1. Scenario-1: f(x)=cosx f(x)=\cos x f(x)=cosx.
If tanθ=34 \tan \theta=\frac{3}{4} tanθ=43 and π<θ<3π2 \pi<\theta<\frac{3 \pi}{2} π<θ<23π, then find the value of cosec(−θ)+sec(−θ) \operatorname{cosec}(-\theta)+\sec (-\theta) cosec(−θ)+sec(−θ).
If f(α)+f(β)=a f(\alpha)+f(\beta)=a f(α)+f(β)=a and f(π2−α)+f(π2−β)=b f\left(\frac{\pi}{2}-\alpha\right)+f\left(\frac{\pi}{2}-\beta\right)=b f(2π−α)+f(2π−β)=b, then prove that, sinα+β2=±ba2+b2 \sin \frac{\alpha+\beta}{2}= \pm \frac{b}{\sqrt{a^{2}+b^{2}}} sin2α+β=±a2+b2b.
Find the area of △ABC \triangle \mathrm{ABC} △ABC using ∠A \angle \mathrm{A} ∠A.
2. Scenario-1: f(x)=sinx f(x)=\sin x f(x)=sinx.
Scenario-2: x=sin(1mlny) x=\sin \left(\frac{1}{m} \ln y\right) x=sin(m1lny).
Find the value of limx→∞7xsina7x \lim _{x \rightarrow \infty} 7^{x} \sin \frac{a}{7^{x}} limx→∞7xsin7xa.
According to the scenario-1, find the derivative using first principle for the function f(π2−2x)f(2x) \frac{f\left(\frac{\pi}{2}-2 x\right)}{f(2 x)} f(2x)f(2π−2x).
According to the scenario-2, prove that, (1−x2)y2− \left(1-x^{2}\right) y_{2}- (1−x2)y2− xy1=m2y x y_{1}=m^{2} y xy1=m2y.
3. Scenario-1: f(x)=x f(x)=\sqrt{x} f(x)=x
Scenario-2: h(x)=sinx h(x)=\sin x h(x)=sinx
Find the derivative with respect to x x x for cos−11−x21+x2⋅ \cos ^{-1} \frac{1-x^{2}}{1+x^{2}} \cdot cos−11+x21−x2⋅
Prove that, sum of intercepts from the axes will be c which is made by the curve f(x)+f(y)=f(c) f(x)+f(y)=f(c) f(x)+f(y)=f(c) at (a,b). (a, b). (a,b).
Find the optimum values for the function y=1+2h(x)+3[1−{h(x)}2] y=1+2 h(x)+3\left[1-\{h(x)\}^{2}\right] y=1+2h(x)+3[1−{h(x)}2] for [0,π2] \left[0, \frac{\pi}{2}\right] [0,2π]
4. A=[2−1131−452−3]B=[p22qr2pq2rp2qr2pq2r] \mathrm{A}=\left[\begin{array}{rrr}2 & -1 & 1 \\ 3 & 1 & -4 \\ 5 & 2 & -3\end{array}\right] B=\left[\begin{array}{lll}\mathrm{p}_{2}^{2} & \mathrm{qr} & 2 \mathrm{p} \\ \mathrm{q}^{2} & \mathrm{rp} & 2 \mathrm{q} \\ \mathrm{r}^{2} & \mathrm{pq} & 2 \mathrm{r}\end{array}\right] A=235−1121−4−3B=p22q2r2qrrppq2p2q2r
If [x−58−1y+3]=[y−18−17] \left[\begin{array}{cc}x-5 & 8 \\ -1 & y+3\end{array}\right]=\left[\begin{array}{cc}y-1 & 8 \\ -1 & 7\end{array}\right] [x−5−18y+3]=[y−1−187], then find the value of ( x,y x, y x,y ).
Find A−1 \mathrm{A}^{-1} A−1.
Prove that, ∣B∣=−2(p−q)(q−r)(p−r)(pq+qr+ |B|=-2(p-q)(q-r)(p-r)(p q+q r+ ∣B∣=−2(p−q)(q−r)(p−r)(pq+qr+ rр).
5. f(x)=sinx f(x)=\sin x f(x)=sinx
g(x,y)=9x2+25y2−225h(x)=x−3 \begin{array}{l} g(x, y)=9 x^{2}+25 y^{2}-225 \\ h(x)=x-3 \end{array} g(x,y)=9x2+25y2−225h(x)=x−3
Find ∫(4−3x)32dx \int(4-3 x)^{\frac{3}{2}} d x ∫(4−3x)23dx,Option A
Find the value of ∫0π2{f(x)}2f(3x)dx \int_{0}^{\frac{\pi}{2}}\{f(x)\}^{2} f(3 x) d x ∫02π{f(x)}2f(3x)dx.
Find the area of smaller part which is bounded by g(x,y)=0 \mathrm{g}(\mathrm{x}, \mathrm{y})=0 g(x,y)=0 and h(x)=0 \mathrm{h}(\mathrm{x})=0 h(x)=0.
