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Question 281 NAT · 1.0 marks
The question of whether total sales in different marketplaces for product A and total sales in
different marketplaces for product B are independent across the different days is being checked.
Then what is the number of degrees of freedom for the test? (Note: Enter the answer **rounded to**
**two decimal places**. For example, if the answer is “1.235”, **then enter it as “1.24”**)
A published solution is not available for this question yet.
This Instructions is just for a reference & not for an evaluation.
Published solution
**1. Identify the entry:** This zero-mark item acknowledges the paper’s instructions rather than asking for a mathematical calculation.
**2. Record the domain convention:** The instructions define \(\mathbb R,\mathbb Q,\mathbb Z,\mathbb N\), explicitly include 0 among natural numbers, and explain how to submit numerical answers. A is the keyed acknowledgment.
**Answer: A — Instructions have been mentioned above.**
**Convention note:** The original option grammar is imperfect. This administrative entry is preserved as exported and carries zero marks.
A: **Correct:** It acknowledges that the instructions have been provided.
B: **Incorrect:** It is not the keyed acknowledgment; the instructions govern answer conventions.
**1. Find the intersection defining the vertex:** Equate \(x^2-4x+1=x^2-6x-3\), giving \(2x=-4\) and \(x=-2\). Substitution gives \(y=4+8+1=13\), so the vertex is \((-2,13)\).
**2. Use vertical-axis vertex form:** A quadratic with that vertex and an axis parallel to the y-axis has \(f(x)=a(x+2)^2+13\).
**3. Apply the y-intercept:** \[f(0)=2\Rightarrow4a+13=2\Rightarrow a=-\frac{11}{4}.\] Hence \(f(x)=-\frac{11}{4}(x+2)^2+13\).
**Answer: A — y=−(11/4)(x+2)²+13.**
A: **Correct:** It has vertex (−2,13) and f(0)=−11+13=2.
B: **Incorrect:** It places the vertex at (2,13), with the wrong x-coordinate.
C: **Incorrect:** It gives f(0)=11+13=24 rather than 2.
D: **Incorrect:** It places the vertex at height −13 rather than 13.
**1. Identify what an intersection means:** A point belongs to both road graphs only when its y-coordinate is the same for the same x. Therefore solve \(r_1(x)=r_2(x)\), not \(r_1(x)=-r_2(x)\).
**2. Distinguish degree from distinct intercepts:** \(r_1=0.02(x-5)(x-7)(x-9)(x-11)(x-13)\), while \(r_2=(x-5)r_1\). Both have the same five distinct x-intercepts 5,7,9,11,13; the repeated root of r₂ at 5 does not create a new intercept.
**3. Verify the actual intersection equation:** \[r_1-r_2=r_1[1-(x-5)]=(6-x)r_1=0.\] This gives the five common intercepts plus x=6: six graph-intersection points, even though each graph has only five x-intercepts.
**Answer: B — Solve r₁(x)=r₂(x).**
A: **Incorrect:** The degree of r₂ is 6, but its repeated root at 5 means it has only five distinct x-intercepts.
B: **Correct:** Equality of the two function values is exactly the intersection condition.
C: **Incorrect:** This generally compares one road with the reflection of the other, producing a different condition.
D: **Incorrect:** Each road has five distinct x-intercepts, not six.
**1. Test the same-level relation R:** Each course shares its level with itself, the condition is unchanged when the two courses are swapped, and courses sharing a level with a common course share a level with each other. Thus R is reflexive, symmetric and transitive.
**2. Test the prerequisite relation S:** A Level-I course is a prerequisite for a Level-II course, but the Level-II course is not a prerequisite for that Level-I course. This disproves symmetry.
**3. Apply the equivalence definition:** R is an equivalence relation; S is not, because equivalence requires symmetry as well as reflexivity and transitivity.
**Answer: A, D — A and D.**
**Convention note:** Whether S includes indirect prerequisites affects its transitivity, but not this answer: the explicit one-way prerequisite pair already disproves symmetry.
A: **Correct:** Sharing a level is reflexive, symmetric and transitive.
B: **Incorrect:** Prerequisite status is not symmetric, so S cannot be an equivalence relation.
C: **Incorrect:** If A and B share a level and B and C share a level, then A and C share that level; R is transitive.
D: **Correct:** Prerequisites are directed: a lower-level prerequisite does not have the higher-level course as its own prerequisite.
**1. Read vertex form correctly:** \(y-6=3(x-10)^2\) is \(y=3(x-10)^2+6\), with vertex \((10,6)\), not \((6,10)\).
**2. Convert each line to slope form:** \[-5x+4y-1=0\Rightarrow y=\frac54x+\frac14,\qquad\frac x4-\frac y5=1\Rightarrow y=\frac54x-5.\]
**3. Compare slopes and intercepts:** Both slopes are 5/4, but the intercepts differ. The lines are distinct and parallel, not perpendicular.
