Three pipes A, B & C fill a tank in 3 hours working simultaneously. The pipe C is twice as faster as B and B is twice as fast as A . The time taken by pipe A alone to fill the tank is :

If \(A, B\) and \(C\) are all the corner points of feasible region (bounded) of an LPP with objective function \(Z\) that needs to be maximized such that \(Z_A>Z_B\) and \(Z_C < Z_A\) (Here \(Z_A\) denotes value of \(Z\) at \(A\) ), then which of the following statements is TRUE ?

If A and B are events such that \(P(A \mid B)=P(B \mid A)\), then :

Consider the following hypothesis test:
\( \begin{array}{l} H_0: \mu \leq 12 \\ H_a: \mu>12 \end{array} \)
A sample of 25 provided a sample mean \(\bar{x}=14\) and a sample standard deviation \(S=4.32\). If \(t_{0.05}=1.711\), then which of the following is correct?
(A) The test statistic is defined as \(t=\frac{\bar{x}-\mu}{S / \sqrt{n}}\).
(B) The value of the test statistic is 1.31 .
(C) At \(\alpha=0.05\), the null hypothesis is rejected.
(D) If the value of the \(t\)-statistic is less than \(t_\alpha\), then null hypothesis is accepted.
Choose the correct answer from the options given below:

If \(V=\frac{4 \pi r^3}{3}\), at what rate in cubic units \(/ sec\) is V increasing when \(r=10\), and \(\frac{d r}{d t}=0.01 ?\)

Match List - I with List - II.

List - I	List - II
(A) If A and B are mutually exclusive events, then \(P ( A \cup B )=\)	(I) \(\frac{ P ( A \cap B )}{ P ( B )}, P ( B ) \neq 0\)
(B) If A and B are independent events, then \(P ( A \cap B )=\)	(II) \(\frac{ P ( A \cap B )}{ P ( A )}, P ( A ) \neq 0\)
(C) If A and B are two events of a sample space of an experiment, then \(P ( A / B )=\)	(III) \(P ( A ) \cdot P ( B )\)
(D) If A and B are two events of a sample space of an experiment, then \(P ( B / A )=\)	(IV) \(P ( A )+ P ( B )\)

Choose the correct answer from the options given below :

The least positive integer \(x\) satisfying \(29 \equiv x(\bmod 5)\) is

Read the passage carefully and answer the questions
The transition metals and their compounds are known for their catalytic activity. This activity is ascribed to their ability to adopt multiple oxidation states and to form complexes. Vanadium(V) oxide (in Contact Process), finely divided iron, and nickel (in Catalytic Hydrogenation) are some of the examples. Catalysts at a solid surface involve the formation of bonds between reactant molecules and atoms of the surface of the catalyst (first row transition metals utilise 3d and 4s electrons for bonding). This has the effect of increasing the concentration of the reactants at the catalyst surface and also weakening of the bonds in the reacting molecules (the activation energy is lowering). Also because the transition metal ions can change their oxidation states, they become more effective as catalysts.
\(TiCl_4 + Al(CH_3)_3\) is used as a catalyst in:

The area bounded by the curve y = 3 cos x and the x-axis between x = 0 and \(x=2 \pi\) is:

Progestasert and LNG-20 are the examples of:

Match List - I with List - II.

List - I (lon)	List - II (Magnetic Moment (BM))
(A) \(V^{2+}\)	(1) 1.73
(A) \(Ni ^{2+}\)	(II) 3.87
(C) \(Cu ^{2+}\)	(III) 4.90
(D) \(Cr ^{2+}\)	(IV) 2.84

Choose the correct answer from the options given below :

The value of parameter k for which the function \(f(x)=1+k x, k \neq 0\) is the inverse of itself is:

The colour of \(\left[ Ni \left( NO _2\right)_6\right]^{4-}\) complex is ______________