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JEE Advanced

A ray of light travellig in water is incident on its surface open to air. The angle of incidence is $θ$, which is less than the critical angle. Then there will be

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Let $f:[0,2]→R$ be a function which is continuous on [0,2] and is differentiable on (0,2) with $f(0)=1$$Let:F(x)=∫_{0}f(t )dtforx∈[0,2]I˙fF_{prime}(x)=f_{prime}(x)$ . for all $x∈(0,2),$ then $F(2)$ equals (a)$e_{2}−1$ (b) $e_{4}−1$(c)$e−1$ (d) $e_{4}$

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover cards numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done isa.$264$ b. $265$ c. $53$ d. $67$

Let $f(x)=xsinπx$, $x>0$ Then for all natural numbers n, f\displaystyle{\left({x}\right)}{v}{a}{n}{i}{s}{h}{e}{s}{a}{t}

Let $f:[−21 ,2]→R$ and $g:[−21 ,2]→R$ be functions defined by $f(x)=[x_{2}−3]$ and $g(x)=∣x∣f(x)+∣4x−7∣f(x)$, where [y] denotes the greatest integer less than or equal to y for $y∈R$. Then,

The value of $(((g)_{2}9)_{2})_{(log)((log)9)1}×(7 )_{(log)71}$is ________.

The value of $∫_{0}4x_{3}{dx_{2}d_{2} (1−x_{2})_{5}}dxis$

Let $−61 <θ<−12π $ Suppose $α_{1}andβ_{1}$, are the roots of the equation $x_{2}−2xsecθ+1=0$ and $α_{2}andβ_{2}$ are the roots of the equation $x_{2}+2xtanθ−1=0$. If $α_{1}>β_{1}$ and $α_{2}>β_{2}$, then $α_{1}+β_{2}$ equals

Let P and Q be distinct points on the parabola $y_{2}=2x$ such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle $ΔOPQ$ is $32$ , then which of the following is (are) the coordinates of $P?$