1. Scenario-1: f(x)=x,g(x)=sinx f(x)=x, g(x)=\sin x f(x)=x,g(x)=sinx
Scenario-2: sin−1x=sin−1y \sin ^{-1} x=\sin ^{-1} y sin−1x=sin−1y
Find the value of limx→∞x4+3x2−13x4+x3−2x \lim _{x \rightarrow \infty} \frac{x^{4}+3 x^{2}-1}{3 x^{4}+x^{3}-2 x} limx→∞3x4+x3−2xx4+3x2−1.
From the scenario-1, find the derivative of {f(x)}2(x) \{\mathrm{f}(\mathrm{x})\}^{2(\mathrm{x})} {f(x)}2(x) +{g(x)}f(x) +\{g(x)\}^{f(x)} +{g(x)}f(x).
From the scenario-2, prove that, (1−x2)y2−xy1+ \left(1-x^{2}\right) y_{2}-x y_{1}+ (1−x2)y2−xy1+ n2y=0 \mathrm{n}^{2} \mathrm{y}=0 n2y=0.
2. Two equation of straight line x−2y+3=0;2x+3y=1 x-2 y+3=0 ; 2 x+3 y=1 x−2y+3=0;2x+3y=1.
Find the point of intersection of two straight lines as 2x−3y+5=0 2 x-3 y+5=0 2x−3y+5=0 and 7x+4y−3=0 7 x+4 y-3=0 7x+4y−3=0.
In stem, two equations represents two sides of a parallelogram respectively and its diagonal intersect at (2,−3) (2,-3) (2,−3), then find the equations of the other two sides.
In stem, find the equation of straight lines parallel to first straight line at 5 \sqrt{5} 5 unit distance from it.
3. Equations of circles as below:
x2+y2+6x+2y+6=0x2+y2+8x+y+10=0 \begin{array}{l} x^{2}+y^{2}+6 x+2 y+6=0 \\ x^{2}+y^{2}+8 x+y+10=0 \end{array} x2+y2+6x+2y+6=0x2+y2+8x+y+10=0
Find the equation of circle whose radius is 3 and concentric with the circle x2+y2−4x−6y=0.x2+y2−4x−6y=0.x2+y2−4x−6y=0. x^{2}+y^{2}-4 x-6 y=0 . x2+y2−4x−6y=0.
In stem, find the equation of the circle whose diameter is the common chord of the circles.
Find the equation of the tangent and normal to the first circle in stem from the point (−3,2) (-3,2) (−3,2).
4. f(x)=sinx f(x)=\sin x f(x)=sinx
Find the value of ∫1e−x+1dx \int \frac{1}{e^{-x}+1} d x ∫e−x+11dx.
Evaluate ∫0π2f′(x)[{f(x)}2−16]{f(x)−3}]dx \int_{0}^{\frac{\pi}{2}} \frac{f^{\prime}(x)}{\left.\left[\{f(x)\}^{2}-16\right]\{f(x)-3\}\right]} d x ∫02π[{f(x)}2−16]{f(x)−3}]f′(x)dx.
From the stem, find the area bounded by the parabola and its latus rectum.
5. Q=[3+x4242+x3234+x] \mathrm{Q}=\left[\begin{array}{ccc}3+\mathrm{x} & 4 & 2 \\ 4 & 2+\mathrm{x} & 3 \\ 2 & 3 & 4+\mathrm{x}\end{array}\right] Q=3+x4242+x3234+x
If [12−1−k] \left[\begin{array}{rr}1 & 2 \\ -1 & -k\end{array}\right] [1−12−k] matrix is singular, find the value of k k k.
If x=7 x=7 x=7, determine Q2−5Q+3I3 Q^{2}-5 Q+3 I_{3} Q2−5Q+3I3 where I3 \mathrm{I}_{3} I3 is identity matrix.
If ∣Q∣=0 |Q|=0 ∣Q∣=0, then find the solution set.
