Question 511 Mark
If $f(x)=2|x|+3 x$, then find$f(-5)$
Answer$ \text { (ii) } f(-5)=2|-5|+3(-5)$
$=2(5)-15 \ldots .[\because|x|=-x, x<0]$
$=10-15$
$=-5$
View full question & answer→Question 521 Mark
If $f(x)=2|x|+3 x$, then find$f(2)$
Answer$ f(x)=2|x|+3 x$
$\text { (i) } f(2)=2|2|+3(2)$
$=2(2)+6 \ldots .[\because|x|=x, x>0]$
$=10$
View full question & answer→Question 531 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}4 x-2, & x \leq-3 \\ 5, & -3<x<3 \\ x^2, & x \geq 3\end{array}\right.$, then find$f(5)$
View full question & answer→Question 541 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}4 x-2, & x \leq-3 \\ 5, & -3<x<3 \\ x^2, & x \geq 3\end{array}\right.$, then find$f(1)$
View full question & answer→Question 551 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}4 x-2, & x \leq-3 \\ 5, & -3$\mathrm{f}(-3)$
Answer$ f(-3)=4(-3)-2$
$=-12-2$
$=-14$
View full question & answer→Question 561 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{cc}4 x-2, & x \leq-3 \\ 5, & -3$f(-4)$
Answer$ f(x)=4 x-2, x \leq-3$
$=5,-3=x^2, x \geq 3$
$\text { (i) } f(-4)=4(-4)-2$
$=-16-2$
$=-18 $
View full question & answer→Question 571 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}x^2+3, & x \leq 2 \\ 5 x+7, & x>2 \end{array}\right.$},then find$f(0)$
View full question & answer→Question 581 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}x^2+3, & x \leq 2 \\ 5 x+7, & x>2 \end{array}\right.$},then find$\mathrm{f}(\mathrm{C})$
Answer$f(2)=2^2+3$
$ =4+3$
$=7 $
View full question & answer→Question 591 Mark
If $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}x^2+3, & x \leq 2 \\ 5 x+7, & x>2 \end{array}\right.$},then find$f(3)$
Answer$ f(x)=x^2+3, x \leq 2$
$=5 x+7, x>2$
$\text { (i) } f(3)=5(3)+7$
$=15+7$
$=22$
View full question & answer→Question 601 Mark
Check if the following functions have an inverse function. If yes, find the inverse function.$f(x)=8$
Answerf(x) = 8 = y (say)
For every value of x, the value of the function f is the same.
∴ f is not one-one i.e. (many-one) function.
∴ f does not have the inverse.
View full question & answer→Question 611 Mark
If $f(x)=2 x^2+3, g(x)=5 x-2$, then findgog
Answer$(g \circ g)(x)=g(g(x))$
$=g(5 x-2)$
$=5(5 x-2)-2$
$=25 x-12 $
View full question & answer→Question 621 Mark
If $f(x)=2 x^2+3, g(x)=5 x-2$, then find fof
Answer$ \text { (fof) }(x)=f(f(x))$
$=f\left(2 x^2+3\right)$
$=2\left(2 x^2+3\right)^2+3$
$=2\left(4 x^4+12 x^2+9\right)+3$
$=8 x^4+24 x^2+21 $
View full question & answer→Question 631 Mark
If $f(x)=2 x^2+3, g(x)=5 x-2$, then findgof
Answer$ (g \circ f)(x)=g(f(x))$
$=g\left(2 x^2+3\right)$
$=5(2 x+3)-2$
$=10 x^2+13 $
View full question & answer→Question 641 Mark
If $f(x)=2 x^2+3, g(x)=5 x-2$, then findfog
Answer$ f(x)=2 x^2+3, g(x)=5 x-2$
$\text { (i) (fog) }(x)=f(g(x))$
$=f(5 x-2)$
$=2(5 x-2)^2+3$
$=2\left(25 x^2-20 x+4\right)+3$
$=50 x^2-40 x+11 $v
View full question & answer→Question 651 Mark
If $f(x)=3 x+5, g(x)=6 x-1$, then find( $\mathrm{f} / \mathrm{g})(\mathrm{x})$ and its domain
Answer$\left(\frac{\mathrm{f}}{\mathrm{g}}\right)(x)=\frac{\mathrm{f}(x)}{\mathrm{g}(x)}=\frac{3 x+5}{6 x-1}, x \neq \frac{1}{6}$ Domain $=R-\left\{\frac{1}{6}\right\}$
View full question & answer→Question 661 Mark
If $f(x)=3 x+5, g(x)=6 x-1$, then find $(fg) (3)$
Answer$ (\mathrm{fg})(3)=f(3) g(3)$
$=[3(3)+5][6(3)-1]$
$=(14)(17)$
$=238 $
View full question & answer→Question 671 Mark
If $f(x)=3 x+5, g(x)=6 x-1$, then find$(f-g)(2)$
Answer$(f-g)(2)=f(2)-g(2)$
$=[3(2)+5]-[6(2)-1]$
$=6+5-12+1$
$=0$
View full question & answer→Question 681 Mark
If $f(x)=3 x+5, g(x)=6 x-1$, then find$(f+g)(x)$
Answer$ f(x)=3 x+5, g(x)=6 x-1$
$(i)(f+g)(x)=f(x)+g(x)$
$=3 x+5+6 x-1$
$=9 x+4$
View full question & answer→Question 691 Mark
Given that $\log 2=a$ and $\log 3=b$, write $\log \sqrt{96}$ terms of $a$ and $b$.
