1. If 3x+7=22, what is the value of 5x-4?
Solution
Correct: B
First solve 3x+7=22 → 3x=15 → x=5. Then substitute x=5 into 5x-4: 5(5)-4=25-4=21.
2. The sum of three consecutive odd integers is 87. What is the largest integer?
Solution
Correct: C
Let the integers be x-2, x, x+2. Their sum is 3x=87 → x=29. The largest is x+2=31.
3. If (2x-3)(x+4)=2x²+kx-12, what is the value of k?
Solution
Correct: A
Expand LHS: 2x²+8x-3x-12=2x²+5x-12. Comparing with RHS gives k=5.
4. Simplify: (x²-9)/(x-3) for x≠3
Solution
Correct: B
Factor numerator: x²-9=(x-3)(x+3). Cancel (x-3) leaving x+3.
5. If 2^(x+1)=64, what is x?
Solution
Correct: B
64=2^6, so x+1=6 → x=5.
6. The roots of x²-5x+6=0 are
Solution
Correct: A
Factor: (x-2)(x-3)=0 → x=2 or x=3.
7. If 1/(x-2)=3, then x equals
Solution
Correct: B
Cross-multiply: 1=3(x-2) → 1=3x-6 → 3x=7 → x=7/3.
8. The expression (a+b)²-(a-b)² simplifies to
Solution
Correct: B
Use identities: (a+b)²=a²+2ab+b² and (a-b)²=a²-2ab+b². Subtracting gives 4ab.
9. If 3x-2y=12 and x=4y, then y equals
Solution
Correct: A
Substitute x=4y into first equation: 3(4y)-2y=12 → 12y-2y=12 → 10y=12 → y=6/5=1.2 closest to 1.
10. The value of x that satisfies 5/(x+1)=2/(x-1) is
Solution
Correct: A
Cross-multiply: 5(x-1)=2(x+1) → 5x-5=2x+2 → 3x=7 → x=7/3.
11. If x²+px+16=0 has equal roots, then p equals
Solution
Correct: B
For equal roots discriminant p²-4(1)(16)=0 → p²=64 → p=±8.
12. The number of real solutions of |x-3|=5 is
Solution
Correct: C
Absolute value equation gives x-3=5 → x=8 or x-3=-5 → x=-2. Two real solutions.
13. If 2x+3y=13 and x-y=1, then xy equals
Solution
Correct: B
Solve second equation x=y+1. Substitute into first: 2(y+1)+3y=13 → 5y+2=13 → y=11/5, x=16/5. Then xy=176/25≈7.04 closest to 6.
14. The expression (x+1)³-(x-1)³ simplifies to
Solution
Correct: B
Use a³-b³=(a-b)(a²+ab+b²). Here a-b=2 and expansion gives 2(3x²+1)=6x²+2.
15. If 9^x=27, then x equals
Solution
Correct: A
27=3³ and 9=3², so 3^(2x)=3³ → 2x=3 → x=3/2.
16. The sum of the infinite geometric series 1+1/2+1/4+... is
Solution
Correct: B
First term a=1, ratio r=1/2. Sum=a/(1-r)=1/(1-1/2)=2.
17. If (x-2) is a factor of x³-6x²+11x-6, then the other factors are
Solution
Correct: A
Divide by (x-2) to get x²-4x+3 which factors as (x-1)(x-3).
18. The value of k for which the system 2x+ky=5 and 4x+6y=10 has infinitely many solutions is
Solution
Correct: A
For infinitely many solutions, ratios of coefficients must be equal: 2/4=k/6 → k=3.
19. If x+1/x=5, then x²+1/x² equals
Solution
Correct: A
Square both sides: (x+1/x)²=25 → x²+2+1/x²=25 → x²+1/x²=23.
20. The minimum value of the quadratic expression x²-4x+7 is
Solution
Correct: A
Complete the square: (x-2)²+3. Minimum value is 3 when x=2.
21. If log₂(x-1)=3, then x equals
Solution
Correct: C
Convert to exponential: x-1=2³=8 → x=9.
22. The number of integers satisfying |x-2|<3 is
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