Determinant

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #81

Prove that: $ \begin{vmatrix} a & a+b & a+b+c \\ 2a & 3a+2b & 4a+3b+2c \\ 3a & 6a+3b & 10a+6b+3c \end{vmatrix} = a^3 $

Question No: #82

Prove that: $ \begin{vmatrix} x^2+y^2+1 & x^2+2y^2+3 & x^2+3y^2+4 \\ y^2+2 & 2y^2+6 & 3y^2+8 \\ y^2+1 & 2y^2+3 & 3y^2+4 \end{vmatrix} = x^2y^2 $

Question No: #83

Prove without expanding. $ \begin{vmatrix} \cos(x-\alpha) & \cos(x+\alpha) & cosx \\ \sin(x+\alpha) & \sin(x-\alpha) & sinx \\ \cos\alpha\ tanx & \cos\alpha\ cotx & \csc\ 2x \end{vmatrix} = 0 $

Question No: #84

Prove that: $ \begin{vmatrix} \cos(x+y) & \sin(x+y) & -\cos(x+y) \\ \sin(x-y) & \cos(x-y) & \sin(x-y) \\ \sin2x & 0 & \sin2y \end{vmatrix} = \sin2(x+y) $

Question No: #85

Prove that $ \begin{vmatrix} 2\cos\theta & 1 & 0 \\ 1 & 2\cos\theta & 1 \\ 0 & 1 & 2\cos\theta \end{vmatrix} = $`\frac{\sin4\theta}{\sin\theta}` [$ \theta \ne n\pi$]

Question No: #86

If $ \Delta = \begin{vmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ 1 & -\sin\theta & 1 \end{vmatrix} $ then show that `2 \le \Delta \le 4`

Question No: #87

Show that $ \begin{vmatrix} 1 & 1 & 1 \\ \sin\alpha & \sin\beta & \sin\gamma \\ \cos\alpha & \cos\beta & \cos\gamma \end{vmatrix} = $`-4sin\frac{\alpha-\beta}{2}sin\frac{\beta-\gamma}{2}sin\frac{\gamma-\alpha}{2}`

Question No: #88

Show that $ \begin{vmatrix} \sin^2A & sinA & \cos^2A \\ \sin^2B & sinB & \cos^2B \\ \sin^2C & sinC & \cos^2C \end{vmatrix} $`= (sinA-sinB)(sinB-sinC)(sinA-sinC)`

Question No: #89

If `A, B, C` are three angles of a triangle, then prove that $ \begin{vmatrix} -1 & \cos\ C & \cos\ B \\ \cos\ C & -1 & \cos\ A \\ \cos\ B & \cos\ A & -1 \end{vmatrix} = 0 $

Question No: #90

If `A, B, C` are three angles of a triangle, then prove that $ \begin{vmatrix} \sin\ (A+B+C) & \sin\ (A+C) & \cos\ C \\ -\sin\ B & 0 & \tan\ A \\ \cos\ (A+B) & \tan\ (B+C) & 0 \end{vmatrix} = 0 $