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 :

  Απάντηση

  (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?

  Απάντηση

  (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

  Απάντηση

  (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

  Απάντηση

  (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 :

  Απάντηση

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

  Απάντηση

  (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

  Απάντηση

  (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 :

  Απάντηση

  (A)

  $$x=y=z$$
• 9

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

  Απάντηση

  (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 :

  Απάντηση

  (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 :

  Απάντηση

  (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 :

  Απάντηση

  (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

  Απάντηση

  (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 :

  Απάντηση

  (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

  Απάντηση

  (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 $$\left[ {0,\,1} \right]$$

  Απάντηση

  (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 $$\left[ {0,\,1} \right]$$

  Απάντηση

  (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
  

  Απάντηση

  (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 :

  Απάντηση

  (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:

  Απάντηση

  (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 :

  Απάντηση

  (C)

  exactly two values