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MATHS Assertion (A) & Reason (B) MCQ

Introduction to Three Dimensional Geometry · Rajasthan Board (RBSE) STD 11 Science (Rajasthan - English Medium). 5 questions.

Questions

Assertion (A) & Reason (B) MCQ

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5 questions · self-marked practice — reveal the answer and mark yourself.

Question 11 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The points A(3, -1, 2), B(1, 2, -4), C(-1, 1, 2) and D(1, -2, 8) are the vertices of a parallelogram.
Reason: Coordinates of mid - point of a line joining the points A(x1, y1, z1) and B(x1, y2, z2) is $\Big(\frac{\text{x}_{1}+\text{x}_{2}}{2},\frac{\text{y}_1+\text{y}_2}{2},\frac{\text{z}_{1}+\text{z}_{2}}{2}\Big).$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

Mid - point of $\text{AC}=\Big(\frac{3-1}{2},\frac{-1+1}{2},\frac{2+2}{2}\Big)=(1,0,2)$

Mid-point of $\text{BD}=\Big(\frac{1+1}{2},\frac{2-2}{2},\frac{-4+8}{2}\Big)=(1,0,2)$

$\because$ Mid - points of AC and BD coincides.

$\therefore$ ABCD is a parallelogram.

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Question 21 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The points A(1, -1, 3), B(2, -4, 5) and C(5, -13, 11) are collinear.
Reason: If AB + BC = AC, then A, B, C are collinear.
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

$\mid\text{AB}\mid=\sqrt{(1)^{2}+(-3)^2+(2)^{2}}=\sqrt{1+9+4}=\sqrt{14}$

$\mid\text{BC}\mid=\sqrt{(3)^{2}+(-9)^2+(6)^{2}}=\sqrt{9+81+36}=3\sqrt{14}$

$\mid\text{AC}\mid=\sqrt{(4)^{2}+(-12)^2+(8)^{2}}=\sqrt{16+144+64}=4\sqrt{14}$

$\because\text{AB}+\text{BC}=4\sqrt{14}=\text{AC}$

$\therefore$ Points A, B and C are collinear.

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Question 31 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: Coordinates of centroid of a triangle formed by the vertices A(3, 2, 0), B(5, 3, 2) and C(0, 2, 4) is $\Big(\frac{8}{3},\frac{8}{3},\frac{8}{3}\Big).$
Reason: Coordinates of centroid of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2) and C(x3, y3, z3) is $\Big(\frac{\text{x}_{1}+\text{x}_{2}+\text{x}_{3}}{3},\frac{\text{y}_{1}+\text{y}_{2}+\text{y}_{3}}{3},\frac{\text{z}_{1}+\text{z}_{2}+\text{z}_{3}}{3}\Big).$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. Assertion is wrong statement but Reason is correct statement.

Solution:

Coordinates of centroid ofa triangle with vertices A(3, 2, 0), B(5, 3, 2) and C(0, 2, 4) is

$\Big(\frac{3+5+0}{3},\frac{2+3+2}{3},\frac{0+2+4}{3}\Big)=\Big(\frac{8}{3},\frac{7}{3},2\Big)$

$\therefore$ Assertion is wrong but Reason is correct.

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Question 41 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The foot of perpendicular drawn from the point A(1, 2, 8) on the xy - plane is (1, 2, 0).
Reason: Equation of xy - plane is y = 0.
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. Assertion is correct statement but Reason is wrong statement.

Solution:

We know that in xy - plane, z - coordinate is 0.

So, coordinate of foot of perpendicular drawn from point A(1, 2, 8) on xy - plane is (1, 2, 0).

Equation of xy - plane is z = 0

$\therefore$ Reason is wrong.

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Question 51 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The distance between the points $ (1+\sqrt{11}, 0, 0) $ and (1, -2, 3) is $2\sqrt{6}$ units.
Reason: Distance between any two points A(x1, y1, z1) and B(x2, y2, z2)
$\mid\text{AB}\mid=\sqrt{(\text{x}_{2}+\text{x}_{1})^{2}+(\text{y}_{2}+\text{y}_{1})^{2}(\text{z}_{2}+\text{z}_{1})^{2}}.$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. Assertion is correct statement but Reason is wrong statement.

Solution:

Let $\text{A}=(1+\sqrt{11}, 0, 0) $ and B = (1, -2, 3)

$\therefore\ \mid\text{AB}\mid=\sqrt{(1-1-\sqrt{11})^{2}+(-2-0)^{2}(3-0)^{2}}$

$=\sqrt{11+4+9}=\sqrt{24}=2\sqrt{6}$ units

$\therefore$ Assertion is correct but Reason is wrong.

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