\(\cos ^3 \theta+\cos ^3\left(120^{\circ}+\theta\right)+\cos ^3\left(\theta-120^{\circ}\right)=\)

If \(f:[0,2) \rightarrow[R\) is defined by
\(f(x)=\left\{\begin{array}{cl}1+\frac{2 x}{k} & \text { for } \quad 0 \leq x < 1 \\ k x & \text { for } 1 \leq x < 2\end{array}\right.\) where \(k>0\), and \(f\) is such that \(\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)\), then

Four cards are drawn at random from a pack of 52 playing cards. The probability of getting all four cards of the same suit is

If the probability for \(A\) to fail in an exam is 0.2 and that for \(B\) is 0.3, then the probability that either \(A\) or \(B\) fails is \(\leq \ldots \ldots.\).

If the percentage error in the radius of circle is 3 , then the percentage error in its area is

If \(\mathrm{t}_{\mathrm{n}}=\frac{1}{4}(\mathrm{n}+2)(\mathrm{n}+3), \mathrm{n} \in \mathrm{N}\), then which one of the following is true?
Assertion (A) : \(\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_{2003}}=\frac{2003}{3009}\)
Reason (R) : \(\frac{1}{\mathrm{t}_1}+\frac{1}{\mathrm{t}_2}+\ldots+\frac{1}{\mathrm{t}_{\mathrm{n}}}=\frac{4 \mathrm{n}}{(2 \mathrm{n}+3)}\)

The length of the common chord of the two circles \(x^2+y^2-4 y=0\) and \(x^2+y^2-8 x\) \(-4 y+11=0\), is

A bag contains 19 red balls and 19 black balls. Two balls are chosen at a time repeatedly and discarded if they are of the same colour, but if they are different, black ball is discarded and red ball is returned to the bag. The probability that this process will terminate with one red ball is

A biker travels \(\frac{1}{3}\) of the distance \(L\) with speed \(v_1\) and \(\frac{2}{3}\) of the distance with speed \(v_2\). Then the average speed is

\(\begin{aligned} & \text { If } \int \frac{3 x+2}{4 x^2+4 x+5} d x=A \log \left(4 x^2+4 x+5\right)+B \operatorname{Tan}^{-1} \\ & \left(x+\frac{1}{2}\right)+C, \text { then }(A, B)=\end{aligned}\)

The general solution of the differential equation \((2 x-y)^2 d y-2(2 x-y)^2 d x-2 d x=0\) is

Using the standard electrode potentials given below identify the correct statements from the following.
\[
\begin{aligned}
\mathrm{Fe}^{2+}+2 e^{-} \longrightarrow \mathrm{Fe} ; E^{\circ} & =-0.44 \mathrm{~V} \\
\mathrm{Cu}^{2+}+2 e^{-} \longrightarrow \mathrm{Cu} ; E^{\circ} & =+0.34 \mathrm{~V} \\
\mathrm{Ag}^{+}+e^{-} \longrightarrow \mathrm{Ag} ; E^{\circ} & =+0.80 \mathrm{~V}
\end{aligned}
\]
(i) Copper can displace iron from \(\mathrm{FeSO}_4\) solution.
(ii) Iron can displace copper from \(\mathrm{CuSO}_4\) solution.
(iii) Silver can displace copper from \(\mathrm{CuSO}_4\) solution.
(iv) Iron can displace silver from \(\mathrm{AgNO}_3\) solution.

Which of the following statement is correct?
(A) Becquerel, who discovered natural radioactivity, belongs to France.
(B) Marconi, who discovered wireless telegraphy, was an American.
(C) Newton was an American, who discovered the laws of motion.
(D) Einstein belongs to England, who simplifies the laws of photoelectric effects.