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 :

  Răspuns

  (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

  Răspuns

  (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 :

  Răspuns

  (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

  Răspuns

  (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 :

  Răspuns

  (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

  Răspuns

  (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

  Răspuns

  (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 :

  Răspuns

  (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

  Răspuns

  (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

  Răspuns

  (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 :

  Răspuns

  (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 :

  Răspuns

  (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

  Răspuns

  (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

  Răspuns

  (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 :

  Răspuns

  (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

  Răspuns

  (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

  Răspuns

  (A)

  $$2\sqrt 2 $$
• 18

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

  Răspuns

  (D)

  is equal to 120