Question

1. if the line passes through the points a (1, 5) & b (4, 2), then what is the direction vector of?

a. 3i+3j b. 3i-3j c. -3i-3j d. 4i+2j

1. the image of the line l:y=3x+1 after rotation of \\(\theta = \frac{\pi}{2}\\) about the point (2,2) is:

a. 3y+x=3 b. 3y-x=7 c. 3y-2x=4 d. 3y+x=1

1. what is the image of p= (1, 1) after reflected in the line y=2x?

a. \\(\left(-\frac{3}{5}, \frac{9}{5}\
ight)\\) b. \\(\left(-\frac{3}{5}, \frac{1}{5}\
ight)\\) c. \\(\left(\frac{3}{5}, \frac{9}{5}\
ight)\\) d. (7,5)

1. equation of the image of the circle \\(x^2 + y^2 - 4x - 2y + 1 = 0\\) after reflected in the line y=x-1 is

a. \\((x-2)^2 + (y-1)^2 = 4\\) b. \\(x^2 + y^2 - 10y - 21 = 0\\) c. \\((x-2)^2 - (y-1)^2 = 1\\) d. \\(x^2 + y^2 - 8x - 8y + 9 = 0\\)

1. the equation of the tangent line to the circle \\(x^2 + y^2 + 10x + 2y + 13 = 0\\) at the point (-3, 2) is:

a. 2x-3y=0 b. 3x-2y=0 c. 2x-3y=13 d. 3x+2y=0

1. the equation of the circle whose end point of a diameter p(2, 3)&q(6, 7) is

a. \\(x^2 + y^2 - 3x - 10y - 15 = 0\\) b. \\(x^2 - y^2 + 3x + 10y + 15 = 0\\) c. \\(x^2 + y^2 - 8x - 4y + 15 = 0\\) d. \\(x^2 - y^2 + 8x + 4y - 15 = 0\\)

1. which one of the following the area of the parallelogram formed by the vectors u= 5i+j and v=2i+3j?

a. 7square units b. 5square units c. 6 square units d. 3 square units

1. the cosine of the angle formed by the vectors u=2i+3j & v=-2i+5j is

a. \\(\frac{8}{13}\\) b. \\(\frac{8}{\sqrt{13}}\\) c. \\(\frac{5}{13}\\) d. \\(\frac{5}{\sqrt{13}}\\)

1. which one of the following data is a qualitative data?

a. number of students b. volume of water in a barrel c. shoe size d. gender

1. if the data 7,9,6,7,11,8, 4,11,7 shows the number of car accident for 9 days. then the median of this

a. 7 b. 9 c. 4 d. 6

1. consider the following frequency distribution table

\

$$\begin{tabular}{|l|l|l|l|l|l|} \\hline value & 5 & 3 & 7 & 6 & 8 \\\\ \\hline frequency & 3 & 4 & 5 & 3 & 5 \\\\ \\hline \\end{tabular}$$

what is the sixth decile?
a. 7 b. 6 c. 7.5 d.5

1. if \\(\vec{a}=(4,3,2)\\) & if \\(\vec{b}=(1,2,-3)\\) are two vectors in space ,then which one following is not correct?

a. \\(\vec{a} + \vec{b} = (5,5,-1)\\) b. \\(\vec{a} + 2\vec{b} = (6,7,-4)\\) c. \\(\vec{b} - \vec{a} = (-3,-2,-3)\\) d. \\(\vec{a} + 2\vec{b} = (2,1,8)\\)

1. one of the following is the \\(7^{\text{th}}\\) term in the expansion \\((3a - 2b)^8\\)?

a. \\(1792a^2b^6\\) b. \\(28 a^6b^2\\) c. \\(16,128 a^2b^6\\) d. \\(252a^6b^2\\)

1. which one of the following is the coefficient of the forth term in the expansion \\((2x + y)^6\\)?

a. 20 b. 160 c. 40 d. 120

1. a data score of 32 students is listed as follows.

52 48 65 70 50 42 40 61
60 45 32 66 76 58 75 63
50 75 44 46 78 74 61 52
76 35 55 58 67 60 55 45
if the data is to be grouped in 6 classes, what has to be the class interval (w)?
a.8 b.6 c. 9 d. 7

Explanation:

Identify the given vectors

Using the 3D Vectors knowledge point

$$ \vec{a} = (4, 3, 2),\quad \vec{b} = (1, 2, -3) $$

Verify vector addition

Using the Vector Operations knowledge point

$$ \vec{a} + \vec{b} = (4+1, 3+2, 2+(-3)) = (5, 5, -1) $$

Verify scalar multiplication and addition

Using the Vector Operations knowledge point

$$
LATEXBLOCK0
$$

Verify vector subtraction

Using the Vector Operations knowledge point

$$ \vec{b} - \vec{a} = (1-4, 2-3, -3-2) = (-3, -1, -5) $$

Identify the incorrect option

Comparing the calculated values with the given options:

• Option A: \(\vec{a} + \vec{b} = (5, 5, -1)\) is correct.
• Option B: \(\vec{a} + 2\vec{b} = (6, 7, -4)\) is correct.
• Option C: \(\vec{b} - \vec{a} = (-3, -1, -5)\) is given as \((-3, 2, -3)\), which is incorrect.
• Option D: \(\vec{a} + 2\vec{b} = (2, 1, 8)\) is also incorrect, but looking closely at the image, Option D is \(\vec{a} - 2\vec{b}\). Let's calculate \(\vec{a} - 2\vec{b}\):

$$ \vec{a} - 2\vec{b} = (4-2, 3-4, 2-(-6)) = (2, -1, 8) $$

The printed option D says \(\vec{a} - 2\vec{b} = (2, 1, 8)\) or similar, but option C is clearly mathematically incorrect since \(\vec{b} - \vec{a} = (-3, -1, -5)\) whereas the option states \((-3, 2, -3)\). Let's re-verify option C:

$$ \vec{b} - \vec{a} = (1-4, 2-3, -3-2) = (-3, -1, -5) eq (-3, 2, -3) $$

Thus, option C is definitely not correct.

Answer:

• (A) \(\vec{a} + \vec{b} = (5, 5, -1)\)
• (B) \(\vec{a} + 2\vec{b} = (6, 7, -4)\)
• (C) \(\vec{b} - \vec{a} = (-3, 2, -3)\) (Correct answer)
• (D) \(\vec{a} - 2\vec{b} = (2, 1, 8)\)