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Chapters
Chapter 2: Inverse Trigonometric Functions
Chapter 3: Matrices
Chapter 4: Determinants
Chapter 5: Continuity and Differentiability
Chapter 6: Application of Derivatives
Chapter 7: Integrals
Chapter 8: Application of Integrals
Chapter 9: Differential Equations
Chapter 10: Vector Algebra
Chapter 11: Three Dimensional Geometry
Chapter 12: Linear Programming
Chapter 13: Probability
Chapter 4: Determinants
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Pages 108 - 109]
Evaluate the determinants in Exercises 1 and 2.
`|(2,4),(-5, -1)|`
Evaluate the determinants in Exercises 1 and 2.
`|(cos theta, -sin theta),(sin theta, cos theta)|`
Evaluate the determinants in Exercises 1 and 2.
`|(x^2-x-1, x -1),(x+1, x+1)|`
if A = `[(1,2),(4,2)]` then show that |2A| = 4|A|
if A=`[(1,0,1),(0,1,2),(0,0,4)]` then show that `|3A| = 27|A|`
Evaluate the determinants `|(3,-1,-2),(0,0,-1),(3,-5,0)|`
Evaluate the determinants `|(0,1,2),(-1,0,-3),(-2,3,0)|`
Evaluate the determinants `|(3,-4,5),(1,1,-2),(2,3,1)|`
Evaluate the determinants
`|(2,-1,-2),(0,2,-1),(3,-5,0)|`
if A = `[(1,1,-2),(2,1,-3),(5,4,-9)]`, Find |A|
Find values of x, if 2451=2x46x
Find values of x, if `|[2,3],[4,5]|=|[x,3],[2x,5]|`
if `|(x, 2),(18, x)| = |(6,2),(18,6)|`, then x is equal to
(A) 6
(B) ±6
(C) −6
(D) 0
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Pages 119 - 121]
Using the property of determinants and without expanding, prove that:
`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`
Using the property of determinants and without expanding, prove that:
`|(a-b,b-c,c-a),(b-c,c-a,a-b),(a-a,a-b,b-c)| = 0`
Using the property of determinants and without expanding, prove that:
`|(2,7,65),(3,8,75),(5,9,86)| = 0`
Using the property of determinants and without expanding, prove that:
`|(1, bc, a(b+c)),(1, ca, b(c+a)),(1, ab, c(a+b))| = 0`
Using the property of determinants and without expanding, prove that:
`|(b+c, q+r, y+z),(c+a, r+p, z +x),(a+b, p+q, x + y )| = 2|(a,p,x),(b,q,y),(c, r,z)|`
By using properties of determinants, show that:
`|(0,a, -b),(-a,0, -c),(b, c,0)| = 0`
By using properties of determinants, show that:
`|(-a^2, ab, ac),(ba, -b^2, bc),(ca,cb, -c^2)| = 4a^2b^2c^2`
By using properties of determinants, show that:
`|(1,a,a^2),(1,b,b^2),(1,c,c^2)| = (a - b)(b-c)(c-a)`
By using properties of determinants, show that:
`|(1,1,1),(a,b,c),(a^3, b^3,c^3)|` = (a-b)(b-c)(c-a)(a+b+c)
By using properties of determinants, show that:
`|(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)| = (x-y)(y-z)(z-x)(xy+yz+zx)`
By using properties of determinants, show that:
`|(x+4,2x,2x),(2x,x+4,2x),(2x , 2x, x+4)| = (5x + 4)(4-x)^2`
By using properties of determinants, show that:
`|(y+k,y, y),(y, y+k, y),(y, y, y+k)| = k^2(3y + k)`
By using properties of determinants, show that:
`|(a-b-c, 2a,2a),(2b, b-c-a,2b),(2c,2c, c-a-b)| = (a + b + c)^2`
By using properties of determinants, show that:
`|(x+y+2z, x, y),(z, y+z+2z,y),(z,x,z+x+2y)| = 2(x+y+z)^3`
By using properties of determinants, show that:
`|(1,x,x^2),(x^2,1,x),(x,x^2,1)| = (1-x^3)^2`
By using properties of determinants, show that:
`|(1+a^2-b^2, 2ab, -2b),(2ab, 1-a^+b^2, 2a),(2b, -2a, 1-a^2-b^2)| = (1+a^2+b^2)`
By using properties of determinants, show that:
`|(a^2+1, ab, ac),(ab, b^2+1, bc),(ca, cb, c^2+1)| = 1+a^2+b^2+c^2`
Let A be a square matrix of order 3 × 3, then | kA| is equal to
(A) k|A|
(B) k2 | A |
(C) k3 | A |
(D) 3k | A |
Which of 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
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Pages 122 - 123]
Find area of the triangle with vertices at the point given in each of the following:
(1, 0), (6, 0), (4, 3)
Find area of the triangle with vertices at the point given in each of the following:
(2, 7), (1, 1), (10, 8)
Find area of the triangle with vertices at the point given in each of the following:
(−2, −3), (3, 2), (−1, −8)
Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.
