=\frac{7}{3}$

$\frac{TE}{mgh}=\frac{7}{3}$

Similarly,

$\frac{TE}{mgh'}=\frac{7}{4}$

Dividing both $\frac{h'}{h}=\frac{4}{3$

$\Rightarrow h'=\frac{4h}{3}$

Discussion Forums -> Mechanics -> A ball of lead and a ball of silver of different masses fall from the -> Go to message

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In both the cases entire Potential energy is converted into Heat energy

$m_1gh=m_1s_1 \Delta T_1$
$m_2gh=m_2s_2 \Delta T_2$

$\left(\frac{\Delta T_1}{\Delta T_2}\right)=\left(\frac{s_1}{s_2} \right)$

Discussion Forums -> Organic Chemistry -> GIVE MECHANISM OF THE REACTION http://www.imaginationcubed.com/loader.php?aDrawingID=258 -> Go to message

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Ring expansion has taken place which suggests the formation of an intermediate,hence it might be a first order reaction.

Since a doube bond is formed, it might be Elimination reaction.

Hence a probable reaction mechanism is

Discussion Forums -> Integral Calculus -> ANANT SIR OR ANY EXPERT SIR PLEASE solve the question If the function integral( -> Go to message

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$\int_{0}^{x} f(t).dt \rightarrow 5$ as $|x|$ $\rightarrow 1$

$\int_{0}^{-1} f(t).dt \rightarrow 5$

$\int_{0}^{1} f(t).dt \rightarrow 5$

$g(x)=2x+{\int_{0}^{x} f(t).dt} -a$

If $g(x)$ has atleast two roots of opposite signs in $(-1,1)$

$g(-1).g(1)>0$ $\Rightarrow (a-3)(a-7)>0$ $\Rightarrow a \in(\infty,3) \cup (7,\infty)$

$g \left(0\right).g(-1)<0$ $\Rightarrow (a)(a-3)<0$ $\Rightarrow a \in(0,3)$

$g \left(0\right).g(1)<0$ $\Rightarrow (a)(a-7)<0$ $\Rightarrow a \in(0,7)$

$(0,3) \cap (0,7) \cap [(\infty,3) \cup (7,\infty)]$

$\Rightarrow a\in (0,3)$

Discussion Forums -> Algebra -> permutation and combinations -> Go to message

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Arranging the letters in alphabetical order,

a,e,g,r,t

No of words beginning with a ... 4!
beginning with e ... 4!
beginning with ga ... 3!
beginning with ge ... 3!
beginning with gra ... 2!

And finally great ... 1

Rank = 2*(4!+3!)+2!+1.

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