JEE MAIN - Mathematics (2018 (Offline))

• 1

  From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

  Trả lời

  (B)

  at least 1000
• 2

  Let S = { $$x$$ $$ \in $$ R : $$x$$ $$ \ge $$ 0 and
  
  $$2\left| {\sqrt x - 3} \right| + \sqrt x \left( {\sqrt x - 6} \right) + 6 = 0$$}. Then S

  Trả lời

  (D)

  contains exactly two elements
• 3

  If $$\alpha ,\beta \in C$$ are the distinct roots of the equation
  x² - x + 1 = 0, then $${\alpha ^{101}} + {\beta ^{107}}$$ is equal to :

  Trả lời

  (D)

  1
• 4

  If the system of linear equations
  
  x + ky + 3z = 0
  3x + ky - 2z = 0
  2x + 4y - 3z = 0
  
  has a non-zero solution (x, y, z), then $${{xz} \over {{y^2}}}$$ is equal to

  Trả lời

  (C)

  10
• 5

  If $$\left| {\matrix{ {x - 4} & {2x} & {2x} \cr {2x} & {x - 4} & {2x} \cr {2x} & {2x} & {x - 4} \cr } } \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2}$$
  
  then the ordered pair (A, B) is equal to :

  Trả lời

  (D)

  (-4, 5)
• 6

  Two sets A and B are as under :
  
  A = {($$a$$, b) $$ \in $$ R $$ \times $$ R : |$$a$$ - 5| < 1 and |b - 5| < 1};
  
  B = {($$a$$, b) $$ \in $$ R $$ \times $$ R : 4($$a$$ - 6)² + 9(b - 5)² $$ \le $$ 36 };
  
  Then

  Trả lời

  (C)

  A $$ \subset $$ B
• 7

  Let $${a_1}$$, $${a_2}$$, $${a_3}$$, ......... ,$${a_{49}}$$ be in A.P. such that
  
  $$\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416$$ and $${a_9} + {a_{43}} = 66$$.
  
  $$a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m$$, then m is equal to

  Trả lời

  (D)

  34
• 8

  A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :

  Trả lời

  (C)

  $${2 \over 5}$$
• 9

  If $$\sum\limits_{i = 1}^9 {\left( {{x_i} - 5} \right)} = 9$$ and
  
  $$\sum\limits_{i = 1}^9 {{{\left( {{x_i} - 5} \right)}^2}} = 45$$, then the standard deviation of the 9 items
  $${x_1},{x_2},.......,{x_9}$$ is

  Trả lời

  (D)

  2
• 10

  Let $$\overrightarrow u $$ be a vector coplanar with the vectors $$\overrightarrow a = 2\widehat i + 3\widehat j - \widehat k$$ and $$\overrightarrow b = \widehat j + \widehat k$$. If $$\overrightarrow u $$ is perpendicular to $$\overrightarrow a $$ and $$\overrightarrow u .\overrightarrow b = 24$$, then $${\left| {\overrightarrow u } \right|^2}$$ is equal to

  Trả lời

  (A)

  336
• 11

  A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is :

  Trả lời

  (D)

  3x + 2y = xy
• 12

  Let y = y(x) be the solution of the differential equation
  
  $$\sin x{{dy} \over {dx}} + y\cos x = 4x$$, $$x \in \left( {0,\pi } \right)$$.
  
  If $$y\left( {{\pi \over 2}} \right) = 0$$, then $$y\left( {{\pi \over 6}} \right)$$ is equal to :

  Trả lời

  (D)

  $$ - {8 \over 9}{\pi ^2}$$
• 13

  The integral
  
  $$\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx$$
  
  is equal to

  Trả lời

  (C)

  $${{ - 1} \over {3\left( {1 + {{\tan }^3}x} \right)}} + C$$
• 14

  The value of $$\int\limits_{ - \pi /2}^{\pi /2} {{{{{\sin }^2}x} \over {1 + {2^x}}}} dx$$ is

  Trả lời

  (A)

  $${\pi \over 4}$$
• 15

  Let g(x) = cosx², f(x) = $$\sqrt x $$ and $$\alpha ,\beta \left( {\alpha < \beta } \right)$$ be the roots of the quadratic equation 18x² - 9$$\pi $$x + $${\pi ^2}$$ = 0. Then the area (in sq. units) bounded by the curve
  y = (gof)(x) and the lines $$x = \alpha $$, $$x = \beta $$ and y = 0 is :

  Trả lời

  (B)

  $${1 \over 2}\left( {\sqrt 3 - 1} \right)$$
• 16

  Let S = { t $$ \in R:f(x) = \left| {x - \pi } \right|.\left( {{e^{\left| x \right|}} - 1} \right)$$$$\sin \left| x \right|$$ is not differentiable at t}, then the set S is equal to

  Trả lời

  (B)

  $$\phi $$ (an empty set)
• 17

  Let $$f\left( x \right) = {x^2} + {1 \over {{x^2}}}$$ and $$g\left( x \right) = x - {1 \over x}$$,
  $$x \in R - \left\{ { - 1,0,1} \right\}$$.
  If $$h\left( x \right) = {{f\left( x \right)} \over {g\left( x \right)}}$$, then the local minimum value of h(x) is

  Trả lời

  (A)

  $$2\sqrt 2 $$
• 18

  For each t $$ \in R$$, let [t] be the greatest integer less than or equal to t.
  
  Then $$\mathop {\lim }\limits_{x \to {0^ + }} x\left( {\left[ {{1 \over x}} \right] + \left[ {{2 \over x}} \right] + ..... + \left[ {{{15} \over x}} \right]} \right)$$

  Trả lời

  (D)

  is equal to 120