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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