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 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 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 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 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
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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
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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.
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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 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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Q. If the sum of the first 5 terms of a geometric series is 31 and the first term is 1, what is the common ratio? (2023)
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Solution
Using the formula for the sum of a geometric series, S_n = a(1 - r^n) / (1 - r), we can solve for r. Here, S_5 = 1(1 - r^5) / (1 - r) = 31, leading to r = 3.
Correct Answer: B — 3
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Q. If the sum of the first 5 terms of an arithmetic progression is 50, what is the value of the first term if the common difference is 2?
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Solution
The sum S_5 = 5/2 * (2a + 4d). Here, 50 = 5/2 * (2a + 8), solving gives a = 10.
Correct Answer: B — 10
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Q. If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n) / (1 - r), what happens to S_n as n approaches infinity when |r| < 1?
A.
S_n approaches 0
B.
S_n approaches infinity
C.
S_n approaches a/(1-r)
D.
S_n approaches a
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Solution
As n approaches infinity and |r| < 1, r^n approaches 0, thus S_n approaches a/(1-r).
Correct Answer: C — S_n approaches a/(1-r)
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Q. If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n)/(1 - r), which of the following is true?
A.
S_n is always positive.
B.
S_n can be negative.
C.
S_n is independent of n.
D.
S_n is always an integer.
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Solution
The sum S_n can be negative if the common ratio r is negative and the first term a is also negative.
Correct Answer: B — S_n can be negative.
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Q. If the sum of the first n terms of a GP is 63 and the first term is 3, what is the common ratio if n = 4?
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Solution
Using the formula S_n = a(1 - r^n) / (1 - r), we have 63 = 3(1 - r^4) / (1 - r). Solving gives r = 2.
Correct Answer: A — 2
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Q. If the sum of the first n terms of an arithmetic progression is given by S_n = 3n^2 + 2n, what is the common difference?
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Solution
The sum of the first n terms S_n = n/2 * (2a + (n-1)d). By differentiating S_n with respect to n, we can find the common difference. The common difference is 3.
Correct Answer: A — 3
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Q. If the sum of the first n terms of an arithmetic progression is given by S_n = 3n^2 + 2n, what is the common difference of the sequence?
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Solution
The common difference can be found by calculating S_n - S_(n-1). S_n = 3n^2 + 2n and S_(n-1) = 3(n-1)^2 + 2(n-1). Simplifying gives the common difference as 6.
Correct Answer: A — 3
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Q. If the sum of the first n terms of an arithmetic series is given by S_n = 3n^2 + 2n, what is the common difference? (2023)
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Solution
The common difference can be found by calculating S_n - S_(n-1). Here, S_n = 3n^2 + 2n and S_(n-1) = 3(n-1)^2 + 2(n-1). The difference simplifies to 4.
Correct Answer: B — 4
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Q. If the sum of the first n terms of an arithmetic series is given by S_n = 5n + 3, what is the 10th term? (2023)
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Solution
The 10th term can be found using T_n = S_n - S_(n-1). Here, S_10 = 5(10) + 3 = 53 and S_9 = 5(9) + 3 = 48. Thus, T_10 = 53 - 48 = 5.
Correct Answer: A — 53
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Q. If the sum of the first n terms of an arithmetic series is given by S_n = 5n + 3, what is the common difference? (2023)
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Solution
The common difference can be found by calculating S_n - S_(n-1). This gives the common difference as 5.
Correct Answer: A — 5
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Q. If the sum of the first n terms of an arithmetic series is given by S_n = n/2(2a + (n-1)d), what does 'a' represent? (2023)
A.
The last term
B.
The first term
C.
The common difference
D.
The number of terms
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Solution
'a' represents the first term of the arithmetic series.
Correct Answer: B — The first term
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Q. If the sum of the first n terms of an arithmetic series is given by S_n = n/2(2a + (n-1)d), what does 'd' represent? (2023)
A.
First term
B.
Common difference
C.
Last term
D.
Number of terms
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Solution
'd' represents the common difference in an arithmetic series.
Correct Answer: B — Common difference
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Q. If the sum of the first three terms of a geometric progression is 14 and the common ratio is 2, what is the first term?
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Solution
Let the first term be a. The sum of the first three terms is a + 2a + 4a = 7a. Setting 7a = 14 gives a = 2.
Correct Answer: B — 3
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Q. If the sum of the first three terms of a GP is 14 and the common ratio is 2, what is the first term?
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Solution
Let the first term be a. The sum of the first three terms is a + 2a + 4a = 7a. Setting 7a = 14 gives a = 2.
Correct Answer: C — 4
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