JEE MAIN - Mathematics (2017 (Offline))

• 1

  Let $$a$$, b, c $$ \in R$$. If $$f$$(x) = ax² + bx + c is such that
  $$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R,$$
  
  then $$\sum\limits_{n = 1}^{10} {f(n)} $$ is equal to

  回答

  (D)

  330
• 2

  A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is:

  回答

  (D)

  485
• 3

  If for a positive integer n, the quadratic equation
  
  $$x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right)$$$$ + .... + \left( {x + \overline {n - 1} } \right)\left( {x + n} \right)$$$$ = 10n$$
  
  has two consecutive integral solutions, then n is equal to :

  回答

  (C)

  11
• 4

  If S is the set of distinct values of 'b' for which the following system of linear equations
  
  x + y + z = 1
  x + ay + z = 1
  ax + by + z = 0
  
  has no solution, then S is :

  回答

  (D)

  a singleton
• 5

  If $$A = \left[ {\matrix{ 2 & { - 3} \cr { - 4} & 1 \cr } } \right]$$,
  
  then adj(3A² + 12A) is equal to

  回答

  (A)

  $$\left[ {\matrix{ {51} & {63} \cr {84} & {72} \cr } } \right]$$
• 6

  Let $$\omega $$ be a complex number such that 2$$\omega $$ + 1 = z where z = $$\sqrt {-3} $$. If
  
  $$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$,
  
  then k is equal to :

  回答

  (D)

  -z
• 7

  The function $$f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$$ defined as
  
  $$f\left( x \right) = {x \over {1 + {x^2}}}$$, is

  回答

  (C)

  surjective but not injective
• 8

  If $$5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$$,
  
  then the value of $$\cos 4x$$ is :

  回答

  (C)

  $$ - {7 \over 9}$$
• 9

  For three events A, B and C,
  
  P(Exactly one of A or B occurs)
  = P(Exactly one of B or C occurs)
  = P (Exactly one of C or A occurs) = $${1 \over 4}$$
  and P(All the three events occur simultaneously) = $${1 \over {16}}$$.
  
  Then the probability that at least one of the events occurs, is :

  回答

  (A)

  $${7 \over {16}}$$
• 10

  Let $$\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k$$ and $$\overrightarrow b = \widehat i + \widehat j$$.
  
  Let $$\overrightarrow c $$ be a vector such that $$\left| {\overrightarrow c - \overrightarrow a } \right| = 3$$,
  
  $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c } \right| = 3$$ and the angle between $$\overrightarrow c $$ and $\overrightarrow a \times \overrightarrow b$ is $$30^\circ $$.
  
  Then $$\overrightarrow a .\overrightarrow c $$ is equal to :

  回答

  (A)

  2
• 11

  Let k be an integer such that the triangle with vertices (k, – 3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point :

  回答

  (C)

  $$\left( {2,{1 \over 2}} \right)$$
• 12

  If $$\left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0$$ and y(0) = 1,
  
  then $$y\left( {{\pi \over 2}} \right)$$ is equal to :

  回答

  (D)

  $${1 \over 3}$$
• 13

  If $$\left( {2 + \sin x} \right){{dy} \over {dx}} + \left( {y + 1} \right)\cos x = 0$$ and y(0) = 1,
  
  then $$y\left( {{\pi \over 2}} \right)$$ is equal to :

  回答

  (D)

  $${1 \over 3}$$
• 14

  The area (in sq. units) of the region
  
  $$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$$ is

  回答

  (C)

  $${5 \over 2}$$
• 15

  Let $${I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).$$
  
  If $${I_4} + {I_6}$$ = $$a{\tan ^5}x + b{x^5} + C$$, where C is a constant of integration,
  
  then the ordered pair $$\left( {a,b} \right)$$ is equal to

  回答

  (A)

  $$\left( {{1 \over 5},0} \right)$$
• 16

  The integral $$\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}} $$ is equal to

  回答

  (A)

  2
• 17

  Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :

  回答

  (B)

  25
• 18

  If for $$x \in \left( {0,{1 \over 4}} \right)$$, the derivatives of
  
  $${\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x^3}}}} \right)$$ is $$\sqrt x .g\left( x \right)$$, then $$g\left( x \right)$$ equals

  回答

  (D)

  $${{9 \over {1 + 9{x^3}}}}$$
• 19

  $$\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\cot x - \cos x} \over {{{\left( {\pi - 2x} \right)}^3}}}$$ equals

  回答

  (A)

  $${1 \over {16}}$$