\( \int(\tan^2(2x) - \cot^2(2x))dx = \)

• (a) \( -\frac{1}{2}(\tan 2x + \cot 2x) + C \)
• (b) \( 2(\tan 2x + \cot 2x) + C \)
• (c) \( \frac{1}{2}(\tan 2x - \cot 2x) + C \)
• (d) \( -\frac{1}{2}(\tan 2x - \cot 2x) + C \)
• (e) \( \frac{1}{2}(\tan 2x + \cot 2x) + C \)

If \( x + 13y = 40 \) is normal to the curve \( y = 5x^2 + ax + \beta \) at the point \( (1,3) \), then the value of \( a\beta \) is equal to:

• (a) 15
• (b) -6
• (c) 6
• (d) 13
• (e) -15

Let \( f(x) = \begin{cases} 3x + 6, & if x \ge c
x^2 - 3x - 1, & if x < c \end{cases} \), where \( x \in \mathbb{R} \) and \( c \) is a constant. The values of \( c \) for which \( f \) is continuous on \( \mathbb{R} \) are:

• (a) -7, 1
• (b) 1, 3
• (c) -1, 7
• (d) -1, 6
• (e) 2, -3

In a box there are four marbles and each of them is marked with distinct number from the set \( \{1, 2, 5, 10\} \). If one marble is randomly selected four times with replacement and the number on it noted, then the probability that the sum of numbers equals 18 is:

• (a) \( \frac{1}{64} \)
• (b) \( \frac{3}{16} \)
• (c) \( \frac{5}{32} \)
• (d) \( \frac{3}{32} \)
• (e) \( \frac{1}{32} \)

Three fair dice are rolled simultaneously. Let \( a, b, c \) be the numbers on the top of the dice. Then the probability that \( \min(a, b, c) = 6 \) is:

• (a) \( \frac{1}{216} \)
• (b) \( \frac{1}{36} \)
• (c) \( \frac{1}{6} \)
• (d) \( \frac{11}{216} \)
• (e) \( \frac{5}{6} \)

Oxidation number of potassium in \(K_2O\), \(K_2O_2\) and \(KO_2\), respectively, is:

• (a) \(+2\), \(+1\) and \(+\frac{1}{2}\)
• (b) \(+1\), \(+1\) and \(+1\)
• (c) \(+1\), \(+4\) and \(+2\)
• (d) \(+1\), \(+2\) and \(+4\)

Which one of the following is not an allylic halide?

• (a) 4-bromopent-2-ene
• (b) 3-bromo-2-methylbut-1-ene
• (c) 1-bromobut-2-ene
• (d) 4-bromobut-1-ene
• (e) 3-bromo-2-methylpropene

A neutral molecule \(XF_3\) has a zero dipole moment. The element X is most likely:

• (a) chlorine
• (b) boron
• (c) nitrogen
• (d) carbon
• (e) bromine

Among the following species, identify the pair having same bond order \(CN^-\), \(O_2^-\), \(NO^+\), \(CN^+\):

• (a) \(CN^-\) and \(O_2^-\)
• (b) \(O_2^-\) and \(NO^+\)
• (c) \(CN^-\) and \(NO^+\)
• (d) \(CN^-\) and \(CN^+\)
• (e) \(NO^+\) and \(CN^+\)

Which one of the following conditions will favour maximum formation of the product in the reaction, \(A_2(g) + B_2(g) \rightleftharpoons X_2(g) \quad \Delta_rH = -x kJ\):

• (a) Low temperature and high pressure
• (b) Low temperature and low pressure
• (c) High temperature and low pressure
• (d) High temperature and high pressure

A light ray enters from medium 1 to medium 2. Its velocity in medium 1 is \(2 \times 10^8\) m/s and in medium 2 is \(1.5 \times 10^8\) m/s. The critical angle for the pair of media is:

• (a) \(\sin^{-1}(0.75)\)
• (b) \(\sin^{-1}(0.5)\)
• (c) \(\sin^{-1}(0.66)\)
• (d) \(\sin^{-1}(0.8)\)

A container with a pin hole at the bottom is filled with water and kerosene (specific gravity 0.8). The height of the water layer is 10 cm and the kerosene layer is 20 cm. The velocity of efflux of water is:

• (a) 2.3 m/s
• (b) 4.5 m/s
• (c) 1.5 m/s
• (d) 3.2 m/s

In a Young's double slit experiment, the intensity at a point where the path difference is \(\frac{\lambda}{6}\) (\(\lambda\) being the wavelength of light used) is \(I\). If \(I_0\) denotes the maximum intensity, \(I/I_0\) is equal to:

• (a) 1/2
• (b) \(\sqrt{3}/2\)
• (c) 3/4
• (d) 1/4

A particle of mass \(m\) is moving in a circular path of constant radius \(r\) such that its centripetal acceleration \(a_c\) is varying with time \(t\) as \(a_c = k^2 r t^2\), where \(k\) is a constant. The power delivered to the particle by the forces acting on it is:

• (a) \(2\pi mk^2 r^2 t\)
• (b) \(m k^2 r^2 t\)
• (c) \(\frac{1}{3} m k^4 r^2 t^5\)
• (d) 0

A source of frequency \(f\) gives 5 beats/sec when sounded with a source of frequency 200 Hz. The second harmonic of \(f\) gives 10 beats/sec when sounded with a source of frequency 420 Hz. The value of \(f\) is:

• (a) 195 Hz
• (b) 205 Hz
• (c) 190 Hz
• (d) 210 Hz