The general solution of the differential equation \((1+y) d x-2 x d y=0\) is :

If \(f(x)=\left\{\begin{array}{ll}\frac{\tan \left(\frac{\pi}{4}-x\right)}{\cot 2 x}, & x \neq \frac{\pi}{4} \\ K, & x=\frac{\pi}{4}\end{array}\right.\) is continuous at \(x=\frac{\pi}{4}\), then the value of \(K\) is :

Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white balls. A bag is selected at random and then a ball is drawn from it. The probability that the ball drawn is black is :

The function \(f(x)=2 \log _e(x-2)-x^2+4 x+1,(x>2)\) is increasing on the interval :

Consider the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\), then which of the following statements are correct?
(A) It is homogeneous
(B) It has \(\sec x\) as its integrating factor
(C) Its general solution is \(y \sec x=\tan x+c\), where \(c\) is an arbitrary constant.
(D) Its degree is not defined
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The relation \(R\) on the set \(A\) of points in a plane, given by \(R=\{(P, Q)\) : Distance of the point \(P\) from the origin is same as the distance of

Match List-I with List-II

List-I	List-II
(A) The maximum value of \(f(x)=\sin (3 x)+6\)	(I) 2
(B) The maximum value of \(f(x)=-|x+2|+4\)	(II) 5
(C) The maximum value of \(f(x)=(3 x+1)^2+5\)	(III) 7
(D) The maximum value of \(f(x)=2 \cos x+4\)	(IV) 4

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The value of \(\int_0^1 x e^x d x\) is :

Let \(f: R \rightarrow R\) be defined as \(f(x)=10 x\). Then (Where R is the set of real numbers):

Answer the question on basis of passage given below :
P block elements are placed in groups 13 to 18 of the periodic table. Their valence shell electronic configuration is \(ns^2 np^{1-6}\). Group 16 of the p block elements are known as the group of chalcogens having \(ns^2 np^4\) as their general electronic configuration. They exhibit number of oxidation states but the stability of \(- 2\) oxidation state decreases down the group. They are sometimes also known as group of chalcogens as the name is derived from the Greek word for brass and points to the association of sulphur and its congeners with copper. Their ionisation enthalpy decreases down the group. Oxygen shows anomalous behaviour due to its small size and high electronegativity.
Given below are the Hydrides formed by group 15 elements. Choose the correct decreasing order of basicity of the following hydrides :
(A) \(AsH_3\)
(B) \(PH_3\)
(C) \(NH_3\)
(D) \(BiH_3\)
(E) \(SbH_3\)
Choose the correct answer from the options given below :

Which of the following is always upright in an ecosystem?

The function \(f: R \rightarrow R , f(x)=x^2\) is

Match List-I with List-II

List-I (Function)	List-II (Property)
(A) \(f(x)=\left\{\begin{array}{ll}\frac{x}{|x|} & : x \neq 0 \\ 0 & : x=0\end{array}\right.\)	(I) continuous but not differentiable at  \(x=0\)
(B) \(f ( x )=| x |\)	(II) continuous but not differentiable at \(x=1\)
(C) \(f(x)=\left|x^2-1\right|\)	(III) discontinuous at \(x=0\)
(D) \(f(x)=|x-1|\)	(IV) continuous but not differentiable at \(x=1,-1\)

Choose the correct answer from the options given below: