Three mangoes and three apples are kept in a box. If two fruits are selected at random from the box, the probability that the selection will contain one mango and one apple is :

The general solution of the differential equation \(\frac{d y}{d x}=1+x+y+x y\) is given by:

If \(\left[\begin{array}{cc}2 & -1 \\ 1 & 0 \\ -3 & 4\end{array}\right] A=\left[\begin{array}{ccc}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]\), then matrix \(A\) is equal to:

The solution of the differential equation \(\frac{d y}{d x}=-\frac{x}{y}\) is:

Match List-I with List-II

List-I	List-II
(A) Maximum value of \(f(x)=\sin ^2 x-\cos ^2 x \forall x \in(\pi, 2 \pi)\)	(I) \(0\)
(B) Minimum value of \(f(x)=\sin x \cdot \cos x\)	(II) 1
(C) Point of Minima of \(f(x)=x^x(x>0)\)	(III) \(-\frac{1}{2}\)
(D) Maximum value of \(f(x)=-x^{2026}\)	(IV) \(\frac{1}{e}\)

Choose the correct answer from the options given below :

If \(y=\sin ^{-1} \sqrt{\frac{x}{x+1}}+\sec ^{-1} \sqrt{\frac{x+1}{x}}\), then \(\frac{d y}{d x}\) is

The value of \(x\) for which the slope of the curve \(y=-x^3+3 x^2+x^2+9 x-27\) is maximum, is:

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.
Heroin, commonly called smack is chemically __________ which is a white, odourless, bitter crystalline compound.

In a college, 30 % students fail in Physics, 25 % fail in Mathematics and 10 % fail in both. One student is chosen at random. The probability that she fails in Physics if she has failed in Mathematics is

The value of the integral \(\int_2^4 \frac{x}{x^2+1} d x\) is :

The compound does not respond to iodoform test is :

Which of the statements given below are correct with reference to corrosion:
(A) Corrosion covers the surfaces of metallic objects with oxides or other salts.
(B) In corrosion, a metal is oxidised by the loss of electrons to oxygen and results in the formation of oxides.
(C) Rust is hydrated ferric oxide.
(D) A sacrificial electrode is used to save an object from corrosion.
Choose the correct answer from the options given below.

Two parallel plate capacitors of capacitances \(2 \mu F\) and \(3 \mu F\) are joined in series and connected to a battery of \(V\) volts. The values of potential across the two capacitors V1 and V2 and energy stored in them U1 and U2 are related as :