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 - Mathematics

प्रश्न

Using properties of determinants, prove that

$| \begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix} | = 0$

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Properties of Determinants

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Q 15 Q 14 Q 16

पाठ 4: Determinants - Exercise 4.7 [पृष्ठ १४२]

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एनसीईआरटी Mathematics [English] Class 12

पाठ 4 Determinants
Exercise 4.7 | Q 15 | पृष्ठ १४२

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संबंधित प्रश्‍न

Using the properties of determinants, prove the following:

$| \begin{matrix} 1 & x & x + 1 \\ 2 x & x (x - 1) & x (x + 1) \\ 3 x (1 - x) & x (x - 1) (x - 2) & x (x + 1) (x - 1) \end{matrix} | = 6 x^{2} (1 - x^{2})$

By using properties of determinants, show that:

$| \begin{matrix} 1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{matrix} | = (a - b) (b - c) (c - a)$

By using properties of determinants, show that:

$| \begin{matrix} a - b - c & 2 a & 2 a \\ 2 b & b - c - a & 2 b \\ 2 c & 2 c & c - a - b \end{matrix} | = {(a + b + c)}^{2}$

By using properties of determinants, show that:

$| \begin{matrix} 1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{matrix} | = {(1 - x^{3})}^{2}$

Using properties of determinants, prove that

$| \begin{matrix} b + c & a & a \\ b & a + c & b \\ c & c & a + b \end{matrix} |$ = 4abc

Using properties of determinants, prove that:

$| \begin{matrix} a^{2} + 1 & a b & a c \\ b a & b^{2} + 1 & b c \\ c a & c b & c^{2} + 1 \end{matrix} | = a^{2} + b^{2} + c^{2} + 1$

Using properties of determinant prove that

$| \begin{matrix} b + c & a & a \\ b & c + a & b \\ c & c & a + b \end{matrix} |$ = 4abc

Evaluate the following determinants:

$| \begin{matrix} x - 1 & x & x - 2 \\ 0 & x - 2 & x - 3 \\ 0 & 0 & x - 3 \end{matrix} | = 0$

Using properties of determinants, show that $| \begin{matrix} a + b & a & b \\ a & a + c & c \\ b & c & b + c \end{matrix} |$ = 4abc.

Solve the following equation: $| \begin{matrix} x + 2 & x + 6 & x - 1 \\ x + 6 & x - 1 & x + 2 \\ x - 1 & x + 2 & x + 6 \end{matrix} |$ = 0

Without expanding the determinant, find the value of $| \begin{matrix} 10 & 57 & 107 \\ 12 & 64 & 124 \\ 15 & 78 & 153 \end{matrix} |$ .

Without expanding the determinants, show that $| \begin{matrix} b + c & bc & b^{2} c^{2} \\ c + a & ca & c^{2} a^{2} \\ a + b & ab & a^{2} b^{2} \end{matrix} |$ = 0

Without expanding the determinants, show that $| \begin{matrix} l & m & n \\ e & d & f \\ u & v & w \end{matrix} | = | \begin{matrix} n & f & w \\ l & e & u \\ m & d & v \end{matrix} |$

Without expanding the determinants, show that $| \begin{matrix} 0 & a & b \\ - a & 0 & c \\ - b & - c & 0 \end{matrix} |$ = 0

Without expanding evaluate the following determinant:

$| \begin{matrix} 2 & 3 & 4 \\ 5 & 6 & 8 \\ 6 x & 9 x & 12 x \end{matrix} |$

Solve the following equation:

$| \begin{matrix} x + 2 & x + 6 & x - 1 \\ x + 6 & x - 1 & x + 2 \\ x - 1 & x + 2 & x + 6 \end{matrix} |$ = 0

Without expanding determinants show that

$| \begin{matrix} 1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12 \end{matrix} | + 4 | \begin{matrix} 2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6 \end{matrix} | = 10 | \begin{matrix} 1 & 2 & 1 \\ 3 & 1 & 7 \\ 3 & 2 & 6 \end{matrix} |$

Select the correct option from the given alternatives:

Which of the following is correct

The value of $| \begin{matrix} 1 & 1 & 1 \\ ^{n} C_{1} & ^{n + 2} C_{1} & ^{n + 4} C_{1} \\ ^{n} C_{2} & ^{n + 2} C_{2} & ^{n + 4} C_{2} \end{matrix} |$ is 8.

Evaluate: $| \begin{matrix} x^{2} - x + 1 & x - 1 \\ x + 1 & x + 1 \end{matrix} |$

Evaluate: $| \begin{matrix} x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4 \end{matrix} |$

The determinant $| \begin{matrix} b^{2} - ab & b - c & bc - ac \\ ab - a^{2} & a - b & b^{2} - ab \\ bc - ac & c - a & ab - a^{2} \end{matrix} |$ equals ______.

The value of the determinant $| \begin{matrix} x & x + y & x + 2 y \\ x + 2 y & x & x + y \\ x + y & x + 2 y & x \end{matrix} |$ is ______.

$| \begin{matrix} x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c \end{matrix} |$ = 0, where a, b, c are in A.P.

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

If the ratio of the H.M. and GM. between two numbers a and bis 4 : 5, then a: b is

The value of the determinant $| \begin{matrix} 1 & \cos (β - α) & \cos (γ - α) \\ \cos (α - β) & 1 & \cos (γ - β) \\ \cos (α - γ) & \cos (β - γ) & 1 \end{matrix} |$ is equal to ______.

In a triangle the length of the two larger sides are 10 and 9, respectively. If the angles are in A.P., then the length of the third side can be ______.

By using properties of determinant prove that $| \begin{matrix} x + y & y + z & z + x \\ z & x & y \\ 1 & 1 & 1 \end{matrix} | = 0$

By using properties of determinant prove that

$| \begin{matrix} x + y & y + z & z + x \\ z & x & y \\ 1 & 1 & 1 \end{matrix} |$ = 0

Without expanding determinants find the value of $| \begin{matrix} 10 & 57 & 107 \\ 12 & 64 & 124 \\ 15 & 78 & 153 \end{matrix} |$

Evaluate the following determinant without expanding:

$| \begin{matrix} 5 & 5 & 5 \\ a & b & c \\ b + c & c + a & a + b \end{matrix} |$

By using properties of determinant prove that

$| \begin{matrix} x + y & y + z & z + x \\ z & x & y \\ 1 & 1 & 1 \end{matrix} | = 0$

Without expanding determinant find the value of $| \begin{matrix} 10 & 57 & 107 \\ 12 & 64 & 124 \\ 15 & 78 & 153 \end{matrix} |$

By using properties of determinants, prove that

$| \begin{matrix} x + y & y + z & z + x \\ z & x & y \\ 1 & 1 & 1 \end{matrix} |$ = 0

Without expanding evaluate the following determinant.

$| \begin{matrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{matrix} |$

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of C²?

Without expanding determinant find the value of $| \begin{matrix} 10 & 57 & 107 \\ 12 & 64 & 124 \\ 15 & 78 & 153 \end{matrix} |$

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 - Mathematics

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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 - Mathematics

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