KCET 2021 Mathematics Questions with Answers Key Solutions
Solution:
Slope $m=\frac{d y}{d x}=-3 x^{2}+6 x+2$
$\frac{d m}{d x}=0 ; -6 x+6=0$
$x =1 ; m =-3+6+2=5$
Solution:
$\text{Tan}^{-1} x^{4}=t ; \frac{4 x^{3}}{1+x^{8}} d x=d t$
$I=\frac{1}{4} \int \sin t d t=\frac{-1}{4} \cos t+c$
$=\frac{-1}{4} \cos \left(\text{Tan}^{-1} x^{4}\right)+c$
Solution:
$x ^{3}= t \,\,\,3 x ^{2} d x = dt$
$I =\frac{1}{3} \int \frac{1}{\sqrt{ t ^{2}+\left( a ^{3}\right)^{2}}} dt$
$=\frac{1}{3} \log \left[ t +\sqrt{ t ^{2}+ a ^{6}}\right]$
$=\frac{1}{3} \log \left[ x ^{3}+\sqrt{ x ^{6}+ a ^{6}}\right]+ c$
Solution:
$\int \frac{(x+1-1) e^{x}}{(1+x)^{2}} d x$
$=\int e^{x}\left(\frac{1}{1+x}-\frac{1}{(1+x)^{2}}\right) d x$
$=\frac{e^{x}}{1+x}+c$
Solution:
$\int e ^{ x }\left(\frac{1+2 \sin \frac{ x }{2} \cos \frac{ x }{2}}{2 \cos ^{2} \frac{ x }{2}}\right) dx$
$=\int e ^{ x }\left(\frac{1}{2} \sec ^{2} \frac{ x }{2}+\text{Tan} \frac{ x }{2}\right) d x$
$= e ^{ x } \text{Tan} \frac{ x }{2}+ c$
Solution:
$I _{ n }+ I _{ n -2}=\frac{1}{ n -1}$
Solution:
$\int\limits_{a}^{b} \frac{f(x)}{f(x)+f(a +b-x)} d x=\frac{b-a}{2}$
Solution:
$x^{2}+y^{2}=16$
$\frac{1}{2} \pi(4)^{2}=8 \pi$
Solution:
$\frac{ x ^{2}}{25}+\frac{ y ^{2}}{\lambda^{2}}=1$
$\pi ab =\pi \cdot 5 \cdot|\lambda|=20\, \pi$
$|\lambda|=4 \Rightarrow \lambda=\pm 4$
Solution:
$x d y=y d x\,\, \frac{d y}{d x}=\frac{y}{x}$
$m=\frac{y}{x}$
$y=m x$
Solution:
One solution
Solution:
$\overline{ a }=\hat{ i }+\hat{ j }+\hat{ k }$
$\frac{\overline{ b } \cdot \overline{ a }}{|\overline{ a }|}=\frac{5+7-1}{\sqrt{3}}$
$=\frac{11}{\sqrt{3}}$
Solution:
$\overline{ a }=\frac{\overline{ d }_{1}+\overline{ d }_{2}}{2}=\frac{2 i +4 j -10 k }{2}$
$= i +2 j -5 k$
$|\overline{ a }|=\sqrt{30}$
$|\overline{ b }|=\frac{\overline{ d }_{1}-\overline{ d }_{2}}{2}=\frac{4 i +8 j +6 k }{2}$
$=2 i +4 j +3 k$
$|\overline{ b }|=\frac{2}{\sqrt{4+16+9}}=\sqrt{29}$
Q14. If $\vec{a} \cdot \vec{b}=0$ and $\vec{a}+\vec{b}$ makes an angle $60^{\circ}$ with $\vec{a} $ then
Solution:
$\cos 60=\frac{(\vec{a}+\vec{b}) \cdot \vec{a}}{|\vec{a}+\vec{b}||\vec{a}|}=\frac{|\vec{a}|^{2}+0}{\sqrt{|\vec{a}|^{2}+|\vec{b}|^{2}}}=$
$\frac{1}{2}=\frac{|\vec{a}|}{\sqrt{|\vec{a}|^{2}+|\vec{b}|^{2}}}$
$|\vec{a}|^{2}+|\vec{b}|^{2}=4|\vec{a}|^{2}$
$|\vec{b}|^{2}=3|\vec{a}|^{2}$
$|\vec{b}|=\sqrt{3}|\vec{a}|$
Solution:
$|\vec{ a } \times \vec{ b }|=15$
$|(3 \vec{ a }+2 \vec{ b }) \times(\vec{ a }+3 \vec{ b })|$
$=|9(\vec{ a }+\vec{ b }) \times 2(\vec{ b }+\vec{ a })|$
$=|7(\vec{ a } \times \vec{ b })|=7 \times 15=105$
Solution:
$A (-3,4,11) B (1,-2,7)$ Dr's of $AB$
$( a , b , c )=1-(-3),(-2,-4) 7,-11$
$4,-6,-4$
$=-2,3,2$
Solution:
$\cos \theta=\left|\frac{\sqrt{3}}{4} \times \frac{\sqrt{3}}{4}+\frac{1}{4} \times \frac{1}{4}-\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}\right|=\left(\frac{-1}{2}\right), \theta$
$=\frac{\pi}{3}$
Solution:
$(1,2,3)=\left(\frac{ a }{3}, \frac{ b }{3}, \frac{ c }{3}\right), a =3, b =6, c =9$
Solution:
$\frac{1}{2}(\overline{ AC } \times \overline{ BD })=9$ sq. units
Solution:
$x \leq 6, y \leq 3,5 x+4 y \geq 20$
Solution:
$\frac{1}{2}= P \frac{( A \cap B )}{\frac{3}{5}}$
