List I List II (P) Let y(x) = cos (3cos–1x), x \in , \left[\right. - 1 , 1 \left]\right. , x \neq \pm \frac{\sqrt{3}}{2}. Then \frac{1}{y \left(\right. x \left.\right)} \textrm{ } \left{\right. \left(\right. x^{2} - 1 \left.\right) \textrm{ } \frac{d^{2} y \left(\right. x \left.\right)}{\left(dx\right)^{2}} + x \frac{dy \left(\right. x \left.\right)}{dx} \left.\right} equals (1) 1 (Q) Let A1, A2, ……., An (n > 2) be the vertices of a regular polygon of n sides with its centre at the origin. Let \overset{\rightarrow}{a_{k}} be the position vector of the point Ak, k = 1, 2, ……., n. If \left|\right. \textrm{ } \sum_{k = 1}^{n - 1} \left(\right. \overset{\rightarrow}{a_{k}} \times \overset{\rightarrow}{a_{k + 1}} \left.\right) \textrm{ } \left|\right. = \left|\right. \textrm{ } \sum_{k = 1}^{n - 1} \left(\right. \overset{\rightarrow}{a_{k}} \textrm{ } \cdot \textrm{ } \overset{\rightarrow}{a_{k + 1}} \left.\right) \textrm{ } \left|\right., then the minimum value of n is (2) 2 (R) If the normal from the point P (h, 1) on the ellipse \frac{x^{2}}{6} + \frac{y^{2}}{3} = 1 is perpendicular to the line x + y = 8, then the value of h is (3) 8 (S) Number of positive solutions satisfying the equation \left(tan\right)^{- 1} \left(\right. \frac{1}{2 x + 1} \left.\right) + \left(tan\right)^{- 1} \left(\right. \frac{1}{4 x + 1} \left.\right) = \left(tan\right)^{- 1} \left(\right. \frac{2}{x^{2}} \left.\right) is (4) 9

Engineering

Mathematics

Vector Product of Two Vectors

Variable Separable Differential Equation

Question

List I	List II
(P) Let y(x) = cos (3cos^–1x), $x \in, [- 1, 1], x \neq \pm \frac{\sqrt{3}}{2}$ . Then $\frac{1}{y (x)} {(x^{2} - 1) \frac{d^{2} y (x)}{{dx}^{2}} + x \frac{dy (x)}{dx}}$ equals	(1) 1
(Q) Let A₁, A₂, ……., A_n (n > 2) be the vertices of a regular polygon of n sides with its centre at the origin. Let $\vec{a_{k}}$ be the position vector of the point A_k, k = 1, 2, ……., n. If $\| \sum_{k = 1}^{n - 1} (\vec{a_{k}} \times \vec{a_{k + 1}}) \| = \| \sum_{k = 1}^{n - 1} (\vec{a_{k}} \cdot \vec{a_{k + 1}}) \|$ , then the minimum value of n is	(2) 2
(R) If the normal from the point P (h, 1) on the ellipse $\frac{x^{2}}{6} + \frac{y^{2}}{3} = 1$ is perpendicular to the line x + y = 8, then the value of h is	(3) 8
(S) Number of positive solutions satisfying the equation $\tan^{- 1} (\frac{1}{2 x + 1}) + \tan^{- 1} (\frac{1}{4 x + 1}) = \tan^{- 1} (\frac{2}{x^{2}})$ is	(4) 9

P → 2

Q → 4

R → 3

S → 1

P → 2

Q → 4

R → 1

S → 3

P → 4

Q → 3

R → 1

S → 2

P → 4

Q → 3

R → 2

S → 1

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(P) $\frac{1}{y (x)} {(x^{2} - 1) \frac{d^{2} y (x)}{{dx}^{2}} + x \frac{dy (x)}{dx}} = 9$

(R) h = 2