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JEE Advanced 2016 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

Let $$\widehat u = {u_1} \widehat i + {u_2}\widehat j + {u_3}\widehat k$$ be a unit vector in $${{R^3}}$$ and
$$\widehat w = {1 \over {\sqrt 6 }}\left( {\widehat i + \widehat j + 2\widehat k} \right).$$ Given that there exists a vector $${\overrightarrow v }$$ in $${{R^3}}$$ such that $$\left| {\widehat u \times \overrightarrow v } \right| = 1$$ and $$\widehat w.\left( {\widehat u \times \overrightarrow v } \right) = 1.$$ Which of the following statement(s) is (are) correct?

There is exactly one choice for such $${\overrightarrow v }$$

There are infinitely many choices for such $${\overrightarrow v }$$

If $$\widehat u$$ lies in the $$xy$$-plane then $$\left| {{u_1}} \right| = \left| {{u_2}} \right|$$

If $$\widehat u$$ lies in the $$xz$$-plane then $$2\left| {{u_1}} \right| = \left| {{u_3}} \right|$$

JEE Advanced 2016 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

Let
$$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},$$ for

all $$x>0.$$ Then

$$f\left( {{1 \over 2}} \right) \ge f\left( 1 \right)$$

$$f\left( {{1 \over 3}} \right) \le f\left( {{2 \over 3}} \right)$$

$$\,f'\left( 2 \right) \le 0$$

$$\,{{f'\left( 3 \right)} \over {f\left( 3 \right)}} \ge {{f'\left( 2 \right)} \over {f\left( 2 \right)}}$$

JEE Advanced 2016 Paper 2 Offline

MCQ (Single Correct Answer)

-1

Let $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$ and I be the identity matrix of order 3. If $$Q = [{q_{ij}}]$$ is a matrix such that $${P^{50}} - Q = I$$ and $${{{q_{31}} + {q_{32}}} \over {{q_{21}}}}$$ equals

103

201

205

JEE Advanced 2016 Paper 2 Offline

MCQ (Single Correct Answer)

-1

Let b_i > 1 for I = 1, 2, ......, 101. Suppose log_eb₁, log_eb₂, ......., log_eb₁₀₁ are in Arithmetic Progression (A.P.) with the common difference log_e2. Suppose a₁, a₂, ......, a₁₀₁ are in A.P. such that a₁ = b₁ and a₅₁ = b₅₁. If t = b₁ + b₂ + .... + b₅₁ and s = a₁ + a₂ + ..... + a₅₁, then

s > t and a₁₀₁ > b₁₀₁

s > t and a₁₀₁ < b₁₀₁

s < t and a₁₀₁ > b₁₀₁

s < t and a₁₀₁ < b₁₀₁

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