Grade 12 challenge/math

Math sample Q&A

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Q1.Which one of the following is true about signum, absolute value and greatest integer functions?

A) sgn(x) = ± ∣ x ∣, for all x ∈ ℝ.

B) ∣ X ∣ = Xsgn(x), for all x ∈ ℝ.

C) sgn(x) ≤ ∣ x ∣, for all x ≤ 0.

D) sgn(x) ≤ ∣ x ∣, for all x ≥ 0

Let’s examine each option one by one:

A) sgn(x) = ± ∣ x ∣, for all x ∈ ℝ.

This statement is not entirely correct. The signum function is defined as:

sgn(x) = -1 if x < 0, 0 if x = 0, 1 if x > 0.

It does not equal ± ∣ x ∣ for all x ∈ ℝ.

B) ∣ X ∣ = Xsgn(x), for all x ∈ ℝ.

This statement is also not entirely accurate. The absolute value of x is defined as:

|X| = x if x ≥ 0, |X| = -x if x < 0.

It is not equal to Xsgn(x) for all x ∈ ℝ.

C) sgn(x) ≤ ∣ x ∣, for all x ≤ 0.

This is true. For any x less than or equal to 0, the signum function is -1 or 0, whereas the absolute value of x is non-negative. So, for x ≤ 0, sgn(x) is less than or equal to |x|.

D) sgn(x) ≤ ∣ x ∣, for all x ≥ 0.

This statement is not true because when x is positive, sgn(x) equals 1, whereas |x| is also 1, making them equal.

Therefore, the correct statement from the given options is C) sgn(x) ≤ ∣ x ∣, for all x ≤ 0.

Q2.What is the equation of the line that passes through (1, 1) and is parallel to the line 3y – x = 1?

A) 3x – y = 2

B) x + 3y = 4

C) 3y – x + 2 = 0

D) x – 3y + 2 = 0

Q3. Which one of the following is an equation of the circle whose end points of a diameter are (0, -2) and (2, 2)?

A) x² + y² – 2x – 4 = 0

B) x² + y² = 4

C) (x – 1)² + y² = 4

D) x² + y² – 2y – 4 = 0

Q4.What is the maximum value of the function f(x)= x⁴ – 2x² on [-2, 1] ?

A) 8

B) 12

C) 24

D) 40

Q5.If the truth value of (p ∧ ¬p) ⇔ [(q ∨ ¬q) ⇒ r] is True, then which one of the following must be True?

A) p

B) q

C) ¬q

D) ¬r

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Q6. Let A and B be two events. Suppose that the probability that neither event occurs is 3/8. What is the probability that at least one of the event occur?

A) 5/8

B) 1/4

C) 1/8

D) 3/4

Q7.If A is a square matrix of order 3 and det (A) =5, then what is the value of det(A.adj(A)) ? A) 3 B) 5 C) 25 D) 125

Q8.A salesman sold items x₁, x₂ and x₃, with different rates of commissions as shown in the table below.

What are the rates of commission on items x₁, x₂ and x₃, respectively.

A) 4, 2 and 11

B) 4, 11 and 2

C) 2, 4 and 11

D) 11, 2 and 4

Q9.The population of a certain city is increasing at a rate of 3% per year. If the population was 100,000 in 2010 E.C., then what will be the population in 2020 E.C?
(Given: (1.03)⁹ =1.30, (1.03)¹⁰= 1.34, (1.3)⁹ = 10.60, (1.3)¹⁰=13 .78)

A) 130,000

B) 134,000

C) 1,060,000

D) 1,378,000

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Q10.What is the slope of the tangent line to the graph of f(x)= 3e^x + sinx + 2 at the point (0, 5)?

A) 2

B) 3

C) 4

D) 5

Q11.What is the area of the region enclosed by the graph of y²= x + 1 and y² = -x + 1?

