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Important Short Questions for Chatper 03

February 22, 2024February 7, 2024 by Mathematics Knowledge for All

Fsc part 02

Important Short Questions for Chatper 03

Find $\partial{y}$ and $dy$ if $y=x^2-1$ when x changes from 3 to 3.02
Find $\partial{y}$ and $dy$ if $y=\sqrt{x}$ when x changes from 4 to 4.41
Find approximate increasse in the volume of a cube if the length of its each edge changes from 5 to 5.02
Evaluate $\int{x(\sqrt{x}+1)dx}$
Evaluate $\int{ (2x+3)^\frac{1}{2} dx}$
Evaluate $\int{(\sqrt{x}+1)^2 dx}$
Evaluate $\int{\frac{(3x+2)}{\sqrt{x}} dx}$
Evaluate $\int{\frac{(\sqrt{θ}-1)^2}{\sqrt{θ}} dθ}$
Evaluate $\int{\frac{(1-\sqrt{x})^2}{\sqrt{x}} dx}$
Evaluate $\int{(a-2x)^{\frac{3}{2}} dx}$
Evaluate $\int{\sqrt{1-cos2x} dx }$
Evaluate $\int{(lnx)× \frac{1}{x} dx}$
Evaluate $\int{\frac{1}{1+cosx} dx}$
Evaluate $\int{cos3x sin2x dx}$
Evaluate $\int{tan^{2}⁡x dx}$
Evaluate $\int{\frac{x}{\sqrt{4+x^2 }} dx}$
Evaluate $\int{\frac{x^2}{4+x^2 } dx}$
Evaluate $\int{\frac{dx}{x^2+4x+13}}$
Evaluate $\int{\frac{1}{xlnx} dx}$
Evaluate $\int{\frac{sec^2⁡x}{\sqrt{tanx}} dx}$
Evaluate $\int{sin^{2}⁡x dx}$
Evaluate $\int{\frac{1}{(1+x^2 )} tan^{-1}⁡x dx}$
Evaluate $\int{\frac{sinθ}{1+cos^{2⁡}θ } dθ}$
Evaluate $\int{cosx\frac{(ln sinx)}{sinx} dx}$
Evaluate $\int{\frac{ \sqrt{2}{sinx+cosx} dx}$
Evaluate $\int{xcosx dx}$
Evaluate $\int{lnx dx}$
Evaluate $\int{x lnx dx}$
Evaluate $\int{tan^{-1}⁡x dx}$
Evaluate $\int{x^2 sinx dx}$
Evalluate $\int{x tan^{-1}{⁡x} dx}$
Evaluate $\int{e^x (\frac{1}{x}+lnx)dx }$
Evaluate $\int{e^x (cosx+sinx) dx }$
Evaluate $\int{e^{2x} (-sinx+2cosx)dx}$
Evaluate $\int{e^{-x} (cosx-sinx)dx}$
Evaluate $\int{\frac{e^{mtan^{-1} x}}{1+x^2} dx}$
Evaluate $\int_{-1}^{3}({x^3+3x^2})dx$

Evaluate $\int_{-1}^{2}{(x+|x|)}dx$
Evaluate $\int_{1}^{e}{x lnx} dx$
Evaluate $\int_{-1}^{1}{(x^\frac{1}{3}+1)}dx$
Evaluate $\int_{-2}^{0}{\frac{1}{(2x-1)^2}} dx$
Evaluate $\int_{-6}^{2}{ \sqrt{3-x}} dx$
Evaluate $\int_{2}^{ \sqrt{5}}{x \sqrt{x^2-1}} dx$
Evaluate $\int_{1}^{2}{\frac{x}{x^2+2}} dx$
Evaluate $\int{0}^{3}{\frac{dx}{x^2+9}$
Evaluate $\int_{1}^{2}{lnx} dx$
Evaluate $\int_{0}^{2}{e^{\frac{x}{2}}-e^-{\frac{x}{ 2}} }dx$
Evaluate $\int{0}^{\frac{π}{3}}{cos^2⁡θ dθ}$
Evaluate $\int_{-1}^{5}{|x-3| } dx$
Evaluate $\int_{\frac{1}{8}}^{1}{\frac{(x^{\frac{1}{3}}+2)^2}{x^{\frac{2}{3}}}} dx$
Evaluate $\int_{0}^{\frac{π}{4}}{\frac{sinx-1}{cos^2⁡x}} dx$

Find the area between the x-axis and the curve $y=4x-x^2$
Solve the differential equation $\frac{dy}{dx}=\frac{1-x}{y}$
Solve the differential siny cosecx $\frac{dy}{dx}=1$
Solve the differential equation $\frac{1}{x}\frac{ dy}{dx}-2y=0$

Importan short Questions for chap # 02 click here

Important Short Questions for Chatper 02 Math

February 7, 2024January 24, 2024 by Mathematics Knowledge for All

FSc Part 2

Q.No. 1. Differentiate from the first principles if $y = x^{m}$

Q.No. 2.Differentiate from first principles: $y = x^{3 / 2}$

Math Statements

Q no. 3.Find the derivative of $Y = (2 \sqrt{x} + 2) (x - \sqrt{x})$ with respect to x.

Q no.4 Differentiate $x^{- 3} + 2 x^{- \frac{3}{2}} + 3$ with respect to x.

