JEE Advance - Mathematics (2009 - Paper 1 Offline)

• 1

  Let $$z = \,\cos \,\theta \, + i\,\sin \,\theta $$ . Then the value of $$\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\theta \, = {2^ \circ }$$ is

  پاسخ دهید

  (D)

  $${1 \over {4\sin \,{2^ \circ }}}$$
• 2

  Let $$z = x + iy$$ be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation $$\overline z {z^3} + z{\overline z ^3} = 350$$ is

  پاسخ دهید

  (A)

  48
• 3

  Then the value of $$\mu $$ for which the vector $${\overrightarrow {PQ} }$$ is parallel to the plane $$x - 4y + 3z = 1$$ is :

  پاسخ دهید

  (A)

  $${1 \over 4}$$
• 4

  If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are unit vectors such that $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$$ and $$\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$$, then

  پاسخ دهید

  (C)

  $$\overrightarrow b \,,\overrightarrow d $$ are non-parallel
• 5

  The probability that $$X\ge3$$ equals :

  پاسخ دهید

  (B)

  $${{25} \over {36}}$$
• 6

  The conditional probability that $$X\ge6$$ given $$X>3$$ equals :

  پاسخ دهید

  (D)

  $${{25} \over {36}}$$
• 7

  The probability that X = 3 equals

  پاسخ دهید

  (A)

  $${{25} \over {216}}$$
• 8

  Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is

  پاسخ دهید

  B

  C

  D
• 9

  Let $$f$$ be a non-negative function defined on the interval $$[0,1]$$.
  
  If $$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,0 \le x \le 1} } $$, and $$f(0) = 0$$, then

  پاسخ دهید

  (C)

  $$f\left( {{1 \over 2}} \right) < {1 \over 2}$$ and $$f\left( {{1 \over 3}} \right) < {1 \over 3}$$
• 10

  Match the conics in Column I with the statements/expressions in Column II :

  	Column I		Column II
  (A)	Circle	(P)	The locus of the point ($$h,k$$) for which the line $$hx+ky=1$$ touches the circle $$x^2+y^2=4$$.
  (B)	Parabola	(Q)	Points z in the complex plane satisfying $$|z+2|-|z-2|=\pm3$$.
  (C)	Ellipse	(R)	Points of the conic have parametric representation $$x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),y = {{2t} \over {1 + {t^2}}}$$
  (D)	Hyperbola	(S)	The eccentricity of the conic lies in the interval $$1 \le x \le \infty $$.
  		(T)	Points z in the complex plane satisfying $${\mathop{\rm Re}\nolimits} {(z + 1)^2} = |z{|^2} + 1$$.

  پاسخ دهید

  (B)

  (A)$$\to$$(P); (B)$$\to$$(S), (T); (C)$$\to$$(R); (D)$$\to$$(Q), (S)
• 11

  If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then

  پاسخ دهید

  B

  C
• 12

  The line passing through the extremity $$A$$ of the major axis and extremity $$B$$ of the minor axis of the ellipse $${x^2} + 9{y^2} = 9$$ meets its auxiliary circle at the point $$M$$. Then the area of the triangle with vertices at $$A$$, $$M$$ and the origin $$O$$ is

  پاسخ دهید

  (D)

  $${{27} \over {10}}$$
• 13

  Tangents drawn from the point P (1, 8) to the circle
  $${x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0$$
  touch the circle at the points A and B. The equation of the cirumcircle of the triangle PAB is

  پاسخ دهید

  (B)

  $${x^2}\, + \,{y^2}\, - \,4x\,\, - 10y\, + 19 = 0$$
• 14

  The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

  پاسخ دهید

  (C)

  77
• 15

  If $${{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},$$ then

  پاسخ دهید

  A

  B
• 16

  Match the statements/expressions in Column I with the open intervals in Column II :

  	Column I		Column II
  (A)	Interval contained in the domain of definition of non-zero solutions of the differential equation $${(x - 3)^2}y' + y = 0$$	(P)	$$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$
  (B)	Interval containing the value of the integral $$\int\limits_1^5 {(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)dx} $$	(Q)	$$\left( {0,{\pi \over 2}} \right)$$
  (C)	Interval in which at least one of the points of local maximum of $${\cos ^2}x + \sin x$$ lies	(R)	$$\left( {{\pi \over 8},{{5\pi } \over 4}} \right)$$
  (D)	Interval in which $${\tan ^{ - 1}}(\sin x + \cos x)$$ is increasing	(S)	$$\left( {0,{\pi \over 8}} \right)$$
  		(T)	$$( - \pi ,\pi )$$

  پاسخ دهید

  (A)

  (A)$$\to$$(P), (Q), (S); (B)$$\to$$(P), (T), (S); (C)$$\to$$(P), (Q), (R), (T); (D)$$\to$$(S)
• 17

  Let $$L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0$$. If L is finite, then

  پاسخ دهید

  A

  C
• 18

  The number of matrices in A is

  پاسخ دهید

  (A)

  12
• 19

  The number of matrices A in A for which the system of linear equations $$A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$$ has a unique solution, is

  پاسخ دهید

  (B)

  at least 4 but less than 7
• 20

  The number of matrices A in A for which the system of linear equations $$A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$$ is inconsistent, is

  پاسخ دهید

  (B)

  more than 2