A) 8/3 sq.units

B) 4/3 sq. units

C) 3/8 sq. units

D) 3/4 sq. units

Q12.What is the area of the triangle (in sq. units) formed by the lines joining the vertex of the parabola x² = -36y to the end points of the latus rectum?

A) 126

B) 162

C) 216

D) 261

Q13.Let A be a 3�3 invertible matrix and B any be any 3�3 matrix. If |A|= a and, |B|= b, then which one of the following is NOT true?

A) |A^T A|= a²

B) If b = 0, then B is not invertible.

A) |KA|=K³ |A|, for any K ∈ ℝ

D) |A^-1 B|= ab

Q14. Which one of the following is NOT true about the function f(x)= 3x⁴ – 4x³?

A) (0, 0) is point of inflection of f.

B) 0 and 1 are critical numbers of f.

C) f is decreasing on (-∞, 1) and increasing on (1,∞).

D) f is concave upward on (0, 2/3) and concave downward on (-∞, 0) and on (2/3, ∞)

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Q15. Suppose the following are the premises of an argument.
He is healthy and he is not angry.
He is angry or he is plan fails.
His plan does not fail if he does not travel abroad.
Given that the premises are true, which one of the following can be a conclusion that makes the argument valid?

A) He travels abroad.

B) His plan fails and he is angry.

C) His plan does not fail.

D) His plans fails and he does not travel abroad.

Let’s analyze the premises one by one:

“He is healthy and he is not angry.”
“He is angry or his plan fails.”
“His plan does not fail if he does not travel abroad.”

Now, let’s consider the conclusions:

A) “He travels abroad.” We cannot directly infer this from the given premises.

B) “His plan fails and he is angry.” This conclusion does not directly follow from the premises. While it is true that his plan fails if he is not traveling abroad, there is no direct connection with his anger.

C) “His plan does not fail.” This conclusion is not necessarily valid because we do not have enough information to infer whether his plan fails or not.

D) “His plan fails and he does not travel abroad.” This conclusion is valid based on the third premise. It states that his plan does not fail if he does not travel abroad, meaning if his plan fails, he must have traveled abroad.

Therefore, the conclusion that makes the argument valid is D) “His plan fails and he does not travel abroad.”

Q16.The age distribution of students in a certain class is given below

What is the modal value of the distribution?

A) 17.38

B) 17.83

C) 18.37

D) 18.73

The modal value is the value that appears most frequently in a data set. In this case, the modal value would be the midpoint of the class interval with the highest frequency. To find the modal value, we first identify the class interval with the highest frequency.

Here is the calculation for the class intervals:

10 – 14: Frequency = 2 15 – 19: Frequency = 10 20 – 24: Frequency = 6 25 – 29: Frequency = 7

The class interval with the highest frequency is 15 – 19 with a frequency of 10. The midpoint of this interval is:

Midpoint=Lower limit+Upper limit2=15+192=342=17Midpoint=2Lower limit+Upper limit=215+19=234=17

Therefore, the modal value of the distribution is 17.

However, none of the provided options matches this result. It’s possible that there might have been a mistake in the options, or there could be a misunderstanding of the question. Please check the options or the question again for any discrepancies.

Q17.Let A={1, 2, 3, 4, 5, 6, 7}, B={7, 8, 9} and C={8, 9, 10}. If one of the number is deleted randomly from each of these sets, what is the probability that all the three deleted numbers are even or are multiples of 3?

A) 1/9

B) 2/21

C) 8/63

D) 4/5

To find the probability that all three deleted numbers are even or multiples of 3, we first find the number of elements in each set that are even or multiples of 3.

Set A: {1, 2, 3, 4, 5, 6, 7} Even numbers: 2, 4, 6 Multiples of 3: 3, 6

Set B: {7, 8, 9} Even numbers: 8 Multiples of 3: 9

Set C: {8, 9, 10} Even numbers: 8, 10 Multiples of 3: 9

Total elements that are even or multiples of 3: 8

Total possible choices for deleting one element from each set: 7 × 3 × 3 = 63

Therefore, the probability that all three deleted numbers are even or multiples of 3 is:

P = 8/63

Therefore, the correct answer is C) 8/63.

