Question

PLEASE ANSWER ALL 6 OF THEM!!

Answer 1

1.

$c = x {y}^{2} \\ d = {x}^{2} y$
H.C.F.(c, d) =>
$c = x \times y \times y \\ d = x \times x \times y$
Common Factors = x, y
Therefore, H.C.F. (c, d) = xy

2.
$\alpha + \beta = \frac{ - 17}{5} \\ \alpha \beta = \frac{7}{5} \\ {x}^{2} - ( \alpha + \beta ) x + \alpha \beta = 0 \\ {x }^{2} + \frac{17}{5} x + \frac{7}{5} = 0 \\ 5 {x}^{2} + 17x + 7 = 0$
3.
${4x}^{2} - \frac{2}{3} x - \frac{1}{16} = 0 \\ a = 4 \\ b = - \frac{2}{3} \\ c = - \frac{1}{16} \\ discriminant \\ = {b}^{2} - 4ac \\ = { (- \frac{2}{3} )}^{2} - 4 \times 4 \times ( - \frac{1}{16} ) \\ = \frac{4}{9} + 1 = \frac{4 + 9}{9} = \frac{13}{9}$
4.
A(5, -6), B(-1, 4), P(x, 0)
Let the ratio be k:1
Section Formula
$= ( \frac{m_{2} x_{1} + m_{1}x_{2}}{m_{1} + m_{2}} \: \frac{m_{2} y_{1} + m_{1}y_{2}}{m_{1} + m_{2}} )$
$(x \: 0) = ( \frac{5 - k}{k + 1} \: \frac{ - 6 + 4k}{k + 1} )$
$\frac{ - 6 + 4k}{k + 1} = 0 \\ - 6 + 4k = 0 \\ 4k = 6 \\ k = \frac{6}{4} = \frac{3}{2}$
Ratio
$= k : 1 \\ = \frac{3}{2} : 1 \\ = 3 : 2$
5.
Ratio of height of vertical tower and shadow
$= 1 : \sqrt{3}$
Height of the tower = 1x
Length if shadow
$= \sqrt{3} x$
$\tan(o) = \frac{1}{ \sqrt{3} } \\ \tan(o) = \tan(30) \\ o = 30$
Angle of elevation of sun = 30°
Altitude of sun will be according to its distance from the tower.

6.
Total number of outcomes = 36
Outcomes with their total as 9 or 11:
(3, 6), (4, 5), (5, 4), (6, 3), (5, 6), (6, 5)
No. of favourable outcomes = 6
P(getting a total of 9 or 11)
$= \frac{no \: of \: favourable \: outcomes}{total \: no \: of \: outcomes} \\ = \frac{6}{36} \\ = \frac{1}{6}$
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