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 :

  คำตอบ

  (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?

  คำตอบ

  (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

  คำตอบ

  (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

  คำตอบ

  (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 :

  คำตอบ

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

  คำตอบ

  (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

  คำตอบ

  (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 :

  คำตอบ

  (A)

  $$x=y=z$$
• 9

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

  คำตอบ

  (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 :

  คำตอบ

  (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 :

  คำตอบ

  (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 :

  คำตอบ

  (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

  คำตอบ

  (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 :

  คำตอบ

  (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

  คำตอบ

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

  คำตอบ

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

  คำตอบ

  (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
  

  คำตอบ

  (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 :

  คำตอบ

  (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:

  คำตอบ

  (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 :

  คำตอบ

  (C)

  exactly two values