# NCERT solutions for Mathematics Exemplar Class 12 chapter 4 - Determinants [Latest edition]

## Chapter 4: Determinants

Solved ExamplesExercise
Solved Examples [Pages 69 - 77]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 4 DeterminantsSolved Examples [Pages 69 - 77]

Solved Examples | Q 1 | Page 69

If |(2x, 5),(8, x)| = |(6, 5),(8, 3)|, then find x

Solved Examples | Q 2 | Page 69

If Δ = |(1, x, x^2),(1, y, y^2),(1, z, z^2)|, Δ1 = |(1, 1, 1),(yz, zx, xy),(x, y, z)|, then prove that ∆ + ∆1 = 0.

Solved Examples | Q 3 | Page 70

Without expanding, show that Δ = |("cosec"^2theta, cot^2theta, 1),(cot^2theta, "cosec"^2theta, -1),(42, 40, 2)| = 0

Solved Examples | Q 4 | Page 70

Show that Δ = |(x, "p", "q"),("p", x, "q"),("q", "q", x)| = (x - "p")(x^2 + "p"x - 2"q"^2)

Solved Examples | Q 5 | Page 71

If Δ = |(0, "b" - "a", "c" - "a"),("a" - "b", 0, "c" - "b"),("a" - "c", "b" - "c", 0)|, then show that ∆ is equal to zero.

Solved Examples | Q 6 | Page 71

Prove that (A–1)′ = (A′)–1, where A is an invertible matrix.

Solved Examples | Q 7 | Page 71

If x = – 4 is a root of Δ = |(x, 2, 3),(1, x, 1),(3, 2, x)| = 0, then find the other two roots.

Solved Examples | Q 8 | Page 72

In a triangle ABC, if |(1, 1, 1),(1 + sin"A", 1 + sin"B", 1 + sin"C"),(sin"A" + sin^2"A", sin"B" + sin^2"B", sin"C" + sin^2"C")| = 0, then prove that ∆ABC is an isoceles triangle.

Solved Examples | Q 9 | Page 73

Show that if the determinant ∆ = |(3, -2, sin3theta),(-7, 8, cos2theta),(-11, 14, 2)| = 0, then sinθ = 0 or 1/2.

#### Objective Type Questions from 10 and 11

Solved Examples | Q 10 | Page 74

Let ∆ = |("A"x, x^2, 1),("B"y, y^2, 1),("C"z, z^2, 1)|and ∆1 = |("A", "B", "C"),(x, y, z),(zy, zx, xy)|, then ______.

• 1 = – ∆

• ∆ ≠ ∆1

• ∆ – ∆1 = 0

• None of these

Solved Examples | Q 11 | Page 74

If x, y ∈ R, then the determinant ∆ = |(cosx, -sinx, 1),(sinx, cosx, 1),(cos(x + y), -sin(x + y), 0)| lies in the interval.

• [-sqrt(2), sqrt(2)]

• [–1, 1]

• [-sqrt(2), 1]

• [-1, -sqrt(2)]

#### Fill in the blanks in the Examples 12 to 14

Solved Examples | Q 12 | Page 75

If A, B, C are the angles of a triangle, then ∆ = |(sin^2"A", cot"A", 1),(sin^2"B", cot"B", 1),(sin^2"C", cot"C", 1)| = ______.

Solved Examples | Q 13 | Page 75

The determinant ∆ = |(sqrt(23) + sqrt(3), sqrt(5), sqrt(5)),(sqrt(15) + sqrt(46), 5, sqrt(10)),(3 + sqrt(115), sqrt(15), 5)| is equal to ______.

Solved Examples | Q 14 | Page 75

The value of the determinant ∆ = |(sin^2 23^circ, sin^2 67^circ, cos180^circ),(-sin^2 67^circ, -sin^2 23^circ, cos^2 180^circ),(cos180^circ, sin^2 23^circ, sin^2 67^circ)| = ______.

#### State whether the following is True or False: s 15 to 18

Solved Examples | Q 15 | Page 76

The determinant ∆ = |(cos(x + y), -sin(x + y), cos2y),(sinx, cosx, siny),(-cosx, sinx, cosy)| is independent of x only.

• True

• False

Solved Examples | Q 16 | Page 76

The value of |(1, 1, 1),(""^"n""C"_1, ""^("n" + 2)"C"_1, ""^("n" + 4)"C"_1),(""^"n""C"_2, ""^("n" + 2)"C"_2, ""^("n" + 4)"C"_2)| is 8.

