1. P=∣2x−S2x2x2y2y−S2y2z2z2z−S∣,B=∣a1b1a1a2b2c2a3b3c3∣ P=\left|\begin{array}{ccc} 2 x-S & 2 x & 2 x \\ 2 y & 2 y-S & 2 y \\ 2 z & 2 z & 2 z-S \end{array}\right|, B=\left|\begin{array}{lll} a_{1} & b_{1} & a_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array}\right| P=2x−S2y2z2x2y−S2z2x2y2z−S,B=a1a2a3b1b2b3a1c2c3
If 2[1x23]=x2 2\left[\begin{array}{ll}1 & x \\ 2 & 3\end{array}\right]=x^{2} 2[12x3]=x2, then find x x x.
If the cofactors of the elements of second row of determinant B B B are A2,B2 A_{2}, B_{2} A2,B2 and C2 C_{2} C2 respectively, then determine the value of a3 A2+b3 B2+c3C2 \mathrm{a}_{3} \mathrm{~A}_{2}+\mathrm{b}_{3} \mathrm{~B}_{2}+\mathrm{c}_{3} \mathrm{C}_{2} a3 A2+b3 B2+c3C2.
If x+y+z=S x+y+z=S x+y+z=S, then show that, P=S P=S P=S.
2. Scenario-1:Scenario-2: secθ=m−ncosφmcosφ−n,m=P+Q2,n=P−Q2 \sec \theta=\frac{m-n \cos \varphi}{m \cos \varphi-n}, m=\frac{P+Q}{2}, n=\frac{P-Q}{2} secθ=mcosφ−nm−ncosφ,m=2P+Q,n=2P−Q.
Determine value of sinθ2+cosθ21+sinθ \frac{\sin \frac{\theta}{2}+\cos \frac{\theta}{2}}{\sqrt{1+\sin \theta}} 1+sinθsin2θ+cos2θ
From scenario-1 show that, asin(A2+B)=(b+c) a \sin \left(\frac{A}{2}+B\right)=(b+c) asin(2A+B)=(b+c) sinA2 \sin \frac{A}{2} sin2A.
From scenario-2 prove that, tanθ2P=tanQ2Q \frac{\tan \frac{\theta}{2}}{\sqrt{P}}=\frac{\tan \frac{Q}{2}}{\sqrt{Q}} Ptan2θ=Qtan2Q.
3. (i) y=acos(lnx)+bsin(lnx) y=a \cos (\ln x)+b \sin (\ln x) y=acos(lnx)+bsin(lnx).
(ii) f(x)=2x3−3x2−12x+30 f(x)=2 x^{3}-3 x^{2}-12 x+30 f(x)=2x3−3x2−12x+30.
If f(x)=lnx f(x)=\ln x f(x)=lnx, then find f′′(x) f^{\prime \prime}(x) f′′(x).
From (i) prove that, x2y2+xy1+y=0 x^{2} y_{2}+x y_{1}+y=0 x2y2+xy1+y=0.
Find the extreme values of f(x) f(\mathrm{x}) f(x).
4. P=(x−4)2(x−3) P=(x-4)^{2}(x-3) P=(x−4)2(x−3) and g(x,y)=x2+y2 g(x, y)=x^{2}+y^{2} g(x,y)=x2+y2.
Determine ∫dx1−4x2 \int \frac{d x}{\sqrt{1-4 x^{2}}} ∫1−4x2dx.
Determine ∫xPdx \int \frac{\mathrm{x}}{\mathrm{P}} \mathrm{dx} ∫Pxdx.
Determine the area of the smallest region bounded by g(x,y)=100 g(x, y)=100 g(x,y)=100 and x=5 x=5 x=5.
5.
Express x2+y2−3y=0 x^{2}+y^{2}-3 y=0 x2+y2−3y=0 into polar form.
Determine the co-ordinate of point D D D.
Find the equation of the bisector of the angle ∠ACB \angle \mathrm{ACB} ∠ACB.
Scenario-2: 