Match the major products obtained in the reactions given in List-I with the corresponding structures in List-II and choose the correct option.
| List-I | List-II |
|---|---|
(P)
|
(1)
|
(Q)
|
(2)
|
(R)
|
(3)
|
(S)
|
(4)
|
(5)
|
Consider the function $f : (0, \infty) \to (-\infty, \infty)$ given by
$f(x) = \sqrt{x} \log_e(x) - x + 1$.
Then which one of the following statements is TRUE?
Let $P$ be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point $P$ is $4$. Let $Q$ be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to the circle at the point $Q$ is $-1$. Let $R$ be the point in the first quadrant lying on the ellipse $x^2 + 4y^2 = 8$ such that the slope of the tangent to the ellipse at the point $R$ is $-\frac{1}{2}$. Then the radius of the circle passing through the points $P, Q$ and $R$ is
Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?
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