Mth102(5)

List of Questions

Latex formatted questions may not properly render

Q1 Given f(x) = $$(7x^{4} - 5x^{3})$$, evaluate \[\frac{d f(x)}{d x}\]
 $$7x^{4}-5x^{3}$$
 $$2x^{3}-15x^{2}$$
 $$28x^{2}-15x^{2}$$
 $$28x^{3}-15x^{2}$$

Q2 Evaluate \[\int x^{2} e^{3x} dx\]
 \[\frac{e^{3x}}{3}\left(x^{2}-\frac{2x}{3}+\frac{2}{9}\right)+c\]
 \[-\frac{e^{3x}}{3}\left(x^{2}+\frac{2x}{3}-\frac{2}{9}\right)+c\]
 \[\frac{e^{2x}}{3}\left(x^{3}-\frac{x}{4}+\frac{2}{9}\right)+c\]
 \[\frac{e^{x}}{3}\left(x^{2}+\frac{2x}{3}-\frac{2}{5}\right)+c\]

Q3 Find the volume of a sphere generated by a semicircle \[y=\sqrt(r^{2}-x^{2})\] revolving around the x-axis
 \[-\pi\frac{-r^{3}}{2}\]
 \[4\pi\frac{r^{3}}{2}\]
 \[\pi\frac{r^{3}}{4}\]
 \[4\pi\frac{r^{3}}{3}\]

Q4 Determine \[\int\frac{x^{2}+1}{(x+2)^{3}}\]
 \[ln (x+2)+\frac{4}{x+2}-\frac{5}{2(x+3)^{2}}+c\]
 \[ln (x+2)-\frac{4}{x+2}-\frac{5}{2(x+3)^{2}}+c\]
 \[-ln (x+2)-\frac{4}{x+2}-\frac{5}{2(x+3)^{2}}+c\]
 \[ln (x-2)+\frac{4}{x-2}-\frac{5}{2(x+3)^{2}}+c\]

Q5 Evaluate \[\int \frac{x+1}{x^{2}-3x+2} dx\]
 \[3ln(x+2)-2ln(x+1)+c\]
 \[3ln(x-2)-2ln(x-1)+c\]
 \[-3ln(x-2)-2ln(x-1)+c\]
 \[3ln(x-2)+2ln(x+1)+c\]

Q6 Find $$ \int {\sec^3 x}{\tan x}dx$$
 $$\sin 2x+c$$
 $$\cos 2x+c$$
 $$\frac{\sec^3 x}{3}+c$$
 $$\frac{\cos^3x}{2} +c$$

Q7 Find the $$\int {\tan^3 x}{\sec^3 x}dx$$
 $$\tan^2 x+1$$
 $$\cot^2 x+1$$
 $$\sec^2 x+1$$
 $$\sec^2 x$$

Q8 Find $$\int x\cos ax^2dx$$ with respect to x
 $$\cos^3 x+c$$
 $$\sin 2x+c$$
 $$\sec^2 x+1$$
 $$\frac{1}{2}a\sin ax^2+c$$

Q9 Evaluate \[\int x e^{6x} dx\]
 \[\frac{x}{6}e^{6x}-1\frac{1}{36}e^{6x}+c\]
 \[\frac{x}{3}e^{6x}+1\frac{1}{6}e^{6x}+c\]
 \[\frac{x}{6}e^{6x}+1\frac{1}{36}e^{6x}+c\]
 \[-\frac{x}{6}e^{6x}+1\frac{1}{36}e^{6x}+c\]

Q10 Evaluate $$\int e^{4x}dx$$
 $$\frac{1}{4}e^{4x}+c$$
 $$ex+c$$
 $$3e^\frac{x}{3}+c$$
 $$\frac{1}{3}ex+c$$

Q11 Integrate with respect to x :$$ \int_{1}^{4}\frac{x+1}{\sqrt{x}}dx$$
 $$\frac{20}{3}$$
 20
 $$\frac{3}{20}$$
 -20

Q12 Integrate with respect to x :$$ \int_{-1}^{3}\frac{x}{\sqrt{7+x^2}}dx$$
 $$2\sqrt{2}$$
 $$4\sqrt{2}$$
 $$4 - 2\sqrt{2}$$
 $$\sqrt{2}$$