Find values of k if area of triangle is 4 square units and vertices are (k, 0), (4, 0), (0, 2)
Find values of k if area of triangle is 4 square units and vertices are (−2, 0), (0, 4), (0, k)
Find equation of line joining (1, 2) and (3, 6) using determinants
Find equation of line joining (3, 1) and (9, 3) using determinants
If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is
A. 12
B. −2
C. −12, −2
D. 12, −2
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Page 126]
Write Minors and Cofactors of the elements of following determinants:
`|(2,-4),(0,3)|`
Write Minors and Cofactors of the elements of following determinants:
`|(a,c),(b,d)|`
Write Minors and Cofactors of the elements of following determinants:
`|(1,0,0),(0,1,0),(0,0,1)|`
Write Minors and Cofactors of the elements of following determinants:
`|(1,0,4),(3,5,-1),(0,1,2)|`
Using Cofactors of elements of second row, evaluate `triangle = |(5,3,8),(2,0,1),(1,2, 3)|`
Using Cofactors of elements of third column, evaluate `triangle = |(1,x,yz),(1,y,zx),(1,z,xy)|`
if `triangle = |(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` and Aij is Cofactors of aij, then value of Δ is given by
(A) a11 A31+ a12 A32 + a13 A33
(B) a11 A11+ a12 A21 + a13 A31
(C) a21 A11+ a22 A12 + a23 A13
(D) a11 A11+ a21 A21 + a31 A31
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Pages 131 - 132]
Find adjoint of each of the matrices.'
`[(1,2),(3,4)]`
Find adjoint of each of the matrices.
`[(1,-1,2),(2,3,5),(-2,0,1)]`
Verify A (adj A) = (adj A) A = |A|I
`[(2,3),(-4,-6)]`
Verify A (adj A) = (adj A) A = |A|I
`[(1,-1,2),(3,0,-2),(1,0,3)]`
Find the inverse of each of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of each of the matrices (if it exists).
`[(-1,5),(-3,2)]`
Find the inverse of each of the matrices (if it exists).
`[(1,2,3),(0,2,4),(0,0,5)]`
Find the inverse of each of the matrices (if it exists).
`[(1,0,0),(3,3,0),(5,2,-1)]`
Find the inverse of each of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
Find the inverse of each of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
Find the inverse of each of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
Let `A =[(3,7),(2,5)] and B = [(6,8),(7,9)]`. Verify that `(AB)^(-1) = B^(-1)A^(-1)`
if A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = O. Hence find A–1.
For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI = O.
For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A3 − 6A2 + 5A + 11 I = O. Hence, find A−1.
if A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A3 − 6A2 + 9A − 4I = O and hence find A−1
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to
(A) |A |
(B) | A|2
(C) | A|3
(D) 3|A|
If A is an invertible matrix of order 2, then det (A−1) is equal to
(A) det (A)
(B) 1/det (A)
(C) 1
(D) 0
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Pages 136 - 137]
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Examine the consistency of the system of equations.
2x − y = 5
x + y = 4
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
Examine the consistency of the system of equations.
x + y + z = 1
2x + 3y + 2z = 2
ax + ay + 2az = 4
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
Examine the consistency of the system of equations.
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
Solve system of linear equations, using matrix method.
5x + 2y = 4
7x + 3y = 5
Solve system of linear equations, using matrix method.
2x – y = –2
3x + 4y = 3
Solve system of linear equations, using matrix method.
4x – 3y = 3
3x – 5y = 7
Solve system of linear equations, using matrix method.
5x + 2y = 3
3x + 2y = 5
Solve system of linear equations, using matrix method.
