List of top Questions asked in KEAM

The rod PQ slides along 2 parallel rails as shown in the figure. It has a length of 20 cm and is perpendicular to the 2 rails. It performs simple harmonic motion with amplitude 5 cm and frequency 10 Hz. The magnetic field is 10-4T and is directed perpendicular to the plane of paper. What is the peak induced electro-magnetic force?

\(∫\dfrac{1}{cosx(sinx+2cosx)} dx=\)
Let \(f:\R\rightarrow\R\) be defined by
\(f(x) =   \begin{cases} x & \text{if}\ x\leq1 \\  -x+2 & \text{if}\ x\gt1\end{cases}\)
Then \(\int^{2}_{0}f(x)dx=\)
\(∫xlog(1+x^2)dx=\)?
\(\int\frac{2\tan x+3}{\sin^2x+2\cos^2x}dx=\)
\(∫\dfrac{x+5}{x^{2}+1}dx=\)

\(∫sin^2 πx dx =\)?

\(\int\frac{1}{x^2-2x+2}dx=\)
\(\lim_{x\to0}\dfrac{ln(1+(ln5)x)}{5^{x}}-1\)
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 ?