VITEEE 2019 :: Mathematics :: General questions

• Mathematics - General questions

The equation of the plane which bisects the angle between the planes 3x - 6y + 2z + 5 = 0 and 4x - 12y + 3z- 3 = 0 which contains the origin is

Answer:

Explanation:

Equation of plane bisecting the angle containing origin is (making constant term of same sign)

$\frac{-3x + 6y - 2z - 5 }{\sqrt{3^{2}+6^{2}+2^{2}}}$

= $+\frac{4x - 12y + 3z- 3}{\sqrt{4^{2}+12^{2}+3^{2}}}$

or $\frac{-3x + 6y - 2z - 5 }{7}=\frac{4x - 12y + 3z- 3}{13}$

or 67x - 162y+ 47z + 44 = 0

If $(\overrightarrow{a}\times\overrightarrow{b})^{2}+(\overrightarrow{a}.\overrightarrow{b})^{2}$= 676 |$\overrightarrow{b}$| = 2 then |$\overrightarrow{a}$| is equal to

Answer:

Explanation:

Since,

$(\overrightarrow{a}\times\overrightarrow{b})^{2}+(\overrightarrow{a}.\overrightarrow{b})^{2}$ = 676

(|$\overrightarrow{a}$|.|$\overrightarrow{b}$|$\sin\theta\hat{n}$)² + (|$\overrightarrow{a}$|.|$\overrightarrow{b}$|$\cos\theta$)² = 676

|$\overrightarrow{a}$|² .|$\overrightarrow{b}$|² ($\sin^{2}\theta+\cos^{2}\theta$) = 676

|$\overrightarrow{a}$|² (2)² = 676 = |$\overrightarrow{a}$|²  =169

|$\overrightarrow{a}$|

=13

The conic represented by x = 2 (cos t + sin t), y = 5 (cos t - sin t) is

Answer:

Explanation:

From given equations

x/2 = cos t + sin t

y/5 = cos t - sint 

Eliminating t from these equations, we have 

$\frac{x^{2}}{4}+\frac{y^{2}}{25}$ = 2

$\frac{x^{2}}{8}+\frac{y^{2}}{50}$ = 1 is an ellipse

The area under the curve y = |cos x - sin x|, 0≤x≤$\frac{\pi}{2}$

Answer:

Explanation:

 y = |cos x - sin x|

Required area = $2\int_{0}^{\frac{\pi}{4}} (\cos x -\sin x)dx$

$2 \left[\sin x+\cos x\right]_0^\frac{\pi}{4}$ = $2 \left[\frac{2}{\sqrt{2}}-1\right]$

$2\sqrt{2}-2$

$\int_{}^{} (27e^{9x}+e^{12x})^{1/3}dx $ is equal to

Answer:

Explanation:

Let I = $\int_{}^{} (27e^{9x}+e^{12x})^{1/3}dx$

= $\int_{}^{}e^{3x} (27+e^{3x})^{1/3}dx $

Put $27+e^{3x}$ = t    3$e^{3x}$dx = dt

I = $\frac{1}{3}\int_{}^{}t^{1/3}dt$

= $\frac{1}{3}\frac{t^{4/3}}{4/3}+C $

= $(1/4)(27+e^{3x})^{4/3}+C$

• Mathematics - General questions