Q. If the product of two numbers is a multiple of 15, which of the following must be true?
A.
At least one of the numbers is a multiple of 3.
B.
At least one of the numbers is a multiple of 5.
C.
Both numbers are even.
D.
Both numbers are odd.
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Solution
For the product to be a multiple of 15, at least one of the numbers must be a multiple of 5.
Correct Answer:
B
— At least one of the numbers is a multiple of 5.
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Q. If the quadratic equation ax^2 + bx + c = 0 has roots p and q, which of the following is correct?
A.
p + q = -b/a and pq = c/a
B.
p + q = b/a and pq = -c/a
C.
p + q = c/a and pq = -b/a
D.
p + q = -c/a and pq = b/a
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Solution
According to Vieta's formulas, the sum of the roots p and q is -b/a and the product is c/a.
Correct Answer:
A
— p + q = -b/a and pq = c/a
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Q. If the quadratic equation x^2 + kx + 16 = 0 has real roots, what is the condition on k?
A.
k^2 >= 64
B.
k^2 < 64
C.
k > 16
D.
k < 16
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Solution
For real roots, the discriminant must be non-negative: k^2 - 4*1*16 >= 0, leading to k^2 >= 64.
Correct Answer:
A
— k^2 >= 64
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Q. If the quadratic equation x^2 - 4x + k = 0 has equal roots, what is the value of k?
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Solution
For equal roots, the discriminant must be zero: (-4)^2 - 4*1*k = 0, leading to k = 4.
Correct Answer:
B
— 4
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Q. If the quadratic equation x² - 4x + k = 0 has one real root, what must be the value of k?
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Solution
For the equation to have one real root, the discriminant must be zero. Thus, k must equal 4.
Correct Answer:
A
— 4
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Q. If the quadratic equation x² - 5x + 6 = 0 is factored, which of the following pairs of numbers represents the roots?
A.
2 and 3
B.
1 and 6
C.
0 and 6
D.
3 and 2
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Solution
Factoring the equation gives (x - 2)(x - 3) = 0, thus the roots are 2 and 3.
Correct Answer:
A
— 2 and 3
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Q. If the radius of a circle is increased by 50%, by what percentage does the area of the circle increase?
A.
25%
B.
50%
C.
75%
D.
100%
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Solution
The area of a circle is given by A = πr². If the radius is increased by 50%, the new radius is 1.5r. The new area is A' = π(1.5r)² = 2.25πr². The increase in area is (2.25 - 1) = 1.25 times the original area, which is a 125% increase.
Correct Answer:
C
— 75%
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Q. If the radius of a circle is increased by 50%, what happens to the area of the circle?
A.
It increases by 50%.
B.
It increases by 100%.
C.
It increases by 125%.
D.
It increases by 150%.
Show solution
Solution
If the radius increases by 50%, the new radius is 1.5r, and the area becomes (1.5r)² = 2.25r², which is a 125% increase.
Correct Answer:
C
— It increases by 125%.
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Q. If the radius of a sphere is halved, how does its volume change?
A.
It remains the same
B.
It doubles
C.
It halves
D.
It reduces to one-eighth
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Solution
Volume of a sphere = (4/3)πr³. If radius is halved, volume becomes (4/3)π(1/2)³ = (4/3)π(1/8) = (1/6)π, which is one-eighth of the original volume.
Correct Answer:
D
— It reduces to one-eighth
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Q. If the ratio of consecutive terms in a geometric series is constant, what can be inferred about the series? (2023)
A.
It is increasing.
B.
It is decreasing.
C.
It is exponential.
D.
It is linear.
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Solution
A constant ratio of consecutive terms indicates that the series is exponential.
Correct Answer:
C
— It is exponential.
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Q. If the ratio of consecutive terms in a geometric series is constant, what is this constant called? (2023)
A.
Common difference
B.
Common ratio
C.
Term factor
D.
Sequence multiplier
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Solution
The constant ratio of consecutive terms in a geometric series is called the common ratio.
Correct Answer:
B
— Common ratio
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Q. If the ratio of consecutive terms in a geometric series is constant, what is this ratio called? (2023)
A.
Common difference
B.
Common ratio
C.
Term ratio
D.
Sequence ratio
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Solution
The constant ratio of consecutive terms in a geometric series is called the common ratio.
Correct Answer:
B
— Common ratio
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Q. If the ratio of the ages of A and B is 5:3 and the sum of their ages is 64 years, what is A's age?
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Solution
Let A's age be 5x and B's age be 3x. Then, 5x + 3x = 64, which gives 8x = 64. Therefore, x = 8, and A's age is 5x = 40 years.
Correct Answer:
A
— 40
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Q. If the ratio of the lengths of two sides of a triangle is 7:5 and the perimeter is 48 cm, what is the length of the longer side?
A.
28 cm
B.
20 cm
C.
24 cm
D.
16 cm
Show solution
Solution
Let the lengths of the sides be 7x and 5x. Then, 7x + 5x = 48, which gives 12x = 48. Thus, x = 4, and the longer side is 7x = 28 cm.
