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9 months ago

Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eleven

If $l_{1}, m_{1}, n_{1}$ ; $l_{2}, m_{2}, n_{2}$ , and $l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to $l_{1} + l_{2} + l_{3}$ , $m_{1} + m_{2} + m_{3}$ , $n_{1} + n_{2} + n_{3}$ and makes equal angles with them.

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Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} L e t \vec{a}, \vec{b}, \vec{c} a n d \vec{d} a r e s u c h t h a t \\ \vec{a} = l_{1} \hat{i} + m_{1} \hat{j} + n_{1} \hat{k} \\ \vec{b} = l_{2} \hat{i} + m_{2} \hat{j} + n_{2} \hat{k} \\ \vec{c} = l_{3} \hat{i} + m_{3} \hat{j} + n_{3} \hat{k} \\ a n d \vec{d} = (l_{1} + l_{2} + l_{3}) \hat{i} + (m_{1} + m_{2} + m_{3}) \hat{j} + (n_{1} + n_{2} + n_{3}) \hat{k} \\ Since t h e g i v e n d' cosines a r e m u t u a l l y p e r p e n d i c u l a r t h e n \\ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0 \\ l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} = 0 \\ l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3} = 0 \\ L e t α, β a n d γ b e t h e a n g l e s b e t w e e n \vec{a} a n d \vec{d}, \vec{b} a n d \vec{d}, \vec{c} a n d \vec{d} r e s p e c t i v e l y . \\ ∴ c o s α = l_{1} (l_{1} + l_{2} + l_{3}) + m_{1} (m_{1} + m_{2} + m_{3}) + n_{1} (n_{1} + n_{2} + n_{3}) \\ = l_{1}^{2} + l_{1} l_{2} + l_{1} l_{3} + m_{1}^{2} + m_{1} m_{2} + m_{1} m_{3} + n_{1}^{2} + n_{1} n_{2} + n_{1} n_{3} \\ = (l_{1}^{2} + m_{1}^{2} + n_{1}^{2}) + (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}) + (l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3}) \\ = 1 + 0 + 0 = 1 \\ ∴ c o s β = l_{2} (l_{1} + l_{2} + l_{3}) + m_{2} (m_{1} + m_{2} + m_{3}) + n_{2} (n_{1} + n_{2} + n_{3}) \\ = l_{1} l_{2} + l_{2}^{2} + l_{2} l_{3} + m_{1} m_{2} + m_{2}^{2} + m_{2} m_{3} + n_{1} n_{2} + n_{2}^{2} + n_{2} n_{3} \\ = (l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) + (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}) + (l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3}) \\ = 1 + 0 + 0 = 1 \\ S i m i l a r l y, \\ ∴ c o s γ = l_{3} (l_{1} + l_{2} + l_{3}) + m_{3} (m_{1} + m_{2} + m_{3}) + n_{3} (n_{1} + n_{2} + n_{3}) \\ = l_{1} l_{3} + l_{2} l_{3} + l_{3}^{2} + m_{1} m_{3} + m_{2} m_{3} + m_{3}^{2} + n_{1} n_{3} + n_{2} n_{3} + n_{3}^{2} \\ = (l_{3}^{2} + m_{3}^{2} + n_{3}^{2}) + (l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3}) + (l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3}) \\ = 1 + 0 + 0 = 1 \\ ∴ c o s α = c o s β = c o s γ = 1 \Rightarrow α = β = γ w h i c h i s t h e r e q u i r e d r e s u l t . \end{array}$

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9 months ago

ILAM - Jagannath University, Delhi Jaipur National University

What is the eligibility criteria for the courses at ILAM - Jaipur National University?

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saurya snehal

Contributor-Level 10

The eligibility criteria for the courses at ILAM - Jaipur National University are as follows:

Courses	Eligibility
MBA/PGDM (9 courses)	Graduation : 50 %
Executive MBA/PGDM (2 courses)	Graduation : 50 %
M.Sc. (1 course)	Graduation : 50 %
BBA (3 courses)	Class 12 : 50 % Exams : CBSE 12thISC+1 More
B.Sc. (1 course)	Class 12 : 50 % Exams : CBSE 12thRBSE 12th

Candidates who meet the eligibility criteria for the programmes can participate in the admission process by filling out the application form.

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9 months ago

University College of Engineering University College

What are the courses offered at University College of Engineering?

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Nishtha Hazarika

Contributor-Level 10

University College of Engineering offers three courses. These programmes are offered in full-time mode and the duration vary depending on the course. Students need to satisfy the admission criteria of the offered course before applying for the same. Check out the list to know about the University College of Engineering courses:

BTech
MTech
BTech + MBA

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9 months ago

Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eleven

Show that the straight lines whose direction cosines are given by $2 l + 2 m - n = 0 and m n + n l + l m = 0$ are at right angles.

