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This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{x\to 0}{lim}\frac{2sin x-sin 2x}{x^3}\\ =\underset{x\to 0}{lim}\frac{2sin x-2sin xcosx}{x^3}=\underset{x\to 0}{lim}\frac{2sin x\left(1-cosx\right)}{x^3}\\ =\underset{x\to 0}{lim}\frac{2sin x}{x}\left(\frac{1-cosx}{x^2}\right)=\underset{x\to 0}{lim}2\left(\frac{sin x}{x}\right)\left(\frac{2si{n}^{2}x/2}{x^2}\right)\\ =\underset{x\to 0}{lim}2\left(\frac{sin x}{x}\right)\left(2\frac{si{n}^{2}x/2}{\frac{x^2}{4}}\times \frac{1}{4}\right)=\underset{x\to 0}{lim}2\left(\frac{sin x}{x}\right)2{\left(\frac{sinx/2}{\frac{x}{2}}\right)}^2\times \frac{1}{4}\\ =\underset{x\to 0}{lim}\frac{4}{4}\left(\frac{sin x}{x}\right)\underset{\frac{x}{2}\to 0}{lim}{\left(\frac{sinx/2}{\frac{x}{2}}\right)}^2=1.1.{\left(1\right)}^2=1\left[?\underset{x\to 0}{lim}\frac{sin x}{x}=1\right]\end{array}$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{x\to 0}{lim}\frac{1-cos2x}{x^2}\\ =\underset{x\to 0}{lim}\frac{2si{n}^{2}x}{x^2}\left[cos2x=1-2si{n}^{2}x\right]\\ =\underset{x\to 0}{lim}2{\left(\frac{sinx}{x}\right)}^2=2\times 1=2\left[?\underset{x\to 0}{lim}\frac{sinx}{x}=1\right]\end{array}$

Hi, it is not mandatory for you to give an entrance exam for getting admission to Engineering diploma courses in any of the specialisations. There are a lot of popular engineering colleges in India that will admit you on the basis of your performance in the Class 10/12 board examination. The minimum eligibility for admission to Diploma courses is normally around 45 to 50% marks depending on the specialisation you're interested in.

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This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{x\to 0}{lim}\frac{si{n}^{2}2x}{si{n}^{2}4x}\\ =\underset{x\to 0}{lim}\frac{si{n}^{2}2x}{si{n}^{2}2\left(2x\right)}\\ =\underset{x\to 0}{lim}\frac{si{n}^{2}2x}{4si{n}^{2}2x.co{s}^{2}2x}\left[sin2x=2sinxcosx\right]\\ =\underset{x\to 0}{lim}\frac{1}{4co{s}^{2}2x}\\ Taking\text{\hspace{0.17em}limit},wehave\\ =\frac{1}{4.co{s}^{2}0}=\frac{1}{4}\end{array}$

Hi, below mentioned are the PG Diploma in Engineering colleges in Mumbai:

The fees for the courses offered by  Vidya Prasarak Mandal's- Institute of Management Studies (VPM's - IMS) are as follows:

• PGDM: INR 3.75 lakh
• PGCM: INR 1.75 lakh

The courses are offered to candidates who meet the eligibility criteria of the programme. Candidates are required to pay the course fee to secure their seat in the institute. 

Hi, given below are the top UG Diploma colleges in Mumbai for Engineering:

Hope this solved your query!

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l}Quantifiermeansaphraselike\text{'}thereexists\text{'},\text{'}forall\text{'}and\text{'}forevery\text{'}etc.\\ \left(i\right)Thereexists\left(ii\right)Forall\\ \left(iii\right)Thereexists\left(iv\right)Forevery\\ \left(v\right)Forall\left(vi\right)Thereexists\\ \left(vii\right)Forall\left(viii\right)Thereexists\\ \left(ix\right)Thereexists\left(x\right)Thereexists\end{array}$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{x\to 0}{lim}\frac{sin 3x}{sin 7x}\\ =\underset{x\to 0}{lim}\frac{\frac{sin 3x}{3x}\times 3x}{\frac{sin 7x}{7x}\times 7x}=\frac{\underset{3x\to 0}{lim}\left(\frac{sin 3x}{3x}\right)}{\underset{3x\to 0}{lim}\left(\frac{sin 7x}{7x}\right)}\times \frac{3}{7}\\ =\frac{1}{1}\times \frac{3}{7}=\frac{3}{7}\left[?\underset{x\to 0}{lim}\frac{sin x}{x}=1\right]\end{array}$

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l}\left(i\right)IftherectangleR isrhombus,thenitisasquare.\\ \left(ii\right)IftomorrowisTuesday,thentodayisMonday.\\ \left(iii\right)IfyoumustvisittheTajMahal,thenyougotoAgra.\\ \left(iv\right) Ifthetriangleisrighttriangle,thenthesumofthesquaresoftwosidesofatriangle\\ isequaltothesquareofthethirdsideofthetriangle.\\ \left(v\right)Ifthetriangleisequilateral,thenallthreeanglesofatriangleareequal.\\ \left(vi\right)If2x=3ythenx:y=3:2.\\ \left(vii\right)Iftheoppositeanglesofaquadrilateralaresupplementary,thenSiscyclic.\\ \left(viii\right)Ifxis neitherpositivenornegativethenx=0.\\ \left(ix\right)Iftheratioofcorrespondingsidesoftwotrianglesareequal,thentrianglesaresimilar.\end{array}$

This is a Short Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Giventhat\underset{x\to 2}{lim}\frac{x^n-2^n}{x-2}=80\\ n.{\left(2\right)}^{n-1}=80\left[?\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n.a^{n-1}\right]\\ \Rightarrow n\times 2^{n-1}=5\times {\left(2\right)}^{5-1}\\ \therefore n=5\end{array}$

Yes, you can surely go for Diploma in Engineering courses in Mumbai after your 10th classes:

Hope this helps!

This is a short answer type question as classified in NCERT Exemplar

$\begin{array}{l}Weknowthatthecontrapositiveofp\to qis\left(\sim q\right)\to \left(\sim p\right)\\ \left(i\right)Ifx\ne 3,thenx\ne yory\ne 3.\\ \left(ii\right)Ifnisnotaninteger,thenitisnotanaturalnumber.\\ \left(iii\right) Ifthetriangleisnotequilateral,thenallthreesidesofthetrianglearenotequal. \\ \left(iv\right) Ifxyisnotpositiveinteger,thenxoryisnotnegativeinteger.\\ \left(v\right)Ifnaturalnumber\text{'}n\text{'}isnotdivisibleby2or3,thennisnotdivisibleby6.\\ \left(vi\right)Theweatherwillnotbecold,ifitdoesnotsnow.\\ \left(vii\right)If{x}^2>1thenxisnotrealnumbersuchthat0<x<1.\end{array}$