Q18. Let A be a 3�3 matrix and|A|= -2. Then what is the value of |adj(A)|?

A) 4

B) -2

C) -1/2

D) -8

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Q19. If the sum of the first three consecutive terms of an arithmetic progression {A_n}, with A_n > 0 for all n, is 9 and the sum of their squares is 35, then what is the sum s_n of the first n terms?

A) n² + 1

B) n² – 1

C) 2n² + 1

D) n²

Let’s denote the first term of the arithmetic progression as a, and the common difference as d.

The sum of the first three consecutive terms of an arithmetic progression can be expressed as:

Q20.20.Let {a_n} be a sequence with a₁ = a₁, a₂ = f(a₁) = f(a), a₃ = f(a₂ ) = f(f(a)), �, a_n+1 = f(a_n), where f is a continuous function.

A) 5ⁿ

B) 5

C) 5^n-1

D) 1

Q21. If f(x)=1/3 x³ + cx² + ax + 5 has a local minimum value at x = 1, then which one of the following is true about the possible value of a and c?

A) a = 3, c = -2

B) a = -2c – 1, c < -1

C) a = -2c – 1, c > -1

D) a = -2c – 1, c any real number.

Q22.A private college has 1000 students. 60% of these students are males. 45% of these students pay their payment by credit card including 175 females. What is the probability that the student is a male or a credit card user?

A) 0.675

B) 0.225

C) 0.775

D) 0.325

Let’s break down the problem step by step.

Number of male students: 60% of 1000 = 0.60 * 1000 = 600 students.
Number of female students: 1000 – 600 = 400 students.
Number of students paying by credit card: 45% of 1000 = 0.45 * 1000 = 450 students.

Now, we need to find the probability that the student is a male or a credit card user. Let’s denote M as the event of the student being male and C as the event of the student using a credit card.

Using the formula for the probability of the union of two events:

Q23.If z = (1 + √3i)(1 + i), then which one of the following is the polar representation of z?

A) z= 2√2 (cos(105⁰) + i sin(105⁰))

B) z = 2√2 (cos(15⁰) + isin(15⁰))

C) z = 4(cos(105⁰) + isin(105⁰))

D) z = 4(cos(75⁰) + i sin(75⁰))

Q24. Let f(x) = ln(x√x). Then what is f ‘(x) equal to?

A) 2x/3

B) √x/2

C) 3/2x

D) 2/x√x

Q25. What is the maximum possible area of a rectangle in square units with diagonal of length 16 units?

A) 128

B) 64

C) 48

D) 256

Q26. A cylindrical tank whose inner diameter is 2m contains 4π m³ oil. If the oil discharged from the tank at the rate of (2π/3) m³/min, then how long (in min) does it take for the tank to be empty?

A) 4/3

B) 4

C) 6

D) 12

Q27. The variance of 20 observation is 5. If each observation is multiplied by 2, then what is the variance of the resulting observations?

A) 5

B) 10

C) 40

D) 20

The variance of a data set is a measure of how much the observations differ from the mean. It is defined as the average of the squared differences from the mean.

If each observation is multiplied by a constant, the variance changes by the square of that constant.

Given that the variance of the original 20 observations is 5, if each observation is multiplied by 2, the variance of the resulting observations will be:

(22)×5=4×5=20

Therefore, the variance of the resulting observations is 20. Hence, the correct answer is D) 20.

Q28. 51. If f(x) = klnx + e^sinx and f ”(π) = -1, then what is the value of k?

A) π

B) 2π²

C) π²

D) 2π

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Q29. Consider the following assertion:
P+2ⁿ is an odd number for any prime p and any n ∈ N.
Which one of the following is correct about a prove or disprove of the assertion?

A) There is a counter example that disproves the assertion.

