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(i) LHS = $\left|\begin{array}{ccc}\hfill a-b-c\hfill & \hfill 2a\hfill & \hfill 2a\hfill \\ \hfill 2b\hfill & \hfill b-c-a\hfill & \hfill 2b\hfill \\ \hfill 2c\hfill & \hfill 2c\hfill & \hfill c-a-b\hfill \end{array}\right|$

$=\left|\begin{array}{ccc}\hfill a-b-c+2b+2c\hfill & \hfill 2a+b-c-a+2c\hfill & \hfill 2a+2b+c-ab\hfill \\ \hfill 2b\hfill & \hfill b-c-a\hfill & \hfill 2b\hfill \\ \hfill 2c\hfill & \hfill 2c\hfill & \hfill c-a-b\hfill \end{array}\right|$R_1→ R₁ + R₂ + R₃

$=\left|\begin{array}{ccc}\hfill a+b+c\hfill & \hfill a+b+c\hfill & \hfill a+b+c\hfill \\ \hfill 2b\hfill & \hfill b-c-a\hfill & \hfill 2b\hfill \\ \hfill 2c\hfill & \hfill 2c\hfill & \hfill c-a-b\hfill \end{array}\right|$

= (a + b + c) $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 2b\hfill & \hfill b-c-a\hfill & \hfill 2b\hfill \\ \hfill 2c\hfill & \hfill 2c\hfill & \hfill c-a-b\hfill \end{array}\right|$ Taking (a + b + c) common from R₁

= (a + b+ c) $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1-1\hfill & \hfill 1-1\hfill \\ \hfill 2b\hfill & \hfill b-c-a-2b\hfill & \hfill 2b-2b\hfill \\ \hfill 2c\hfill & \hfill 2c-2c\hfill & \hfill c-a-b-2c\hfill \end{array}\right|\begin{array}{cc}\hfill {}_{}\to {}_2-{}_1\hfill & \hfill {}_{}\to {}_3-{}_1\hfill \end{array}$

= (a + b + c) $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 2b\hfill & \hfill -b-c-a\hfill & \hfill 0\hfill \\ \hfill 2c\hfill & \hfill 0\hfill & \hfill -c-a-b\hfill \end{array}\right|.$

= (a + b + c) × 1. $\left|\begin{array}{cc}\hfill -(a+b+c)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -(a+b+c)\hfill \end{array}\right|$ Expand along R₁

= (a + b + c){(a + b + c)²- 0}

= (a + b +c)³ = R.H.S

(ii) LHS = ${\left|\begin{array}{ccc}\hfill x+y+2z\hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill z\hfill & \hfill y+z+2x\hfill & \hfill y\hfill \\ \hfill z\hfill & \hfill x\hfill & \hfill z+x+2y\hfill \end{array}\right|}_{\cdot }$

$=|\begin{array}{ccc}\hfill x+y+2z+x+y\hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill z+y+z+2x+y\hfill & \hfill y+z+2x\hfill & \hfill y\hfill \\ \hfill z+x+z+x+2y\hfill & \hfill x\hfill & \hfill z+x+2y\hfill \end{array}$
C_1→ C₁ + C₂ + C₃.

$={\left|\begin{array}{ccc}\hfill 2(x+y+z)\hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill 2(x+y+z)\hfill & \hfill y+z+2x\hfill & \hfill y\hfill \\ \hfill 2(x+y+z)\hfill & \hfill x\hfill & \hfill z+x+2y\hfill \end{array}\right|}_{.}$

= 2 (k + y + z) $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill x\hfill & \hfill y\hfill \\ \hfill 1\hfill & \hfill y+z+2x\hfill & \hfill y\hfill \\ \hfill 1\hfill & \hfill x\hfill & \hfill z+x+2y\hfill \end{array}\right|$ Taking 2

As per popularity basis, listed below are the top MSW colleges in Mumbai along with their tuition fees:

Disclaimer: This information is sourced from the official website.

