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Amongst the given options, the compound(s) in which all the atoms are in the one plane in all the possible conformation (if any), is (are)

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Let $P$be a point in the first octant, whose image $Q$in the plane $x+y=3$(that is, the line segment $PQ$is perpendicular to the plane $x+y=3$and the mid-point of $PQ$lies in the plane $x+y=3)$lies on the z-axis. Let the distance of $P$from the x-axis be 5. If $R$is the image of $P$in the xy-plane, then the length of $PR$is _______.

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60

Let $XandY$be two arbitrary, $3×3$, non-zero, skew-symmetric matrices and $Z$be an arbitrary $3×3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?a.$Y_{3}Z_{4}Z_{4}Y_{3}$b. $x_{44}+Y_{44}$c. $X_{4}Z_{3}−Z_{3}X_{4}$d. $X_{23}+Y_{23}$

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}

A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the tune, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?

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)

Let $S$be the circle in the $xy$-plane defined by the equation $x_{2}+y_{2}=4.$(For Ques. No 15 and 16)Let $P$be a point on the circle $S$with both coordinates being positive. Let the tangent to $S$at $P$intersect the coordinate axes at the points $M$and $N$. Then, the mid-point of the line segment $MN$must lie on the curve$(x+y)_{2}=3xy$(b) $x_{2/3}+y_{2/3}=2_{4/3}$(c) $x_{2}+y_{2}=2xy$(d) $x_{2}+y_{2}=x_{2}y_{2}$