Q. Find the coordinates of the point that divides the segment joining (2, 3) and (4, 7) in the ratio 1:3.
A.
(3, 5)
B.
(2.5, 4)
C.
(3.5, 5.5)
D.
(3, 6)
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Solution
Using the section formula: P(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)), where m=1, n=3, x1=2, y1=3, x2=4, y2=7. P = ((1*4 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (3, 5).
Correct Answer:
A
— (3, 5)
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Q. Find the coordinates of the point that divides the segment joining (2, 3) and (8, 7) in the ratio 1:3.
A.
(5, 5)
B.
(4, 5)
C.
(6, 5)
D.
(3, 4)
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Solution
Using the section formula: P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) where m=1, n=3. P = ((1*8 + 3*2)/(1+3), (1*7 + 3*3)/(1+3)) = (5, 5).
Correct Answer:
A
— (5, 5)
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Q. Find the distance between the points (-1, -1) and (2, 3).
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Solution
Using the distance formula: d = √((2 - (-1))² + (3 - (-1))²) = √((3)² + (4)²) = √(9 + 16) = √25 = 5.
Correct Answer:
C
— 5
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Q. Find the midpoint of the line segment joining the points (1, 2) and (5, 6).
A.
(3, 4)
B.
(4, 3)
C.
(2, 5)
D.
(5, 2)
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Solution
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((1 + 5)/2, (2 + 6)/2) = (3, 4).
Correct Answer:
A
— (3, 4)
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Q. Find the roots of the equation 3x^2 + 6x + 3 = 0.
A.
x = -1
B.
x = -3
C.
x = 1
D.
x = 3
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Solution
This can be simplified to x^2 + 2x + 1 = 0, which factors to (x + 1)(x + 1) = 0. Thus, the root is x = -1.
Correct Answer:
A
— x = -1
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Q. Find the roots of the equation 3x^2 - 12 = 0.
A.
x = 2, -2
B.
x = 4, -4
C.
x = 2, 4
D.
x = -4, 2
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Solution
Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.
Correct Answer:
A
— x = 2, -2
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Q. Find the roots of the equation 4x^2 - 12x + 9 = 0.
A.
x = 1.5
B.
x = 3
C.
x = 0
D.
x = -3
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Solution
Factoring gives (2x - 3)(2x - 3) = 0. Thus, x = 3.
Correct Answer:
B
— x = 3
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Q. Find the roots of the equation x^2 + 2x - 8 = 0.
A.
x = 2, -4
B.
x = -2, 4
C.
x = 4, -2
D.
x = -4, 2
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Solution
Factoring gives (x + 4)(x - 2) = 0. Thus, the roots are x = 4 and x = -2.
Correct Answer:
C
— x = 4, -2
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Q. Find the roots of the equation x^2 - 8x + 16 = 0.
A.
x = 4
B.
x = -4
C.
x = 8
D.
x = 0
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Solution
This is a perfect square: (x - 4)² = 0. Thus, x - 4 = 0, so x = 4.
Correct Answer:
A
— x = 4
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Q. Find the roots of the polynomial equation x^2 + 5x + 6 = 0.
A.
x = -2, -3
B.
x = 2, 3
C.
x = -1, -6
D.
x = 1, -6
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Solution
Factoring gives (x + 2)(x + 3) = 0. Thus, the roots are x = -2 and x = -3.
Correct Answer:
A
— x = -2, -3
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Q. Find the roots of the quadratic equation 3x^2 + 6x + 3 = 0.
A.
x = -1
B.
x = -3
C.
x = 1
D.
x = 3
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Solution
Dividing the equation by 3 gives x^2 + 2x + 1 = 0, which factors to (x + 1)(x + 1) = 0. Thus, x = -1.
Correct Answer:
A
— x = -1
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Q. Find the roots of the quadratic equation 3x^2 - 12x = 0.
A.
x = 0, 4
B.
x = 3, 4
C.
x = 0, 3
D.
x = 1, 2
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Solution
Factoring gives 3x(x - 4) = 0. Thus, x = 0 and x = 4.
Correct Answer:
A
— x = 0, 4
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Q. Find the roots of the quadratic equation 4x^2 - 12x + 9 = 0.
A.
x = 1.5
B.
x = 3
C.
x = 0
D.
x = -3
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Solution
This can be factored as (2x - 3)(2x - 3) = 0. Thus, x = 3.
Correct Answer:
B
— x = 3
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Q. Find the solution for x in the equation 2(x + 3) = 16.
A.
x = 4
B.
x = 5
C.
x = 6
D.
x = 7
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Solution
Step 1: Divide both sides by 2: x + 3 = 8. Step 2: Subtract 3 from both sides: x = 5.
Correct Answer:
B
— x = 5
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Q. Find the solution set for the inequality: 3x + 2 > 5.
A.
x > 1
B.
x < 1
C.
x ≥ 1
D.
x ≤ 1
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Solution
Step 1: Subtract 2 from both sides: 3x > 3. Step 2: Divide by 3: x > 1.
Correct Answer:
A
— x > 1
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Q. Find the solution set for the inequality: 4x - 7 ≤ 9.
A.
x ≤ 4
B.
x ≥ 4
C.
x < 4
D.
x > 4
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Solution
Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.
Correct Answer:
A
— x ≤ 4
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Q. Find the solution set for the inequality: x^2 + 2x - 8 > 0.
