JEE MAIN - Mathematics (2009)

• 1

  Statement - 1 : The variance of first n even natural numbers is $${{{n^2} - 1} \over 4}$$
  
  Statement - 2 : The sum of first n natural numbers is $${{n\left( {n + 1} \right)} \over 2}$$ and the sum of squares of first n natural numbers is $${{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 6}$$

  答え

  (D)

  Statement-1 is false, Statement-2 is true
• 2

  If the mean deviation of number 1, 1 + d, 1 + 2d,........, 1 + 100d from their mean is 255, then the d is equal to

  答え

  (B)

  10.1
• 3

  For real x, let f(x) = x³ + 5x + 1, then

  答え

  (C)

  f is one-one and onto R
• 4

  Let $$f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1$$
  
  Statement - 1 : The set $$\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}$$.
  
  Statement - 2 : $$f$$ is a bijection.

  答え

  (C)

  Statement - 1 is true, Statement - 2 is false
• 5

  If $$\,\left| {z - {4 \over z}} \right| = 2,$$ then the maximum value of $$\,\left| z \right|$$ is equal to :

  答え

  (A)

  $$\sqrt 5 + 1$$
• 6

  Let $$y$$ be an implicit function of $$x$$ defined by $${x^{2x}} - 2{x^x}\cot \,y - 1 = 0$$. Then $$y'(1)$$ equals

  答え

  (D)

  $$-1$$
• 7

  One ticket is selected at random from $$50$$ tickets numbered $$00, 01, 02, ...., 49.$$ Then the probability that the sum of the digits on the selected ticket is $$8$$, given that the product of these digits is zer, equals :

  答え

  (D)

  $${1 \over 14}$$
• 8

  $$\int\limits_0^\pi {\left[ {\cot x} \right]dx,} $$ where $$\left[ . \right]$$ denotes the greatest integer function, is equal to:

  答え

  (C)

  $$ - {\pi \over 2}$$
• 9

  The area of the region bounded by the parabola $${\left( {y - 2} \right)^2} = x - 1,$$ the tangent of the parabola at the point $$(2, 3)$$ and the $$x$$-axis is :

  答え

  (B)

  $$9$$
• 10

  $$\left| {\matrix{ a & {a + 1} & {a - 1} \cr { - b} & {b + 1} & {b - 1} \cr c & {c - 1} & {c + 1} \cr } } \right| + \left| {\matrix{ {a + 1} & {b + 1} & {c - 1} \cr {a - 1} & {b - 1} & {c + 1} \cr {{{\left( { - 1} \right)}^{n + 2}}a} & {{{\left( { - 1} \right)}^{n + 1}}b} & {{{\left( { - 1} \right)}^n}c} \cr } } \right| = 0$$
  
  then the value of $$n$$ :

  答え

  (B)

  any odd integer
• 11

  Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only
  real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( { - 1} \right) < P\left( 1 \right),$$ then in the interval $$\left[ { - 1,1} \right]:$$

  答え

  (A)

  $$P(-1)$$ is not minimum but $$P(1)$$ is the maximum of $$P$$
• 12

  Let $$f\left( x \right) = x\left| x \right|$$ and $$g\left( x \right) = \sin x.$$
  Statement-1: gof is differentiable at $$x=0$$ and its derivative is continuous at that point.
  Statement-2: gof is twice differentiable at $$x=0$$.

  答え

  (B)

  Statement-1 is true, Statement-2 is false
• 13

  The ellipse $${x^2} + 4{y^2} = 4$$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that passes through the point $$(4,0)$$. Then the equation of the ellipse is :

  答え

  (A)

  $${x^2} + 12{y^2} = 16$$
• 14

  Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point $$(1, 0)$$ to the distance from the point $$(-1, 0)$$ is equal to $${1 \over 3}$$. Then the circumcentre of the triangle ABC is at the point :

  答え

  (A)

  $$\left( {{5 \over 4},0} \right)$$
• 15

  The lines $$p\left( {{p^2} + 1} \right)x - y + q = 0$$ and $$\left( {{p^2} + 1} \right){}^2x + \left( {{p^2} + 1} \right)y + 2q$$ $$=0$$ are perpendicular to a common line for :

  答え

  (A)

  exactly one values of $$p$$
• 16

  The shortest distance between the line $$y - x = 1$$ and the curve $$x = {y^2}$$ is :

  答え

  (D)

  $${{3\sqrt 2 } \over 8}$$
• 17

  The remainder left out when $${8^{2n}} - {\left( {62} \right)^{2n + 1}}$$ is divided by 9 is :

  答え

  (A)

  2
• 18

  From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is :

  答え

  (C)

  at least 1000
• 19

  If the roots of the equation $$b{x^2} + cx + a = 0$$ imaginary, then for all real values of $$x$$, the expression $$3{b^2}{x^2} + 6bcx + 2{c^2}$$ is :

  答え

  (B)

  greater than $$-4ab$$
• 20

  Let A and B denote the statements

  A: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$

  B: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$

  If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - {3 \over 2},$$ then:

  答え

  (B)

  both A and B are true
• 21

  The projections of a vector on the three coordinate axis are $$6,-3,2$$ respectively. The direction cosines of the vector are :

  答え

  (B)

  $${6 \over 7 },{{ - 3} \over 7},{2 \over 7}$$
• 22

  If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then :

  答え

  (B)

  $B=C$