class 12

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JEE Advanced

For the identification of $β$-naphthol using dye test, it is necessary to use

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Let $S$be the set of all column matrices $[b_{1}b_{2}b_{3}]$such that $b_{1},b_{2},b_{3}∈R$and the system of equations (in real variable)$−x+2y+5z=b_{1}$$2x−4y+3z=b_{2}$$x−2y+2z=b_{3}$has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $[b_{1}b_{2}b_{3}]∈S$?(a) $x+2y+3z=b_{1},4y+5z=b_{2}$and $x+2y+6z=b_{3}$(b) $x+y+3z=b_{1},5x+2y+6z=b_{2}$and $−2x−y−3z=b_{3}$(c) $−x+2y−5z=b_{1},2x−4y+10z=b_{2}$and $x−2y+5z=b_{3}$(d) $x+2y+5z=b_{1},2x+3z=b_{2}$and $x+4y−5z=b_{3}$

The common tangents to the circle $x_{2}+y_{2}=2$ and the parabola $y_{2}=8x$ touch the circle at $P,Q$ andthe parabola at $R,S$. Then area of quadrilateral $PQRS$ is

Let $O$be the origin and let PQR be an arbitrary triangle. The point S is such that$OPO˙Q+ORO˙S=ORO˙P+OQO˙S=OQ$.$OR+OPO˙S$Then the triangle PQ has S as its:circumcentre (b) orthocentre (c) incentre (d) centroid

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is

Let $XandY$be two events that $P(X)=31 ,P(X|Y)=21 andP(Y|X)=52 $then:$P(Y)=154 $ (b) $P(X∪Y)=52 $$P(X_{prime}|Y)=21 $ (d) $P(X∩Y)=51 $

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. $67$

If $f(x)∣cos(2x)cos(2x)sin(2x)−cosxcosx−sinxsinxsinxcosx∣,then:$$f_{prime}(x)=0$at exactly three point in $(−π,π)$$f_{prime}(x)=0$at more than three point in $(−π,π)$$f(x)$attains its maximum at $x=0$$f(x)$attains its minimum at $x=0$

From a point $P(λ,λ,λ)$, perpendicular PQ and PR are drawn respectively on the lines $y=x,z=1$ and $y=−x,z=−1$.If P is such that $∠QPR$ is a right angle, then the possible value(s) of $λ$ is/(are)