List of top Questions asked in KEAM

\(∫\dfrac{x+5}{x^{2}+1}dx=\)

\(∫sin^2 πx dx =\)?

\(\int\frac{1}{x^2-2x+2}dx=\)
\(\lim_{x\to0}\dfrac{ln(1+(ln5)x)}{5^{x}}-1\)
If \(y=x^{e^x}+x^{e} for x>0\),the  \( \dfrac{dy}{dx}\) is equal to ?
The equation of the line passing through origin which is parallel to the tangent of the curve \(y=\dfrac{x-2}{x-3}\) at \(x=4\) is 

Let  \(f:R→R\)  be defined by  \(f(x)=\){\(2x+3,x≤5 3x+α,x>5\) .Then the value of \(α\) so that f is continuous on \(R\) is 

 

 

Let \(f(x)=αsin^23x\) .If  \(f ′(\dfrac{π}{12})\) = \(−3\),then the value of \(α \) is

The minimum of \(f(x)=\sqrt{(10-x^2)}\) in the interval \([-3,2]\) is

Let \(f:R→R\) be a function defined by
\(f(x) =   \begin{cases}     3e^x       & \quad \text {if}\  {x<0}\\     x^2+3x+3  & \quad \text {if}\  0≤x<1   \\ x^2-3x-3 & \quad \text {if} \ x≥1\end{cases}\)
Let \( f(x)=(1-1/x)2,x>0\). Then
Let [.]denote the greatest integer function and f(x)=[x]+[2-x],-1≤x≤4.Then
\(\lim_{x\to0} e^{x}-\dfrac{1}{3}×(1-e^{2x})\)\(=\)?
Let \(f(x)\)={-\(5,x≤0 x-5,x>0\) and \(g(x)=If(x)+2f(IxI)\).Then \(g(-2)\) will be 

The projection of the vector \(a=2i-3j+4k \)on the vector \(b=i+2j+2k\) is ?

The value of λ for which the vectors \(2i-3j+4k\) and \(-4i+λj-8k\) are collinear is 
Let x be a real number and a be any non-zero vector such \(|(4-x)a|<|3a|\).Then which of the following options is correct?
Let \(a\) and \(b\) be perpendicular vectors such that \(IaI=√104\) and \(IbI=6\).Then the value of\( Ia-bI\) is ?
The position vectors of two points P and Q are given by \(OP=2a-b\) and \(OQ=a+3b\) ,respectively.If a point R divides the line joining P and Q internally in the ratio \(1:2\) ,then the position vector of the point R is ?
The value of \(λ\) for which the vectors \(i+j-k\)and \(λi+3j+k\) are perpendicular is ?
The direction cosines of the vector $\vec{a} = -2\hat{i} + \hat{j} - \hat{k}$ are

Let \(a=i+j+2k\) and \(b=i-2j+3k\) be two vectors. Then the unit vector in the direction of \(a-b\) is

The value of \(\tan^{-1}(√3)-\sec^{-1}(\dfrac{2}{√3})\) is ?

Let, \(a=i-j+2k\). Then the vector in the direction of a with magnitude \(5\) units is ?

The value of \(\tan^{-1}(\dfrac{1}{2})+\tan^{-1}(\dfrac{2}{5})\) is ?