Answer$\log 2=a \text { and } \log 3=b$
$\log \sqrt{ } 96=\frac{1}{2} \log (96)$
$=\frac{1}{2} \log \left(2^5 \times 3\right)$
$=\frac{1}{2}\left(\log 2^5+\log 3\right) \ldots . .[\because \log m n=\log m+\log n]$
$=\frac{1}{2}(5 \log 2+\log 3) \ldots \ldots\left[\cdot\left[\log m^n=n \log m\right]\right.$
$=\frac{5 a+b}{2} $
View full question & answer→Question 701 Mark
Write the following expressions as a single logarithm.$\ln (x+2)+\ln (x-2)-3 \ln (x+5)$
Answer$\ln (x+2)+\ln (x-2)-3 \ln (x+5)$
$=\ln [(x+2)(x-2)]-\ln (x+5)^3$
$\ldots\left[\begin{array}{l} \log m+\log n=\log m n \\ n \log m=\log m^n \end{array}\right]$
$=\ln \left(x^2-4\right)-\ln (x+5)^3$
$=\ln \left(\frac{x^2-4}{(x+5)^3}\right) \ldots\left[\log m-\operatorname{logn}=\log \frac{m}{n}\right]$
View full question & answer→Question 711 Mark
Write the following expressions as a single logarithm.$\frac{1}{3} \log (x-1)+\frac{1}{2} \log (x)$
Answer$ \frac{1}{3} \log (x-1)+\frac{1}{2} \log x$
$=\log \left((x-1)^{\frac{1}{3}}\right)+\log \left(x^{\frac{1}{2}}\right)$
$=\log (\sqrt[3]{x-1} \sqrt{x}) $
$\ldots\left[n \log m=\log m^n\right]$
$=\log (\sqrt[3]{x-1} \sqrt{x})$$[\log m+\log n=\log m n]$
View full question & answer→Question 721 Mark
Write the following expressions as a single logarithm.
$5 \log x+7 \log y-\log z$
Answer$5 \log x+7 \log y-\log z$
$=\log \left(x^5\right)+\log \left(y^7\right)-\log z$
$\ldots\left[n \log m=\log m^n\right]$
$=\log \left(x^5 y^7\right)-\log \mathrm{z}$
$\ldots[\log m+\log n=\log m n]$
$=\log \left(\frac{x^5 y^7}{z}\right)$
$\ldots\left[\log m-\log n=\log \frac{m}{n}\right]$
View full question & answer→Question 731 Mark
Write the following expressions as sum or difference of logarithms:$\ln \left(\frac{a^3(a-2)^2}{\sqrt{b^2+5}}\right)$
Answer$ \ln \left(\frac{a^3(a-2)^2}{\sqrt{b^2+5}}\right)$
$=\ln \left(a^3(a-2)^2\right)-\ln \sqrt{b^2+5}$
$\quad \ldots\left[\log \frac{m}{n}=\log m-\log n\right]$
$ =\ln a^3+\ln (a-2)^2-\ln \left(b^2+5\right)^{\frac{1}{2}}$
$\qquad \ldots[\log m n=\log m+\log n]$
$=3 \ln a+2 \ln (a-2)-\frac{1}{2} \ln \left(b^2+5\right) $
$\ldots\left[\log m^n=n \log m\right]$
View full question & answer→Question 741 Mark
Write the following expressions as sum or difference of logarithms:$\log (\sqrt{x} \sqrt[3]{y})$
Answer$
\begin{aligned}
\log (\sqrt{x} \sqrt[3]{y})= & \log (\sqrt{x})+\log (\sqrt[3]{y}) \\
& \ldots[\log mn =\log m +\log n ] \\
= & \log x^{\frac{1}{2}}+\log y^{\frac{1}{3}} \\
= & \frac{1}{2} \log x+\frac{1}{3} \log y
\end{aligned}
$
... [ $\left.\log m ^{ n }= nlog m \right]$
View full question & answer→Question 751 Mark
Write the following expressions as sum or difference of logarithms:
$\log \left(\frac{p q}{r s}\right)$
Answer$\log \left(\frac{ pq }{ rs }\right)=\log ( pq )-\log ( rs )$
$\left[\log \frac{m}{n}=\log m-\log n\right]$
$ =\log p+\log q-(\log r+\log s)$
$\quad \ldots[\log m n=\log m+\log n]$
$=\log p+\log q-\log r-\log s$
View full question & answer→Question 761 Mark
Find the domain of$f(x)=\log 10\left(x^2-5 x+6\right)$
Answer$f(x)=\log _{10}\left(x^2-5 x+6\right)$
$
x^2-5 x+6=(x-2)(x-3)
$
$f$ is defined, when $(x-2)(x-3)>0$
