1. f(x)=secx f(x)=\sec x f(x)=secx and g(x)=tanx g(x)=\tan x g(x)=tanx
∫f(x)dx=? \int f(x) d x=? ∫f(x)dx=?
ln∣tan(π4−x2)∣+c \ln \left|\tan \left(\frac{\pi}{4}-\frac{x}{2}\right)\right|+c lntan(4π−2x)+c
ln∣tan(π4+x2)∣+c \ln \left|\tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\right|+c lntan(4π+2x)+c
ln∣secx−tanx∣+c \ln |\sec x-\tan x|+c ln∣secx−tanx∣+c
ln∣tanx−secx∣+c \ln |\tan x-\sec x|+c ln∣tanx−secx∣+c
2. ∫011−x1+xdx=? \int_{0}^{1} \frac{1-x}{1+x} d x=? ∫011+x1−xdx=?
ln2−2 \ln 2-2 ln2−2
1−2ln2 1-2 \ln 2 1−2ln2
2ln2−1 2 \ln 2-1 2ln2−1
2ln2+2 2 \ln 2+2 2ln2+2
3. If A=[15−32] A=\left[\begin{array}{rr}1 & 5 \\ -3 & 2\end{array}\right] A=[1−352] and I=[1001],IA=? I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], I A=? I=[1001],IA=?
[600−1] \left[\begin{array}{rr}6 & 0 \\ 0 & -1\end{array}\right] [600−1]
[1−352] \left[\begin{array}{rr}1 & -3 \\ 5 & 2\end{array}\right] [15−32]
[500−6] \left[\begin{array}{rr}5 & 0 \\ 0 & -6\end{array}\right] [500−6]
[15−32] \left[\begin{array}{rr}1 & 5 \\ -3 & 2\end{array}\right] [1−352]
4. What is the value of ∫01xex2dx \int_{0}^{1} \mathrm{xe}^{\mathrm{x}^{2}} d x ∫01xex2dx ?
1−2e 1-\frac{2}{\mathrm{e}} 1−e2
1
12(e−1) \frac{1}{2}(e-1) 21(e−1)
14e \frac{1}{4} \mathrm{e} 41e
5. If limx→af(x)=l \lim _{x \rightarrow a} f(x)=l limx→af(x)=l and limx→ag(x)=m \lim _{x \rightarrow a} g(x)=m limx→ag(x)=m.
i. limx→a[f(x)−g(x)]=l−m \lim _{\mathrm{x} \rightarrow \mathrm{a}}[\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})]=l-\mathrm{m} limx→a[f(x)−g(x)]=l−m
ii. limx→ag(x)f(x)=ml \lim _{x \rightarrow a} g(x) f(x)=m l limx→ag(x)f(x)=ml
iii. limx→ag(x)f(x)=l m \lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x}) \mathrm{f}(\mathrm{x})=\frac{l}{\mathrm{~m}} limx→ag(x)f(x)= ml
Which one is correct?
i and ii
i and iii
ii and iii
i, ii and iii