• True

• False

Solved Examples | Q 17 | Page 76

If A = [(x, 5, 2),(2, y, 3),(1, 1, z)], xyz = 80, 3x + 2y + 10z = 20, ten A adj. A = [(81, 0, 0),(0, 81, 0),(0, 0, 81)]

• True

• False

Solved Examples | Q 18 | Page 77

If A = [(0, 1, 3),(1, 2, x),(2, 3, 1)], A–1 = [(1/2, -4, 5/2),(-1/2, 3, -3/2),(1/2, y, 1/2)] then x = 1, y = –1.

• True

• False

Exercise [Pages 77 - 85]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 4 DeterminantsExercise [Pages 77 - 85]

#### Using the properties of determinants in 1 to 6 short Answer

Exercise | Q 1 | Page 77

Evaluate: |(x^2 - x + 1, x - 1),(x + 1, x + 1)|

Exercise | Q 2 | Page 77

Evaluate: |("a" + x, y, z),(x, "a" + y, z),(x, y, "a" + z)|

Exercise | Q 3 | Page 77

Evaluate: |(0, xy^2, xz^2),(x^2y, 0, yz^2),(x^2z, zy^2, 0)|

Exercise | Q 4 | Page 77

Evaluate: |(3x, -x + y, -x + z),(x - y, 3y, z - y),(x - z, y - z, 3z)|

Exercise | Q 5 | Page 77

Evaluate: |(x + 4, x, x),(x, x + 4, x),(x, x, x + 4)|

Exercise | Q 6 | Page 77

Evaluate: |("a" - "b" - "c", 2"a", 2"a"),(2"b", "b" - "c" - "a", 2"b"),(2"c", 2"c", "c" - "a" - "b")|

#### Using the proprties of determinants in 7 to 9

Exercise | Q 7 | Page 77

Prove that: |(y^2z^2, yz, y + z),(z^2x^2, zx, z + x),(x^2y^2, xy, x + y)| = 0

Exercise | Q 8 | Page 77

Prove that: |(y + z, z, y),(z, z + x, x),(y, x, x + y)| = 4xyz

Exercise | Q 9 | Page 78

Prove that: |("a"^2 + 2"a", 2"a" + 1, 1),(2"a" + 1, "a" + 2, 1),(3, 3, 1)| = ("a" - 1)^3

Exercise | Q 10 | Page 78

If A + B + C = 0, then prove that |(1, cos"c", cos"B"),(cos"C", 1, cos"A"),(cos"B", cos"A", 1)| = 0

Exercise | Q 11 | Page 78

If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x1, y1), (x2, y2), (x3, y3), then |(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4

Exercise | Q 12 | Page 78

Find the value of θ satisfying [(1, 1, sin3theta),(-4, 3, cos2theta),(7, -7, -2)] = 0

Exercise | Q 13 | Page 78

If [(4 - x, 4 + x, 4 + x),(4 + x, 4 - x, 4 + x),(4 + x, 4 + x, 4 - x)] = 0, then find values of x.

Exercise | Q 14 | Page 78

If a1, a2, a3, ..., ar are in G.P., then prove that the determinant |("a"_("r" + 1), "a"_("r" + 5), "a"_("r" + 9)),("a"_("r" + 7), "a"_("r" + 11), "a"_("r" + 15)),("a"_("r" + 11), "a"_("r" + 17), "a"_("r" + 21))| is independent of r.

Exercise | Q 15 | Page 78

Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

Exercise | Q 16 | Page 78

Show that the ∆ABC is an isosceles triangle if the determinant

Δ = [(1, 1, 1),(1 + cos"A", 1 + cos"B", 1 + cos"C"),(cos^2"A" + cos"A", cos^2"B" + cos"B", cos^2"C" + cos"C")] = 0

Exercise | Q 17 | Page 79

Find A–1 if A = [(0, 1, 1),(1, 0, 1),(1, 1, 0)] and show that A–1 = ("A"^2 - 3"I")/2.

Exercise | Q 18 | Page 79

If A = [(1, 2, 0),(-2, -1, -2),(0, -1, 1)], find A–1. Using A–1, solve the system of linear equations x – 2y = 10 , 2x – y – z = 8 , –2y + z = 7.

Exercise | Q 19 | Page 79

Using matrix method, solve the system of equations
3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.