$
\therefore x <2 \text { or } x >3
$
Solution of $(x-a)(x-b)>0$ is $x<a$ or $x>b$ where $a<b$
$\therefore$ Domain of $f=(-\infty, 2) \cup(3, \infty)$
View full question & answer→Question 771 Mark
Find the domain of$f(x)=\ln (x-5)$
Answer$f(x)=\ln (x-5)$
$f$ is defined, when $x-5>0$
$
\therefore x >5
$
$\therefore$ Domain of $f=(5, \infty)$
View full question & answer→Question 781 Mark
Express the following logarithmic equations in exponential form:$\ln \frac{1}{2}=-0.693$
Answer$\ln \left(\frac{1}{2}\right)=-0.693$
$\therefore \quad \frac{1}{2}= e ^{-0.693}$,i.e., $e ^{-0.693}=\frac{1}{2}$
View full question & answer→Question 791 Mark
Express the following logarithmic equations in exponential form:$\ln e =1$
Answer$\quad \ln e =1$
$\therefore \quad e = e ^1$,i.e., $e ^{ l }= e$
View full question & answer→Question 801 Mark
Express the following logarithmic equations in exponential form:$\ln 1=0$
Answer$\quad \ln 1=0 \quad$
$\therefore \quad 1= e ^0$, i.e., $e ^0=1$
View full question & answer→Question 811 Mark
Express the following logarithmic equations in exponential form:$\log _{\frac{1}{2}}(8)=-3$
Answer$\log _{\frac{1}{2}}(8)=-3$
$\therefore \quad 8=\left(\frac{1}{2}\right)^{-3}$,i.e., $\left(\frac{1}{2}\right)^{-3}=8$
View full question & answer→Question 821 Mark
Express the following logarithmic equations in exponential form:$\log _{10} 0.001=-3$
Answer$\log _{10}(0.001)=-3$
$\therefore \quad 0.001=10^{-3}$, i.e., $10^{-3}=0.001$
View full question & answer→Question 831 Mark
Express the following logarithmic equations in exponential form:$\log _5 \frac{1}{25}=-2$
Answer$\quad \log _5\left(\frac{1}{25}\right)=-2$
$\therefore \quad \frac{1}{25}=5^{-2}$,i.e., $5^{-2}=\frac{1}{25}$
View full question & answer→Question 841 Mark
Express the following logarithmic equations in exponential form:$\log _2 64=6$
Answer$\log _2 64=6$
$\therefore 64=2^6$, i.e., $2^6=64$
View full question & answer→Question 851 Mark
Express the following exponential equations in logarithmic form:$e^{-x}=6$
Answer$\quad e ^{-x}=6$
$\therefore \quad-x=\log _e 6$
[By definition of logarithm]
i.e., $\log _e 6=-x$
View full question & answer→Question 861 Mark
Express the following exponential equations in logarithmic form:$e^{\frac{1}{2}}=1.6487$
Answer$e ^{\frac{1}{2}}=1.6487$
$\therefore \quad \frac{1}{2}=\log _e(1.6487) \ldots$ [By definition of logarithm]
i.e., $\log _e(1.6487)=\frac{1}{2}$
View full question & answer→Question 871 Mark
Express the following exponential equations in logarithmic form:$e ^2=7.3890$
Answer$e ^2=7.3890$
$\therefore \quad 2=\log _e(7.3890) \ldots$ [By definition of logarithm] i.e., $\log _e(7.3890)=2$
(e is a mathematical constant, whose value is approximately 2.71828 )
View full question & answer→Question 881 Mark
Express the following exponential equations in logarithmic form:$10^{-2}=0.01$
Answer$10^{-2}=0.01$
$\therefore \quad-2=\log _{10}(0.01) \ldots$ [By definition of logarithm] i.e., $\log _{10}(0.01)=-2$
View full question & answer→Question 891 Mark
Express the following exponential equations in logarithmic form:$3^{-4}=\frac{1}{81}$