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Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} G i v e n t h a t 2 l + 2 m - n = 0 \dots (i) \\ a n d m n + n l + l m = 0 \dots (i i) \\ Eliminating m f r o m e q . (i) a n d (i i) w e g e t, \\ m = \frac{n - 2 l}{2} [F r o m (i)] \\ \Rightarrow (\frac{n - 2 l}{2}) n + n l + l (\frac{n - 2 l}{2}) = 0 \\ \Rightarrow \frac{n^{2} - 2 n l + 2 n l + n l - 2 l^{2}}{2} = 0 \\ \Rightarrow n^{2} + n l - 2 l^{2} = 0 \\ \Rightarrow n^{2} + 2 n l - n l - 2 l^{2} = 0 \\ \Rightarrow n (n + 2 l) - l (n + 2 l) = 0 \\ \Rightarrow (n - l) (n + 2 l) = 0 \\ \Rightarrow n = - 2 l a n d n = l \\ ∴ m = \frac{- 2 l - 2 l}{2}, m = \frac{l - 2 l}{2} \\ \Rightarrow m = - 2 l, m = \frac{- l}{2} \\ T h e r e f o r e, t h e d i r e c t i o n r a t i o s a r e p r o p o r t i o n a l t o l, - 2 l, - 2 l a n d l, \frac{- l}{2}, l . \\ \Rightarrow 1, - 2, - 2 a n d 2, - 1, 2 \\ I f t h e t w o l i n e s a r e p e r p e n d i c u l a r t o e a c h o t h e r t h e n \\ 1 (2) - 2 (- 1) - 2 \times 2 = 0 \\ 2 + 2 - 4 = 0 \\ 0 = 0 \\ H e n c e, t h e t w o l i n e s a r e p e r p e n d i c u l a r . \end{array}$

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9 months ago

Exam Study Abroad

Where is the ACT test conducted in India?

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Abhishek Khanna

Contributor-Level 8

The ACT test is conducted at various locations in the major cities of India. Some of these cities are Ahmedabad, New Delhi, Pune, Noida, Jaipur, Kodaikanal, Vadodara, Mumbai, Hyderabad and Chennai. Indian students willing to take the ACT exam to study abroad in the USA or other eligible locations can find complete details about test centres by logging into their My ACT accounts.

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9 months ago

Exemplar Solution Maths NCERT Exemplar Solutions Class 12th Chapter Eleven

AB = 3 $\hat{i}$ - $\hat{j}$ + $\hat{k}$ and $C D$ = -3 $\hat{i}$ + 2 $\hat{j}$ + 4 $\hat{k}$ are two vectors. The position vectors of the points A and C are 6 $\hat{i}$ + 7 $\hat{j}$ + 4 $\hat{k}$ and -9 $\hat{i}$ + 2 $\hat{j}$ + $\hat{k,}$ respectively. Find the position vector of a point P on the line AB and a point Q on the line CD such that $\vec{P Q}$ is perpendicular to both $\vec{A B}$ and $\vec{C D}$ both.

