12th Mathematics - Book Back Answers - Chapter 6 - English Medium Guides ~ Kalvi Tips

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Tamil Nadu Board 12th Standard Mathematics - Chapter 6: Book Back Answers and Solutions

This post covers the book back answers and solutions for Chapter 6 from the Tamil Nadu State Board 12th Standard Mathematics textbook. These detailed answers have been carefully prepared by our expert teachers at KalviTips.com.

We have explained each answer in a simple, easy-to-understand format, highlighting important points step by step under the relevant subtopics. Students are advised to read and memorize these subtopics thoroughly. Once you understand the main concepts, you’ll be able to connect other related points with real-life examples and confidently present them in your tests and exams.

By going through this material, you’ll gain a strong understanding of Chapter 6 along with the corresponding book back questions and answers (PDF format).

Question Types Covered:

1 Mark Questions: Choose the correct answer, Fill in the blanks, Identify the correct statement, Match the following
2 Mark Questions: Answer briefly
3, 4, and 5 Mark Questions: Answer in detail

All answers are presented in a clear and student-friendly manner, focusing on key points to help you score full marks.

All the best, Class 12 students! Prepare well and aim for top scores. Thank you!

I. Multiple Choice Questions

1. If $\vec{a}$ and $\vec{b}$ are parallel vectors, then [ $\vec{a}$ , $\vec{c}$ , $\vec{b}$ ] is equal to
(1) 2
(2) −1
(3) 1
(4) 0
Answer Key:
(4) 0

2. If $\vec{α}$ vector lies in the plane of $\vec{β}$ and then
(1) [

\vec{α}

\vec{γ}

\vec{β}

] =1
(2) [

\vec{α}

\vec{γ}

\vec{β}

] = −1
(3) [

\vec{α}

\vec{γ}

\vec{β}

] = 0

(4) [

\vec{α}

\vec{γ}

\vec{β}

] = 2
Answer Key:
(3) [

\vec{α}

\vec{γ}

\vec{β}

] = 0

3. If $\vec{a}$ . $\vec{b}$ = $\vec{b}$ . $\vec{c}$ = $\vec{c}$ . $\vec{a}$ = $0,$ then the value of [ $\vec{a}$ , $\vec{b}$ , $\vec{c}$ ]
(1) | $\vec{a}$ | | $\vec{b}$ | | $\vec{c}$ |
(2)

\frac{1}{3}

| $\vec{a}$ | | $\vec{b}$ | | $\vec{c}$ |

(3) 1
(4) −1
Answer Key:
(1) | $\vec{a}$ | | $\vec{b}$ | | $\vec{c}$ |

4. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three unit vectors such that $\vec{a}$ is perpendicular to $\vec{b}$ , and is parallel to $\vec{c}$ then $\vec{a}$ x( $\vec{b}$ x $\vec{c}$ ) is equal to
(1)

\vec{a}

(2)

\vec{b}

(3)

\vec{c}

(4)

\vec{0}

Answer Key:
(2)

\vec{b}

5. If [ $\vec{a}$ , $\vec{b}$ , $\vec{c}$ ] =1, then the value of $\frac{\vec{a} . (\vec{b} \times \vec{c)}}{(\vec{c} \times \vec{a}) . \vec{b}} + \frac{\vec{b} . (\vec{c} \times \vec{a})}{(\vec{a} \times \vec{b}) . \vec{c}} + \frac{\vec{c} . (\vec{a} \times \vec{b})}{(\vec{c} \times \vec{b}) . \vec{a}}$ is
(1) 1
(2) −1
(3) 2
(4) 3
Answer Key:
(1) 1

6. The volume of the parallelepiped with its edges represented by the vectors
$\hat{i}$ + $\hat{j}$ , $\hat{i}$ +2 $\hat{j}$ , $\hat{j}$ $+$ $𝞹$ $\hat{k}$
(1)

\frac{π}{2}

(2)

\frac{π}{3}

(3)

π

(4)

\frac{π}{4}

Answer Key:
(3)

π

7. If $\vec{a}$ and $\vec{b}$ are unit vectors such that $[$ $\vec{a}$ , $\vec{b}$ , $\vec{a} \times \vec{b}$ $] =$ $\frac{1}{4 ,}$ then the angle between $\vec{a}$ and $\vec{b}$ is
(1)

\frac{π}{6}

(2)

\frac{π}{4}

(3)

