Match the statements/expressions given in Column I with the values given in Column II. Column I (A) Root(s) of the equation \(2 \sin ^{2} \theta+\sin ^{2} 2 \theta=2\) (B) Points of discontinuity of the function \(f(x)=\left[\frac{6 x}{\pi}\right] \cos \left[\frac{3 x}{\pi}\right]\), where \([y]\) denotes the largest integer less than or equal to \(y\) (C) Volume of the parallelopiped with its edges represented by the vectors \(\hat{i}+\hat{j}, \quad \hat{i}+2 \hat{j}\) and \(\hat{i}+\hat{j}+\pi \hat{k}\) (D) Angle between vectors \(\vec{a}\) and \(\vec{b}\) where \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors satisfying \(\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}\) Column II (p) \(\frac{\pi}{6}\) (q) \(\frac{\pi}{4}\) (r) \(\frac{\pi}{3}\) (s) \(\frac{\pi}{2}\) (t) \(\pi\)

(A) simplifies to , giving , so . Matches (q), (s). (B) The function is discontinuous where either floor function jumps. For , discontinuities occur at . Matches (p), (r), (s), (t). (C) Volume = . Matches (t). (D) From , we get . Since , we have , so and . Matches (r).

JEE Advanced · 2009 · 12 Apr 2009 · Paper 2 · Vector Algebra

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Match the statements/expressions given in Column I with the values given in Column II.

Column I
(A) Root(s) of the equation

2 sin^{2} θ + sin^{2} 2 θ = 2

(B) Points of discontinuity of the function

f (x) = [\frac{6 x}{π}] cos [\frac{3 x}{π}]

, where

[y]

denotes the largest integer less than or equal to

y

\hat{i} + \hat{j}, \hat{i} + 2 \hat{j}

and

\hat{i} + \hat{j} + π \hat{k}

(D) Angle between vectors

a

and

b

where

a, b

and

c

are unit vectors satisfying

a + b + 3 c = 0

Column II
(p)

\frac{π}{6}

(q)

\frac{π}{4}

(r)

\frac{π}{3}

(s)

\frac{π}{2}

(t)

π

Answer: A → II, A → IV, B → I, B → III, B → IV, B → V, C → V, D → III

Solution

(A)

2 sin^{2} θ + 4 sin^{2} θ cos^{2} θ = 2

simplifies to

sin^{2} θ (3 - 2 sin^{2} θ) = 1

, giving

sin θ = \pm 1/ 2, \pm 1

, so

θ = π /4, π /2

. Matches (q), (s).

(B) The function

f (x) = [6 x / π] cos [3 x / π]

is discontinuous where either floor function jumps. For

x \in [π /6, π]

, discontinuities occur at

x = π /6, π /3, π /2, π

. Matches (p), (r), (s), (t).

(C) Volume =

∣ det ∣ = 111121 00 π = π

. Matches (t).

(D) From

a + b + 3 c = 0

, we get

∣ a + b ∣^{2} = 3

. Since

∣ a ∣ = ∣ b ∣ = 1

, we have

2 + 2 cos α = 3

, so

cos α = 1/2

and

α = π /3

. Matches (r).

JEE Advanced · 2009 · 12 Apr 2009 · Paper 2 · Match the columns

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JEE Advanced · 2009 · 12 Apr 2009 · Paper 2 · Vector Algebra

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