1. f(x,y)=y2−4x−6y+20,g(x)=x3−6x2+9x+1 f(x, y)=y^{2}-4 x-6 y+20, g(x)=x^{3}-6 x^{2}+9 x+1 f(x,y)=y2−4x−6y+20,g(x)=x3−6x2+9x+1.
Determine ddx{(xx)x} \frac{\mathrm{d}}{\mathrm{dx}}\left\{\left(\mathrm{x}^{\mathrm{x}}\right)^{\mathrm{x}}\right\} dxd{(xx)x}.
Determine the equation of the tangent and the normal of the curve f(x,y)=0 f(x, y)=0 f(x,y)=0 at the point (3,2) (3,2) (3,2).
Determine the mximum and minimum values of the function g(x) \mathrm{g}(\mathrm{x}) g(x).
2. f(x)=sinx f(x)=\sin x f(x)=sinx
Evaluate limx→π2f′(x) \lim _{x \rightarrow \frac{\pi}{2} f^{\prime}(x)} limx→2πf′(x).
Determine the derivative of 1f(3x) \frac{1}{f(3 x)} f(3x)1 with respect to x x x from the first principle.
If y=f(asin−1x) y=f\left(a \sin ^{-1} x\right) y=f(asin−1x), then show that, (1−x2)d2ydx2−xdydx+ \left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+ (1−x2)dx2d2y−xdxdy+ a2y=0 \mathrm{a}^{2} \mathrm{y}=0 a2y=0
3. A=(1401),B=(1 m0n),C=(012120204) \mathrm{A}=\left(\begin{array}{ll}1 & 4 \\ 0 & 1\end{array}\right), \mathrm{B}=\left(\begin{array}{ll}1 & \mathrm{~m} \\ 0 & \mathrm{n}\end{array}\right), \mathrm{C}=\left(\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 0 \\ 2 & 0 & 4\end{array}\right) A=(1041),B=(10 mn),C=012120204 and f(x)=x2+5x+6 f(x)=x^{2}+5 x+6 f(x)=x2+5x+6.
If P=(123) \mathrm{P}=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right) P=123 and Q=(450) \mathrm{Q}=\left(\begin{array}{lll}4 & 5 & 0\end{array}\right) Q=(450), then determine (PQ)⊤ (\mathrm{PQ})^{\top} (PQ)⊤.
If AB=I2 \mathrm{AB}=\mathrm{I}_{2} AB=I2, then determine the value of m \mathrm{m} m and n \mathrm{n} n.
Determine f(C) \mathrm{f}(\mathrm{C}) f(C).
4. f(x)=sinx;g(x,y)=25x2+36y2−900 f(x)=\sin x ; g(x, y)=25 x^{2}+36 y^{2}-900 f(x)=sinx;g(x,y)=25x2+36y2−900.
Determine ∫tan−1xdx \int \tan ^{-1} \mathrm{xdx} ∫tan−1xdx.
Evalute ∫0π2 f(π2−x)9−0 {f(x)}2dxEvalute\ \int_0^{\frac{\pi}{2\ }}\frac{f\left(\frac{\pi}{2}-x\right)}{9-0\ \left\{f\left(x\right)\right\}^2}dxEvalute ∫02 π9−0 {f(x)}2f(2π−x)dx
Determine the area of smaller portion enclosed by the curve g(x,y)=0 \mathrm{g}(\mathrm{x}, \mathrm{y})=0 g(x,y)=0 and the line x=3 \mathrm{x}=3 x=3.
5. P(1,2)P(1,2) P(1,2) and Q(2,3) \mathrm{Q}(2,3) Q(2,3) be two points and x2+y2−6x−4y+1=0 x^{2}+y^{2}-6 x-4 y+1=0 x2+y2−6x−4y+1=0 be an equation of a circle.
Determine the radius of the circle 3x2+3y2−6x− 3 x^{2}+3 y^{2}-6 x- 3x2+3y2−6x− 12y+1=0 12 y+1=0 12y+1=0.
Determine the equation of a circle with centre at P P P and which passes through the centre of the given circle.
Determine the equation of a circle which passes through the point P P P and Q Q Q and touches y y y-axis.
