VITEEE 2019 :: Mathematics :: General questions

• Mathematics - General questions

If the coordinates at one end of the diameter of the circle x² + y² - 8x- 4y+ c = 0 are (-3, 2), then the coordinates at the other end are

Answer:

Explanation:

The centre of the given circle is C = (4, 2) Let A = (-3,2)

If (α, β) are the coordinates of the other end of the diameter. then, as the middle point of the diameter is the centre,

$\frac{\alpha-3}{2}=4$ and $\frac{\beta+2}{2}=2$

$\Rightarrow\alpha=11,\beta= 2$

Thus, the coordinates of the other end of diameter are (11,2)

If  $z=x+iy, z^{1/3} = a - ib,$ then $\frac{x}{a}-\frac{y}{b} = k(a^{2}-b^{2})$ where k is equal to 

Answer:

Explanation:

$z^{1/3} = a-ib\Rightarrow z = (a -ib)^{3}$

$x +iy = a^{3} + ib^{3} - 3ia^{2}b -3ab^{2}$

$Then x= a^{3} -3ab^{2}\Rightarrow \frac{x}{a} = a^{2}-3b^{2}$

$y =b^{3} -3a^{2}b\Rightarrow \frac{y}{b} = b^{2}-3a^{2}$

$So, \frac{x}{a}-\frac{y}{b} = 4(a^{2}-b^{2})$

If tan θ = √n for some non-square natural number n, then sec 2θ is :

Answer:

Explanation:

$\cos2\theta =\frac{1-\tan^{2}\theta}{1+\tan^{2}\theta}$

= $\frac{1-n}{1+n}$→ $\sec2\theta =\frac{1+n}{1-n}$

Since, n is non - square natural number, n ≠ 1

.'. $\sec2\theta =\frac{1+n}{1-n}$ = rational number

• Mathematics - General questions