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Payal Gupta

Contributor-Level 10

This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} P o s i t i o n v e c t o r o f A i s 6 \hat{i} + 7 \hat{j} + 4 \hat{k} a n d \vec{A B} = 3 \hat{i} - \hat{j} + \hat{k} \\ S o, e q u a t i o n o f a n y l i n e passing t h r o u g h A a n d p a r a l l e l t o \vec{A B} \\ \vec{r} = (6 \hat{i} + 7 \hat{j} + 4 \hat{k}) + λ (3 \hat{i} - \hat{j} + \hat{k}) \dots (i) \\ N o w a n y point P o n \vec{A B} = (6 + 3 λ, 7 - λ, 4 + λ) \\ S i m i l a r l y, p o s i t i o n v e c t o r o f C i s - 9 \hat{j} + 2 \hat{k} a n d \vec{C D} = - 3 \hat{i} + 2 \hat{j} + 4 \hat{k} \\ S o, e q u a t i o n o f a n y l i n e passing t h r o u g h C a n d p a r a l l e l t o \vec{C D} i s \\ \vec{r} = (- 9 \hat{j} + 2 \hat{k}) + μ (- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}) \dots (i i) \\ A n y point Q o n \vec{C D} = (- 3 μ, - 9 + 2 μ, 2 + 4 μ) \\ d' r a t i o s o f \vec{P Q} a r e \\ (6 + 3 λ + 3 μ, 7 - λ + 9 - 2 μ, 4 + λ - 2 - 4 μ) \\ \Rightarrow (6 + 3 λ + 3 μ), (16 - λ - 2 μ), (2 + λ - 4 μ) \\ N o w \vec{P Q} i s ⊥ t o e q . (i), t h e n \\ 3 (6 + 3 λ + 3 μ) - 1 (16 - λ - 2 μ) + 1 (2 + λ - 4 μ) = 0 \\ \Rightarrow 18 + 9 λ + 9 μ - 16 + λ + 2 μ + 2 + λ - 4 μ = 0 \\ \Rightarrow 11 λ + 7 μ + 4 = 0 \dots (i i i) \\ \vec{P Q} i s ⊥ t o e q . (i i), t h e n \\ - 3 (6 + 3 λ + 3 μ) + 2 (16 - λ - 2 μ) + 4 (2 + λ - 4 μ) = 0 \\ \Rightarrow - 18 - 9 λ - 9 μ + 32 - 2 λ - 4 μ + 8 + 4 λ - 16 μ = 0 \\ \Rightarrow - 7 λ - 29 μ + 22 = 0 \\ \Rightarrow 7 λ + 29 μ - 22 = 0 \dots (i v) \\ S o l v i n g e q n . (i i i) a n d (i v) w e g e t \\ \Rightarrow 77 λ + 49 μ + 28 = 0 \\ 77 λ + 319 μ - 242 = 0 \\ \underline{(-) (-) (+)} \\ - 270 μ + 270 = 0 ∴ μ = 1 \\ P u t t i n g t h e v a l u e o f μ i n e q n . (i v) w e g e t, \\ 7 λ + 29 - 22 = 0 \Rightarrow λ = - 1 \\ ∴ P o s i t i o n v e c t o r o f P = [6 + 3 (- 1), 7 + 1, 4 - 1] = (3, 8, 3) \\ a n d P o s i t i o n v e c t o r o f Q = [- 3 (1), - 9 + 2 (1), 2 + 4 (1)] = (- 3, - 7, 6) \\ H e n c e, t h e p o s i t i o n v e c t o r o f \\ P = 3 \hat{i} + 8 \hat{j} + 3 \hat{k} a n d Q = - 3 \hat{i} - 7 \hat{j} + 6 \hat{k} \end{array}$

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9 months ago

Principal K.M. Kundnani College of Pharmacy Kirori Mal College, University of Delhi

What is the minimum cutoff for Principal K.M. Kundnani College of Pharmacy?

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9 months ago

NEET Counselling

Can candidates submit objections against the seat allotment result of Kerala NEET counselling?

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Sreetama Datta Roy

Contributor-Level 10

Yes, candidates are given an opportunity to submit objections or challenges against the Kerala NEET seat allotment result. At first, CEE Kerala releases the provisional seat allotment result. Based on that, candidates can submit objections through email within the stipulated schedule. The counselling authorities will review the objections and release the final seat allotment result.

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9 months ago

Vanderbilt University Study Abroad

What is the Vanderbilt University acceptance rate for international students?

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Ridhi

Contributor-Level 10

For the admission cycle of 2024-25. Vanderbilt University received over 8,300 applications from prospective international students, of which the university offered the opportunity to enroll at the Vanderbilt campus to about 360 students. In the end, almost 200 students ended up enrolling. These statistics suggests that the Vanderbilt University acceptance rate for international rate stands at a low of 4.34%, meaning that the admission criteria of the university are extrmely selective as out of every 100 students who apply to the Vanderbilt university's admission process, the university is likely to reject around 96 of them.

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9 months ago

Vidyavardhini's College of Engineering and Technology Colleges in Mumbai

What are the job profiles that I can get after completing MMS at Vidyavardhini's College of Engineering and Technology?

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Himanshi Pandey

Contributor-Level 10

Vidyavardhini's College of Engineering and Technology (VCET) offers a Master of Management Science at the PG level. The course is AICTE-approved. Pursuing the course can lead to various managerial-level and consulting jobs across several domains and industries. Some of the job profiles that MMS degree holders of VCET Mumbai are tabulated below:

Job Profile	Average Salary
Management Consultant	INR 7 LPA - INR 42.7 LPA
Business Consultant	INR 2.3 LPA - INR 26 LPA
Project Manager	INR 4.2 LPA - INR 30 LPA
HR Manager	INR 2.9 LPA - INR 19 LPA
Marketing Manager	INR 2.4 LPA - INR 25 LPA

Note: The average salary has been taken from Ambitionbox.com.

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9 months ago

Kerala NEET Counselling Kerala NEET

How to make corrections in the Kerala NEET counselling application form?

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Aishwarya Aggarwal

Beginner-Level 5

Students can make changes or corrections in the Kerala NEET 2025 counselling using the process shared below-

Go to cee.kerala.gov.in.
Click on KEAM 2025-Candidate Portal.
Enter the Application Number and Password.
Click on Application Form.
Verify and make necessary changes or upload documents.

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9 months ago

Vanderbilt University Study Abroad

What is the Vanderbilt University acceptance rate?

