Practice Test GeeksAMC12 - American Mathematics Competition 12 Practice Test
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Algebra Flashcards

6 cards from real AMC12 practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 6 Algebra flashcards as text

  1. If log(xy)=5 and log(x/y)=1, what is log x?

    Answer: 3

    Adding the two equations gives 2log(x)=6, so log(x)=3.

  2. How many real solutions does x^4 - 5x^2 + 4 = 0 have?

    Answer: 4

    Substituting u=x^2 gives (u-1)(u-4)=0, yielding x=+-1 or x=+-2, so 4 real solutions.

  3. If a and b are roots of 2x^2-3x+1=0, what is a/b + b/a?

    Answer: 5/2

    a/b+b/a=(a^2+b^2)/ab=((a+b)^2-2ab)/ab=(9/4-1)/(1/2)=5/2.

  4. What is the range of f(x) = 3/(x^2+1)?

    Answer: (0,3]

    Since x^2+1>=1, we have 0<3/(x^2+1)<=3, with maximum 3 at x=0.

  5. If 2^(x+1) = 3^x, what is x?

    Answer: log(2)/log(3/2)

    Taking logs: (x+1)log2=x*log3, so x(log3-log2)=log2, giving x=log2/log(3/2).

  6. How many ordered pairs (x,y) of integers satisfy x^2+y^2=25?

    Answer: 12

    The pairs are (0,+-5),(+-5,0),(+-3,+-4),(+-4,+-3) giving 4+4+4=12 ordered pairs.