Q no.5 Differentiate $\binom{a + x}{a - x}$ with respect to x.

Q no. 6 Differentiate ${(\sqrt{x} - \binom{1}{\sqrt{x}})}^{2}$ with respect to x.

Q no. 7 If $y = \sqrt{x} - \binom{1}{\sqrt{x}}$ , show that $2 x \binom{dy}{dx} + y = 2 \sqrt{x}$ .

Q no. 8 $If \( y = x^4 + 2x^2 + 2 \), prove that \( \frac{dy}{dx} = 4x\sqrt{y-1} \).$

Q no. 9 $Find \( \frac{dy}{dx} \) if \( x = at^2 \) and \( y = 2at \).$

Q no.10 $Find \( \frac{dy}{dx} \) if \( x^2 + y^2 = 4 \).$

Q no.11 Find if $\frac{dy}{dx} = \frac{-2y}{2y + 4 - 2x}$

Q no.12 Find $\(\frac{dy}{dx}\) if y^2-xy-x^2+4=0$

Q no.13 Find $\frac{dy}{dx}$ by making a suitable substitution, if $y = \sqrt{x + \sqrt{x}}$

Q no.14 Find $\frac{dy}{dx}$ by making a suitable substitution, if $y=(3x^2-2x+7)^6$

Q no.15 Find $\(\frac{dy}{dx}\) if 3x+4y+7=0$

Q no.16 Find $\(\frac{dy}{dx}\) if xy+y^2=2$

Q No.17 Find $\(\frac{dy}{dx}\) if x^2-4xy-5y=0$

Q No.18 Find $\frac{dy}{dx}$ if $x=\theta+\frac{1}{\theta}$ and $y=\theta+1$

Q.No.19. Differentiate $((1+x)^2)^n$ with respect to $x^2$

Q No. 20 If $y=tan(2{\tan^{-1}⁡\frac{x}{2}})$ , show that $\frac{dy}{dx}= \frac{(4(1+y^2))}{(4+x^2 )}$

Q No. 21 Differentiate $x^2 sec4x$ w.r.t involved variale

Q No. 22 Differentiate $\tan^{3}{\theta} \sec^{2}{\theta}$ w.r.t involved variale

Q No. 23 Find $\(\frac{dy}{dx}\) if y =xcosx$

Q No. 24 Find $\(\frac{dy}{dx}\) if x =ysiny$

Q No.25 Differentiate $sinx$ w.r.t $cosx$

Q No.26 If $tany(1+tanx)=1-tanx$ , show that $\frac{dy}{dx}=1$

Q No.27 If $y\sqrt{(tanx+\sqrt{(tanx+\sqrt{tanx) ) +… }}}\infty$ Prove that $(2y-1)\frac{dy}{dx}=sec^2⁡x$

Q No.28 Differentiate $cot^{-1}⁡{\frac{x}{a}}$ w.r.t $x$

Q No.29 Differentiate $\frac{1}{a}sin^{-1}⁡{\frac{a}{x}}$ w.r.t $x$

Q No.30 If $y=tan⁡(ptan^{-1}{ x})$ , Show that $(1+x^2 )y_{1}-p(1+y^2 )=0$

Q No.31 Differentiate $y = a^x$ w.r.t $x$

Q No.32 Differentiate $(\ln{x})^x$ w.r.t $x$

Q No.33 Find $f '(x)$ if $f(x)=x^3 e^\frac{1}{x}$

Q No.34 Find $f '(x)$ if $f(x)=e^{x}(1+ln{x})$

Q No.35 Find $f '(x)$ if $f(x)=ln{x}(e^x +e^{-x})$

Q No.36 Find $f '(x)$ if $f(x)=ln\sqrt{e^{2x}+e^{-2x}}$

Q No. 37 Find $\frac{dy}{dx}$ if $y =x\sqrt{ln{x}}$

Q No. 38 Find $\frac{dy}{dx}$ if $y = \frac{x}{ln{x}}$

Q No. 39 Find $\frac{dy}{dx}$ if $y = ln⁡(9-x^2 )$

Q No. 40 Find $\frac{dy}{dx}$ if $y = e^{-2x} sin2x$

Q No. 41 Find $\frac{dy}{dx}$ if $y = xe^{sinx}$

Q No. 42 Find $\frac{dy}{dx}$ if $y = 5e^{3x-4}$

Q No. 43 Find $\frac{dy}{dx}$ if $y = (x+1)^x$

Q No. 44 Find $\frac{dy}{dx}$ if $y = tanh^{-1}⁡(sinx)$

Q No. 45 Find $\frac{dy}{dx}$ if $y = \sqrt{x} +\frac{1}{\sqrt{x}}$

Q No. 46 Find $y_{2}$ if $x^2+y^2=a^2$

Q No. 47 Find $y_{2}$ if $x=at^2 ,y=bt^4$

Q No.48 Apply Maclaurin series to prove $e^x=1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + …$

Q No.49 Define increasing and decreasing function

Q No.50 Define Point of inflection

Q No.51 Define Stationary point and critical point

Q No.52 Find extreme value for $f(x)=1-x^2$

Q No.53 Find extreme value for $f(x)= 3x^2$

Q No.54 Find extreme value for $f(x) =5 + 3x - x^3$

Q No.55 Find extreme value for $f(x) =x^2-x-2$

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Important short questions for Chapter 03 click here

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