Exercise | Q 20 | Page 79

Given A = [(2, 2, -4),(-4, 2, -4),(2, -1, 5)], B = [(1, -1, 0),(2, 3, 4),(0, 1, 2)], find BA and use this to solve the system of equations y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.

Exercise | Q 21 | Page 79

If a + b + c ≠ 0 and |("a", "b","c"),("b", "c", "a"),("c", "a", "b")| 0, then prove that a = b = c.

Exercise | Q 22 | Page 79

Prove tha |("bc" - "a"^2, "ca" - "b"^2, "ab" - "c"^2),("ca" - "b"^2, "ab" - "c"^2, "bc" - "a"^2),("ab" - "c"^2, "bc" - "a"^2, "ca" - "b"^2)| is divisible by a + b + c and find the quotient.

Exercise | Q 23 | Page 80

If x + y + z = 0, prove that |(x"a", y"b", z"c"),(y"c", z"a", x"b"),(z"b", x"c", y"a")| = xyz|("a", "b", "c"),("c", "a", "b"),("b", "c", "a")|

#### Objective Type Questions from 24 to 37

Exercise | Q 24 | Page 80

If |(2x, 5),(8, x)| = |(6, -2),(7, 3)|, then value of x is ______.

• 3

• ±3

• ±6

• 6

Exercise | Q 25 | Page 80

The value of determinant |("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")| is ______.

• a3 + b3 + c3

• 3bc

• a3 + b3 + c3 – 3abc

• None of these

Exercise | Q 26 | Page 80

The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq.units. The value of k will be ______.

• 9

• 3

• – 9

• 6

Exercise | Q 27 | Page 80

The determinant |("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)| equals ______.

• abc (b–c) (c – a) (a – b)

• (b–c) (c – a) (a – b)

• (a + b + c) (b – c) (c – a) (a – b)

• None of these

Exercise | Q 28 | Page 81

The number of distinct real roots of |(sinx, cosx, cosx),(cosx, sinx, cosx),(cosx, cosx, sinx)| = 0 in the interval pi/4  x ≤ pi/4 is ______.

• 0

• 2

• 1

• 3

Exercise | Q 29 | Page 81

If A, B and C are angles of a triangle, then the determinant |(-1, cos"C", cos"B"),(cos"C", -1, cos"A"),(cos"B", cos"A", -1)| is equal to ______.

• 0

• – 1

• 1

• None of these

Exercise | Q 30 | Page 81

Let f(t) = |(cos"t","t", 1),(2sin"t", "t", 2"t"),(sin"t", "t", "t")|, then lim_("t" - 0) ("f"("t"))/"t"^2 is equal to ______.

• 0

• – 1

• 2

• 3

Exercise | Q 31 | Page 81

The maximum value of Δ = |(1, 1, 1),(1, 1 + sin theta, 1),(1 + cos theta, 1, 1)| is ______. (θ is real number)

• 1/2

• sqrt(3)/2

• sqrt(2)

• (2sqrt(3))/4

Exercise | Q 32 | Page 82

If f(x) = |(0, x - "a", x - "b"),(x + "b", 0, x - "c"),(x + "b", x + "c", 0)|, then ______.

• f(a) = 0

• f(b) = 0

• f(0) = 0

• f(1) = 0

Exercise | Q 33 | Page 82

If A = [(2, lambda, -3),(0, 2, 5),(1, 1, 3)], then A–1 exists if ______.

• λ = 2

• λ ≠ 2

• λ ≠ – 2

• None of these

Exercise | Q 34 | Page 82

If A and B are invertible matrices, then which of the following is not correct?

• det(A)–1 = [det(A)]–1

• (AB)–1 = B–1A–1

• (A + B)–1 = B–1 + A–1

Exercise | Q 35 | Page 82

If x, y, z are all different from zero and |(1 + x, 1, 1),(1, 1 + y, 1),(1, 1, 1 + z)| = 0, then value of x–1 + y–1 + z–1 is ______.

• x y z

• x–1 y–1 z–1

• – x – y – z

• –1

Exercise | Q 36 | Page 82

The value of the determinant |(x , x + y, x + 2y),(x + 2y, x, x + y),(x + y, x + 2y, x)| is ______.

• 9x2(x + y)

• 9y2(x + y)

• 3y2(x + y)

• 7x2(x + y)

Exercise | Q 37 | Page 83

There are two values of a which makes determinant, ∆ = |(1, -2, 5),(2, "a", -1),(0, 4, 2"a")| = 86, then sum of these number is ______.