A.
(-∞, -4) ∪ (2, ∞)
B.
(-4, 2)
C.
(-2, 4)
D.
(-∞, 2) ∪ (4, ∞)
Show solution
Solution
Step 1: Factor the quadratic: (x - 2)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (2, ∞).
Correct Answer:
A
— (-∞, -4) ∪ (2, ∞)
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Q. Find the solution set for the inequality: x^2 + 3x - 4 > 0.
A.
(-∞, -4) ∪ (1, ∞)
B.
(-4, 1)
C.
(-∞, 1) ∪ (4, ∞)
D.
(-4, ∞)
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Solution
Step 1: Factor the quadratic: (x - 1)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (1, ∞).
Correct Answer:
A
— (-∞, -4) ∪ (1, ∞)
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Q. Find the solution set for the inequality: x^2 - 4 > 0.
A.
(-∞, -2) ∪ (2, ∞)
B.
(-2, 2)
C.
(2, -2)
D.
(-2, ∞)
Show solution
Solution
Step 1: Factor the inequality: (x - 2)(x + 2) > 0. Step 2: The solution set is (-∞, -2) ∪ (2, ∞).
Correct Answer:
A
— (-∞, -2) ∪ (2, ∞)
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Q. Find the solution set for the inequality: x^2 - 6x + 8 > 0.
A.
x < 2 or x > 4
B.
2 < x < 4
C.
x > 2
D.
x < 4
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Solution
Step 1: Factor: (x - 2)(x - 4) > 0. Step 2: Critical points are x = 2 and x = 4. Step 3: Test intervals: valid for x < 2 or x > 4.
Correct Answer:
A
— x < 2 or x > 4
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Q. Find the solution to the inequality: -3x + 6 > 0.
A.
x < 2
B.
x > 2
C.
x ≤ 2
D.
x ≥ 2
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Solution
Step 1: Subtract 6 from both sides: -3x > -6. Step 2: Divide by -3 (reverse the inequality): x < 2.
Correct Answer:
B
— x > 2
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Q. Find the solution to the inequality: 4x + 1 ≥ 2x + 5.
A.
x ≥ 2
B.
x ≤ 2
C.
x ≥ 4
D.
x ≤ 4
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Solution
Step 1: Subtract 2x from both sides: 2x + 1 ≥ 5. Step 2: Subtract 1: 2x ≥ 4. Step 3: Divide by 2: x ≥ 2.
Correct Answer:
A
— x ≥ 2
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Q. Find the solution to the inequality: 4x - 1 ≤ 3x + 2.
A.
x ≤ 3
B.
x ≤ 1
C.
x ≥ 1
D.
x ≥ 3
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Solution
Step 1: Subtract 3x from both sides: x - 1 ≤ 2. Step 2: Add 1 to both sides: x ≤ 3.
Correct Answer:
B
— x ≤ 1
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Q. Find the solution to the inequality: 4x - 7 ≤ 9.
A.
x ≤ 4
B.
x ≥ 4
C.
x ≤ 2
D.
x ≥ 2
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Solution
Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.
Correct Answer:
A
— x ≤ 4
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Q. Find the solution to the inequality: 4x - 7 ≥ 5.
A.
x < 3
B.
x > 3
C.
x ≤ 3
D.
x ≥ 3
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Solution
Step 1: Add 7 to both sides: 4x ≥ 12. Step 2: Divide by 4: x ≥ 3.
Correct Answer:
D
— x ≥ 3
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Q. Find the solution to the inequality: 5 - 2x > 3.
A.
x < 1
B.
x > 1
C.
x < -1
D.
x > -1
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Solution
Step 1: Subtract 5 from both sides: -2x > -2. Step 2: Divide by -2 (reverse inequality): x < 1.
Correct Answer:
A
— x < 1
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Q. Find the solution to the inequality: x^2 + 4x < 5.
A.
(-5, 1)
B.
(1, -5)
C.
(1, 5)
D.
(-5, 5)
Show solution
Solution
Step 1: Rearrange: x^2 + 4x - 5 < 0. Step 2: Factor: (x + 5)(x - 1) < 0. Step 3: Test intervals: solution is (-5, 1).
Correct Answer:
A
— (-5, 1)
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Q. Find the solution to the inequality: x^2 - 9 > 0.
A.
(-∞, -3) ∪ (3, ∞)
B.
(-3, 3)
C.
(-3, ∞)
D.
(-∞, 3)
Show solution
Solution
Step 1: Factor the inequality: (x - 3)(x + 3) > 0. Step 2: The critical points are x = -3 and x = 3. Step 3: Test intervals: The solution set is (-∞, -3) ∪ (3, ∞).
Correct Answer:
A
— (-∞, -3) ∪ (3, ∞)
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Q. Find the value of k for which the equation x^2 + kx + 9 = 0 has one real solution.
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Solution
For the equation to have one real solution, the discriminant must be zero: k^2 - 4*1*9 = 0. Thus, k^2 = 36, giving k = ±6. The correct answer is -9.
Correct Answer:
B
— -9
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Q. Find the value of x in the equation 3x^2 + 12x + 12 = 0.
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Solution
Dividing the entire equation by 3 gives x^2 + 4x + 4 = 0. Factoring gives (x + 2)(x + 2) = 0, so x = -2.
Correct Answer:
B
— -4
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