MHT CET 2020 :: Mathematics :: General questions

• Mathematics - General questions

 The equation of the directrix  of the parabola 3x²=16y is 

Answer:

Explanation:

 Given  parabola,   3x²=16y

$x^{2}=\frac{16}{3}y$

 Here,    $4a=\frac{16}{3}$

 $a=\frac{4}{3}$

 equation of directrix of parabola,

    x² = 4ay is

 y+a=0

 $\therefore$     Equation of directris of parabola,.

 $x^{2}=\frac{16}{3}y$

  $y+\frac{4}{3}=0$

 3y+4=0

 The cofactors of the elements of the first column of the matrix 

$A=\begin{bmatrix}2 & 0 &-1\\3 & 1&2\\ -1 &1 & 2 \end{bmatrix}$  are

Answer:

Explanation:

 Given matrix  

$A=\begin{bmatrix}2 & 0 &-1\\3 & 1&2\\ -1 &1 & 2 \end{bmatrix}$

 cofactor of the elements of first column of the matrix

$A_{11}=\begin{bmatrix}1 & 2 \\1 &2 \end{bmatrix}=2-2=0$

$A_{21}=-\begin{bmatrix}0 & -1 \\1 &2 \end{bmatrix}=-(0+1)=-1$

$A_{31}=\begin{bmatrix}0 & -1 \\1 &2 \end{bmatrix}=0+1=1$

 $\therefore$   cofactor is (0,-1,1)

 The principal solutions  of  $\cot x=\sqrt{3}$ are

Answer:

Explanation:

 we have,

  $\cot x=\sqrt{3}$ 

   $\cot x=\cot\frac{\pi}{6}$ or   $ \cot\left(\pi+\frac{\pi}{6}\right)$

 $\therefore$   $ x=\frac{\pi}{6} ,\frac{7\pi}{6}$

 $\therefore$ principal solution is   $\frac{\pi}{6},\frac{7\pi}{6}$

$\int \frac{dx}{x^{2}+4x+13}=$

Answer:

Explanation:

 We have,

 l= $\int \frac{dx}{x^{2}+4x+13}$

 l=$\int \frac{dx}{x^{2}+4x+4+9}$=$\int \frac{dx}{(x+2)^{2}+(3)^{2}}$

 l=  $\frac{1}{3}\tan^{-1}\left(\frac{x+2}{3}\right)+C$

 The equation of planes parallel to the plane x+2y+2z+8=0 , which are at a distance of 2 units from the point (1,1,2) are

Answer:

Explanation:

 Equation  of plane parallel  to the plane

x+2y+2z+8=0 is x+2y+2z+λ=0

 Given distance from the  point (1,1,2)  to the plane.

 x+2y+2z +$\lambda$=0 is 2 units

$\therefore$             $2=|\frac{1+2+4+\lambda}{\sqrt{1+4+4}}|$

$\Rightarrow$     $ \pm2=\frac{\lambda+7}{3}$

$\Rightarrow \lambda+7=\pm6$

 $\therefore$      $\lambda$  = -13,-1

 Hence , equation of plane is

x+2y+2z-13=0 or x+2y+2z-1=0

• Mathematics - General questions