1.
Find the slope of the straight line AB \mathrm{AB} AB.
Find the foot point of the perpendicular on AB A B AB from P P P.
Find the equations of the straight lines passing through the point P \mathrm{P} P which make an angle 30∘ 30^{\circ} 30∘ witl the straight line y=x y=x y=x.
2. X=[xyz],A=[2−1−11323−1−5],B=[611] \mathrm{X}=\left[\begin{array}{l}x \\ \mathrm{y} \\ z\end{array}\right], \mathrm{A}=\left[\begin{array}{rrr}2 & -1 & -1 \\ 1 & 3 & 2 \\ 3 & -1 & -5\end{array}\right], \mathrm{B}=\left[\begin{array}{l}6 \\ 1 \\ 1\end{array}\right] X=xyz,A=213−13−1−12−5,B=611 and C= \mathrm{C}= C=
[pqrp2q2r2p3−1q3−1r3−1] \left[\begin{array}{ccc} p & q & r \\ p^{2} & q^{2} & r^{2} \\ p^{3}-1 & q^{3}-1 & r^{3}-1 \end{array}\right] pp2p3−1qq2q3−1rr2r3−1
Find the value of the determinant ∣y+zx1z+xy1x+yz1∣ \left|\begin{array}{lll}y+z & x & 1 \\ z+x & y & 1 \\ x+y & z & 1\end{array}\right| y+zz+xx+yxyz111 without expansion.
Show that, ∣C∣=(pqr−1)(p−q)(q−r)(r−p) |C|=(p q r-1)(p-q)(q-r)(r-p) ∣C∣=(pqr−1)(p−q)(q−r)(r−p).
If AX=B A X=B AX=B, find the matrix X X X by the help of determinant.
3. F(x)=ln(x) \mathrm{F}(\mathrm{x})=\ln (\mathrm{x}) F(x)=ln(x).
Evaluate: limx→0sinaxtanbx \lim _{x \rightarrow 0} \frac{\sin a x}{\tan b x} limx→0tanbxsinax.
Find the derivative of e2 F(x)+(xx)x \mathrm{e}^{2 \mathrm{~F}(\mathrm{x})}+\left(\mathrm{x}^{\mathrm{x}}\right)^{\mathrm{x}} e2 F(x)+(xx)x with respect to
Find the minimum value of eF(x)F(x) \frac{e^{F(x)}}{F(x)} F(x)eF(x).
4. A=[3−42−210−1−11] A=\left[\begin{array}{rrr}3 & -4 & 2 \\ -2 & 1 & 0 \\ -1 & -1 & 1\end{array}\right] A=3−2−1−41−1201.
If [x2x−32] \left[\begin{array}{cc}x^{2} & x \\ -3 & 2\end{array}\right] [x2−3x2] is a singular matrix, find the value of x x x.
If AB=BA=I3 A B=B A=I_{3} AB=BA=I3, find the matrix B B B where B B B is a 3x 3 x 3x 3 matrix.
If C=A′ C=A^{\prime} C=A′, find C2−5C+6I C^{2}-5 C+6 I C2−5C+6I.
5.
Find the area of the triangle ABE.
Find the bisector of ∠AOE \angle \mathrm{AOE} ∠AOE.
If CD⊥AB \mathrm{CD} \perp \mathrm{AB} CD⊥AB and AC:BC=2:3 \mathrm{AC}: \mathrm{BC}=2: 3 AC:BC=2:3 then find the equation of the straight line CD.