2x + y + z = 1
x – 2y – z = 3/2
3y – 5z = 9
Solve system of linear equations, using matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
Solve system of linear equations, using matrix method.
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3
Solve system of linear equations, using matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A−1. Using A−1 solve the system of equations
2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3
The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.
NCERT solutions for Class 12 Maths Chapter 4 Determinants [Pages 141 - 143]
Prove that the determinant `|(x,sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1, x)|` is independent of θ.is independent of θ.
Without expanding the determinant, prove that
`|(a, a^2,bc),(b,b^2, ca),(c, c^2,ab)| = |(1, a^2, a^3),(1, b^2, b^3),(1, c^2, c^3)|`
Evaluate `|(cos alpha cos beta, cos alpha sin beta, -sin alpha),(-sin beta, cos beta, 0),(sin alpha cos beta, sin alpha sin beta,cos alpha )|`
If a, b and c are real numbers, and triangle =`|(b+c, c+a, a+b),(c+a,a+b, b+c),(a+b, b+c, c+a)|` = 0 Show that either a + b + c = 0 or a = b = c.
Solve the equations `|(x+a,x,x),(a,x+a,x),(x,x,x+a)| = 0, a != 0`
Prove that `|(a^2, bc, ac+c^2),(a^2+ab, b^2, ac),(ab, b^2+bc, c^2)| = 4a^2b^2c^2`
if A^(-1) =`[(3,-1,1),(-15,6,-5),(5,-2,2)] and B = [(1,2,-2),(-1,3,0),(0,-2,1)] " find " (AB)^(-1)`
Let A = `[(1,-2,1),(-2,3,1),(1,1,5)]` verify that
1) [adj A]–1 = adj (A–1)
2) (A–1)–1 = A
Evaluate `|(x, y, x+y),(y, x+y, x),(x+y, x, y)|`
Evaluate `|(1,x,y),(1,x+y,y),(1,x,x+y)|`
Using properties of determinants, prove that:
`|(alpha, alpha^2,beta+gamma),(beta, beta^2, gamma+alpha),(gamma, gamma^2, alpha+beta)|` = (β – γ) (γ – α) (α – β) (α + β + γ)
Using properties of determinants, prove that:
`|(x, x^2, 1+px^3),(y, y^2, 1+py^3),(z, z^2, 1+pz^2)|` = (1 + pxyz) (x – y) (y – z) (z – x), where p is any scalar.
Using properties of determinants, prove that:
`|(3a, -a+b, -a+c),(-b+a, 3b, -b+c),(-c+a, -c+b, 3c)|`= 3(a + b + c) (ab + bc + ca)
Using properties of determinants, prove that:
`|(1, 1+p, 1+p+q),(2, 3+2p, 4+3p+2q),(3,6+3p,10+6p+3q)| = 1`
Using properties of determinants, prove that
`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`
Solve the system of the following equations
`2/x+3/y+10/z = 4`
`4/x-6/y + 5/z = 1`
`6/x + 9/y - 20/x = 2`
Choose the correct answer.
If a, b, c, are in A.P., then the determinant
`|(x+2, x+3,x +2a),(x+3,x+4,x+2b),(x+4,x+5,x+2c)|`
A. 0
B. 1
C. x
D. 2x
Choose the correct answer.
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is
A) `[(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))]`
B) `xyz[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`
c) `1/xyz[(x,0,0),(0,y,0),(0,0,z)]`
D) `1/xyz [(1,0,0),(0,1,0),(0,0,1)]`
Choose the correct answer.
Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ≤ 2π, then
A. Det (A) = 0
B. Det (A) ∈ (2, ∞)
C. Det (A) ∈ (2, 4)
D. Det (A)∈ [2, 4]
Chapter 4: Determinants
NCERT solutions for Class 12 Maths chapter 4 - Determinants
NCERT solutions for Class 12 Maths chapter 4 (Determinants) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.
Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.
Concepts covered in Class 12 Maths chapter 4 Determinants are Applications of Determinants and Matrices, Elementary Transformations, Inverse of a Square Matrix by the Adjoint Method, Properties of Determinants, Determinant of a Square Matrix, Determinants of Matrix of Order One and Two, Determinant of a Matrix of Order 3 × 3, Rule A=KB, Introduction of Determinant, Minors and Co-factors, Area of a Triangle.
Using NCERT Class 12 solutions Determinants exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.
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