Deen Dayal Upadhyay Hospital is a government-run hospital in Delhi. The hospital offers affordable healthcare services. DDU Hospital also offers a wide range of services such as emergency rooms, general wards, trauma block, OPD services, laboratories, etc.  The hospital is equipped with modern diagnostic and tratement facilities. DDU Hospital also offers DNB courses across several specialisations. Further, Deen Dayal Upadhyay Hospital is known for its efficient and dedicated faculty and staff.

About 6 top MSW colleges are there in Mumbai. Of these, 2 colleges are privately owned, 1 college is owned by public/government organisations, and public-private entities own 1 college. MAH CET is the top most accepting by the top MSW colleges in Mumbai. Some of the top popular colleges include University of Mumbai, Amity University, College of Social Work, Nirmala Niketan, YES Education, Maharshi Dayanand College of Arts Science and Commerce, Sewa Community College, Walchand College of Arts and Science, etc.

MSc admission at UHS Baglkot takes place through the ICAR AIEEA PG exam. Candidates applying for admission into the programme will need to have a minimum aggregate of 60% in Class 12. Candidates who have secured the eligibility criteria for admission into the MSc programme can apply through the institute website. 

There are many specializations that the students can pursue at MMS colleges in Mumbai. Some of these are listed in the table below, along with the number of colleges offering them:

Disclaimer: This information is sourced from the official website and may vary.

(i) LHS = $\left|\begin{array}{ccc}\hfill x+4\hfill & \hfill 2x\hfill & \hfill 2x\hfill \\ \hfill 2x\hfill & \hfill x+4\hfill & \hfill 2x\hfill \\ \hfill 2x\hfill & \hfill 2x\hfill & \hfill x+4\hfill \end{array}\right|$

$=\left|\begin{array}{ccc}\hfill x+4+2x+2x\hfill & \hfill 2x+x+4+2x\hfill & \hfill 2x+2x+x+4\hfill \\ \hfill 2x\hfill & \hfill x+4\hfill & \hfill 2x\hfill \\ \hfill 2x\hfill & \hfill 2x\hfill & \hfill x+4\hfill \end{array}\right|$ R_1→R₁ + R₂ + R₃

$=\left|\begin{array}{ccc}\hfill 5x+4\hfill & \hfill 5x+4\hfill & \hfill 5x+4\hfill \\ \hfill 2x\hfill & \hfill x+4\hfill & \hfill 2x\hfill \\ \hfill 2x\hfill & \hfill 2x\hfill & \hfill x+4\hfill \end{array}\right|$

Taking (5x + 4) common from R₁.

$=(5x+4)\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 2x\hfill & \hfill x+4\hfill & \hfill 2x\hfill \\ \hfill 2x\hfill & \hfill 2x\hfill & \hfill x+4\hfill \end{array}\right|.$

= (5x + 4)[0 - (4 -x)(x- 4)]

= (5x + 4)(4 -x)(4 -x)

= (5x + 4)(4 -x)² = R.H.S.

= $\left|\begin{array}{ccc}\hfill y+k+y+y\hfill & \hfill y+y+k+y\hfill & \hfill y+y+y+k\hfill \\ \hfill y\hfill & \hfill y+k\hfill & \hfill y\hfill \\ \hfill y\hfill & \hfill y\hfill & \hfill y+k\hfill \end{array}\right|$ R_1→R₁ + R₂+ R₃

$=\left|\begin{array}{ccc}\hfill 3y+k\hfill & \hfill 3y+k\hfill & \hfill 3y+k\hfill \\ \hfill y\hfill & \hfill y+k\hfill & \hfill y\hfill \\ \hfill y\hfill & \hfill y\hfill & \hfill y+k\hfill \end{array}\right|$

= (3y + k) $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill y\hfill & \hfill y+k\hfill & \hfill y\hfill \\ \hfill y\hfill & \hfill y\hfill & \hfill y+k\hfill \end{array}\right|$

Yes, applications are open at Sri Venkateswara College of Engineering, Kanchipuram for various courses. The mode of application is online. Aspirants need to apply at the official website for admission to the various programme. Candidates can fill out the application form on the college's official website.

