Q. If the number 36 is expressed in terms of its prime factors, which of the following is correct?
A.
2^2 * 3^2
B.
2^3 * 3^1
C.
3^2 * 4^1
D.
6^2
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Solution
The prime factorization of 36 is 2^2 * 3^2, as 36 = 2 * 2 * 3 * 3.
Correct Answer: A — 2^2 * 3^2
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Q. If the perimeter of a regular hexagon is 72 cm, what is the length of each side?
A.
10 cm
B.
12 cm
C.
14 cm
D.
8 cm
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Solution
The perimeter of a regular hexagon is the sum of the lengths of its 6 equal sides. Therefore, each side is 72 cm / 6 = 12 cm.
Correct Answer: B — 12 cm
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Q. If the perimeter of a regular octagon is 64 cm, what is the length of each side? (2023)
A.
6 cm
B.
8 cm
C.
10 cm
D.
12 cm
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Solution
The perimeter of a regular octagon is 8 times the length of one side. Therefore, each side is 64 cm / 8 = 8 cm.
Correct Answer: B — 8 cm
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Q. If the perimeter of an equilateral triangle is 36 cm, what is the length of each side?
A.
9 cm
B.
12 cm
C.
15 cm
D.
18 cm
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Solution
In an equilateral triangle, all sides are equal. Therefore, if the perimeter is 36 cm, each side is 36/3 = 12 cm.
Correct Answer: B — 12 cm
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Q. If the perimeter of an isosceles triangle is 24 cm and the lengths of the two equal sides are 10 cm each, what is the length of the base?
A.
4 cm
B.
6 cm
C.
8 cm
D.
10 cm
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Solution
The perimeter is the sum of all sides. Therefore, base = 24 - (10 + 10) = 4 cm.
Correct Answer: B — 6 cm
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Q. If the polynomial g(x) = x^2 + bx + c has a double root, what can be inferred about its discriminant?
A.
It is greater than zero.
B.
It is less than zero.
C.
It is equal to zero.
D.
It can be any value.
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Solution
A polynomial has a double root when its discriminant is equal to zero.
Correct Answer: C — It is equal to zero.
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Q. If the polynomial P(x) = x^2 + bx + c has roots 3 and -2, what is the value of b?
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Solution
Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.
Correct Answer: B — 5
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Q. If the polynomial P(x) = x^2 - 5x + 6 has roots r1 and r2, what is the value of r1 + r2?
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Solution
According to Vieta's formulas, the sum of the roots r1 + r2 of the polynomial x^2 - 5x + 6 is equal to the coefficient of x (which is -(-5)) = 5.
Correct Answer: A — 5
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Q. If the polynomial P(x) = x^3 - 3x^2 + 4 has a local maximum at x = 1, what is the value of P(1)?
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Solution
Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.
Correct Answer: A — 2
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Q. If the price of a book is increased by 10% and then decreased by 10%, what is the net change in price?
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Solution
Let the original price be $100. After a 10% increase, the price becomes $110. After a 10% decrease, the price becomes $110 - $11 = $99. The net change is -1%, so the answer is 0%.
Correct Answer: A — 0%
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Q. If the price of a book is increased by 20% and then decreased by 20%, what is the net change in price?
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Solution
Let the original price be Rs. 100. After a 20% increase, price = 120. After a 20% decrease, price = 120 - 24 = 96. Net change = (96 - 100)/100 * 100 = -4%.
Correct Answer: B — 4%
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Q. If the price of a book is increased by 20% and then decreased by 20%, what is the net change in the price?
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Solution
Let the original price be $100. After a 20% increase, the price becomes $120. After a 20% decrease, the price becomes $120 - 0.20 × 120 = $96. The net change is (96 - 100)/100 × 100 = -4%.
Correct Answer: C — 5%
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Q. If the principal amount is $2000 and the total amount after 3 years at a certain rate of simple interest is $2400, what is the rate of interest? (2000)
A.
