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What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?
find slope:
m = y2-y1/x2-x1
Once the slope and y-intercept are known, the slope intercept form is convenient to use.
y=mx+b.
For some angle @, scs @ = -8/5. What is cos 2@?
Use the cosine double angle formula.
cos 2@ = 1-2sin^2 @
=1-2(1/csc @)^2
What is the rectangle form of the following polar equation?
r^2=1-tan^2(@)
Use the identities relating r and @ to x and y.
r=(x^2+y^2) and
@=tan^-1(y/x)
For three matrices A, B, C, which of the following statements is not necessarily true?
A+(B+C)=(A+B)+C
A(B+C)=AB+AC
(B+C)A=AB+AC
A+(B+C)=C+(A+B)
Matrix addition is both associative and commutative, so choices A and D are true. Multiplication is distributive B, but not commutative C.
For the three vectors A B C, what is the product A product (B*C)?
A=6i+8j+10k
B=i+2j+3k
C=3i+4j+5k
Find the cross product B, C. The augmented matrix method is the easiest approach.
B cross C=i(2)(5)+j(3)(3)+k(1)(4)-i(4)(3)-j(5)(1)-k(3)(2)
Now calcuate the dot product:
A product (B cross product C) =6*(-2)+8*4+10*-2=0.
The second and sixth terms of a geometric progession are 3/10 and 243/160. What is the first term of this sequence?
Apply the formula for a geometric sequence. Solve this problem by dividing the expression for the two given terms.
ln=ar^n-1
l6/l4=ar^6-1/ar^2-1
l2=ar
A marksman can hit a bull's-eye from 100 m with three out of four shots. What is the probability the he will hit a bull's-eye with at least one of his next three shots?
The easiest way to find the probability of making at least one bull's-eye is actually to solve for its complementary probability, that of making zero bull's-eyes.
P(miss) = 1-P(hit)
=1-(3/4)
P(at least one)=1-P(none)
=1-(P(miss)*P(miss)*P(miss))
=1-.25*.25*.25
The final scores of students in a graduate course are distributed normally with a mean of 72 and a standard deviation of 10. What is the probability that a student's score will be between 65 and 78?
Calculate standard normal values for the points of interest, 65 and 78.
x=X0-mu/stress
x65=65-72/10
x78=78-72/10
The probability of a score falling between 65 and 78 is equal to the area under the unit normal curve between these two standard normal values.
F(x)=R(-x)
F(x)=F(x78)-F(x65)
=F(.6)-R(.7)
=.72-.24
=.4837
Evaluate the following limit.
lim(1-e^3x/4x)
x-->0.
This limit has the indeterminate form 0/0, so use L'Hopital's rule.
lim (f(x)/g(x))=lim(f'(x)/g'(x))
x->a x->a
lim(1-e^3x/4x)
x->0
lim(-3e^3x/4)
x->0