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DMA101


Diploma / Sem I / DMA101 (Midterm)
2026
ENGINEERING MATHEMATICS II
Full Marks: 50
Time: Two (2) hours
The figures in the margin indicate full marks for the questions.
Answer any five (5) questions.

1.

  • a) If \(\sin \theta = \frac{4}{5}\), find the value of \(\frac{4 \tan \theta - 5 \cos \theta}{\sec \theta + 4 \cot \theta}\) (5)
  • b) Evaluate (any two): (\(2.5 \times 2 = 5\))
    • (i) \(\sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ\)
    • (ii) \(\tan 30^\circ \sec 45^\circ + \tan 60^\circ \sec 30^\circ\)
    • (iii) \(\tan 60^\circ \tan 45^\circ + \tan 60^\circ \tan 45^\circ\)

2.

  • a) Show that (any two): (\(2.5 \times 2 = 5\))

    • (i) \(\frac{\sin A + \sin B}{\cos A + \cos B} = \tan \frac{A + B}{2}\)
    • (ii) \(\frac{\sin A + \sin B}{\cos B - \cos A} = \cot \frac{A + B}{2} \cot \frac{A - B}{2}\)
    • (iii) \(\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ = \frac{1}{16}\)

  • b) Prove that (any two): (\(2.5 \times 2 = 5\))

    • (i) \(\frac{\cot A - \tan A}{\cot A + \tan A} = \cos 2A\)
    • (ii) \(\frac{\sin A + \sin 2A}{1 + \cos A + \cos 2A} = \tan A\)
    • (iii) \(\frac{\cos A - \sin A}{\cos A + \sin A} = \sec 2A - \tan 2A\)

3.

  • a) Express the following in the form of \(A + iB\) (any one): (5)

    • (i) \(\frac{3 + 2i}{4 - 5i} + \frac{3 - 2i}{4 + 5i}\)
    • (ii) \(\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}\)

  • b) If \(x = 3 + 2i\) and \(y = 3 - 2i\), then find the values of \(x^3 + y^3\). (5)

4.

  • a) If \(x + iy = \frac{a + ib}{a - ib}\), then show that \(x^2 + y^2 = 1\). (6)

  • b) Find the square root of the following (any one): (4)

    • (i) \(16 - 30i\)
    • (ii) \(-8 + 6i\)
    • (iii) \(-i\)

5.

  • a) Find the conjugate of the following (any one): (5)

    • (i) \(\frac{1 + 3i}{5 - 4i} + \frac{2 - 3i}{5 + 4i}\)
    • (ii) \(\frac{(1 + 2i)(2 - i)}{(3 - 2i)(2 + 3i)}\)

  • b) Use De-Moivre's theorem to compute the following (any one): (5)

    • (i) \((\sqrt{3} + 3i)^6\)
    • (ii) \((-\sqrt{2} + i\sqrt{2})^6\)

6.

  • a) For what real values of \(x\) and \(y\) are the complex numbers \(-3 + ix^2y\) and \(x^2 + y + 4i\) conjugate of each other? (5)

  • b) Resolve the following into partial fractions (any one): (5)

    • (i) \(\frac{7x - 1}{6x^2 - 5x + 1}\)
    • (ii) \(\frac{x^2 + 1}{x(x^2 - 1)}\)

7.

  • a) Resolve the following into partial fractions (any one): (10)
    • (i) \(\frac{x^2 + 3}{x^3(x + 3)}\)
    • (ii) \(\frac{x^2 - 2}{(x^2 + 3x)(x^2 + 2)}\)