JEE MAIN - Mathematics (2013 (Offline))

• 1

  Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A $$ \times $$ B having 3 or more elements is :

  Одговорити

  (A)

  219
• 2

  All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given?

  Одговорити

  (C)

  variance
• 3

  $$\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}}$$ is equal to

  Одговорити

  (D)

  2
• 4

  At present, a firm is manufacturing $$2000$$ items. It is estimated that the rate of change of production P w.r.t. additional number of workers $$x$$ is given by $${{dp} \over {dx}} = 100 - 12\sqrt x .$$ If the firm employs $$25$$ more workers, then the new level of production of items is

  Одговорити

  (C)

  $$3500$$
• 5

  The area (in square units) bounded by the curves $$y = \sqrt {x,} $$ $$2y - x + 3 = 0,$$ $$x$$-axis, and lying in the first quadrant is :

  Одговорити

  (A)

  $$9$$
• 6

  Statement-1 : The value of the integral
  $$\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}} $$ is equal to $$\pi /6$$

  Statement-2 : $$\int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)} dx.$$

  Одговорити

  (D)

  Statement-1 is false; Statement-2 is true.
• 7

  If $$\int {f\left( x \right)dx = \psi \left( x \right),} $$ then $$\int {{x^5}f\left( {{x^3}} \right)dx} $$ is equal to

  Одговорити

  (C)

  $${1 \over 3}{x^3}\psi \left( {{x^3}} \right) - \int {{x^2}\psi \left( {{x^3}} \right)dx} + C$$
• 8

  If $$x, y, z$$ are in A.P. and $${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$$ and $${\tan ^{ - 1}}z$$ are also in A.P., then :

  Одговорити

  (A)

  $$x=y=z$$
• 9

  If $$y = \sec \left( {{{\tan }^{ - 1}}x} \right),$$ then $${{{dy} \over {dx}}}$$ at $$x=1$$ is equal to :

  Одговорити

  (A)

  $${1 \over {\sqrt 2 }}$$
• 10

  The equation of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having centre at $$(0,3)$$ is :

  Одговорити

  (A)

  $${x^2} + {y^2} - 6y - 7 = 0$$
• 11

  The $$x$$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $$(0, 1) (1, 1)$$ and $$(1, 0)$$ is :

  Одговорити

  (B)

  $$2 - \sqrt 2 $$
• 12

  A ray of light along $$x + \sqrt 3 y = \sqrt 3 $$ gets reflected upon reaching $$X$$-axis, the equation of the reflected ray is :

  Одговорити

  (B)

  $$\sqrt 3 y = x - \sqrt 3 $$
• 13

  The term independent of $$x$$ in expansion of
  $${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$$ is

  Одговорити

  (C)

  210
• 14

  Let $${T_n}$$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $${T_{n + 1}} - {T_n}$$ = 10, then the value of n is :

  Одговорити

  (B)

  5
• 15

  If the equations $${x^2} + 2x + 3 = 0$$ and $$a{x^2} + bx + c = 0,$$ $$a,\,b,\,c\, \in \,R,$$ have a common root, then $$a\,:b\,:c\,$$ is

  Одговорити

  (A)

  $$1:2:3$$
• 16

  The real number $$k$$ for which the equation, $$2{x^3} + 3x + k = 0$$ has two distinct real roots in $$\left[ {0,\,1} \right]$$

  Одговорити

  (D)

  does not exist.
• 17

  The real number $$k$$ for which the equation, $$2{x^3} + 3x + k = 0$$ has two distinct real roots in $$\left[ {0,\,1} \right]$$

  Одговорити

  (D)

  does not exist.
• 18

  The number of values of $$k$$, for which the system of equations : $$$\matrix{ {\left( {k + 1} \right)x + 8y = 4k} \cr {kx + \left( {k + 3} \right)y = 3k - 1} \cr } $$$
  has no solution, is
  

  Одговорити

  (B)

  1
• 19

  If z is a complex number of unit modulus and argument $$\theta $$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$ equals :

  Одговорити

  (C)

  $$\theta \,$$
• 20

  The expression $${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$$ can be written as:

  Одговорити

  (B)

  $$\,\sec {\rm A}\,\cos ec{\rm A} + 1$$
• 21

  If the lines $${{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}$$ and $${{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}$$ are coplanar, then $$k$$ can have :

  Одговорити

  (C)

  exactly two values