Q13 Integrate with respect to x :$$ \int_{-1}^{2}\frac{x^2}{(x^3+4)^2}dx$$
 12
 $$\frac{1}{2}$$
 6
 $$\frac{5}{12}$$

Q14 Evaluate \[\int_{-1}^{2} y^{2}+y^{-2} dy\]
 \[\frac{7}{16}\]
 \[\frac{3}{16}\]
 \[\frac{17}{16}\]
 \[\frac{5}{16}\]

Q15 Find the integral with respect to x $$ \int \cos x\sin x dx$$
 $$\frac{\sin{^2} x}{2} +c$$
 $$\sin 2x+c$$
 $$\frac{\cos^{2} x}{2}+c$$
 \[\sin x\]

Q16 Evalute \[\int x^{2}(3-10x^{3})dx\]
 \[\frac{1}{150}(3-10x^{3})^5)+c\]
 \[\frac{1}{10}(1-10x^{2})^5)+c\]
 \[\frac{1}{15}(3-20x^{3})^5)+c\]
 \[\frac{1}{100}(3-2x^{3})^5)+c\]

Q17 Evaluate \[\int 3e^{x}+5\cos (x) -10 \sec^{2}(x) dx\]
 \[3e^{2}+5\sin x-10\tan x+c\]
 \[3e^{x}+\cos x-10\tan x+c\]
 \[3e^{x}+5\sin x-10\sec x+c\]
 \[2e^{x}- x-10\tan x+c\]

Q18 Evaluate \[\int \cos (6x+4)dx\]
 \[\frac{\sin (6x+4)}{6}+c\]
 \[\frac{\cos(6x+4)}{6}+c\]
 \[\frac{\tan(6x+4)}{6}+c\]
 \[\frac{\sec (6x+4)}{6}+c\]

Q19 Evaluate \[\int(3x-2)^{6} dx\]
 \[\frac{(3x+2)^{7}}{2}+c\]
 \[\frac{(3x+2)^{7}}{21}+c\]
 \[\frac{(3x-2)^{7}}{21}+c\]
 \[\frac{3(3x-2)^{7}}{2}+c\]

Q20 Integrate \[\int (x^{3}+3x^{2}+2x+4)\]
 \[\frac{x^{4}}{4}+x^{3}+x^{2}+4x+c\]
 \[\frac{x^{4}}{2}-x^{3}+x^{2}+4x+c\]
 \[3\frac{x^{4}}{4}+2x^{3}+x+c\]
 $$6x^{4}-3x^{2}+c$$

Q21 Differentiate \[y=3\sqrt(x^2)(2x-x^{2})\] with respect to x
 \[y=\frac{10 x^{\frac{2}{3}}}{3}-\frac{8 x^{\frac{5}{3}}}{3}\]
 \[y=\frac{10 x^{\frac{2}{3}}}{3}+\frac{8 x^{\frac{5}{3}}}{3}\]
 \[y=\frac{5 x^{\frac{2}{3}}}{3}-\frac{4 x^{\frac{5}{3}}}{3}\]
 \[y=\frac{5 x^{\frac{2}{3}}}{3}+\frac{4 x^{\frac{5}{3}}}{3}\]

Q22 Differentiate with respect to x: \[f(x) = (ax{^3} + bx)\]
 $$3a - b$$
 $$ax^{2} + b$$
 $$3x^{2} + 1$$
 $$3ax^{2} + b$$

Q23 Given \[y(x)=x^{4} - 4x^{3} + 3x^{2} -5x \], evaluate \[\frac{d^{4} y}{d x^{4}}\]
 30
 42
 24
 22

Q24 Given \[\frac{2x^{5}+x^{2}-5}{t^{2}}\], find \[\frac{d y}{d x}\] by using the first principle
 c\[-t^{-2}+8t^{-3}\]
 \[6t+7t^{-3}\]
 \[t^{2}+5t^{-3}\]
 \[6t^{2}+10t^{-3}\]

Q25 Find the derivative \[f(x)=2x^{2}-16x+35\] by using first principle
 \[x+16\]
 \[4x-16\]
 \[3x-5\]
 \[2x-8\]