\frac{π}{3}

(4)

\frac{π}{2}

Answer Key:
(1)

\frac{π}{6}

8. If $\overset{\to=}{a}$ $\hat{i}$ + $\hat{j}$ + $\hat{k}$ $,$ $\vec{b}$ = $\hat{i}$ + $\hat{j}$ $,$ $\vec{c}$ = $\hat{i}$ and $(\vec{a} \times \vec{b})$ $\times$ $\vec{c}$ = λ $\vec{a}$ +𝛍 $\vec{b}$ , then the value of λ+μ is
(1) 0
(2) 1
(3) 6
(4) 3
Answer Key:
(1) 0

9. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are non-coplanar, non-zero vectors such that [ $\vec{a}$ , $\vec{b}$ , $\vec{c}$ ] =3, then {[ $\vec{a} \times \vec{b}$ , $\vec{b} \times$ $\vec{c}$ , $\vec{c} \times \vec{a}$ ]}² is equal to
(1) 81
(2) 9
(3) 27
(4)18
Answer Key:
(1) 81

10. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three non-coplanar unit vectors such that $\vec{a} \times$ ( $\vec{b} \times$ $\vec{c}$ )= $\frac{\vec{b} + \vec{c}}{\sqrt{2}}$ , then the angle between $\vec{a}$ and $\vec{b}$ is
(1)

\frac{π}{2}

(2)

\frac{3 π}{4}

(3)

\frac{π}{4}

(4)

π

Answer Key:
(2)

\frac{3 π}{4}

11. If the volume of the parallelepiped with $\vec{a} \times \vec{b}$ , $\vec{b} \times$ $\vec{c}$ , $\vec{c} \times \vec{a}$ as coterminous edges is 8 cubic units, then the volume of the parallelepiped with ( $\vec{a} \times \vec{b}$ )×( $\vec{b} \times$ $\vec{c}$ ),( $\vec{b} \times$ $\vec{c}$ )×( $\vec{c} \times \vec{a}$ ) and ( $\vec{c} \times \vec{a}$ )×( $\vec{a} \times \vec{b}$ ) as coterminous edges is,
(1) 8 cubic units
(2) 512 cubic units
(3) 64 cubic units
(4) 24 cubic units
Answer Key:
(3) 64 cubic units

12. Consider the vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ , $\vec{d}$ such that ( $\vec{a} \times \vec{b}$ )×( $\vec{c}$ × $\vec{d}$ ) = $\vec{0}$ . Let P₁ and P₂be the planes determined by the pairs of vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ , $\vec{d}$ respectively. Then the angle P₁between P₂ is
(1) 0°
(2) 45°
(3) 60°
(4) 90°
Answer Key:
(1) 0°

13. If $\vec{a} \times$ ( $\vec{b} \times$ $\vec{c}$ ) = ( $\vec{a} \times \vec{b}$ ) $\times$ $\vec{c}$ , where $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are any three vectors such that $\vec{b} .$ $\overset{\to\neq}{c}$ 0
and $\vec{a} .$ $\vec{b}$ $\neq$ 0, then $\vec{a}$ and $\vec{c}$ are
(1) perpendicular
(2) parallel
(3) inclined at an angle

\frac{π}{3}

(4) inclined at an angle

\frac{π}{6}

Answer Key:
(2) parallel

14. If $\vec{a}$ = 2 $\hat{i}$ + 3 $\hat{j}$ − $\hat{k}$ , $\vec{b}$ = $\hat{i}$ + 2 $\hat{j}$ − 5 $\hat{k}$ , $\vec{c}$ = 3 $\hat{i}$ + 5 $\hat{j}$ − $\hat{k}$ , then a vector perpendicular to $\vec{a}$ and lies in the plane containing $\vec{b}$ and $\vec{c}$ is
(1) −17 $\hat{i}$ + 21 $\hat{j}$ − 97 $\hat{k}$
(2) 17 $\hat{i}$ + 21 $\hat{j}$ −123 $\hat{k}$
(3) −17 $\hat{i}$ − 21 $\hat{j}$ + 97 $\hat{k}$
(4) −17 $\hat{i}$ − 21 $\hat{j}$ − 97 $\hat{k}$
Answer Key:
(4) −17 $\hat{i}$ − 21 $\hat{j}$ − 97 $\hat{k}$

