Curious to know how well you have learned Vector Algebra? The NCERT Solutions for Class 12 maths chapter 10 miscellaneous exercise are ideal to test your knowledge with mixed and challenging questions. These 19 questions cover all that you have learned till now in this chapter, and help you revise and increase your confidence level before exams. These solutions are prepared by Careers360 experts, these solutions are according to the latest CBSE syllabus.
Attempt to solve the Class 12 Maths Chapter 10 Miscellaneous Exercise of NCERT yourself to gauge how well you have understood the concepts. These questions are a combination of all important topics and assist you in analyzing what needs to be revised. The NCERT solution for Class 12 Maths are aimed at building your foundation through step-by-step solutions.
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Question 1: Write down a unit vector in XY-plane, making an angle of
Answer:
As we know
a unit vector in XY-Plane making an angle
Hence for
Answer- the unit vector in XY-plane, making an angle of
Question 2: Find the scalar components and magnitude of the vector joining the points
Answer:
Given in the question
And we need to finrd the scalar components and magnitude of the vector joining the points P and Q
Magnitiude of vector PQ
Scalar components are
Answer:
As the girl walks 4km towards west
Position vector =
Now as she moves 3km in direction 30 degree east of north.
hence final position vector is;
Question 4: If
Answer:
No, if
the condition
also, in a triangle,
Since, the condition
Question 5: Find the value of x for which
Answer:
Given in the question,
a unit vector,
We need to find the value of x
The value of x is
Question 6: Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
Answer:
Given two vectors
Resultant of
Now, a unit vector in the direction of
Now, a unit vector of magnitude in direction of
Hence the required vector is
Question 7: If
Answer:
Given in the question,
Now,
let vector
Now, a unit vector in direction of
Now,
A unit vector parallel to
OR
Answer:
Given in the question,
points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)
now let's calculate the magnitude of the vectors
As we see that AB = BC + AC, we conclude that three points are colinear.
we can also see from here,
Point B divides AC in the ratio 2 : 3.
Answer:
Given, two vectors
the point R which divides line segment PQ in ratio 1:2 is given by
Hence position vector of R is
Now, Position vector of the midpoint of RQ
which is the position vector of Point P . Hence, P is the mid-point of RQ
Question 10: The two adjacent sides of a parallelogram are
Answer:
Given, two adjacent sides of the parallelogram
The diagonal will be the resultant of these two vectors. so
resultant R:
Now unit vector in direction of R
Hence unit vector along the diagonal of the parallelogram
Now,
Area of parallelogram
Hence the area of the parallelogram is
Question 11: Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
Answer:
Let a vector
let direction cosines of this vector be
Now
Hence direction cosines are:
Question 12: Let
Answer:
Given,
Let
now, since it is given that d is perpendicular to
And
here we got 2 equation and 3 variable. one more equation will come from the condition:
so now we have three equations and three variables,
On solving this three equation we get,
Hence required vector :
Question 13: The scalar product of the vector
Answer:
Let, the sum of vectors
unit vector along
Now, the scalar product of this with
Question 14: If
Answer:
Given
Now, let vector
Now,
Now, Since,
Hence vector
Question 15: Prove that
Answer:
Given in the question,
= RHS
LHS ie equal to RHS
Hence proved.
Question 16: Choose the correct answer If
Answer:
Given in the question
this will satisfy when
Hence option B is the correct answer.
Question 18: The value of
(A) 0
(B) –1
(C) 1
(D) 3
Answer:
To find the value of
Hence option C is correct.
Question 19: Choose the correct. If
is equal to
(A) 0
(B)
( C )
( D )
Answer:
Given in the question
To find the value of
Hence, option D is correct.
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Frequently Asked Questions (FAQs)
Vectors is an important chapter of Class 12 Maths NCERT syllabus. The concepts studied in this chapter are used not only on mathematics but also to solve problems in Class 11 and 12 physics problems also. Also Vectors is an important chapter for CBSE Board exam. Solving the NCERT questions and referring to the NCERT exemplar will be beneficial for board exam preparation.
The topics covered are vectors basic concepts, algebra of vectors, direction cosine and ratios, dot product and cross product of vectors.
On Question asked by student community
Hello Pruthvi,
Taking a drop year to reappear for the Karnataka Common Entrance Test (KCET) is a well-defined process. As a repeater, you are fully eligible to take the exam again to improve your score and secure a better rank for admissions.
The main procedure involves submitting a new application for the KCET through the official Karnataka Examinations Authority (KEA) website when registrations open for the next academic session. You must pay the required application fee and complete all formalities just like any other candidate. A significant advantage for you is that you do not need to retake your 12th board exams. Your previously secured board marks in the qualifying subjects will be used again. Your new KCET rank will be calculated by combining these existing board marks with your new score from the KCET exam. Therefore, your entire focus during this year should be on preparing thoroughly for the KCET to achieve a higher score.
For more details about the KCET Exam preparation,
CLICK HERE.
I hope this answer helps you. If you have more queries, feel free to share your questions with us, and we will be happy to assist you.
Thank you, and I wish you all the best in your bright future.
Yes, you can switch from Science in Karnataka State Board to Commerce in CBSE for 12th. You will need a Transfer Certificate from your current school and meet the CBSE school’s admission requirements. Since you haven’t studied Commerce subjects like Accountancy, Economics, and Business Studies, you may need to catch up before or during 12th. Not all CBSE schools accept direct admission to 12th from another board, so some may ask you to join Class 11 first. Make sure to check the school’s rules and plan your subject preparation.
Hello
For the 12th CBSE Hindi Medium board exam, important questions usually come from core chapters like “Madhushala”, “Jhansi ki Rani”, and “Bharat ki Khoj”.
Questions often include essay writing, letter writing, and comprehension passages. Grammar topics like Tenses, Voice Change, and Direct-Indirect Speech are frequently asked.
Students should practice poetry questions on themes and meanings. Important questions also cover summary writing and translation from Hindi to English or vice versa.
Previous years’ question papers help identify commonly asked questions.
Focus on writing practice to improve handwriting and presentation. Time management during exams is key to answering all questions effectively.
Hello,
If you want to improve the Class 12 PCM results, you can appear in the improvement exam. This exam will help you to retake one or more subjects to achieve a better score. You should check the official website for details and the deadline of this exam.
I hope it will clear your query!!
For the 2025-2026 academic session, the CBSE plans to conduct board exams from 17 February 2026 to 20 May 2026.
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