Answer$\quad 3^{-4}=\frac{1}{81}$
$\therefore \quad-4=\log _3\left(\frac{1}{81}\right) \ldots$ [By definition of logarithm] i.e., $\log _3\left(\frac{1}{81}\right)=-4$
View full question & answer→Question 901 Mark
Express the following exponential equations in logarithmic form:$9^{\frac{3}{2}}=27$
Answer$\quad 9^{\frac{3}{2}}=27$
$\therefore \quad \frac{3}{2}=\log _9 27$
..[By definition of logarithm]
i.e., $\log _9 27=\frac{3}{2}$
View full question & answer→Question 911 Mark
Express the following exponential equations in logarithmic form:$23^1=23$
Answer$23^1=23$
$\therefore \quad 1=\log _{23} 23$
[By definition of logarithm]
i.e., $\log _{23} 23=1$
View full question & answer→Question 921 Mark
Express the following exponential equations in logarithmic form:$54^0=1$
Answer$54^0=1$
$\therefore \quad 0=\log _{54} 1$
..[By definition of logarithm]
i.e., $\log _{54} 1=0$
View full question & answer→Question 931 Mark
Express the following exponential equations in logarithmic form:$2^5=32$
Answer$\begin{array}{ll}
& 2^5=32 \\
\therefore & 5=\log _2 32 \quad \ldots \text { [By definition of logarithm] } \\
& \text { i.e., } \log _2 32=5
\end{array}
$
View full question & answer→Question 941 Mark
If $f(x)=3\left(4^{x+1}\right)$, find $f(-3)$.
Answer$ f(x)=3\left(4^{x+1}\right)$
$\therefore f(-3)=3\left(4^{-3+1}\right)$
$=3\left(4^{-2}\right)$
$=\frac{3}{16} $
View full question & answer→Question 951 Mark
Express the area $\mathrm{A}$ of a square as a function of itsperimeter $\mathrm{P}$
Answerperimeter $(P)=4 \mathrm{~s}$
$\therefore \mathrm{s}=\frac{\mathrm{P}}{4}$
Area $(\mathrm{A})=s^2=\left(\frac{\mathrm{P}}{4}\right)^2$
$\therefore \mathrm{A}=\frac{\mathrm{P}^2}{16}$
View full question & answer→Question 961 Mark
Express the area $\mathrm{A}$ of a square as a function of itsside $s$
View full question & answer→Question 971 Mark
Find the domain and range of the following functions.$f(x)=\sqrt[3]{x+1}$
Answer$
f(x)=\sqrt[3]{x+1}
$
$f$ is defined for all real $x$ and the values of $f(x) \in R$
$\therefore$ Domain of $f=R$, Range of $f=R$
View full question & answer→Question 981 Mark
If $\mathrm{f}(\mathrm{x})=\frac{a-x}{b-x}, \mathrm{f}(2)$ is undefined, and $\mathrm{f}(3)=5$, find $a$ and $\mathrm{b}$.
Answer$f(x)=\frac{a-x}{b-x}$
Given that,
$f(2)$ is undefined
$ b-2=0$
$\therefore b=2 \ldots .(i)$
$f(3)=5$
$\therefore \frac{a-3}{b-3}=5$
$\therefore \frac{a-3}{2-3}=5 \ldots . .[\text { [From (i)] }$
$\therefore a-3=-5$
$\therefore a=-2$
$\therefore a=-2, b=2 $
View full question & answer→Question 991 Mark
Find $x$, if $g(x)=0$ where
$g(x)=x^3-2 x^2-5 x+6$
Answer$ g(x)=x^3-2 x^2-5 x+6$
$=(x-1)\left(x^2-x-6\right)$
$=(x-1)(x+2)(x-3)$
$g(x)=0$
$\therefore(x-1)(x+2)(x-3)=0$
$\therefore x-1=0 \text { or } x+2=0 \text { or } x-3=0$
$\therefore x=1,-2,3$
View full question & answer→Question 1001 Mark
Find $x$, if $g(x)=0$ where
$g(x)=6 x^2+x-2$
Answer$g(x)=6 x^2+x-2$
$g(x)=0$
$\therefore 6 x^2+x-2=0$
$\therefore(2 x-1)(3 x+2)=0$
$\therefore 2 x-1=0 \text { or } 3 x+2=0$
$\therefore x=\frac{1}{2} \text { or } x=\frac{-2}{3}$
View full question & answer→