Correct Answer:
A
— 28 cm
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Q. If the ratio of the number of cats to dogs in a shelter is 2:3 and there are 30 dogs, how many cats are there?
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Solution
The ratio of cats to dogs is 2:3. If there are 30 dogs, the number of cats can be calculated as (2/3) * 30 = 20 cats.
Correct Answer:
A
— 20
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Q. If the roots of the equation x^2 - 5x + 6 = 0 are p and q, what is the value of p + q?
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Solution
According to Vieta's formulas, the sum of the roots p + q is equal to -(-5) = 5.
Correct Answer:
A
— 5
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Q. If the roots of the quadratic equation ax^2 + bx + c = 0 are p and q, which of the following is correct?
A.
p + q = -b/a and pq = c/a
B.
p + q = c/a and pq = -b/a
C.
p - q = -b/a and pq = c/a
D.
p * q = -b/a and p + q = c/a
Show solution
Solution
According to Vieta's formulas, the sum of the roots p + q = -b/a and the product pq = c/a.
Correct Answer:
A
— p + q = -b/a and pq = c/a
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Q. If the roots of the quadratic equation x² + px + q = 0 are both negative, which of the following must be true?
A.
p > 0 and q > 0
B.
p < 0 and q < 0
C.
p < 0 and q > 0
D.
p > 0 and q < 0
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Solution
For both roots to be negative, the sum (p) must be positive and the product (q) must also be positive.
Correct Answer:
A
— p > 0 and q > 0
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Q. If the second term of a geometric progression is 12 and the common ratio is 3, what is the first term?
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Solution
Let the first term be a. The second term is a * r = a * 3 = 12, thus a = 12/3 = 4.
Correct Answer:
A
— 4
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Q. If the second term of a GP is 12 and the common ratio is 2, what is the first term?
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Solution
Let the first term be a. The second term is a * r = a * 2 = 12, thus a = 12/2 = 6.
Correct Answer:
A
— 6
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Q. If the second term of a GP is 12 and the common ratio is 3, what is the first term?
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Solution
Let the first term be a. The second term is a * r = a * 3 = 12, thus a = 12/3 = 4.
Correct Answer:
A
— 4
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Q. If the second term of a GP is 6 and the common ratio is 3, what is the first term?
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Solution
Let the first term be a. The second term is a * r = a * 3 = 6. Thus, a = 6/3 = 2.
Correct Answer:
A
— 2
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Q. If the second term of a GP is 8 and the fourth term is 32, what is the common ratio?
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Solution
Let the first term be a and the common ratio be r. Then, 8 = ar and 32 = ar^3. Dividing these gives r^2 = 4, so r = 2.
Correct Answer:
A
— 2
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Q. If the sides of a triangle are in the ratio 3:4:5, which type of triangle is it?
A.
Equilateral
B.
Isosceles
C.
Scalene
D.
Right-angled
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Solution
A triangle with sides in the ratio 3:4:5 is a right-angled triangle, as it satisfies the Pythagorean theorem.
Correct Answer:
D
— Right-angled
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Q. If the simple interest on a certain sum for 3 years at 5% per annum is $150, what is the principal amount?
A.
$1000
B.
$1200
C.
$1500
D.
$1800
Show solution
Solution
Using SI = PRT, we have 150 = P * 0.05 * 3. Solving for P gives P = 1000.
Correct Answer:
A
— $1000
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Q. If the simple interest on a sum of money for 5 years is $300 at a rate of 4% per annum, what is the principal? (2000)
A.
$1200
B.
$1500
C.
$1800
D.
$2000
Show solution
Solution
Using SI = PRT, we have 300 = P * 4/100 * 5. Solving for P gives P = $1500.
Correct Answer:
A
— $1200
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Q. If the solution to the linear equation 4x + 5y = 20 is (2, 0), what is the value of y when x = 2?
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Solution
Substituting x = 2 into the equation gives 4(2) + 5y = 20, leading to y = 0.
Correct Answer:
A
— 0
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Q. If the statement 'All squares are rectangles' is true, which of the following must also be true?
A.
All rectangles are squares.
B.
Some rectangles are not squares.
C.
No rectangles are squares.
D.
Some squares are not rectangles.
Show solution
Solution
If all squares are rectangles, it implies that there are rectangles that are not squares, thus some rectangles are not squares.
Correct Answer:
B
— Some rectangles are not squares.
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Q. If the sum of an infinite geometric series is 20 and the common ratio is 1/4, what is the first term?
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Solution
The sum S of an infinite GP is given by S = a / (1 - r). Here, 20 = a / (1 - 1/4) = a / (3/4). Thus, a = 20 * (3/4) = 15.
Correct Answer:
A
— 25
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Q. If the sum of an infinite GP is 10 and the common ratio is 1/3, what is the first term?
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Solution
The sum of an infinite GP is given by S = a / (1 - r). Here, S = 10 and r = 1/3. Thus, 10 = a / (1 - 1/3) = a / (2/3) => a = 10 * (2/3) = 20.
Correct Answer:
B
— 20
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