JEE Advance - Mathematics (2008 - Paper 1 Offline)

• 1

  Let z be any point in $$A \cap B \cap C$$

  Then, $${\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}$$ lies between :

  Risposta

  (C)

  35 and 39
• 2

  The number of elements in the set $$A \cap B \cap C$$ is

  Risposta

  (B)

  1
• 3

  Let $${L_1},$$ $${L_2},$$ $${L_3}$$ be the lines of intersection of the planes $${P_2}$$ and $${P_3},$$ $${P_3}$$ and $${P_1},$$ $${P_1}$$ and $${P_2},$$ respectively.
  
  STATEMENT - 1Z: At least two of the lines $${L_1},$$ $${L_2}$$ and $${L_3}$$ are non-parallel and
  
  STATEMENT - 2: The three planes doe not have a common point.

  Risposta

  (D)

  STATEMENT - 1 is False, STATEMENT - 2 is True
• 4

  The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ such that $$\widehat a\,.\,\widehat b = \widehat b\,.\,\widehat c = \widehat c\,.\,\widehat a = {1 \over 2}.$$ Then, the volume of the parallelopiped is :

  Risposta

  (A)

  $${1 \over {\sqrt 2 }}$$
• 5

  Consider the system of equations $$ax+by=0; cx+dy=0,$$
  where $$a,b,c,d$$ $$ \in \left\{ {0,1} \right\}$$
  
  STATEMENT - 1 : The probability that the system of equations has a unique solution is $${3 \over 8}.$$ and
  
  STATEMENT - 2 : The probability that the system of equations has a solution is $$1.$$

  Risposta

  (B)

  STATEMENT - 1 is True, STATEMENT - 2 is True; STATEMENT - 2 is NOT a correct explanation for STATEMENT - 1
• 6

  $$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $$

  Risposta

  (D)

  $$2g(1)$$
• 7

  The area of the region bounded by the curve $$y=f(x),$$ the
  $$x$$-axis, and the lines $$x=a$$ and $$x=b$$, where $$ - \infty < a < b < - 2,$$ is :

  Risposta

  (A)

  $$\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)$$
• 8

  If $$f\left( { - 10\sqrt 2 } \right) = 2\sqrt 2 ,$$ then $$f''\left( { - 10\sqrt 2 } \right) = $$

  Risposta

  (B)

  $$-{{4\sqrt 2 } \over {{7^3}{3^2}}}$$
• 9

  Let $$f(x)$$ be a non-constant twice differentiable function defined on $$\left( { - \infty ,\infty } \right)$$
  
  
  such that $$f\left( x \right) = f\left( {1 - x} \right)$$ and $$f'\left( {{1 \over 4}} \right) = 0.$$ Then,

  Risposta

  A

  B

  C
• 10

  If $$0 < x < 1$$, then
  
  $$\sqrt {1 + {x^2}} {\left[ {{{\left\{ {x\cos \left( {{{\cot }^{ - 1}}x} \right) + \sin \left( {{{\cot }^{ - 1}}x} \right)} \right\}}^2} - 1} \right]^{1/2}} = $$

  Risposta

  (C)

  $$x\sqrt {1 + {x^2}} $$
• 11

  Consider the two curves $${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then,

  Risposta

  (B)

  $${C_1}$$ and $${C_2}$$ touch each other exactly at two points
• 12

  Let z be any point $$A \cap B \cap C$$ and let w be any point satisfying $$\left| {w - 2 - i} \right| < 3\,$$. Then, $$\left| z \right| - \left| w \right| + 3$$ lies between :

  Risposta

  (D)

  - 3 and 9
• 13

  Let $${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}} $$ and $${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \over {{n^2} + kn + {k^2}}}} $$ for $$n$$ $$=1, 2, 3, ............$$ Then,

  Risposta

  A

  D
• 14

  A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then :

  Risposta

  B

  D
• 15

  The equation of circle C is

  Risposta

  (D)

  $${\left( {x\, - \sqrt 3 \,} \right)^2} + {(y - 1)^2} = 1$$
• 16

  Points E and F are given by

  Risposta

  (A)

  $$\left( {{{\,\sqrt 3 } \over 2},\,{3 \over 2}} \right),\,\left( {\sqrt 3 ,\,0} \right)$$
• 17

  Equations of the sides QR, RP are

  Risposta

  (D)

  $$y = \sqrt 3 \,x,\,\,y = \,0$$
• 18

  Let $$P\left( {{x_1},{y_1}} \right)$$ and $$Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,$$ be the end points of the latus rectum of the ellipse $${x^2} + 4{y^2} = 4.$$ The equations of parabolas with latus rectum $$PQ$$ are :

  Risposta

  B

  C
• 19

  STATEMENT - 1: $$\mathop {\lim }\limits_{x \to 0} \,\,\left[ {g\left( x \right)\cot x - g\left( 0 \right)\cos ec\,x} \right] = f''\left( 0 \right)$$ and
  
  STATEMENT - 2: $$f'\left( 0 \right) = g\left( 0 \right)$$

  Risposta

  (A)

  Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1
• 20

  Let a and b be non-zero real numbers. Then, the equation
  
  $$(a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0$$ represents :

  Risposta

  (B)

  two straight lines and a circle, when a = b, and c is of sign opposite to that of a
• 21

  Let $$g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 < x < 2,m$$ and $$n$$ are integers, $$m \ne 0,n > 0$$, and let $$p$$ be the left hand derivative of $$|x - 1|$$ at $$x = 1$$. If $$\mathop {\lim }\limits_{x \to {1^ + }} g(x) = p$$, then

  Risposta

  (C)

  $$n = 2,m = 2$$
• 22

  The total number of local maxima and local minima of the function
  
  $$f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.$$ is

  Risposta

  (C)

  2
• 23

  Consider the system of equations:

  $$x-2y+3z=-1$$

  $$-x+y-2z=k$$

  $$x-3y+4z=1$$

  Statement - 1 : The system of equations has no solution for $$k\ne3$$.

  and

  Statement - 2 : The determinant $$\left| {\matrix{ 1 & 3 & { - 1} \cr { - 1} & { - 2} & k \cr 1 & 4 & 1 \cr } } \right| \ne 0$$, for $$k \ne 3$$.

  Risposta

  (A)

  Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1