**Answer: A, D — A and D.**
A: **Correct:** It correctly reads the vertex as (10,6).
B: **Incorrect:** It reverses the horizontal and vertical vertex coordinates.
C: **Incorrect:** The equal nonzero slopes do not have product −1, so the lines are not perpendicular.
D: **Correct:** Equal slopes with unequal intercepts give distinct parallel lines.
**1. Recover the correct quadratic:** Vertex \((3,-2)\) gives \(f(x)=a(x-3)^2-2\). Its y-intercept 4 gives \(9a-2=4\), so \(a=2/3\). Expanding gives \(b=-6a=-4\) and \(c=4\).
**2. Use the sum-of-roots formula:** For \(ax^2+bx+c=0\), the sum is \(-b/a\). The correct sum is \(4/(2/3)=6\). With the mistaken coefficient b=5, the recorded equation has sum \(-5/(2/3)=-7.5\).
**3. Take the absolute difference:** \[|{-7.5}-6|=13.5.\] To two decimal places this is 13.50.
**Answer: 13.50.**
In an online degree program, it was found that in one term, 72 students took Mathematics, 85
students took Statistics, and 30 students took both Mathematics and Statistics. How many
students took only one of the two subjects?
Published solution
**1. Remove the overlap from each subject:** Mathematics-only has 72−30=42 students. Statistics-only has 85−30=55 students.
**2. Add the disjoint only-one groups:** \[|M\setminus S|+|S\setminus M|=42+55=97.\]
**3. Distinguish union from exactly one:** Adding 72+85 and subtracting the overlap once would count everyone taking at least one subject, including the 30 taking both. Exactly one instead removes that overlap from both subject totals.
**Answer: 97.**
**1. Respect the stated natural-number convention:** Here \(S=\{0,1,\ldots,10\}\). For \(R_1\), the condition \(y=3x\le10\) gives \[R_1=\{(0,0),(1,3),(2,6),(3,9)\}.\]
**2. List the second relation:** The condition \(y=x^2\le10\) gives \[R_2=\{(0,0),(1,1),(2,4),(3,9)\}.\]
**3. Remove common ordered pairs:** \(R_1\cap R_2=\{(0,0),(3,9)\}\). Hence \(R_1\setminus R_2=\{(1,3),(2,6)\}\), with cardinality 2.
**Answer: 2.**
**1. Find the intercept endpoints:** For \(y=-\frac23x+4\), the x-intercept is \(M=(6,0)\) and the y-intercept is \(N=(0,4)\).
**2. Apply internal division in the stated order:** With \(MP:PN=2:1\), \[P=\frac{1M+2N}{3}=\left(2,\frac83\right).\] The larger weight is on N because P is closer to N.
**3. Calculate the slope from the origin:** \[m_{OP}=\frac{8/3-0}{2-0}=\frac43\approx1.33.\]
**Answer: 1.33.**
**1. Write the intercept-form line:** Since \(l_1\) meets the axes at (4,0) and (0,2), \[\frac x4+\frac y2=1\quad\Rightarrow\quad x+2y=4.\]
**2. Find its intersection with x+y=5:** Subtracting the latter equation from the former gives \(y=-1\), then \(x=6\). The intersection is \((6,-1)\).
**3. Use the direction of l₂:** A line parallel to the x-axis through that point is \(y=-1\). Its y-intercept is therefore −1.
**Answer: -1.**
**1. Read the vertex of the quadratic:** \(q(x)=-(x-7)^2-9\) has vertex \((7,-9)\). The polynomial graph must pass through that point.
**2. Substitute x=7 into the polynomial:** \[p(7)=a(7-4)(7-6)(7-8)(7-10)=a(3)(1)(-1)(-3)=9a.\]
**3. Match the required height:** \(9a=-9\), hence \(a=-1\). This uses the vertex height, not a root of q.
**Answer: -1.**
**1. Compute predictions:** For \(\hat y=2x+3\) at x=1,2,3,4,5, predictions are 5,7,9,11,13.
**2. Use the known residual squares:** The first four observed values are 6,6,10,11. Their squared errors total \[(6-5)^2+(6-7)^2+(10-9)^2+(11-11)^2=3.\]
**3. Recover the fifth response:** Total SSE=10 leaves \((y_5-13)^2=7\). Since actual sales exceeded the prediction, choose \(y_5=13+\sqrt7\approx15.64575\), not the negative-residual solution.
**4. Apply the requested rounding:** To the nearest integer, the fifth-day sales are 16.
**Answer: 16.**
**Convention note:** The exact response recovered from the stated SSE is 13+√7. Because drinks are ordinarily counted in integers, the problem’s data use an approximation: the rounded value 16 would itself give SSE=12 if substituted literally. The requested nearest-integer answer is nevertheless 16.