• 4

• 5

• – 4

• 9

#### Fill in the blanks

Exercise | Q 38 | Page 83

If A is a matrix of order 3 × 3, then |3A| = ______.

Exercise | Q 39 | Page 83

If A is invertible matrix of order 3 × 3, then |A–1| ______.

Exercise | Q 40 | Page 83

If x, y, z ∈ R, then the value of determinant |((2x^2 + 2^(-x))^2, (2^x - 2^(-x))^2, 1),((3^x + 3^(-x))^2, (3^x -3^(-x))^2, 1),((4^x + 4^(-x))^2, (4^x - 4^(-x))^2, 1)| is equal to ______.

Exercise | Q 41 | Page 83

If cos2θ = 0, then |(0, costheta, sin theta),(cos theta, sin theta,0),(sin theta, 0, cos theta)|^2 = ______.

Exercise | Q 42 | Page 83

If A is a matrix of order 3 × 3, then (A2)–1 = ______.

Exercise | Q 43 | Page 83

If A is a matrix of order 3 × 3, then number of minors in determinant of A are ______.

Exercise | Q 44 | Page 83

The sum of the products of elements of any row with the co-factors of corresponding elements is equal to ______.

Exercise | Q 45 | Page 83

If x = – 9 is a root of |(x, 3, 7),(2, x, 2),(7, 6, x)| = 0, then other two roots are ______.

Exercise | Q 46 | Page 83

|(0, xyz, x - z),(y - x, 0, y  z),(z - x, z - y, 0)| = ______.

Exercise | Q 47 | Page 84

If f(x) = |((1 + x)^17, (1 + x)^19, (1 + x)^23),((1 + x)^23, (1 + x)^29, (1 + x)^34),((1 +x)^41, (1 +x)^43, (1 + x)^47)| = A + Bx + Cx2 + ..., then A = ______.

#### State whether the following is True or False:

Exercise | Q 48 | Page 84

(A3)–1 = (A–1)3, where A is a square matrix and |A| ≠ 0.

• True

• False

Exercise | Q 49 | Page 84

("aA")^-1 = 1/"a"  "A"^-1, where a is any real number and A is a square matrix.

• True

• False

Exercise | Q 50 | Page 84

|A–1| ≠ |A|–1, where A is non-singular matrix.

• True

• False

Exercise | Q 51 | Page 84

If A and B are matrices of order 3 and |A| = 5, |B| = 3, then |3AB| = 27 × 5 × 3 = 405.

• True

• False

Exercise | Q 52 | Page 84

If the value of a third order determinant is 12, then the value of the determinant formed by replacing each element by its co-factor will be 144.

• True

• False

Exercise | Q 53 | Page 84

|(x + 1, x + 2, x + "a"),(x + 2, x + 3, x + "b"),(x + 3, x + 4, x + "c")| = 0, where a, b, c are in A.P.

• True

• False

Exercise | Q 54 | Page 84

|adj. A| = |A|2, where A is a square matrix of order two.

• True

• False

Exercise | Q 55 | Page 84

The determinant |(sin"A", cos"A", sin"A" + cos"B"),(sin"B", cos"A", sin"B" + cos"B"),(sin"C", cos"A", sin"C" + cos"B")| is equal to zero.

• True

• False

Exercise | Q 56 | Page 84

If the determinant |(x + "a", "p" + "u", "l" + "f"),("y" + "b", "q" + "v", "m" + "g"),("z" + "c", "r" + "w", "n" + "h")| splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K is 8.

• True

• False

Exercise | Q 57 | Page 85

Let Δ = |("a", "p", x),("b", "q", y),("c", "r", z)| = 16, then Δ1 = |("p" + x, "a" + x, "a" + "p"),("q" + y, "b" + y, "b" + "q"),("r" + z, "c" + z, "c" + "r")| = 32.

• True

• False

Exercise | Q 58 | Page 85

The maximum value of |(1, 1, 1),(1, (1 + sintheta), 1),(1, 1, 1 + costheta)| is 1/2

• True

• False

## Chapter 4: Determinants

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 12 chapter 4 - Determinants

NCERT solutions for Mathematics Exemplar Class 12 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 Mathematics Exemplar Class 12 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 Mathematics Exemplar Class 12 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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