Balbharati solutions for Mathematics and Statistics 1 (Commerce) [English] 11 Standard Maharashtra State Board chapter 6 - Determinants [Latest edition]

Evaluate the following determinant:

`|(4, 7),(-7, 0)|`

Evaluate the following determinant:

`|(3, -5, 2),(1, 8, 9),(3, 7, 0)|`

Evaluate the following determinants: `|(1, "i", 3),("i"^3, 2, 5),(3, 2, "i"^4)|`

Evaluate the following determinants: `|(5, 5, 5),(5, 4, 4),(5, 4, 8)|`

Evaluate the following determinants: `|(2"i", 3),(4, -"i")|`

Evaluate the following determinants: `|(3, -4, 5),(1, 1, -2),(2, 3, 1)|`

Evaluate the following determinant:

`|("a", "h", "g"),("h", "b", "f"),("g", "f","c")|`

Evaluate the following determinant:

`|(0, "a", -"b"),(-"a", 0, -"c"),("b", "c", 0)|`

Find the value(s) of x, if `|(2, 3),(4, 5)| = |(x, 3),(2x, 5)|`

Find the value(s) of x, if `|(2, 1, x + 1),(-1, 3, -4),(0, -5, 3)|` = 0

Evaluate the following determinants:

`|(x - 1, x, x - 2),(0, x - 2, x - 3),(0, 0, x - 3)| = 0`

Solve the following equation : `|(x, 2, 2),(2, x, 2),(2, 2, x)| = 0`

Solve the following equation : `|(1, 4, 20),(1, -2, 5),(1, 2x, 5x^2)| = 0`

Find the value of x, if `|(x, -1, 2),(2x, 1, -3),(3, -4, 5)|` = 29

Find x and y if `|(4"i", "i"^3, 2"i"),(1, 3"i"^2, 4),(5, -3, "i")|` = x + iy, where i = `sqrt(-1)`.

Without expanding evaluate the following determinant:

`|(1, "a", "b" + "c"),(1, "b", "c" + "a"),(1, "c", "a" + "b")|`

Without expanding evaluate the following determinant:

`|(2, 3, 4),(5, 6, 8),(6x, 9x, 12x)|`

Without expanding evaluate the following determinant:

`|(2, 7, 65),(3, 8, 75),(5, 9, 86)|`

Using properties of determinants, show that `|("a" + "b", "a", "b"),("a", "a" + "c", "c"),("b", "c", "b" + "c")|` = 4abc.

Solve the following equation: `|(x + 2, x + 6, x - 1),(x + 6, x - 1,x + 2),(x - 1, x + 2, x + 6)|` =  0

If `|(4 + x, 4 - x, 4 - x),(4 - x, 4 + x, 4 - x),(4 - x, 4 - x, 4 + x)|` = 0, then find the values of x.

Without expanding determinants, show that

`|(1, 3, 6),(6, 1, 4),(3, 7, 12)| + |(2, 3, 3),(2, 1, 2),(1, 7, 6)| = 10|(1, 2, 1),(3, 1, 7),(3, 2, 6)|`

Without expanding the determinant, find the value of `|(10, 57, 107),(12, 64, 124),(15, 78, 153)|`.

Without expanding determinants, find the value of `|(2014, 2017, 1),(2020, 2023, 1),(2023, 2026, 1)|`

Without expanding determinants, prove that `|("a"_1, "b"_1, "c"_1),("a"_2, "b"_2, "c"_2),("a"_3, "b"_3, "c"_3)| = |("b"_1, "c"_1, "a"_1),("b"_2, "c"_2, "a"_2),("b"_3, "c"_3, "a"_3)| = |("c"_1, "a"_1, "b"_1),("c"_2, "a"_2, "b"_2),("c"_3, "a"_3, "b"_3)|` 

Without expanding determinants, prove that `|(1, yz, y + z),(1, zx, z + x),(1, xy, x + y)| = |(1, x, x^2),(1, y, y^2),(1, z, z^2)|`.

