HSC Arts (English Medium) इयत्ता ११ वी - Maharashtra State Board Question Bank Solutions

In ΔABC, A + B + C = π show that

sin²A + sin²B − sin²C = 2 sin A sin B cos C

In ΔABC, A + B + C = π show that

`sin^2  "A"/2 + sin^2  "B"/2 - sin^2  "C"/2 = 1 - 2cos  "A"/2  cos  "B"/2 sin  "C"/2`

In ΔABC, A + B + C = π show that

`tan  "A"/2 tan  "B"/2 + tan  "B"/2 tan  "C"/2 + tan  "C"/2tan  "A"/2` = 1

In ΔABC, A + B + C = π show that

`cot  "A"/2 + cot  "B"/2 + cot  "C"/2 = cot  "A"/2  cot  "B"/2 cot  "C"/2`

In ΔABC, A + B + C = π show that

tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C

In ΔABC, A + B + C = π show that

cos²A +cos²B – cos²C = 1 – 2 sin A sin B cos C

Select the correct option from the given alternatives :

In ∆ABC if cot A cot B cot C > 0 then the triangle is _________

Prove the following:

If sin α sin β − cos α cos β + 1 = 0 then prove cot α tan β = −1

Prove the following:

`cos  (2pi)/15 cos  (4pi)/15cos  (8pi)/15cos  (16pi)/15 = 1/16`

Prove the following:

`(1 + cos  pi/8)(1 + cos  (3pi)/8)(1 + cos  (5pi)/8)(1 + cos  (7pi)/8) = 1/8`

Prove the following:

If A + B + C = `(3pi)/2`, then cos 2A + cos 2B + cos 2C = 1 − 4 sin A sin B sin C

Prove the following:

In any triangle ABC, sin A − cos B = cos C then ∠B = `pi/2`.

Prove the following:

In ∆ABC, ∠C = `(2pi)/3`, then prove that cos²A + cos²B − cos A cos B = `3/4`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(3, -2, 4),(0, 0, -5),(0, 0, 0)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(0, 4, 7),(-4, 0, -3),(-7, 3, 0)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(5),(4),(-3)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[9   sqrt(2)  -3]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(6, 0),(0, 6)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(2, 0, 0),(3, -1, 0),(-7, 3, 1)]`

Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:

`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`