1
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
In $${R^3},$$ consider the planes $$\,{P_1}:y = 0$$ and $${P_2}:x + z = 1.$$ Let $${P_3}$$ be the plane, different from $${P_1}$$ and $${P_2}$$, which passes through the intersection of $${P_1}$$ and $${P_2}.$$ If the distance of the point $$(0,1, 0)$$ from $${P_3}$$ is $$1$$ and the distance of a point $$\left( {\alpha ,\beta ,\gamma } \right)$$ from $${P_3}$$ is $$2,$$ then which of the following relations is (are) true?
A
$$2\alpha + \beta + 2\gamma + 2 = 0$$
B
$$2\alpha - \beta + 2\gamma + 4 = 0$$
C
$$2\alpha + \beta - 2\gamma - 10 = 0$$
D
$$2\alpha - \beta + 2\gamma - 8 = 0$$
2
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
In $${R^3},$$ let $$L$$ be a straight lines passing through the origin. Suppose that all the points on $$L$$ are at a constant distance from the two planes $${P_1}:x + 2y - z + 1 = 0$$ and $${P_2}:2x - y + z - 1 = 0.$$ Let $$M$$ be the locus of the feet of the perpendiculars drawn from the points on $$L$$ to the plane $${P_1}.$$ Which of the following points lie (s) on $$M$$?
A
$$\left( {0, - {5 \over 6}, - {2 \over 3}} \right)$$
B
$$\left( { - {1 \over 6}, - {1 \over 3},{1 \over 6}} \right)$$
C
$$\left( { - {5 \over 6},0,{1 \over 6}} \right)$$
D
$$\left( { - {1 \over 3},0,{2 \over 3}} \right)$$
3
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$\Delta PQR$$ be a triangle. Let $$\vec a = \overrightarrow {QR} ,\vec b = \overrightarrow {RP}$$ and $$\overrightarrow c = \overrightarrow {PQ} .$$ If $$\left| {\overrightarrow a } \right| = 12,\,\,\left| {\overrightarrow b } \right| = 4\sqrt 3 ,\,\,\,\overrightarrow b .\overrightarrow c = 24,$$ then which of the following is (are) true?
A
$${{{{\left| {\overrightarrow c } \right|}^2}} \over 2} - \left| {\overrightarrow a } \right| = 12$$
B
$${{{{\left| {\overrightarrow c } \right|}^2}} \over 2} + \left| {\overrightarrow a } \right| = 30$$
C
$$\left| {\overrightarrow a \times \overrightarrow b + \overrightarrow c \times \overrightarrow a } \right| = 48\sqrt 3$$
D
$$\overrightarrow a .\overrightarrow b = - 72$$
4
JEE Advanced 2015 Paper 1 Offline
+4
-0
Match the following :

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$
(A)$$\,\,\,\,$$ In $${R^2},$$ If the magnitude of the projection vector of the vector $$\alpha \widehat i + \beta \widehat j$$ on $$\sqrt 3 \widehat i + \widehat j$$ and If $$\alpha = 2 + \sqrt 3 \beta ,$$ then possible value of $$\left| \alpha \right|$$ is/are
(B)$$\,\,\,\,$$ Let $$a$$ and $$b$$ be real numbers such that the function $$f\left( x \right) = \left\{ {\matrix{ { - 3a{x^2} - 2,} & {x < 1} \cr {bx + {a^2},} & {x \ge 1} \cr } } \right.$$ if differentiable for all $$x \in R$$. Then possible value of $$a$$ is (are)
(C)$$\,\,\,\,$$ Let $$\omega \ne 1$$ be a complex cube root of unity. If $${\left( {3 - 3\omega + 2{\omega ^2}} \right)^{4n + 3}} + {\left( {2 + 3\omega - 3{\omega ^2}} \right)^{4n + 3}} + {\left( { - 3 + 2\omega + 3{\omega ^2}} \right)^{4n + 3}} = 0,$$ then possible value (s) of $$n$$ is (are)
(D)$$\,\,\,\,$$ Let the harmonic mean of two positive real numbers $$a$$ and $$b$$ be $$4.$$ If $$q$$ is a positive real nimber such that $$a, 5, q, b$$ is an arithmetic progression, then the value(s) of $$\left| {q - a} \right|$$ is (are)

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$II$$
(p)$$\,\,\,\,$$ $$1$$
(q)$$\,\,\,\,$$ $$2$$
(r)$$\,\,\,\,$$ $$3$$
(s)$$\,\,\,\,$$ $$4$$
(t)$$\,\,\,\,$$ $$5$$

A
$$\left( A \right) \to p, q;\,\,\left( B \right) \to p,q;\,\,\left( C \right) \to p,q,s,t;\,\,\left( D \right) \to q,t$$
B
$$\left( A \right) \to q;\,\,\left( B \right) \to q;\,\,\left( C \right) \to p,q,s,t;\,\,\left( D \right) \to q,t$$
C
$$\left( A \right) \to q;\,\,\left( B \right) \to p,q;\,\,\left( C \right) \to p,t;\,\,\left( D \right) \to q,t$$
D
$$\left( A \right) \to q;\,\,\left( B \right) \to p,q;\,\,\left( C \right) \to p,q,s,t;\,\,\left( D \right) \to q$$
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