Solve the following equations using Cramer’s Rule:

x + 2y – z = 5, 2x – y + z = 1, 3x + 3y = 8

Solve the following equations using Cramer’s Rule:

2x – y + 6z = 10, 3x + 4y – 5z = 11, 8x – 7y – 9z = 12

Solve the following equations using Cramer’s Rule:

11x – y – z = 31, x – 6y + 2z = –26, x + 2y – 7z = –24

Solve the following equations using Cramer’s Rule:

`1/x + 1/y + 1/z = - 2,  1/x - 2/y + 1/z = 3,  2/x - 1/y + 3/z` = -1

Solve the following equations using Cramer’s Rule:

`2/x - 1/y + 3/z = 4, 1/x - 1/y + 1/z = 2, 3/x + 1/y - 1/z ` = 2

An amount of ₹ 5,000 is invested in three plans at rates 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹ 350. If the total annual income from first two investments is ₹ 70 more than the income from the third, find the amount invested in each plan by using Cramer’s Rule.

Show that the following equations are consistent: 2x + 3y + 4 = 0, x + 2y + 3 = 0, 3x + 4y + 5 = 0

Find k, if the following equations are consistent: x + 3y + 2 = 0, 2x + 4y – k = 0, x – 2y – 3k = 0

Find k, if the following equations are consistent:

(k – 1)x + (k – 1)y = 17, (k – 1)x + (k – 2)y = 18, x + y = 5

Find the area of the triangle whose vertices are: (4, 5), (0, 7), (–1, 1)

Find the area of the triangle whose vertices are: (3, 2), (–1, 5), (–2, –3)

Find the area of the triangle whose vertices are: (0, 5), (0, – 5), (5, 0)

Find the value of k, if the area of the triangle with vertices at A(k, 3), B(–5, 7), C(–1, 4) is 4 square units.

Find the area of the quadrilateral whose vertices are A(–3, 1), B(–2, –2), C(4, 1), D(2, 3).

By using determinant, show that the following points are collinear: P(5, 0), Q(10, –3), R(–5, 6)

The sum of three numbers is 15. If the second number is subtracted from the sum of first and third numbers, then we get 5. When the third number is subtracted from the sum of twice the first number and the second number, we get 4. Find the three numbers.

Evaluate: `|(2, -5, 7),(5, 2, 1),(9, 0, 2)|`

Evaluate: `|(1, -3, 12),(0, 2, -4),(9, 7, 2)|`

Find the value (s) of x, if `|(1, 4, 20),(1, -2, -5),(1, 2x, 5x^2)|` = 0

Find the value (s) of x, if `|(1, 2x, 4x),(1, 4, 16),(1, 1, 1)|` = 0

By using properties of determinants, prove that `|(x + y, y + z, z + x),(z, x, y),(1, 1, 1)|` = 0.

Without expanding the determinants, show that `|("b" + "c", "bc", "b"^2"c"^2),("c" + "a", "ca", "c"^2"a"^2),("a" +  "b", "ab", "a"^2"b"^2)|` = 0

Without expanding the determinants, show that `|(x"a", y"b", z"c"),("a"^2, "b"^2, "c"^2),(1, 1, 1)| = |(x, y, z),("a", "b", "c"),("bc", "ca", "ab")|`

Without expanding the determinants, show that `|(l, "m", "n"),("e", "d", "f"),("u", "v", "w")| = |("n", "f", "w"),(l, "e", "u"),("m", "d", "v")|`

Without expanding the determinants, show that `|(0, "a", "b"),(-"a", 0, "c"),(-"b", -"c", 0)|` = 0

Solve the following linear equations by Cramer’s Rule:  

2x –  y + z = 1, x + 2y + 3z = 8, 3x + y – 4z = 1

Solve the following equations using Cramer’s Rule:

`1/x + 1/y + 1/z = - 2,  1/x - 2/y + 1/z = 3,  2/x - 1/y + 3/z` = -1

Solve the following linear equations by Cramer’s Rule:

x – y + 2z = 7, 3x + 4y – 5z = 5, 2x – y + 3z = 12

Find the value (s) of k, if the following equations are consistent: 3x + y – 2 = 0, kx + 2y – 3 = 0 and 2x – y = 3

Find the value (s) of k, if the following equations are consistent: kx + 3y + 4 = 0, x + ky + 3 = 0, 3x + 4y + 5 = 0

Find the area of triangles whose vertices are A(−1, 2), B(2, 4), C(0, 0).

Find the area of triangles whose vertices are P(3, 6), Q(−1, 3), R(2, −1)

Find the area of triangles whose vertices are L(1, 1), M(−2, 2), N(5, 4)

Find the value of k, if area of ΔPQR is 4 square units and vertices are P(k, 0), Q(4, 0), R(0, 2).

Find the value of k, if area of ΔLMN is `33/2` square units and vertices are L(3, − 5), M(− 2, k), N(1, 4).