JEE MAIN - Mathematics (2015 (Offline))

• 1

  If the function.
  
  $$g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr } } \right.$$
  
  is differentiable, then the value of $$k+m$$ is :

  отговор

  (C)

  $$2$$
• 2

  The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is :

  отговор

  (B)

  14.0
• 3

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

  отговор

  (A)

  2
• 4

  Let $$\alpha $$ and $$\beta $$ be the roots of equation $${x^2} - 6x - 2 = 0$$. If $${a_n} = {\alpha ^n} - {\beta ^n},$$ for $$n \ge 1,$$ then the value of $${{{a_{10}} - 2{a_8}} \over {2{a_9}}}$$ is equal to :

  отговор

  (A)

  $$3$$
• 5

  If $$12$$ different balls are to be placed in $$3$$ identical boxes, then the probability that one of the boxes contains exactly $$3$$ balls is :

  отговор

  (C)

  $${{55} \over 3}{\left( {{2 \over 3}} \right)^{11}}$$
• 6

  Let $$y(x)$$ be the solution of the differential equation
  
  $$\left( {x\,\log x} \right){{dy} \over {dx}} + y = 2x\,\log x,\left( {x \ge 1} \right).$$ Then $$y(e)$$ is equal to :

  отговор

  (A)

  $$2$$
• 7

  The area (in sq. units) of the region described by
  
  $$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$ and $$\left. {y \ge 4x - 1} \right\}$$ is :

  отговор

  (B)

  $${{9} \over {32}}$$
• 8

  The integral
  $$\int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx} $$ is equal to :

  отговор

  (A)

  $$1$$
• 9

  The integral $$\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}} $$ equals :

  отговор

  (B)

  $$ - {\left( {{{{x^4} + 1} \over {{x^4}}}} \right)^{{1 \over 4}}} + c$$
• 10

  If $$A = \left[ {\matrix{ 1 & 2 & 2 \cr 2 & 1 & { - 2} \cr a & 2 & b \cr } } \right]$$ is a matrix satisfying the equation
  
  $$A{A^T} = 9\text{I},$$ where $$I$$ is $$3 \times 3$$ identity matrix, then the ordered
  
  pair $$(a, b)$$ is equal to :

  отговор

  (B)

  $$(-2, -1)$$
• 11

  The set of all values of $$\lambda $$ for which the system of linear equations:
  
  $$\matrix{ {2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}} \cr {2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}} \cr { - {x_1} + 2{x_2} = \lambda {x_3}} \cr } $$
  
  has a non-trivial solution

  отговор

  (A)

  contains two elements
• 12

  Let $$f(x)$$ be a polynomial of degree four having extreme values
  at $$x=1$$ and $$x=2$$. If $$\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left( x \right)} \over {{x^2}}}} \right] = 3$$, then f$$(2)$$ is equal to :

  отговор

  (A)

  $$0$$
• 13

  Let $${\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right),$$
  where $$\left| x \right| < {1 \over {\sqrt 3 }}.$$ Then a value of $$y$$ is :

  отговор

  (C)

  $${{3x - {x^3}} \over {1 - 3{x^2}}}$$
• 14

  Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the ratio $$1:3$$, then locus of $$P$$ is :

  отговор

  (B)

  $${{x^2} = 2y}$$
• 15

  Locus of the image of the point $$(2, 3)$$ in the line $$\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,\,k \in R,$$ is a :

  отговор

  (A)

  circle of radius $$\sqrt 2 $$.
• 16

  The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $$(0, 0)$$ $$(0, 41)$$ and $$(41, 0)$$ is :

  отговор

  (B)

  780
• 17

  If m is the A.M. of two distinct real numbers l and n $$(l,n > 1)$$ and $${G_1},{G_2}$$ and $${G_3}$$ are three geometric means between $$l$$ and n, then $$G_1^4\, + 2G_2^4\, + G_3^4$$ equals:

  отговор

  (D)

  $$4\,l\,{m^2}n$$
• 18

  The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:

  отговор

  (D)

  192
• 19

  A complex number z is said to be unimodular if $$\,\left| z \right| = 1$$. Suppose $${z_1}$$ and $${z_2}$$ are complex numbers such that $${{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }}$$ is unimodular and $${z_2}$$ is not unimodular. Then the point $${z_1}$$ lies on a :

  отговор

  (A)

  circle of radius 2.
• 20

  Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set A $\times$ B , each having atleast three elements are

  отговор

  (A)

  219