List of top Questions asked in KEAM

If $ y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right), $ then $ \frac{dy}{dx} $ is equal to

Let $ {{a}_{n}}={{i}^{{{(n+1)}^{2}}}}, $ where $ i=\sqrt{-1} $ and $ n=1,2,3..... $ . Then the value of $ {{a}_{1}}+{{a}_{3}}+{{a}_{5}}+...+{{a}_{25}} $ is

If $*$ is defined by $a*b$ = $a - b^2$ and $\oplus$ is defined by $\oplus$ = $a^2 + b$, where a and b are integers, then ($3 \oplus 4) * 5$ is equal to

The number of functions that can be defined from the set $A \,= \{a, b, c, d\}$ into the set $B\, =\{1,2,3\}$ is equal to

The length of the transverse axis of a hyperbola is $2 \,\cos \,\alpha$ . The foci of the hyperbola are the same as that of the ellipse $9x^{2}+16y^{2}=144$ . The equation of the hyperbola is

The perpendicular distance from the point $(1, -1)$ to the line $x + 5y - 9 = 0$ is equal to

The value of the determinant $\begin{vmatrix}\sin ^{2} 36^{\circ} & \cos ^{2} 36^{\circ} & \cot 135^{\circ} \\ \sin ^{2} 53^{\circ} & \cot 135^{\circ} & \cos ^{2} 53^{\circ} \\ \cot 135^{\circ} & \cos ^{2} 25^{\circ} & \cos ^{2} 65^{\circ}\end{vmatrix}$ is

Two finite sets $A $ and$ B $ have m and n elements respectively. If the total number of subsets of $A $ is 112 more than the total number of subsets of $B$, then the value of m is

For any two statements $p$ and $q$, the statement $\sim\left(p \vee q\right) \vee \left(\sim p \wedge q\right)$ the is equivalent to

The equation of the tangent to the curve $ y={{(1+x)}^{y}}+{{\sin }^{-1}}({{\sin }^{2}}x) $ at $ x=0 $ is:

$ ^{15}{{C}_{0}}{{.}^{5}}{{C}_{5}}{{+}^{15}}{{C}_{1}}{{.}^{5}}{{C}_{4}}{{+}^{15}}{{C}_{2}}{{.}^{5}}{{C}_{3}}{{+}^{15}}{{C}_{3}}{{.}^{5}}{{C}_{2}} $ $ {{+}^{15}}{{C}_{4}}{{.}^{5}}{{C}_{1}} $ is equal to

If $a$ and $b$ are positive numbers such that $ a>b, $ then the minimum value of $ a\sec \theta -b\tan \theta \left( 0

If $ {{c}_{1}},{{c}_{2}},{{c}_{3}},{{c}_{4}},{{c}_{5}} $ and $ {{c}_{6}} $ are constants, then the order of the differential equation whose general solution is given by $ y={{c}_{1}}cos $ $ (x+{{c}_{2}})+{{c}_{3}}\sin (x+{{c}_{4}})+{{c}_{5}}{{e}^{x}}+{{c}_{6}} $

If $\begin{bmatrix}1&x&1\end{bmatrix} \begin{bmatrix}1&3&2\\ 2&5&1\\ 15&3&2\end{bmatrix}\begin{bmatrix}1\\ 2\\ x\end{bmatrix} = 0 $, then x can be

Equation of the plane passing through the intersection of the planes $ x+y+z=6 $ and $ 2x+3y+4z+5=0 $ and the point $(1, 1, 1)$ is

The area of the plane region bounded by the curve $ x={{y}^{2}}-2 $ and the line $ y=-x $ is (in square units)

The chord joining the points $(5, 5)$ and $(11, 227)$ on the curve $y =3x^{2}-11x-15$ is parallel to tangent at a point on the curve. Then the abscissa of the point is

Two distinct numbers $x$ and $y$ are chosen from $1,2,3,4,5$. The probability that the arithmetic mean of $x$ and $y$ is an integer is

A man of $2\,m$ height walks at a uniform speed of $6 \,km/h$ away from a lamp post of $6 \,m$ height. The rate at which the length of his shadow increases is

If $a = e^{i \theta}$ , then $\frac{1 + a}{1-a}$ is equal to

A complete cycle of a traffic light takes $60\, seconds$. During each cycle the light is green for $25\, seconds$, yellow for $5 \,seconds$ and red for $30\, seconds$. At a randomly chosen time, the probability that the light will not be green, is

If tan $\frac{\theta}{2}=\frac{1}{2}$,then the value of sin $\theta$ is

If $A = \begin{pmatrix}1&0\\ 1&1\end{pmatrix}$ , then $A^n + nI$ is equal to

If $ y={{\sin }^{-1}}\sqrt{1-x}, $ then $ \frac{dy}{dx} $ is equal to

An insoluble dye is reduced to a soluble colourless leuco form by an alkaline reducing agent. The fibre is soaked in the dye solution and then exposed to air to develop the colour. The dye is