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Math > SAT 2 Subject test > Flashcards

SAT 2 Subject test Flashcards

1
Q

If polynomial P(x) is divided by x-a, what is the remainder?

A

P(a)

Because P(x)=(x-a)Q(x) + R

at x=a we have P(a)=R

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2
Q

If f(a) = 0 then f(x) has a factor of (x-a)

A

Polynomial f(x) with a factor of (x-a) can be expressed as

f(x) = (x-a)Q(x)

Therefore, f(a) = 0 means that the remainder is 0

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3
Q

P(x) = anxn + an-1xn-1 + an-2xn-2 + ….+ a1x + a0 = 0

What are the sum and product of the roots?

A

Sum of the roots = - an-1/an

Product of the roots = a0/ann

Where n is the degree of the polynomial

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4
Q

Polynomial P(x) has one root a+bi, with a and b real numbers, what is the other root?

A

The conjugate a-bi is also a root of P(x)

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5
Q

What is |a+bi|?

A

(a2 + b2)1/2

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6
Q

What does the discriminant D tell us for:

  1. D > 0
  2. D = 0
  3. D <0
A
  1. Roots are real and unequal
  2. Roots are real and equal
  3. Roots are imaginary (no real roots)
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7
Q

For two Linear functions:

y = m1x + b1 and y = m2x + b2

If m1 = m2 and b1 not= b2

A

The two lines are parallel

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8
Q

For two Linear functions:

y = m1x + b1 and y = m2x + b2

If m1 = m2 and b1 = b2

A

Then the two lines coincide

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9
Q

For two Linear functions:

y = m1x + b1 and y = m2x + b2

If m1.m2 = -1

A

Then these two lines are perpendicular

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10
Q

For two Linear functions:

y = m1x + b1 and y = m2x + b2

If m1 not= m2

A

Then these two lines are intersecting

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11
Q

Distance between point (x1, y1) and a line ax + by + c = 0

A

D = |ax1 + by1 + c|/(a2 + b2)1/2

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12
Q

Distance between (x1, y1) and (x2, y2)

A

D = {(x2-x1)2 + (y2-y1)2}1/2

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13
Q

Distance from the origin to a point (a, b, c)

A

[a2 + b2 +c2]1/2

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14
Q

Standard equation of a circle with center at (h, k) and radius r

A

(x-h)2 + (y-k)2 = r2

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15
Q

Standard equation of an ellipse with center (h, k) and where a > b

A

(x - h)2/a2 + (y - k)2/b2 = 1 Major axis is horizontal

(x - h)2/b2 + (y - k)2/a2 = 1 Major axis is vertical

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16
Q

Length of Major and Minor Axes of ellipse?

A

Major axis = 2a

Minor Axis = 2b

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17
Q

For an ellipse if c is the length from the center to the focus what is its value?

A

c2 = a2 - b2

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18
Q

Standard form of parabola with vertex at (0, 0)

A

Vertical axis: x2 = 4py

Horizontal axis: y2 = 4px

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19
Q

Standard form of a hyperbola, center at (0, 0)

A

Transverse axis horizontal: x2/a2 - y2/b2 = 1

Transverse axis vertical y2/a2 - x2/b2 = 1

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20
Q

Hyperbola Focus for (_+_c, 0)

A

c2 = a2 + b2

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21
Q

Asymptotes, horizontal and vertical axis for a hyperbola

A

Horizontal: y = +(b/a)x

Vertical: y = +(a/b)x

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22
Q

What are Domain and Range

A

Domain is the set of X (input)

Range is the set of Y (output)

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23
Q

What are odd and even functions

A

Even function: f(x) = f(-x)

Odd function: f(x) = -f(-x)

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24
Q

If p is the period of f(x) then:

A

f(x + p) = f(x)

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25
Q

If p is the period of f(x) then what is the period of

y = cf(x)

y = f(cx)

A

Period is p

Period is p/c

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26
Q

If g is the inverse of f, what are the properties?

A

f(g(x)) = x and

g(f(x)) = x

f-1(x) is the reflection of the graph of f in the line y = x

If point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f-1

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27
Q

When does an inverse function exist?

A

When f is increasing on its entire domain

When f is decreasing on its entire domain

Use horizontal line test

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28
Q

If f is continuous on a closed interval [a ,b] and k is any number between f(a) and f(b) then

A

There is at least one number c in [a ,b] such that f(c) = k

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29
Q

If a polynomial has integer coefficients the possible rational zeros of f are:

A

(factors or constant term)/(factors of leading coefficient)

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30
Q

For a polynomial with real coefficients and a0 not= 0

A
  1. The number of positive zeros of f is either equal to the number of variations in sign of f or less than the number by an even integer
  2. The number of negative zeros of f is either equal to the number of variations in sign of f or less than the number by an even integer
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31
Q

x → 0 lim (1 + x)1/x

and

x → infinity lim (1 + 1/x)x

A

e

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32
Q

What is the nth term tn if the first is t1 and the common difference is d

A

tn = t1 + d(n - 1)

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33
Q

The sum of a finite arithmetic sequence with n terms is

A

Sn = n(t1 + tn)/2

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34
Q

What is the nth term if the first term is t1 and the ommon ratio is r?

