The value of \(\tan^{-1}(\sqrt{3}) + \sec^{-1}(-2) - \sin^{-1}\left(-\frac{1}{2}\right)\) is

• (A) \(\frac{\pi}{2}\)
• (B) \(\frac{7\pi}{6}\)
• (C) \(\frac{2\pi}{3}\)
• (D) \(\frac{5\pi}{6}\)

If the statement \((p \wedge q) \rightarrow (r \vee \neg s)\) is False (F), then the truth values of \(p, q, r\) and \(s\) are respectively

• (A) T, T, F, T
• (B) T, F, T, F
• (C) F, F, T, T
• (D) T, T, T, F

In \(\triangle ABC\), if \(2a^2 = b^2 + c^2\), then the value of \(\frac{\cos 3A}{\cos A} + 2\) is

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(3\)

In \(\triangle ABC\), \((b-c)^2 \cos^2 \frac{A}{2} + (b+c)^2 \sin^2 \frac{A}{2} =\)

• (A) \(a\)
• (B) \(a^2\)
• (C) \(b^2 + c^2\)
• (D) \(2a^2\)

If \(\frac{dy}{dx} = y + 5\) and \(y(0) = 4\), then \(y(\log 2)\) is equal to

• (A) \(13\)
• (B) \(15\)
• (C) \(18\)
• (D) \(9\)

The coordinates of the foot of the perpendicular from the origin to the plane \(2x - 3y - 6z = 4\) are

• (A) \(\left(\frac{8}{49}, -\frac{12}{49}, -\frac{24}{49}\right)\)
• (B) \(\left(\frac{2}{7}, -\frac{3}{7}, -\frac{6}{7}\right)\)
• (C) \(\left(\frac{8}{7}, -\frac{12}{7}, -\frac{24}{7}\right)\)
• (D) \(\left(\frac{4}{49}, -\frac{6}{49}, -\frac{12}{49}\right)\)

The solution of the differential equation \(\frac{dy}{dx} = \frac{x+y}{x-y}\) is

• (A) \(\tan^{-1}\left(\frac{y}{x}\right) = \log\sqrt{x^2+y^2} + C\)
• (B) \(\tan^{-1}\left(\frac{x}{y}\right) = \log(x+y) + C\)
• (C) \(x^2 + y^2 = C(x+y)\)
• (D) \(y = x \tan(\log x + C)\)

The monomer used to prepare Orlon is

• (A) Vinyl chloride
• (B) Tetrafluoroethylene
• (C) Acrylonitrile
• (D) Styrene

Identify the product when a ketone reacts with hydrazine \((NH_2-NH_2)\)

• (A) Oxime
• (B) Hydrazone
• (C) Semicarbazone
• (D) Phenylhydrazone

Which of the following reagents is used to prepare alkyl isocyanides from alkyl halides \((RX)\)?

• (A) \(KCN\) (alc.)
• (B) \(AgCN\) (alc.)
• (C) \(NaCN\) (aq.)
• (D) \(HCN\)

Which of the following aqueous solutions will show maximum vapour pressure at 300 K?

• (A) \(0.1\) M Glucose
• (B) \(0.1\) M \(NaCl\)
• (C) \(0.1\) M \(CaCl_2\)
• (D) \(0.1\) M \(AlCl_3\)