JEE MAIN - Mathematics (2016 (Offline))

• 1

  Let $$p = \mathop {\lim }\limits_{x \to {0^ + }} {\left( {1 + {{\tan }^2}\sqrt x } \right)^{{1 \over {2x}}}}$$ then $$log$$ $$p$$ is equal to :

  Odpověď

  (A)

  $${1 \over 2}$$
• 2

  For $$x \in \,R,\,\,f\left( x \right) = \left| {\log 2 - \sin x} \right|\,\,$$
  
  and $$\,\,g\left( x \right) = f\left( {f\left( x \right)} \right),\,\,$$ then :

  Odpověď

  (B)

  $$g'\left( 0 \right) = \cos \left( {\log 2} \right)$$
• 3

  A value of $$\theta \,$$ for which $${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$$ is purely imaginary, is :

  Odpověď

  (B)

  $${\sin ^{ - 1}}\left( {{1 \over {\sqrt 3 }}} \right)\,$$
• 4

  If the standard deviation of the numbers 2, 3, a and 11 is 3.5, then which of the following is true?

  Odpověď

  (B)

  3$$a$$² - 32$$a$$ + 84 = 0
• 5

  Let two fair six-faced dice $$A$$ and $$B$$ be thrown simultaneously. If $${E_1}$$ is the event that die $$A$$ shows up four, $${E_2}$$ is the event that die $$B$$ shows up two and $${E_3}$$ is the event that the sum of numbers on both dice is odd, then which of the following statements is $$NOT$$ true?

  Odpověď

  (D)

  $${E_1},$$ $${E_2}$$ and $${E_3}$$ are independent.
• 6

  If a curve $$y=f(x)$$ passes through the point $$(1,-1)$$ and satisfies the differential equation, $$y(1+xy) dx=x$$ $$dy$$, then $$f\left( { - {1 \over 2}} \right)$$ is equal to :

  Odpověď

  (B)

  $${4 \over 5}$$
• 7

  The area (in sq. units) of the region $$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$$ is :

  Odpověď

  (D)

  $$\pi - {8 \over 3}$$
• 8

  The integral $$\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx$$ is equal to :

  Odpověď

  (D)

  $${{ {x^{10}}} \over {2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + C$$
• 9

  The system of linear equations

  $$\matrix{ {x + \lambda y - z = 0} \cr {\lambda x - y - z = 0} \cr {x + y - \lambda z = 0} \cr } $$

  has a non-trivial solution for :

  Odpověď

  (D)

  exactly three values of $$\lambda .$$
• 10

  If $$A = \left[ {\matrix{ {5a} & { - b} \cr 3 & 2 \cr } } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :

  Odpověď

  (D)

  $$5$$
• 11

  A wire of length $$2$$ units is cut into two parts which are bent respectively to form a square of side $$=x$$ units and a circle of radius $$=r$$ units. If the sum of the areas of the square and the circle so formed is minimum, then:

  Odpověď

  (A)

  $$x=2r$$
• 12

  The eccentricity of the hyperbola whose length of the latus rectum is equal to $$8$$ and the length of its conjugate axis is equal to half of the distance between its foci, is :

  Odpověď

  (A)

  $${2 \over {\sqrt 3 }}$$
• 13

  If one of the diameters of the circle, given by the equation, $${x^2} + {y^2} - 4x + 6y - 12 = 0,$$ is a chord of a circle $$S$$, whose centre is at $$(-3, 2)$$, then the radius of $$S$$ is :

  Odpověď

  (D)

  $$5\sqrt 3 $$
• 14

  Two sides of a rhombus are along the lines, $$x - y + 1 = 0$$ and $$7x - y - 5 = 0$$. If its diagonals intersect at $$(-1, -2)$$, then which one of the following is a vertex of this rhombus?

  Odpověď

  (A)

  $$\left( {{{ 1} \over 3}, - {8 \over 3}} \right)$$
• 15

  If the $${2^{nd}},{5^{th}}\,and\,{9^{th}}$$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

  Odpověď

  (D)

  $${4 \over 3}$$
• 16

  If all the words (with or without meaning) having five letters,formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL is :

  Odpověď

  (D)

  $${58^{th}}$$
• 17

  The sum of all real values of $$x$$ satisfying the equation $${\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}}\, = 1$$ is :

  Odpověď

  (C)

  $$3$$
• 18

  If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$

  Odpověď

  (C)

  contains exactly two elements.