Q26 Evaluate the limit \[\lim {x\rightarrow \infty} \frac{6e^{4x}-e^{-2x}}{8e^{4x}-e^{2x}+3e^{-x}}\]
 \[\frac{3}{4}\]
 \[\frac{1}{4}\]
 \[\frac{1}{2}\]
 \[\frac{3}{5}\]

Q27 Evaluate the limit \[\lim {x\rightarrow -\infty} \frac{x^{2}-5t-9}{2x^{4}+3x^{3}}\]
 4
 2
 0
 1

Q28 Evaluate the limit \[\lim {x\rightarrow \infty} \frac{2x^{4}-x^{2}+8x}{-5x^{4}+7}\]
 \[\frac{1}{3}\]
 \[\frac{2}{3}\]
 \[\frac{1}{2}\]
 \[\frac{3}{4}\]

Q29 Evaluate the limit \[\lim {t\rightarrow 4} \frac{t-\sqrt(3+4)}{4-t}\]
 \[\frac{-3}{8}\]
 \[\frac{-5}{8}\]
 \[\frac{-1}{8}\]
 \[\frac{3}{4}\]

Q30 Evaluate the limit \[\lim_{h\rightarrow 0}\frac{2(-3+h)^{2}-18}{h}\]
 12
 8
 14
 6

Q31 Differentiate \[y=3\sqrt(x^2)(2x-x^{2})\] with respect to x
 \[y=\frac{10 x^{\frac{2}{3}}}{3}-\frac{8 x^{\frac{5}{3}}}{3}\]
 \[y=\frac{10 x^{\frac{2}{3}}}{3}+\frac{8 x^{\frac{5}{3}}}{3}\]
 \[y=\frac{5 x^{\frac{2}{3}}}{3}-\frac{4 x^{\frac{5}{3}}}{3}\]
 \[y=\frac{5 x^{\frac{2}{3}}}{3}+\frac{4 x^{\frac{5}{3}}}{3}\]

Q32 Differentiate with respect to x: \[f(x) = (ax{^3} + bx)\]
 $$3a - b$$
 $$ax^{2} + b$$
 $$3x^{2} + 1$$
 $$3ax^{2} + b$$

Q33 Given \[y(x)=x^{4} - 4x^{3} + 3x^{2} -5x \], evaluate \[\frac{d^{4} y}{d x^{4}}\]
 30
 42
 24
 22

Q34 Given \[\frac{2x^{5}+x^{2}-5}{t^{2}}\], find \[\frac{d y}{d x}\] by using the first principle
 c\[-t^{-2}+8t^{-3}\]
 \[6t+7t^{-3}\]
 \[t^{2}+5t^{-3}\]
 \[6t^{2}+10t^{-3}\]

Q35 Find the derivative \[f(x)=2x^{2}-16x+35\] by using first principle
 \[x+16\]
 \[4x-16\]
 \[3x-5\]
 \[2x-8\]

Q36 Evaluate the limit \[\lim {x\rightarrow \infty} \frac{6e^{4x}-e^{-2x}}{8e^{4x}-e^{2x}+3e^{-x}}\]
 \[\frac{3}{4}\]
 \[\frac{1}{4}\]
 \[\frac{1}{2}\]
 \[\frac{3}{5}\]

Q37 Evaluate the limit \[\lim {x\rightarrow -\infty} \frac{x^{2}-5t-9}{2x^{4}+3x^{3}}\]
 4
 2
 0
 1

Q38 Evaluate the limit \[\lim {x\rightarrow \infty} \frac{2x^{4}-x^{2}+8x}{-5x^{4}+7}\]
 \[\frac{1}{3}\]
 \[\frac{2}{3}\]
 \[\frac{1}{2}\]
 \[\frac{3}{4}\]

Q39 Evaluate the limit \[\lim {t\rightarrow 4} \frac{t-\sqrt(3+4)}{4-t}\]
 \[\frac{-3}{8}\]
 \[\frac{-5}{8}\]
 \[\frac{-1}{8}\]
 \[\frac{3}{4}\]

Q40 Evaluate the limit \[\lim_{h\rightarrow 0}\frac{2(-3+h)^{2}-18}{h}\]
 12
 8
 14

 6