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Ridhi

Contributor-Level 10

Vanderbilt University received over 45,400 applications, of which 2,600 students were offered the opportunity to enroll at the Vanderbilt campus, suggesting that the Vanderbilt University acceptance rate stands at just 5.8%. This makes the admission criteria of the university extremely selective.

Simlarly, when it comes to international students, the university received over 8,300 applications however, just 360 students got admitted, making the Vanderbilt acceptance rate for international students 4.34%.

Particulars	Class of 2028
Student Applications Received	45,409
Students Admitted	2,662
Students Enrolled	1,630
Yield Rate	61.2%
Acceptance Rate	5.86%
International Acceptance Rate	4.34%

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9 months ago

Ajay Kumar Garg Engineering College Colleges in Ghaziabad

What are the key takeaways from the Ajay Kumar Garg Engineering College placements?

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Meitankeisangbam Vasundhara devi

Contributor-Level 10

The Ajay Kumar Garg Engineering College offered a the highest package of INR 30 LPA in 2025. For more information on key takeaways for the AKG Engineering College placements, refer to the table below:

Particulars	Statistics (2024)	Statistics (2025)
Highest Salary	INR 42.75 LPA	INR 30 LPA
Average Salary	INR 5.68 LPA	INR 5.96 LPA
Offers Made	1727	1410
Total Students	1110	1152
Companies Visited	235	322
Total Exclusive Offers	1080	983

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9 months ago

Aviation CBSE 12th

How many years to become a pilot after 12th?

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Muskan Chugh

Contributor-Level 10

Students can start their training to become a Pilot right after they complete their Class 12. The Pilot training is divided into two parts:

The Ground Course: Typically, it lasts 4-6 months.
Flying Training: Usually takes around 1 year.

The complete procedures, from obtaining the medical certificates to obtaining the CPL, can take between 1.5 and 2 years, depending on the institute.

For complete details, Click Here: How to Become Pilot After 12th

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9 months ago

Aviation Pilot

After 12th how to become a pilot?

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Muskan Chugh

Contributor-Level 10

Students who wish to become a Pilot after completing their Class 12 must pass from a recognised board with at least 50-60% marks in Physics, Maths, and English. After completing their Class 12, they are required to enroll in flying schools where they complete their Ground course and flying training. To start their career as a commercial pilot, students must obtain the CPL, after which they can get paid for their services in an airline.

For complete details, Click Here: How to Become Pilot After 12th

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9 months ago

Ajay Kumar Garg Engineering College Colleges in Ghaziabad

What is the reality of Ajay Kumar Garg Engineering College placements?

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Meitankeisangbam Vasundhara devi

Contributor-Level 10

Ajay Kumar Garg Engineering College offers decent placements annually to its BTech, MTech and MCA courses. A total of 322 recruiters participated in 2025 placements and preseted 410 total offers and 983 exclusive offers. The highest and average package offered during AKGEC Ghaziabad placements 2025 stood at INR 30 LPA and INR 5.96 LPA.

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9 months ago

Chitkara University, Chandigarh Colleges in Himachal Pradesh

How does Chitkara University assist students to overcome financial barriers through scholarships?

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Ashwin Yadav

Contributor-Level 10

Chitkara University offers a variety of programs to students in a range of specializations. Some of the prominent scholarships offered by the university are Post-matric Scholarship Scheme for SCs/OBCs Students, Scheme for Minorities Students, Central Sector Scholarship Scheme for College and University Students, and AICTE's Scholarship for J&K Students.

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9 months ago

Chitkara University, Chandigarh Colleges in Solan

How does the campus life at Chitkara University, Chandigarh compare to other universities in India?

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Ashwin Yadav

Contributor-Level 10

Chitkara University campus is one of the advanced campus in its region. It has auditoriums, laboratories, separate hostels, etc. The campus is also well connected by road. For students overall growth, there area clubs, societies, and other extra-curricular activities as well.

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9 months ago

VCET - Vedant College of Engineering & Technology Vidyavardhini's College of Engineering and Technology

What are the MMS specialisations offered VCET Mumbai?

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Himanshi Pandey

Contributor-Level 10

MMS at VCET Mumbai is a two-year programme leading to a PG degree. Candidates pursuing an MMS at the college can choose between one of the five specialisations available at the college. The eligibility and selection criteria for these specialisations are the same. Listed below are the VCET Mumbai MMS specialisations:

Marketing
Finance
IT
Operations
HR

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9 months ago

Chitkara University, Chandigarh Colleges in Solan

How are the placements at Chitkara University?

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Ashwin Yadav

Contributor-Level 10

In the BTech placements 2023, the highest package stood at INR 1.11 crore per annum. The advanced Placement Cell provides students with comprehensive trainings and career events in order to make them industry-ready. The top recruiters at Chitkara University, Chandigarh include Airtel, TCS, Infosys, Amazon, Deloitte, Wipro, HCL, etc.

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