NCERT Solutions: Determinants

# NCERT Solutions Class 12 Maths Chapter 4 - Determinants

## Exercise 4.1

Q1: Evaluate the determinants in Exercises 1 and 2.

Ans:
= 2(−1) − 4(−5) = − 2 + 20 = 18

Q2: Evaluate the determinants in Exercises 1 and 2.

Ans:
(i) = (cos θ)(cos θ) − (−sin θ)(sin θ) = cos2 θ+ sin2 θ = 1

(ii) = (x2 − x + 1)(x + 1) − (x − 1)(x + 1)
= x3 − x2 + x + x2 − x + 1 − (x2 − 1)
= x+ 1 − x2 + 1
= x− x2 + 2

Q3:
Ans:
.

Q4:
Ans: The given matrix is  .

Q5: Evaluate the determinants

Ans: (i) Let
It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation.

Q6: If
Ans:

Q7: Find values of x, if

Q8: If  , then x is equal to
(A) 6
(B) ±6
(C) −6
(D) 0
Ans: B

Hence, the correct answer is B.

## Exercise 4.2

Q1: Find area of the triangle with vertices at the point given in each of the following:
(i) (1, 0), (6, 0), (4, 3)
(ii) (2, 7), (1, 1), (10, 8)
(iii) (−2, −3), (3, 2), (−1, −8)

Ans: (i) The area of the triangle with vertices (1, 0), (6, 0), (4, 3) is given by the relation,
(ii) The area of the triangle with vertices (2, 7), (1, 1), (10, 8) is given by the relation,
(iii)The area of the triangle with vertices (−2, −3), (3, 2), (−1, −8) is given by the relation,
Hence, the area of the triangle is |-15| = 15 sq units.

Q2: Show that points
A (a, b + c), B (b, c + a), C (c, a + b)  are collinear
Ans: Area of ΔABC is given by the relation,
= 0
Thus, the area of the triangle formed by points A, B, and C is zero. Hence, the points A, B, and C are collinear.

Q3: Find values of k if area of triangle is 4 square units and vertices are
(i) (k, 0), (4, 0), (0, 2)
(ii) (−2, 0), (0, 4), (0, k)
Ans: We know that the area of a triangle whose vertices are (x1, y1), (x2, y2), and (x3, y3) is the absolute value of the determinant (Δ), where
(i) The area of the triangle with vertices (k, 0), (4, 0), (0, 2) is given by the relation,
∴ -k + 4 = ±4
When −k + 4 = − 4, k = 8.
When −k + 4 = 4, k = 0.
Hence, k = 0, 8.
(ii) The area of the triangle with vertices (−2, 0), (0, 4), (0, k) is given by the relation,
∴ k - 4 = ±4
When k − 4 = − 4, k = 0.
When k − 4 = 4, k = 8.
Hence, k = 0, 8.

Q4: (i) Find equation of line joining (1, 2) and (3, 6) using determinants
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants
Ans: (i) Let P (x, y) be any point on the line joining points A (1, 2) and B (3, 6). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.
Hence, the equation of the line joining the given points is y = 2x.
(ii) Let P (x, y) be any point on the line joining points A (3, 1) and B (9, 3). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.
Hence, the equation of the line joining the given points is x − 3y = 0.

Q5: 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
Ans:(d)
The area of the triangle with vertices (2, −6), (5, 4), and (k, 4) is given by the relation,
When 5 − k = −7, k = 5 + 7 = 12.
When 5 − k = 7, k = 5 − 7 = −2.
Hence, k = 12, −2.

## Exercise 4.3

Q1: Write Minors and Cofactors of the elements of following determinants:

Ans: (i) The given determinant is Minor of element aij is Mij.

Q2:

Ans:

Q3: Using Cofactors of elements of second row, evaluate Δ =.
Ans: The given determinant is
.We have:

We know that ∆ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
∆ = a21A21 + a22A22 + a23A23 = 2(7) + 0(7) + 1(−7) = 14 − 7 = 7.

Q4: Using Cofactors of elements of third column, evaluate Δ=

Ans: The given determinant is
.
We know that Δ is equal to the sum of the product of the elements of the second row
with their corresponding cofactors.

The document NCERT Solutions Class 12 Maths Chapter 4 - Determinants is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on NCERT Solutions Class 12 Maths Chapter 4 - Determinants

 1. What is the importance of determinants in mathematics?
Ans. Determinants play a crucial role in various mathematical concepts such as solving systems of equations, finding the area of triangles, and calculating the inverse of matrices. They provide valuable information about the properties of matrices and systems of linear equations.
 2. How do determinants help in solving systems of linear equations?
Ans. Determinants can be used to determine whether a system of linear equations has a unique solution, no solution, or infinitely many solutions. By calculating the determinant of the coefficient matrix, we can quickly assess the nature of the system.
 3. Can determinants be negative?
Ans. Yes, determinants can be negative, positive, or zero. The sign of the determinant depends on the arrangement of the elements in the matrix. If the determinant is negative, it indicates a reflection or rotation in the transformation represented by the matrix.
 4. What is the relationship between the determinant of a matrix and its inverse?
Ans. The determinant of a matrix is closely related to its inverse. A matrix is invertible if and only if its determinant is non-zero. Additionally, the determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.
 5. How are determinants useful in calculating the area of triangles?
Ans. The determinant of a 2x2 matrix formed by the coordinates of the vertices of a triangle can be used to calculate the area of the triangle. By taking half of the absolute value of the determinant, we can determine the area of the triangle formed by the given points.

## Mathematics (Maths) for JEE Main & Advanced

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