$\Rightarrow P ( A \cap B )=\frac{3}{10}$
$\frac{4}{5}=\frac{3}{5}+ P ( A )-\frac{3}{10}$
$\Rightarrow P ( A )=\frac{1}{2}$
Solution:
$P(A)=P(B)=P(C)=P$
$P(A) \cdot P(B) \cdot P(C)+3 \cdot P^{2}(1-p)$
$P^{3}+3 p^{2}(1-P)=3 P^{2}-2 P^{3}$
Solution:
$(1,1)(1,2)(1,3)(1,4)(2,1)(2,2)(2,3)$
$(3,1)(3,2)(4,1)$
$P(B)=\frac{10}{36}$
$n ( A )=(1,2)(2,1)$
$P ( A )=\frac{2}{36}$
$P \left(\frac{ B }{ A }\right)=\frac{\frac{2}{36}}{\frac{10}{36}}$
$=\frac{2}{10}=\frac{1}{5}$
Solution:
$=\frac{\frac{70}{100} \times \frac{80}{100}}{\frac{70}{100} \times \frac{80}{100}+\frac{90}{100} \times \frac{30}{100}}$
$=\frac{56}{83}$
Solution:
$x=\frac{65 x}{100}+15000-\frac{15 x}{100}$
$=30,000$
Solution:
$n ( A \times B )=35=7 \times 5,7_{ C _{5}}=7_{ C _{2}}=21$
Solution:
$\cos \left(3 \times 360^{\circ}+120^{\circ}\right)+\tan \left(4 \times 360^{\circ}+45^{\circ}\right)$
$=\frac{1}{2}$
Solution:
$\tan \theta \cdot \cot \theta=1$
Solution:
$\left(\frac{1+i}{1-i}\right)^{x}=1$
$\Rightarrow i^{x}=1$
Solution:
$R(x)-c(x)>0 ; 60 x+2000-20 x-4000>0$
$x>50$
Solution:
${ }^{13} C _{10}-{ }^{6} C _{3}=286-20=266$
Solution:
mid term is $T _{26}=300$
$T _{1}=300-25 d ; T _{51}=300+25 d$
$S =\frac{51}{2}[300-25 d +300+25 d ]$
$\frac{51}{2}[600]=15,300$
Solution:
$\frac{ x }{\sin \theta}-\frac{ y }{\cos \theta}=\frac{ a \cos ^{3} \theta}{\sin \theta}-\frac{ a \sin ^{3} \theta}{\cos \theta}$
$\frac{ x \cos \theta- y \sin \theta}{\sin \theta \cos \theta}=\frac{ a (\cos 2 \theta)}{\sin \theta \cos \theta}$
$x \cos \theta- y \sin \theta= a \cos 2 \theta$
Solution:
$\left(\frac{1+0+2}{3}, \frac{5+4+3}{3}, \frac{-1-2+4}{3}\right)$
$\left(1,4, \frac{1}{3}\right)$
Solution:
statement 1 is true Statement 2 is false
${\left[\frac{a+b+c}{a+b+c}=1\right]} $
$\displaystyle\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2} \text { is }-\frac{1}{4}$
Solution:
$\frac{ d }{ dx }\left(\frac{ a }{ x ^{4}}-\frac{ b }{ x ^{2}}+\cos x \right)=\left(-\frac{4 a }{ x ^{5}}+\frac{2 b }{ x ^{3}}-\sin x \right)$
$= ma + nb - p$
$m =-\frac{4}{ x ^{5}} ; n =\frac{+2}{ x ^{3}} ; p =\sin x$
Solution:
S.D. $=\sqrt{\frac{ n ^{2}-1}{12}}( n =17)$
$=\sqrt{\frac{17^{2}-1}{12}}$
$=2 \sqrt{6}$
Solution:
$P\left(A^{1} \cap B^{1}\right)=1-P(A \cup B)$
$=1-[0.59+0.3-0.21]$
$=0.32$
Solution:
$f(-2)+f(3)+f(4)$
$-6+9+8$
$=11$
Solution:
$f'(x)=\frac{-2}{(x-1)^{2}}<0$
$f$ is s.d. $f$ is one-one
$\frac{2 x}{x-1}=y$
$\Rightarrow x=\frac{y}{y-2} \notin \pi$
for $y=2f$ is not out
Solution:
$f=\sqrt{3} \sin 2 x-\cos 2 x+4=2\left[\sin \left(2 x-\frac{\pi}{6}\right)\right]+4$
$f$ is one-one
$-\frac{\pi}{2} \leq 2 x-\frac{\pi}{6} \leq \frac{\pi}{2}$
$\Rightarrow -\frac{\pi}{6} \leq x \leq \frac{\pi}{3}$
${\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]}$
Solution:
$\left[x^{2}\right]-[x]-6>0 ([x]-3)([x]+2)>0$
$[x]<-2,[x]>3$
$\Rightarrow x \in(-\infty,-2) \cup[4, \infty)$
Solution:
$\cos \left(\pi-\frac{\pi}{6}+\frac{\pi}{6}\right)=\cos \pi=-1$
Solution:
$\left(\frac{\pi}{6}\right)^{2}$
Solution:
$A B=\begin{pmatrix}-3 & -2 \\ 10 & 7\end{pmatrix}$
$(A B)^{T}=\begin{pmatrix}-3 & 10 \\ -2 & 7\end{pmatrix}$
Solution:
$m =\begin{pmatrix} a & 0 \\0 & a\end{pmatrix}$
$m$ is invertible.
