#### Online Mock Tests

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

#### Short Answer

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

If Δ = `|(1, x, x^2),(1, y, y^2),(1, z, z^2)|`, Δ_{1} = `|(1, 1, 1),(yz, zx, xy),(x, y, z)|`, then prove that ∆ + ∆_{1} = 0.

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

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

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

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

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

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.

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

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}= 0None of these

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

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)|` = ______.

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 ______.

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

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

True

False

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3}), then `|(x_1, y_1, 1),(x_2, y_2, 1),(x_3, y_3, 1)|^2 = (3"a"^4)/4`

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

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.

If a_{1}, a_{2}, a_{3}, ..., 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.

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.

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

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

#### Long Answer

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.

Using matrix method, solve the system of equations

3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x – y + z = 2.

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.

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

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.

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

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

3

±3

±6

6

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

a

^{3}+ b^{3}+ c^{3}3bc

a

^{3}+ b^{3}+ c^{3}– 3abcNone of these

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

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

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

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

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

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`

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

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

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

adj A = |A|.A

^{–1}det(A)

^{–1}= [det(A)]^{–1}(AB)

^{–1}= B^{–1}A^{–1}(A + B)

^{–1}= B^{–1}+ A^{–1}

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

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

9x

^{2}(x + y)9y

^{2}(x + y)3y

^{2}(x + y)7x

^{2}(x + y)

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

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

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

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 ______.

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

If A is a matrix of order 3 × 3, then (A^{2})^{–1} = ______.

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

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

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

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

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 + Cx^{2} + ..., then A = ______.

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

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

True

False

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

True

False

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

True

False

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

True

False

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

`|(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

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

True

False

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

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

Let Δ = `|("a", "p", x),("b", "q", y),("c", "r", z)|` = 16, then Δ_{1} = `|("p" + x, "a" + x, "a" + "p"),("q" + y, "b" + y, "b" + "q"),("r" + z, "c" + z, "c" + "r")|` = 32.

True

False

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

True

False

## Chapter 4: Determinants

## 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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