LHS $\left|\begin{array}{ccc}\hfill x\hfill & \hfill x^2\hfill & \hfill yz\hfill \\ \hfill y\hfill & \hfill y^2\hfill & \hfill zx\hfill \\ \hfill z\hfill & \hfill z^2\hfill & \hfill xy\hfill \end{array}\right|$

$=\frac{1}{xyz}$ $\left|\begin{array}{ccc}\hfill x·x\hfill & \hfill x·x^2\hfill & \hfill x·yz\hfill \\ \hfill y·y\hfill & \hfill y·y^2\hfill & \hfill y·zx\hfill \\ \hfill z·z\hfill & \hfill z·z^2\hfill & \hfill z·xy\hfill \end{array}\right|$ Multiplying R1, R2& R3 by x, y&z.

$=\frac{1}{xyz}\left|\begin{array}{ccc}\hfill x^2\hfill & \hfill x^3\hfill & \hfill xyz\hfill \\ \hfill y^2\hfill & \hfill y^3\hfill & \hfill xyz\hfill \\ \hfill z^2\hfill & \hfill z^3\hfill & \hfill xyz\hfill \end{array}\right|.$

$=\frac{xyz}{xyz}\left|\begin{array}{ccc}\hfill x^2\hfill & \hfill x^3\hfill & \hfill 1\hfill \\ \hfill y^2\hfill & \hfill y^3\hfill & \hfill 1\hfill \\ \hfill z^2\hfill & \hfill z^3\hfill & \hfill 1\hfill \end{array}\right|$ Taking xyz common from c3.

$=\left|\begin{array}{ccc}\hfill x^2\hfill & \hfill x^3\hfill & \hfill 1\hfill \\ \hfill y^2-x^2\hfill & \hfill y^3-x^3\hfill & \hfill 1-1\hfill \\ \hfill z^2-x^2\hfill & \hfill 2^3-x^3\hfill & \hfill 1-1\hfill \end{array}\right|\begin{array}{cc}\hfill {}_2\to {}_2-{}_1\hfill & \hfill {}_3\to {}_3-{}_1.\hfill \end{array}$

= $\left|\begin{array}{ccc}\hfill x^2\hfill & \hfill x^3\hfill & \hfill 1\hfill \\ \hfill (y-x)(y+x)\hfill & \hfill (y-x)\left(y^2+x^2+xy\right)\hfill & \hfill 0\hfill \\ \hfill (z-x)(z+x)\hfill & \hfill (2-x)\left(z^2+x^2+2x\right)\hfill & \hfill 0\hfill \end{array}\right|$

Expanding along c3 we get,

= 1 $\left|\begin{array}{cc}\hfill (y-x)(y+x)\hfill & \hfill (y-x)\left(y^2+x^2+xy\right)\hfill \\ \hfill (z-x)(z+x)\hfill & \hfill (z-x)\left(z^2+x^2+zx\right)\hfill \end{array}\right|$

= (y-x)(z-x). $\left|\begin{array}{cc}\hfill y+x\hfill & \hfill y^2+x^2+xy\hfill \\ \hfill z+x\hfill & \hfill z^2+x^2+zx\hfill \end{array}\right|$ Taking (yx) & (zx) common from R₁ and R₂.

= (y-x)(z-x).  Taking (y-x) & (z-x) common from R₁ and R₂.

