1. Scenario: A=[aij]3×3; A=\left[a_{i j}\right]_{3 \times 3} ; A=[aij]3×3; where aij=2i−j a_{i j}=2 i-j aij=2i−j.
I3=[100010001] and f(x)=x2+3x. I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \text { and } f(x)=x^{2}+3 x \text {. } I3=100010001 and f(x)=x2+3x.
For which value of k,[k+3−1kk+2] k,\left[\begin{array}{cc}k+3 & -1 \\ k & k+2\end{array}\right] k,[k+3k−1k+2] will be singular?
Determine: f(A)+2I3 \mathrm{f}(\mathrm{A})+2 \mathrm{I}_{3} f(A)+2I3.
Determine: (A+I3)⋅(AT−I3) \left(A+I_{3}\right) \cdot\left(A^{T}-I_{3}\right) (A+I3)⋅(AT−I3).
2. Scenario-1: f(x)=tanx f(x)=\tan x f(x)=tanx.
Scenario-2: 9x2+25y2=225 9 x^{2}+25 y^{2}=225 9x2+25y2=225.
Determine: ∫lnx3dx \int \ln x^{3} d x ∫lnx3dx.
Integral : ∫dx5+f(π2−x) \int \frac{d x}{5+f\left(\frac{\pi}{2}-x\right)} ∫5+f(2π−x)dx.
Determine the area of enclosed by the straight line x−3= x-3= x−3= 0 and the ellipse of the scenario-2 in the first quadrant.
3.
Determine the polar coordinate of the point (−3,−1) (-\sqrt{3},-1) (−3,−1).
Determine the equation of the bisector from the stem if it is AB=3BC A B=3 B C AB=3BC.
From the stem, determine the co-ordinate of the foot of perpendicular foot drawn on the straight line AB A B AB from the point P P P.
4.
If the distance of the point (a,5) (a, 5) (a,5) from Y Y Y-axis and the point (2,2) (2,2) (2,2) is equal, determine the value of a a a.
From the stem determine the equation of the straight line CD.
Described in the stem, if the area of △OAB \triangle \mathrm{OAB} △OAB is 18 square unit, determine the equation of the straight line AB \mathrm{AB} AB.
5. Scenario-1: x5+3y10+z10=x4+y4=3y7+4z7=1 \frac{x}{5}+\frac{3 y}{10}+\frac{z}{10}=\frac{x}{4}+\frac{y}{4}=\frac{3 y}{7}+\frac{4 z}{7}=1 5x+103y+10z=4x+4y=73y+74z=1.
Scenario-2: Δ=∣(s−x)2x2x2y2(s−y)2y2z2z2(s−z)2∣ \Delta=\left|\begin{array}{ccc}(s-x)^{2} & x^{2} & x^{2} \\ y^{2} & (s-y)^{2} & y^{2} \\ z^{2} & z^{2} & (s-z)^{2}\end{array}\right| Δ=(s−x)2y2z2x2(s−y)2z2x2y2(s−z)2.
Show that, A=[21−2−1] A=\left[\begin{array}{rr}2 & 1 \\ -2 & -1\end{array}\right] A=[2−21−1] is an idempotent matrix.
Solve the system of equation by Cremer's rule described in the scenario-1.
If it is s=x+y+z s=x+y+z s=x+y+z in the scenario-2, prove that Δ= \Delta= Δ= 2xyzs3 2 x y z s^{3} 2xyzs3.
