NCERT Solutions Class 12
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NCERT Solutions-Mathematics
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 1)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 2)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 3)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 4)
- Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 5)
- Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 1)
- Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 2)
- Inverse Trigonometric Function : NCERT Solutions – Class 12 Maths (Ex 3)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 1)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 2)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 3)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 4)
- Matrices : NCERT Solutions – Class 12 Maths (Ex 5)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 1)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 2)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 3)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 4)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 5)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 6)
- Determinants : NCERT Solutions – Class 12 Maths (Ex 7)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 1)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 2)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 3)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 4)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 5)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 6)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 7)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 8)
- Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 9)
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NCERT Solutions-Chemistry
- Aldehydes, Ketones and Carboxylic Acids : NCERT Solutions – Class 12 Chemistry
- Alcohols, Phenols and Ethers : NCERT Solutions – Class 12 Chemistry
- Amines : NCERT Solutions – Class 12 Chemistry
- Biomolecules : NCERT Solutions – Class 12 Chemistry
- Chemical Kinetics : NCERT Solutions – Class 12 Chemistry
- Chemistry in Everyday Life : NCERT Solutions – Class 12 Chemistry
- Coordination Compounds : NCERT Solutions – Class 12 Chemistry
- Electrochemistry : NCERT Solutions – Class 12 Chemistry
- General Principles and Processes of Isolation of Elements : NCERT Solutions – Class 12 Chemistry
- Haloalkanes and Haloarenes : NCERT Solutions – Class 12 Chemistry
- Polymers : NCERT Solutions – Class 12 Chemistry
- Surface Chemistry : NCERT Solutions – Class 12 Chemistry
- The d-and f-Block Elements : NCERT Solutions – Class 12 Chemistry
- The p-Block Elements : NCERT Solutions – Class 12 Chemistry
- The Solid State : NCERT Solutions – Class 12 Chemistry
- Solutions : NCERT Solutions – Class 12 Chemistry
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NCERT Solutions-Biology
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NCERT Solutions-Physics
- Electrostatic Potential And Capacitance : NCERT Solutions – Class 12 Physics
- Electric Charges And Fields : NCERT Solutions – Class 12 Physics
- Semiconductor Electronics: Materials, Devices And Simple Circuits : NCERT Solutions – Class 12 Physics
- Ray Optics And Optical Instruments : NCERT Solutions – Class 12 Physics
- Nuclei : NCERT Solutions – Class 12 Physics
- Moving Charges And Magnetism : NCERT Solutions – Class 12 Physics
- Magnetism And Matter : NCERT Solutions – Class 12 Physics
- Electromagnetic Induction : NCERT Solutions – Class 12 Physics
- Dual Nature Of Radiation And Matter : NCERT Solutions – Class 12 Physics
- Current Electricity : NCERT Solutions – Class 12 Physics
- Communication Systems : NCERT Solutions – Class 12 Physics
- Atoms : NCERT Solutions – Class 12 Physics
- Alternating Current : NCERT Solutions – Class 12 Physics
Determinants : NCERT Solutions – Class 12 Maths (Ex 2)
Exercise 4.2
Using the properties of determinants and without expanding in Exercise 1 to 5, prove that:
1.
Ans. Given:
Operating
0 = 0 [
C1 and C2 are identical]
L.H.S. = R.H.S.
2.
Ans. On Operating
=
= = 0 = R.H.S.
[ All entries of one column here first are zero]
3.
Ans. On operating
= =
= 9 x 0 = 0 [ two columns are identical] Proved.
4.
Ans. Given: =
Operating
=
= = 0 [
two columns are identical] Proved.
5.
Ans. L.H.S. = operating
=
=
=
= [operating
]
= [operating
]
= [operating
]
= [Interchanging
and
]
= [Interchanging
and
]
= = R.H.S.
6.
Ans. Let
= [Taking
common from each row]
Interchanging rows and columns in the determinants on R.H.S.,
Proved.
7.
Ans. L.H.S. =
Taking common from
respectively,
=
= [operating
]
= =
= =
= R.H.S.
8. (i)
(ii)
Ans. (i) L.H.S. = operating
and
,
=
= [Expanding along 1st column]
=
=
= =
= R.H.S. Proved.
(ii) L.H.S. = operating
and
=
=
=
=
=
=
=
=
=
= = R.H.S.
9.
Ans.
L.H.S. = =
[Multiplying
by
respectively]
= =
= [operating
]
=
=
=
=
=
=
=
=
=
=
= = R.H.S.
10. (i)
(ii)
Ans. (i) L.H.S. =
=
=
= [operating
and
]
=
= = R.H.S.
(ii) L.H.S. =
=
=
= [operating
and
]
=
= =
= R.H.S. Proved.
11. (i)
(ii)
Ans. (i) L.H.S. =
=
=
=
=
=
= = R.H.S. Proved.
(ii) L.H.S. =
=
=
=
=
=
= = R.H.S. Proved.
12.
Ans. L.H.S. =
=
=
=
=
=
=
=
=
=
=
= = R.H.S. Proved.
13.
Ans. L.H.S. =
=
=
=
=
=
= = R.H.S.
14.
Ans. L.H.S. =
Multiplying by
respectively and then dividing the determinant by
=
=
=
=
=
=
= = R.H.S. Proved.
Choose the correct answer in Exercises 15 and 16.
15. Let A be a square matrix of order 3 x 3, then is equal to:
(A)
(B)
(C)
(D)
Ans. Let A = be a square matrix of order 3 x 3. ……….(i)
= [From eq. (i)]
Therefore, option (C) is correct.
16. Which is the following is correct:
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these.
Ans. Since, Determinant is a number associated to a square matrix.
Therefore, option (C) is correct.