Solution:
$|3 AB |=3^{3}| AB |$
$=27 \times 3 \times 5$
$=405$
Solution:
$(A+B)^{-1}=B^{-1}+A^{-1}$
Solution:
$f(x)=4 \cos ^{3} x-3 \cos x$
$=\cos 3 x$
$\displaystyle\lim _{x \rightarrow \pi} \cos 3 x=\cos 3 \pi$
$=-1$
Solution:
$(x-2)\left(x^{2}-9\right)=0$
$x=2,3,-3$
$f(x)=|A|=-12 x+60$
Max value at $x=-3$
$\therefore |A|=96$
Solution:
$\displaystyle\lim _{x \rightarrow 1+} x^{3}-1=0$
$\displaystyle\lim _{x \rightarrow 1-}(x-1)=0$
$F$ is continuous
$f(n) =
\begin{cases}
3x^2 & 1 < x < \infty \\
1 & -\infty < x <
\end{cases}$
$f'\left(1^{+}\right)=3, f'\left(1^{-}\right)=1$
$\Rightarrow f$ is not differentiable
Solution:
$\frac{d y}{d x}=2 \cos x^{2} \cdot\left(-\sin x^{2}\right) 2 x$
$=-2 x \sin \left(2 x^{2}\right)$
Solution:
$\frac{d}{d x}\left(x^{x}+x^{a}+a^{x}+a^{a}\right)$
Solution:
$x^{x}(1+\log x)+a x^{a-1}+a^{x} \log _{e}^{a}$
$y=\frac{\log x}{\log 10}+\log x$
$\frac{d y}{d x}=\frac{1}{x \log 10}+\frac{1}{x}$
Solution:
$\frac{ d x }{ d 0}=-\sin \theta+\frac{1}{\tan \left(\frac{\theta}{2}\right)} \cdot \sec ^{2}\left(\frac{\theta}{2}\right) \frac{1}{2}$
$=-\sin \theta+\frac{1}{2 \sin \left(\frac{\theta}{2}\right) \cos \left(\frac{\theta}{2}\right)}$
$=-\sin \theta+\frac{1}{\sin \theta}$
$=\frac{1-\sin ^{2} \theta}{\sin \theta} ; \frac{ dx }{ d \theta}=\frac{\cos ^{2} \theta}{\sin \theta} ; \frac{ dy }{ d \theta}=\cos \theta$
$\frac{ dy }{ dx }=0 ; \tan \theta=0$
$\theta= n \pi, n \in z$
Solution:
$\log y =2 \log ( x -1)+3 \log ( x -2)+5 \log ( x -3)$
$\frac{ dy }{ d x }=( x -1)^{2}( x -2)^{2}( x -3)^{5}\left[\frac{2}{ x -1}+\frac{3}{ x -2}+\frac{5}{ x -3}\right]$
$\left(\frac{ dy }{ d x }\right)_{ x =4}=516$
Solution:
$\frac{ d \theta}{ dt }=\frac{2 t }{20}+\frac{1}{5}$
$=\frac{ t }{10}+\frac{1}{5}$
$\left(\frac{ d \theta}{ dt }\right)_{\varepsilon=4}=\frac{4}{10}+\frac{1}{5}$
$k =\frac{3}{5}$
$5 k =3$
Solution:
$y=\alpha x^{2}-6 x+\beta$ passes through $(0,2)$
$2=\beta$
$\frac{d y}{d x}=2 \alpha x-6$
$\left(\frac{d y}{d x}\right)_{x=\frac{3}{2}}=0$
$2 \alpha\left(\frac{3}{2}\right)-6=0$
$3 \alpha=6$
$\alpha=2$
Solution: