If \(e^y(x+1)=1\) and \(\frac{d^2 y}{d x^2}=k\left(\frac{d y}{d x}\right)^2\), then \(k\) is equal to

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is four. The probability that it is actually four is :

Which of the following statements is/are correct?
(A) \(|x|\) is continuous everywhere in its domain.
\((B)|x|\) is differentiable everywhere in its domain.
(C) \([x]\) is continuous at every integral point.
\((D)[x]\) is discontinuous at every integral point.
Choose the correct answer from the options given below :

If \(\left[\begin{array}{cc}4 x & 5 x-7 \\ 4 x & 2 x+y\end{array}\right]=\left[\begin{array}{cc}x+6 & y \\ 7 y-13 & 7\end{array}\right]\) then the value of \(2 x+y\) is:

The degree of the differential equation
\(\left[1+\left(\frac{d y}{d x}\right)\right]^3=\left(\frac{d^2 y}{d x^2}\right)^2\) is :

If the probability of two successes is 9 times the probability of 3 successes in 3 trials of a binomial distribution, then the probability of success in each trial is:

The value of the determinant \(\left|\begin{array}{ccc}85 & 15 & 30 \\ 1 & 3 & 4 \\ 17 & 3 & 6\end{array}\right|\) is:

Question based on the following passage:
P-Block elements are placed in group 13 to 18 in the periodic table. Group 15 is nitrogen family.
Nitrogen can react with hydrogen, oxygen, halogens etc. It shows multiple oxidation states in its various oxides i.e. \(N_2O, N_2O_3, N_2O_5\) etc. Important allotropic forms of phosphorus are white, red and black phosporus. Various compounds of phosphorus are phosphine, \(PCl_3, PCl_5\) and oxoacids of phosphorus. Oxidation states \(+1\) to \(+5\) are seen in various oxoacids of phosphorus. Group 17 elements are collectively known as halogens. Bond dissociation enthalpy of \(F_2\) is smaller than \(Cl_2\).
The gases present in group 18 are chemically unreactive, they form only few compounds. Neil Bartlett (in 1962) observed the reaction of a noble gas. Since then, many noble gas compounds have been synthesised.
Amongst all noble gases, only Xe can form large number of compounds because :
(A) Its size is similar to that of Pt
(B) Its \(IE_1\) is almost identical with that of \(O_2\)
(C) Xe is not inert
(D) Xe is very large in size
(E) It has great affinity with \(PtF _6^{-}\)
Choose the correct answer from the options given below :

Read the following paragraph carefully and answer the given questions.
Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. For determining the cause of allergy, the patient is exposed to or injected with very small doses of possible allergens, and the reactions studied. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. This is obtained by acetylation of morphine. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal.
Which one of the following drugs can not quickly reduce the symptoms of allergy?

match List - I with List - II

List - I (Differential Equation)	List - II (Degree)
(A) \(x y \frac{d^2 y}{d x^2}+x\left(\frac{d y}{d x}\right)^2-y \frac{d y}{d x}=0\)	(I) 3
(B) \(\frac{d^2 y}{d x^2}+\log \left(\frac{d y}{d x}\right)=0\)	(II) 1
(C) \(\left(\frac{d^2 y}{d x^2}\right)^2+\left(\frac{d y}{d x}\right)^3+\frac{d y}{d x}+1=0\)	(III) not defined
(D) \(2 x^2\left(\frac{d^2 y}{d x^2}\right)^3-5\left(\frac{d y}{d x}\right)^2+y=0\)	(IV) 2

Choose the correct answert from the option given below :

Read the passage given below and answer the question.
The transition elements which give the greatest number of oxidation states occur in or near the middle of the series. Manganese, for example, exhibits all the oxidation states from +2 to +7 . The lesser number of oxidation states at extreme ends stems from either too few electrons to lose or share (Sc, Ti) on too many d-electrons (hence fewer orbitals available in which to share electrons with others) for higher valence \(( Cu , Zn )\). Thus early in the series scandium (II) is virtually unknown and titanium (IV) is more stable than Ti (III) or Ti (II). At the other end, the only oxidation state of zinc is \(+ 2\) (no d electrons are involved). The maximum oxidation states of reasonable stability correspond in value to the sum of the s - and d-electrons upto manganese ( \(Ti ^{I V} O _2, V^V O _2^{+}, Cr ^{V I} O _4^{2-}\), \(Mn ^{\text {VII }} O _4\) ) followed by a rather abrupt decrease in stability of higher oxidation states, so that the typical species to follow are \(Fe ^{I I, I I I}, Co ^{I I, I I I}, Ni ^{I I}, Cu ^{I, I I}, Zn ^{I I}\).
The variability of oxidation states, a characteristic of transition elements, arises out of incomplete filling of d-orbitals in such a way that their oxidation states differ from each other by unity, e.g. \(V ^{I I}, V^{I I I}, V^{I V}, V^V\). This is in contrast with the variability of oxidation states of non transition elements where oxidation states normally differ by a unit of two.
An interesting feature in the variability of oxidation states of the d-block elements is noticed among the groups (groups 4 through 10). Although in the p-block, the lower oxidation states are favoured by the heavier members (due to inert pair effect), the opposite is true in the groups of dblock. For example, in group 6, \(Mo ( VI )\) and \(W ( VI )\) are found to be more stable than \(Cr ( VI )\). The \(Cr ( VI )\) in the form of dichromate in acidic medium is a strong oxidising agent, whereas \(MoO _3\) and \(WO _3\) are not.
Low oxidation states are found when a complex compound has ligands capable of \(\pi\)-acceptor character in addition to the \(\sigma\)-bonding. For example, in \(Ni ( CO )_4\) and \(Fe ( CO )_5\), the oxidation state of nickel and iron is zero.
Most acidic species among the following is :

The core of electromagnets are made of ferromagnetic material like soft iron because the soft iron has-

A transformer has 4000 turns in primary and 100 turns in secondary coils. The primary coil in connected to a 240 V and secondary to a bulb of 1.2 \(\Omega\) resistance. The current in the bulb is :