= (y-x)(z-x)[(y + x)(z² + x² + zx) - (z + x)(y² + x² + xy)]

= (y-x)(z-x)[yz² + yx² + xyz + xz² + x³ + x²z-zy²-zx²-xyz-xy²-x³-x²y]

= y-x)(z-x)[yz²-zy² + xz²-xy²]

= (y-x)(z-x)[yz (z-y) + x(z²-y²)]

= y-x)(z-x)(z-y)[yz + x (z + y)]             

= (- 1)(x-y)(z-x)(- 1)(y-z)[yz+ 1 xz + xy]

= (x-y)(y-z)(z-x)(xy + yz + xz). = R.H.S

The table given below provides details about the various MMS colleges in Mumbai that provide the course within various course fee ranges:

Disclaimer: This information is sourced from the official website and may vary.

=(b-a)(c-a)[(c+a)-(b+a)]

=(b-a)(c-a)[(c+a – b – a)]

=(b-a)(c-a)(c-b)

=(-1)(a-b) ×(-1)(b-c)(c-a)

=(a-b)(b-c)(c-a).

= R.H.S.

$=\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 1-1\hfill & \hfill 1-1\hfill \\ \hfill a\hfill & \hfill b-a\hfill & \hfill c-a\hfill \\ \hfill a^3\hfill & \hfill b^3-a^3\hfill & \hfill c^3-a^3\hfill \end{array}\right|\begin{array}{cc}\hfill c_2\to c_2-c_1\hfill & \hfill c_3\to c_3-c_1\hfill \end{array}$

= $\left|\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill a\hfill & \hfill b-a\hfill & \hfill c-a\hfill \\ \hfill a^3\hfill & \hfill (b-a)\left(b^2+a^2+ab\right)\hfill & \hfill (c-a)\left(c^2+a^2+ab\right)\hfill \end{array}\right|$ $\{\begin{array}{l}?x^3-y^3\\ =(x-y)(x^2+y^2+xy)\end{array}$

Expanding along R1

= $\left|\begin{array}{cc}\hfill b-a\hfill & \hfill c-a\hfill \\ \hfill \left(b-a(b^2+a^2+ab\right)\hfill & \hfill (c-a)\left(c^2+a^2+ab\right)\hfill \end{array}\right|$

= (ba)(ca) $\left|\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill b^2+a^2+ab\hfill & \hfill c^2+a^2+ab\hfill \end{array}\right|$ $\{\begin{array}{l}(b-a)\&(c-a)\\ c_1\&c_3\end{array}$

= (b-a)(c-a) {(c² + a² + ab)- (b² + a² + ab)}

= (b-a) (c -a) {c² + a² + ab-b²-a²-ab}

= (b-a) (c-a)(c²-b² + ac-ab)

= (b-a)(c-a)(c-b)(c + b) + (c-b) a}

= (b-a)(c-a)(c-b)(a + c + b).

= (- 1)(a-b)(c-a)(- 1)(b-c)(a + c + b)

= (a-b)(b-c)(c-a)(a + c + b)

= R.H.S.

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=a²b²c²× (−2) $\left(1\times (-1)-1\times 1\right)$

= a²b²c²× (−2)× (−2)

= 4a²b²c²

= R.H.S.

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Check the method below to calculate scores in KLEE 2026 exam-

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Here are some notable pharmacy colleges located in Chhatrapati Sambhajinagar (formerly Aurangabad), Maharashtra: 

1. Y B Chavan College Of Pharmacy

Location: Dr. Rafiq Zakaria Campus, Rauza Baug, Cidco, Sambhajinagar

Programs: B.Pharm, M.Pharm, Ph.D

2. Dr. Vedprakash Patil Pharmacy College

Location: Gevrai Tanda, Paithan Road, Sambhajinagar

Programs: B.Pharm, M.Pharm.

.3. Rajesh Bhaiyya Tope College Of Pharmacy

Location: Nipani-Bhalgaon, Beed Highway, Sambhajinagar

Programs: B.Pharm

4. Srinath College Of Pharmacy

Location: MIDC Waluj, Sambhajinagar

Programs: B.Pharm, M.Pharm

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