15. The angle between the lines $\frac{x - 2}{3} = \frac{y + 1}{- 2, z=2}$ and $\frac{x - 1}{1} = \frac{2 y + 3}{3} = \frac{z + 5}{2}$ is
(1)

\frac{π}{6}

(2)

\frac{π}{4}

(3)

\frac{π}{3}

(4)

\frac{π}{2}

Answer Key:
(4)

\frac{π}{2}

16. If the line $\frac{x - 2}{3} = \frac{y - 1}{- 5} - \frac{z + 2}{2}$ lies in the plane x + 3y − $α$ z + $β$ = 0, then ( $α$ , $β$ ) is
(1) (−5,5)
(2) (−6,7)
(3) (5,−5)
(4) (6,−7)
Answer Key:
(2) (−6,7)

17. The angle between the line $\vec{r}$ $=$ ( $\hat{i}$ + 2 $\hat{j}$ − 3 $\hat{k}$ )+t(2 $\hat{i}$ + $\hat{j}$ −2 $\hat{k}$ ) and the plane $\vec{r}$ . ( $\hat{i}$ + $\hat{j}$ ) + 4 = 0 is
(1) 0°
(2) 30°
(3) 45°
(4) 90°
Answer Key:
(3) 45°

18. The coordinates of the point where the line $\vec{r}$ $=$ ( $\hat{6i}$ − $\hat{j}$ − 3 $\hat{k}$ )+t(− $\hat{i}$ +4 $\hat{k}$ ) meets the plane
$\vec{r}$ .( $\hat{i}$ + $\hat{j}$ − $\hat{k}$ ) = 3 are
(1) (2,1,0)
(2) (7,−1,−7)
(3) (1,2,−6)
(4) (5,−1,1)
Answer Key:
(4) (5,−1,1)

19. Distance from the origin to the plane 3x − 6y + 2z + 7 = 0 is
(1) 0
(2) 1
(3) 2
(4) 3
Answer Key:
(2) 1

20. The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is
(1)

\frac{\sqrt{7}}{2 \sqrt{2}}

(2)

\frac{7}{2}

(3)

\frac{\sqrt{7}}{2}

(4)

\frac{7}{2 \sqrt{2}}

Answer Key:

(1)

\frac{\sqrt{7}}{2 \sqrt{2}}

21. If the direction cosines of a line are $\frac{1}{c}$ , $\frac{1}{c}$ , $\frac{1}{c}$ , then
(1) c = ±3
(2) c = ±

\sqrt{3}

(3) c > 0
(4) 0 < c <1
Answer Key:
(2) c = ±

\sqrt{3}

22. The vector equation $\vec{r}$ = ( $\hat{i}$ − $\hat{2j}$ − $\hat{k}$ ) + t(6 $\hat{j}$ − $\hat{k}$ ) represents a straight line passing through the
points
(1) (0,6,−1) and (1,−2,−1)
(2) (0,6,−1) and (−1,−4,−2)
(3) (1,−2,−1) and (1,4,−2)
(4) (1,−2,−1) and (0,−6,1)
Answer Key:
(3) (1,−2,−1) and (1,4,−2)

23. If the distance of the point (1,1,1) from the origin is half of its distance from the plane
x + y + z + k = 0 , then the values of k are
(1) ±3
(2) ±6
(3) −3,9
(4) 3,−9
Answer Key:
(4) 3,−9

24. If the planes $\vec{r}$ .(2 $\hat{i}$ − $\hat{λj}$ − $\hat{k}$ ) =3 and $\vec{r}$ .(4 $\hat{i}$ + $\hat{j}$ − $\hat{μk}$ ) = 5 are parallel, then the value of $λ$ and μ are
(1)

\frac{1}{2, -2}

(2)

\frac{-1}{2, 2}

(3)

\frac{-1}{2, -2}

(4)

\frac{1}{2, 2}

Answer Key:
(3)

\frac{-1}{2, -2}

25. If the length of the perpendicular from the origin to the plane 2x + 3y + $λ$ z =1, $λ$ > 0 is
$\frac{1}{5}$ , then the value of $λ$ is
(1) 2

\sqrt{3}

(2) 3

\sqrt{2}

(3) 0
(4) 1
Answer Key:
(1) 2

\sqrt{3}

II.Short Answer Questions.

12th Mathematics

III. Long Answer Questions.

12th Mathematics

IV. Exercise.

12th Mathematics