The particular solution of the differential equation \(x d y=\left(2 x^2+1\right) d x, x \neq 0\), given that \(y=1\) when \(x=1\) is:

The value of the determinant \(\left|\begin{array}{ccc}\cos ^2 \theta & \cos \theta \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right|\) is equal to

The value of a for which \(\int_1^a x^2 d x=\frac{7}{3}\) is:

If a fair coin is tossed 10 times, then the probability of obtaining at least one head is:

A die is thrown three times. If the first throw is a five, the probability of getting 14 as the sum is:

The least positive integer \(x\) satisfying \(28=x(\bmod 6)\) is :

For which value of \(k\), the given system of linear equations is inconsistent?
\(3 x+6 y+3 z=4\)
\(x+k y+2 z=6\)
\(z=7\)

Question based on following passage :
Adsorption is the process of attracting and retaining the adsorbate molecules on the surface of adsorbent. It is of two types i.e. Physisrption and chemisorption. To increase the rate of reaction, the substance called 'catalyst' is added. Promotors enhance the activity of catalyst while poison decrease the catalyst activity. Solutions may be true solution, suspension and colloids. Colloids are heterogenous but quite stable. On the basis of nature of interaction, colloids may be Lyophilic or Lyophobic. On the basis of type of dispersed phase particles, there may be multimolecular colloids, macromlecular colloids and associated colloids. The congulation of colloids may be done by boiling, dialysis or by mixing oppositely charged sols.
Reason for the stability of colloidal solution is :

Three magnetic materials: (A) Paramagnetic, (B) Diamagnetic, (C) Ferromagnetic. The correct order of increasing magnetic susceptibility is :

A boat is moving due east in a region where the Earth's magnetic field is \( 5.0 \times 10^{-5} \text{ NA}^{-1}\text{m}^{-1} \) due north and horizontal. The boat carries a vertical aerial 2 m long. If the speed of the boat is \( 10.5 \text{ m s}^{-1} \), find the induced e.m.f. in the aerial.

Match List-I with List-II

List-I (Differential equation)	List-II (Order and degree)
(A) \(\left(y^{\prime \prime}\right)^3+\left(y^{\prime}\right)^4-6=\left(y^{\prime \prime \prime}\right)^2\)	(I) Order \(=1\), Degree = 2
(B) \(\sqrt{\left(y^{\prime}\right)^2+5}=y^{\prime \prime}\)	(II) Order \(=2\), Degree = 3
(C) \(\left(y^{\prime}\right)^2=\left(2+y^{\prime \prime}\right)^{3 / 2}\)	(III) Order \(=2\), Degree = 2
(D) \(y=x y^{\prime}+\sqrt{a^2\left(y^{\prime}\right)^2+b^2}\)	(IV) Order \(=3\), Degree = 2

Choose the correct answer from the options given below :

A coil of 500 turns is threaded by a magnetic flux of \( 7 \times 10^{-5} \text{ Wb} \). If the flux changes to \( 3 \times 10^{-5} \text{ Wb} \) in 0.01 s, what average induced emf is developed in the coil? If resistance offered by the coil is \( 10 \, \Omega \), then the flow of average induced current will be :

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 of the following technique is not very useful to detect the cancers of internal organs?