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 $a =\hat{ i }-2 \hat{ j }+3 \hat{ k }$. If $b$ is a vector such that $a \cdot b =| b |^{2}$ and $| a - b |=\sqrt{7}$, then $| b |$ 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 $\lambda\left(3\hat{i}+2\hat{j}-6\hat{k}\right)$ is a unit vector, then the values of $\lambda$ are

The area bounded by $y =x^{2} +3$ and $y =2x+3$ 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:

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

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

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}} $

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

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

$ ^{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

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

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

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

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

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

The reaction of $H_2$ is given below $H_2 + CO + R - CH = CH_2 \to R - CH_2 - CH_2 - CHO$ is specifically called as

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

The boiling point of para nitrophenol is greater than ortho nitrophenol, because:

The correct increasing order of the acid strength of benzoic acid $(I), 4$ -nitrobenzoic acid $(II), 3,4$ -dinitrobenzoic acid(III) and $4$-methoxybenzoic acid $(IV)$ is