5%
B.
6.67%
C.
10%
D.
12%
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Solution
The interest earned is $400. Using SI = PRT, we have 400 = 2000 * R * 3. Solving for R gives R = 6.67%.
Correct Answer: B — 6.67%
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Q. If the principal is $2000 and the rate of interest is 4% per annum, what will be the total amount after 5 years at simple interest? (2000)
A.
$2400
B.
$2500
C.
$2600
D.
$2700
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Solution
Using A = P + SI, where SI = P * r * t / 100, we have SI = 2000 * 4 * 5 / 100 = $400. Thus, A = 2000 + 400 = $2400.
Correct Answer: A — $2400
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Q. If the principal is $800 and the rate of interest is 4% per annum, what will be the total amount after 3 years at simple interest?
A.
$840
B.
$850
C.
$860
D.
$870
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Solution
Using the simple interest formula, Total Amount = Principal + SI = 800 + (800 * 4 * 3) / 100 = $840.
Correct Answer: A — $840
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Q. If the principal is $800 and the rate of interest is 6% per annum, what will be the total amount after 4 years at simple interest?
A.
$960
B.
$1040
C.
$800
D.
$840
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Solution
Total amount = Principal + SI = 800 + (800 * 6/100 * 4) = $1040.
Correct Answer: B — $1040
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Q. If the principal is Rs. 5000 and the rate of interest is 12% per annum, what will be the total amount after 3 years under simple interest?
A.
Rs. 5600
B.
Rs. 5800
C.
Rs. 6000
D.
Rs. 6200
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Solution
Simple Interest = P * r * t = 5000 * 0.12 * 3 = 1800. Total Amount = Principal + Interest = 5000 + 1800 = 6800.
Correct Answer: B — Rs. 5800
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Q. If the probability of a student passing an exam is 0.75, what is the probability that the student fails?
A.
0.25
B.
0.5
C.
0.75
D.
0.1
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Solution
The probability of failing is 1 - P(passing) = 1 - 0.75 = 0.25.
Correct Answer: A — 0.25
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Q. If the probability of event A is 0.2 and the probability of event B is 0.5, what is the probability of either A or B occurring if A and B are independent?
A.
0.7
B.
0.6
C.
0.5
D.
0.4
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Solution
The probability of either A or B occurring is P(A) + P(B) - P(A and B) = 0.2 + 0.5 - (0.2 * 0.5) = 0.7.
Correct Answer: A — 0.7
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Q. If the probability of event A is 0.4 and the probability of event B is 0.5, what is the probability of both A and B occurring if they are independent?
A.
0.2
B.
0.4
C.
0.5
D.
0.9
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Solution
For independent events, P(A and B) = P(A) * P(B) = 0.4 * 0.5 = 0.2.
Correct Answer: A — 0.2
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Q. If the probability of event C is 0.2 and the probability of event D is 0.3, what is the probability of either C or D occurring if they are mutually exclusive?
A.
0.5
B.
0.6
C.
0.3
D.
0.2
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Solution
For mutually exclusive events, P(C or D) = P(C) + P(D) = 0.2 + 0.3 = 0.5.
Correct Answer: B — 0.6
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Q. If the probability of rain tomorrow is 0.7, what is the probability that it will not rain?
A.
0.3
B.
0.5
C.
0.7
D.
0.9
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Solution
The probability that it will not rain is 1 - 0.7 = 0.3.
Correct Answer: A — 0.3
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Q. If the product of two consecutive integers is 72, which of the following pairs could represent these integers?
A.
8 and 9
B.
6 and 7
C.
5 and 6
D.
9 and 10
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Solution
6 and 7 are consecutive integers whose product is 42, not 72.
Correct Answer: B — 6 and 7
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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.
Show solution
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
Show solution
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 - 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 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.
Show solution
Solution
A constant ratio of consecutive terms indicates that the series is exponential.
Correct Answer: C — It is exponential.
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