JEE Advance - Mathematics (2012 - Paper 1 Offline)

• 1

  Let z be a complex number such that the imaginary part of z is non-zero and $$a\, = \,{z^2} + \,z\, + 1$$ is real. Then a cannot take the value

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  (D)

  $${3 \over 4}$$
• 2

  If $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors satisfying
  $${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\overrightarrow c - \overrightarrow a } \right|^2} = 9,$$ then $$\left| {2\overrightarrow a + 5\overrightarrow b + 5\overrightarrow c } \right|$$ is

  Відповідь

  3
• 3

  A ship is fitted with three engines $${E_1},{E_2}$$ and $${E_3}$$. The engines function independently of each other with respective probabilities $${1 \over 2},{1 \over 4}$$ and $${1 \over 4}$$. For the ship to be operational at least two of its engines must function. Let $$X$$ denote the event that the ship is operational and Let $${X_1},{X_2}$$ and $${X_3}$$ denote respectively the events that the engines $${E_1},{E_2}$$ and $${E_3}$$ are functioning. Which of the following is (are) true?

  Відповідь

  D

  B
• 4

  The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

  Відповідь

  (B)

  150
• 5

  The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

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  (B)

  150
• 6

  The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle $${x^2}\, + \,{y^2} = 9$$ is

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  (A)

  $$20\,({x^2}\, + \,{y^2}) - \,\,36x\,\, + \,\,45y = 0$$
• 7

  The ellipse $${E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1$$ is inscribed in a rectangle $$R$$ whose sides are parallel to the coordinate axes. Another ellipse $${E_2}$$ passing through the point $$(0, 4)$$ circumscribes the rectangle $$R$$. The eccentricity of the ellipse $${E_2}$$ is

  Відповідь

  (C)

  $${{1 \over 2}}$$
• 8

  The point $$P$$ is the intersection of the straight line joining the points $$Q(2, 3, 5)$$ and $$R(1, -1, 4)$$ with the plane $$5x-4y-z=1.$$ If $$S$$ is the foot of the perpendicular drawn from the point $$T(2, 1, 4)$$ to $$QR,$$ then the length of the line segment $$PS$$ is

  Відповідь

  (A)

  $${{1 \over {\sqrt 2 }}}$$
• 9

  Let $$S$$ be the focus of the parabola $${y^2} = 8x$$ and let $$PQ$$ be the common chord of the circle $${x^2} + {y^2} - 2x - 4y = 0$$ and the given parabola. The area of the triangle $$PQS$$ is

  Відповідь

  4
• 10

  Let $$p(x)$$ be a real polynomial of least degree which has a local maximum at $$x=1$$ and a local minimum at $$x=3$$. If $$p(1)=6$$ and $$p(3)=2$$, then $$p'(0)$$ is

  Відповідь

  9
• 11

  Let $$f:IR \to IR$$ be defined as $$f\left( x \right) = \left| x \right| + \left| {{x^2} - 1} \right|.$$ The total number of points at which $$f$$ attains either a local maximum or a local minimum is

  Відповідь

  5
• 12

  The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant $$K$$)

  Відповідь

  (C)

  $-\frac{1}{(\sec x+\tan x)^{11 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^2\right\}+K$
• 13

  Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then

  Відповідь

  A

  D

  B
• 14

  If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then

  Відповідь

  D

  A
• 15

  A ship is fitted with three engines $${E_1},{E_2}$$ and $${E_3}$$. The engines function independently of each other with respective probabilities $${1 \over 2},{1 \over 4}$$ and $${1 \over 4}$$. For the ship to be operational at least two of its engines must function. Let $$X$$ denote the event that the ship is operational and Let $${X_1},{X_2}$$ and $${X_3}$$ denote respectively the events that the engines $${E_1},{E_2}$$ and $${E_3}$$ are functioning. Which of the following is (are) true?

  Відповідь

  D

  B
• 16

  If $$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$$, then

  Відповідь

  (B)

  a = 1, b = $$-$$4
• 17

  Let $$P = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix and let $$Q = [{b_{ij}}]$$, where $${b_{ij}} = {2^{i + j}}{a_{ij}}$$ for $$1 \le i,j \le 3$$. If the determinant of P is 2, then the determinant of the matrix Q is

  Відповідь

  (D)

  2¹³
• 18

  Let $$f(x) = \left\{ {\matrix{ {{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$

  x$$\in$$R, then f is

  Відповідь

  (B)

  differentiable at x = 0 but not differentiable at x = 2.
• 19

  The function $$f:[0,3] \to [1,29]$$, defined by $$f(x) = 2{x^3} - 15{x^2} + 36x + 1$$, is

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  (B)

  onto but not one-one.
• 20

  The value of $$6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}...} } } } \right)$$ is __________.

  Відповідь

  4