A

tn = t1rn-1

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35
Q

Sum of the sequence?

A

Sn = t1(1-rn)/(1-r)

36
Q

If |r| < 1, the sum S of the infinite series is

A

S = t1/(1-r)

37
Q

A permutation of a set of values is an arangement, where order is important.

The number of permutations of r elements from n elements is

A

nPr = n!/(n-r)!

38
Q

A selection where order is not important is called a combination

The number of combinations of n things taken r at a time

A

nCr = n!/(n-r)!r!

or

nCr = nPr/r!

39
Q

The number of terms of (x + y)n

40
Q

The rth term of the expansion is

A

nCr-1 (x)n-r+1 (y)r-1

41
Q

Multiplication law:

xa*xb

42
Q

Power Law:

(xa)b

43
Q

Division Law:

xa ÷ xb

44
Q

Power of a Product Law:

(xy)a

45
Q

What is the compounding equation? continuous compounding equation?

A

A = A0(1+r/n)nt

continuous compounding:

A = Pert

46
Q

n > 0, a > 0 and a not 1

logan = e

then

47
Q

loga1=

48
Q

logaa =

49
Q

loga ax =

50
Q

alogax =

51
Q

loga (xy)

A

logax + logay

52
Q

loga (x/y)

A

logax - logay

53
Q

logaxn =

54
Q

Sine-Cosine Identity

A

sin2x+cos2x=1

55
Q

What are the domain, range and period of sinx?

A

Domain: All real numbers

Range: -1<=sinx <=1

Period: 2pi, 360 deg

56
Q

What are the domain, range and period of cosx?

A

Domain: All real numbers

Range: -1<=cosx <=1

Period: 2pi, 360 deg

57
Q

What are the domain, range and period of tanx?

A

Domain: All real numbers except x = 180n + 90

Range: all real numbers

Period: pi, 180 deg

58
Q

What are the domain, range and period of csc(x)?

A

Domain: All real numbers except 180n

Range: |csc(x)|>=1

Period: 2pi, 360

59
Q

What are the domain, range and period of sec(x)?

A

Domain: All real numbes except 90 + 180n

Range: |sec(x)| >= 1

Perikod: 2pi, 360

60
Q

What are the domain, range and period of cot(x)?

A

Domain: All real numbers except 180n

Range: All real numbers

Period: pi, 180

61
Q

What is a cofunction?

A

Any trignometric function of an acute angle is equal to the cofunction of its complement

62
Q

If A + B = 90 then

sinA =

63
Q

If A + B = 90 then

what id the cofunction of secA

64
Q

If A + B = 90 then

what is the cofunction of tanA

65
Q

For either

y = asin(bx - c) + d or

y = acos(bx-c) + d

what are the amplitude, Period and Middle line?

A

Amplitude = |a|

Period p = 2pi/b

Middle line: y = d

66
Q

For y = tan(bx)

what is the Period?

A

Period: pi/2

67
Q

What are the domain and range of arcsin(x)?

A

Domain: -1 =<x<= +1

Range: -900 =<y<= +900

68
Q

What are the domain and range of arccos(x)

A

Domain: -1 =<x<= +1

Range: 00 =<y<= 1800

69
Q

What are the domain and range of arctan(x)?

A

Domain: all real numbers

Range: -900 <y< +900

70
Q

sin(A + B) =

A

sinAcosB + cosAsinB

71
Q

cos(A + B) =

A

cosAcosB - sinAsinB

72
Q

tan(A + B) =

A

(tanA + tanB)/(1 - tanAtanB)

73
Q

sin (A - B) =

A

sinAcosB - cosAsinB

74
Q

cos (A - B) =

A

cosAcosB + sinAsinB

75
Q

tan (A - B) =

A

(tanA - tanB)/(1 + tanAtanB)

76
Q

sin2A =

77
Q

cos2A =

A

cos2A - sin2A

1 - 2sin2A

2cos2A - 1

78
Q

tan2A =

A

2tanA/(1 - tan2A)

79
Q

sin(A/2) =

A

+ [(1-cosA)/2]1/2

80
Q

cos(A/2) =

A

+ [(1 + cosA)/2]1/2

81
Q

tan(A/2) =

A

+ [(1 - cosA)/(1 + cosA)]1/2

82
Q

If ABC is a triangle with sides a, b, and c then

A

a/sinA = b/sinB = c/sinC

83
Q

If ABC is a triangle with sides a, b, and c then the area =

A

(bcsinA)/2

(absinC)/2

(acsinB)/2

84
Q

If ABC is a triangle with sides a, b, and c then

what is the Law of Cosines

A

cosA =( b2 + c2 - a2)/2bc

cosB = (a2 + c2 - b2)/2ac

cosC = (a2 + b2 -c2)/2ab

85
Q

For a vector V→ what is a unit vector u→?

A

u→ = (V→)/|V→|

Divide a vector by its length

Math (1 decks)

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