1. Scenario: f(z)=13z3−132z2+42z+1 f(z)=\frac{1}{3} z^{3}-\frac{13}{2} z^{2}+42 z+1 f(z)=31z3−213z2+42z+1.
g(x,y)=x2−2y2−7 g(x, y)=x^{2}-2 y^{2}-7 g(x,y)=x2−2y2−7
Determine the derivative of 12sin−110x1+25x2 \frac{1}{2} \sin ^{-1} \frac{10 \mathrm{x}}{1+25 \mathrm{x}^{2}} 21sin−11+25x210x w.r.tx.
Determine the equation of normal and tangent at the point (3,1) (3,1) (3,1) of the curve g(x,y)=0 g(x, y)=0 g(x,y)=0.
Determine the maximum value and minimum value of the function f(x) f(x) f(x).
2. Secnario:
Determine the co-ordinate of the external divisional point which divides the line segment joining the points (−2,3) (-2,3) (−2,3) and (1,2) (1,2) (1,2) by the ratio 3:2 3: 2 3:2.
Determine the equation of the straight line AB \mathrm{AB} AB.
Determine the equation of the straight line which creates an angle 45∘ 45^{\circ} 45∘ with CD C D CD and passes through the point (4,1) (4,1) (4,1).
3. Scenario:
x+y+z=3x+ay+a2z=1x+a2y+a4z=m},C=∣1230∣,f(x)= \left.\begin{array}{l} x+y+z=3 \\ x+a y+a^{2} z=1 \\ x+a^{2} y+a^{4} z=m \end{array}\right\}, C=\left|\begin{array}{ll} 1 & 2 \\ 3 & 0 \end{array}\right|, f(x)= x+y+z=3x+ay+a2z=1x+a2y+a4z=m⎭⎬⎫,C=1320,f(x)=x2+3x−7 x^{2}+3 x-7 x2+3x−7
Prove that, ∣x+y3(y+z)z+x131z3xy∣=0 \left|\begin{array}{ccc}x+y & 3(y+z) & z+x \\ 1 & 3 & 1 \\ z & 3 x & y\end{array}\right|=0 x+y1z3(y+z)33xz+x1y=0.
Tum the equations into AX=B \mathrm{AX}=\mathrm{B} AX=B and show that, Det (A)=a(a−1)2(a2−1). (A)=a(a-1)^{2}\left(a^{2}-1\right) \text {. } (A)=a(a−1)2(a2−1).
Determine f(C) f(C) f(C).
4. Scenario: f(x)=sinx f(x)=\sin x f(x)=sinx.
Determine the value of limx→π21−f(x)f′(x) \lim _{x \rightarrow \frac{\pi}{2}} \frac{1-f(x)}{f^{\prime}(x)} limx→2πf′(x)1−f(x).
Determine the differential coefficient of f(π2−7x) f\left(\frac{\pi}{2}-7 x\right) f(2π−7x) by first principle.
If y=8+5f(2x) y=\sqrt{8+5 f(2 x)} y=8+5f(2x), determine the value of d2ydx2+(dydx)2+ \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}+ dx2d2y+(dxdy)2+ 2y2 2 y^{2} 2y2.
5. Scenario: f(y)=y,g(x,y)=x2+y2−225 f(y)=y, g(x, y)=x^{2}+y^{2}-225 f(y)=y,g(x,y)=x2+y2−225.
Determine ∫ln7xdx \int \ln 7 x d x ∫ln7xdx.
Determine ∫08f(x2)64−f(x2)dx \int_{0}^{8} f\left(x^{2}\right) \sqrt{64-f\left(x^{2}\right) d x} ∫08f(x2)64−f(x2)dx.
Determine the area of the closed region of the upper part of axis x x x by g(x,y)=0 g(x, y)=0 g(x,y)=0.
