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RJ-113

B. Sc. B. Ed. (Secondary) Examination, 2024

(First Semester)

MATHEMATICS

DC-I

Algebra and Trigonometry

Time : 3 Hours

[Maximum Marks : 60

Note : All questions are compulsory. Attempt any Two part from each question.

1. (a) (5)

Define Inverse of a matrix. If matrix A =

then prove that AX Adj A = Adj A x A = |A| I

(b)

Define Row Rank and column Rank of matrix.

Find the rank of matrix A, where : A =

Define Transpose of a matrix. If A =

Show that the product AA' and A'A are symmetric but not equal.

(d)

Define Eigen values and Eigen vectors. If A =

Then find out the characteristics roots of matrix A.

(e) (4)

Prove that every square matrix satisfies its characteristic equation.

(c)

Find the eigen value of the matrix A =

and verified that Cayley-Hamilton theorem for matrix A.

(b) (5)

Simply :

If tan(θ + φ) = tan α + i sec α.

Prove that :

3. (a) (5)

Find the equation of the lowest degree. with rational coefficient having 1+√3 and 1+√-5 as two of its roots.

(b)

Solve the following equation by Cardon's method :

Find the equation whose roots are equal to the roots of

each diminished by 3.

4. (a)

State and prove De-Moivre's theorem.

5. (a) (5)

Show that :

(b) (5)

Prove that if θ be on angle whose cosine is positive.

To find the sum of a series of sine or cosine of angles in arithmetical progression.