B) The assertion can be proved by direct method; because p is odd and 2ⁿ = 2(2^n-1) is even imply that p + 2ⁿ is odd since the sum of odd and even is odd.

C) The assertion can be proved by the indirect method: because if n ∉ N, then 2ⁿ ∉ N and hence p + 2ⁿ is not odd

D) The assertion can be disproved by the method of contradiction.

D) The assertion can be disproved by the method of contradiction: This option implies that the assertion can be shown to be false by assuming it is true and then deriving a contradiction. However, the given statement seems logically consistent, so proving it false may not be straightforward.

Based on the analysis, option B seems to provide the most reasonable explanation for the validity of the given assertion. However, it lacks a formal proof. Therefore, the most accurate answer might be that the assertion can be considered true but lacks a formal proof, so option B is the most appropriate.

Q30. Let P(n) be an open proposition on the set of natural numbers (N). Which one of the following is a correct application of the principle of mathematical induction?

A) If P(1) is true for n = 1; and if both P(n) and P(n+1) are true for certain n ∈ N, then P(n) is true for all n ∈ N.

B) If P(10) is true; and if p(n) is true implies that p(n+1) is true, then p(n) is true for all n ∈ N.

C) If p(10) is true; and assuming p(n) is true for any n > 10 if it follows that p(n+1) is true, then p(n) is true for all n ≥ 10

D) If p(1) is true; and p(n) ⇒ p(n+1) is true for any n ∈ N, then p(n) is true for all n?N.

The principle of mathematical induction is a method of proving mathematical statements for all natural numbers. It typically involves two steps: the base step and the inductive step.

Let’s analyze the options provided:

A) If P(1) is true for n = 1; and if both P(n) and P(n+1) are true for certain n ∈ N, then P(n) is true for all n ∈ N. This statement does not accurately represent the principle of mathematical induction. It seems to suggest that both P(n) and P(n+1) must be true for a certain value of n, which is not the standard form of the principle.

B) If P(10) is true; and if P(n) is true implies that P(n+1) is true, then P(n) is true for all n ∈ N. This option also does not accurately represent the principle of mathematical induction. It is more like the principle of deduction or logical inference.

C) If P(10) is true; and assuming P(n) is true for any n > 10 if it follows that P(n+1) is true, then P(n) is true for all n ≥ 10. This option is not a correct representation of the principle of mathematical induction. It introduces the assumption of truth for n > 10, which is not necessary for the principle of mathematical induction.

D) If P(1) is true; and P(n) ⇒ P(n+1) is true for any n ∈ N, then P(n) is true for all n ∈ N. This option accurately represents the principle of mathematical induction. It presents the base step and the inductive step, as required by the principle.

Therefore, the correct application of the principle of mathematical induction is represented in option D.

Q31. If the image of the line 2x – 3y = 7 under a translation is 2x – 3y = 0, which one of the following is a translation vector of the translation line?

A) U = (-2, 1)

B) U = (1, -2)

C) U = (-1, 2)

D) U = (2, -1)

Q32. Let P = (1, α, α) and Q=(α – 1, 1, 1) be two points in space and the distance between P and Q is 3. Then what is the value(s) of α?

A) α = -3, α = 1?3

B) &alpha = 1, α = -9

C) α = -1, α =9

D) α = 3, α = -1/3

Q33. What is the image of the circle x² + y² – 4x – 6y + 12 = 0 when it is reflected with respect to the line y = -x?

A) (x – 2)² + (y – 3)² = 1

B) (x + 3)² + (y + 2)² = 1

C) (x + 2)² + (y + 3)² = 1

D) (x – 3)² + (y – 2)² = 1

Q34. Let l be the line given by the vector equation (x, y)=(-2, 1)+λ(1,1), λ∈R, which one of the following is the equation of the image of l after being translated by the vector u=(2, -1) followed by a rotation through 45° about the origin?

A) y = -2√2 x

B) y = √2

C) y = √2 x

D) x = 0