3 points each
6.) What are the roots of x^5 + x^4 - 25x^3 + 35x^2 + 24x - 36 = 0?
Hint: Rational Root Theorem. 😂 Katamad isove.
7.) Find the area of a triangle that has sides of 3cm, 8cm and 9cm?
Herons formula
A=sqrt[s(s-a)(s-b)(s-c)]
Where s=(a+b+c)/2
First find s
s=(3+8+9)/2
s=20/2
s=10
A=sqrt[10(10-3)(10-8)(10-9)]
A=sqrt[10(7)(2)(1)]
A=sqrt(140)
A=sqrt(4*35)
A=2sqrt(35) cm^2
8.) S is a point outside the rectangle HAWN. If SH = 7cm, SW = 10cm, SA = 12cm, find SN.
Apply British Flag Theorem.
10^2+7^2=12^2+x^2
149=144+x^2
x^2=5
x=sqrt(5) cm
9.) If the first term of a geometric sequence is 7, and the common ratio is 3, what is the sum of the first 5 terms?
The formula for the finite sum of a geometric series is
S_n=[a_1(1-r^n)]/(1-r)
where a_1=first term
r=common ratio
n=number of terms
S_5=[7(1-3^5)]/(1-3)
S_5=[7(-242)]/(-2)
S_5=(-1694)/(-2)
S_5=847
10.) If the first term of a geometric sequence is 8/7 and the 4th term is 392, what term is 2,259,801,992?
Use the formula
r=(a_n/a_k)^[1/(n-k)]
r=[392/(8/7)]^[1/(4-1)]
r=(343)^(1/3)
r=7
Multiply 392 by 7 then count the terms to know which term is 2,259,801,992.
11.) A circle passes through the points (2,3),(6,1) and (-4,3). Find the equation of the line, in general form, that is tangent to the circle at (2,3)?
Since (-4,3) and (2,3) are collinear,
(2-h)^2=(-4-h)^2
h^2-4h+4=h^2+8h+16
-4h-8h=12
-12h=12
h=-1
(2+1)^2+(3-k)^2=(6+1)^2+(1-k)^2
9+k^2-6k+9=49+k^2-2k+1
18+k^2-6k=k^2-2k+50
-6k+2k=32
-4k=32
k=-8
The center is at (-1,-8).
Find the slope of the line passing through (-1,-8) and (2,3).
m=(3+8)/(2+1)
m=11/3
Take its negative reciprocal which is -3/11.
m=-3/11 is the slope of the tangent line.
The equation of the line is
y=mx+b
y=(-3/11)x+b
Substitute (2,3) --> x=2, y=3
3=(-3/11)(2)+b
3=-6/11+b
-6/11+b=3
b=3+6/11
b=33/11 + 6/11
b=39/11
The tangent line is y=(-3/11)x+39/11.
Multiply both sides by 11.
11y=-3x+39
3x+11y-39=0 --> in general form.
12.) Safeway bus left the station at 1pm with a constant speed of 80kph. 1 hour later, the Nova bus left the same station with a constant speed of 100kph. Assuming they have the same destination, at what time will the Nova bus be 50km ahead over Safeway bus?
Let t be the time. Since ung first bus my 1 hour advantage, yung time niya is 80+80t since constant ung speed. Since kakasimula palang nung una, ang speed niya ay 100t. Bale kukunin natin ung t kung kailan 50 km na naoovertaken ni Nova si Safeway. Subtract mo distances nila.
100t - (80+80t) = 50
20t-80=50
20t=130
t=6.5 hours --> that is 6 hours and 30 minutes.
So, 2:00+6 hours and 30 minutes = 8:30 PM
2 PM kasi umalis sa istasyon ung Nova.
13.) Find the height of the flagpole if the distance of its base from the base of the building is 2m and the half height of the flagpole forms 17 degrees with respect to the base of the building. (Given tan73 = 3.27). Use the given as your basis in your computation.
tan(73)=x/4
x=(3.27)(4)=13.08 m. --> I will further elaborate in the comments
14.) If x^3 + y^3 = 288 and x+y = 12, find x-y
x^3+y^3=(x+y)(x^2-xy+y^2)
Since x^2+y^2=(x+y)^2-2xy
x^2+y^2=144-2xy
288=12(144-3xy)
288=1728-36xy
1440=36xy
xy=40
Since (x-y)^2=x^2+y^2-4xy,
(x-y)^2=144-4(40)
(x-y)^2=-16
x-y=+/-4i
15.) The perimeter of a rectangle is 44cm and the length of one diagonal is sqrt(274)cm. Find its area.
Let x be the length of the rectangle
y be the width
2x+2y=44
x+y=22
By the Pythagorean Theorem,
x^2+y^2=274
We have to find xy (which is the area) in terms of x+y and x^2+y^2. From the identity,
(x+y)^2=x^2+2xy+y^2
22^2=274+2xy
484=274+2xy
2xy=210
xy=105
A=105 cm^2
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