JEE MAIN - Mathematics (2021 - 17th March Morning Shift)

• 1

  In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation of the perpendicular bisector of PR is 2x $$-$$ y + 2 = 0, then the centre of the circumcircle of the $$\Delta$$PQR is :

  Хариулт

  (D)

  ($$-$$2, $$-$$2)
• 2

  The value of
  $$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [ x ] denotes the greatest integer $$ \le $$ x is :

  Хариулт

  (C)

  $${\pi \over 2}$$
• 3

  Which of the following statements is correct for the function g($$\alpha$$) for $$\alpha$$ $$\in$$ R such that
  
  $$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $$

  Хариулт

  (B)

  $$g(\alpha )$$ is an even function
• 4

  Which of the following is true for y(x) that satisfies the differential equation
  
  $${{dy} \over {dx}}$$ = xy $$-$$ 1 + x $$-$$ y; y(0) = 0 :

  Хариулт

  (B)

  y(1) = e^{$$-$$$${1 \over 2}$$} $$-$$ 1
• 5

  The sum of possible values of x for
  
  tan^$$-$$1(x + 1) + cot^$$-$$1$$\left( {{1 \over {x - 1}}} \right)$$ = tan^$$-$$1$$\left( {{8 \over {31}}} \right)$$ is :

  Хариулт

  (A)

  $$-$$$${{{32} \over 4}}$$
• 6

  The value of $$4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$$ is :

  Хариулт

  (A)

  2 + $${2 \over 5}\sqrt {30} $$
• 7

  The inverse of $$y = {5^{\log x}}$$ is :

  Хариулт

  (B)

  $$x = {y^{{1 \over {\log 5}}}}$$
• 8

  Two dies are rolled. If both dices have six faces numbered 1, 2, 3, 5, 7 and 11, then the probability that the sum of the numbers on the top faces is less than or equal to 8 is :

  Хариулт

  (D)

  $${17 \over {36}}$$
• 9

  If the fourth term in the expansion of $${(x + {x^{{{\log }_2}x}})^7}$$ is 4480, then the value of x where x$$\in$$N is equal to :

  Хариулт

  (D)

  2
• 10

  The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k² has no solution if k is equal to :

  Хариулт

  (C)

  $$-$$2
• 11

  In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement?
  
  

  Хариулт

  (B)

  None of these
• 12

  If cot^$$-$$1($$\alpha$$) = cot^$$-$$1 2 + cot^$$-$$1 8 + cot^$$-$$1 18 + cot^$$-$$1 32 + ...... upto 100 terms, then $$\alpha$$ is :

  Хариулт

  (C)

  1.01
• 13

  Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

  Хариулт

  (C)

  4
• 14

  The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

  Хариулт

  (B)

  $${1 \over 2}$$| z |²
• 15

  If (2021)³⁷⁶² is divided by 17, then the remainder is __________.

  Хариулт

  4
• 16

  Let there be three independent events E₁, E₂ and E₃. The probability that only E₁ occurs is $$\alpha$$, only E₂ occurs is $$\beta$$ and only E₃ occurs is $$\gamma$$. Let 'p' denote the probability of none of events occurs that satisfies the equations
  ($$\alpha$$ $$-$$ 2$$\beta$$)p = $$\alpha$$$$\beta$$ and ($$\beta$$ $$-$$ 3$$\gamma$$)p = 2$$\beta$$$$\gamma$$. All the given probabilities are assumed to lie in the interval (0, 1).
  
  Then, $$\frac{Probability\ of\ occurrence\ of\ E_{1}}{Probability\ of\ occurrence\ of\ E_{3}} $$ is equal to _____________.

  Хариулт

  6
• 17

  The maximum value of z in the following equation z = 6xy + y², where 3x + 4y $$ \le $$ 100 and 4x + 3y $$ \le $$ 75 for x $$ \ge $$ 0 and y $$ \ge $$ 0 is __________.

  Хариулт

  904
• 18

  If [ . ] represents the greatest integer function, then the value of
  
  
  $$\left| {\int\limits_0^{\sqrt {{\pi \over 2}} } {\left[ {[{x^2}] - \cos x} \right]dx} } \right|$$ is ____________.

  Хариулт

  1
• 19

  If $$f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)$$ and its first derivative with respect to x is $$ - {b \over a}{\log _e}2$$ when x = 1, where a and b are integers, then the minimum value of | a² $$-$$ b² | is ____________ .

  Хариулт

  481
• 20

  If the function $$f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is continuous at each point in its domain and $$f(0) = {1 \over k}$$, then k is ____________.

  Хариулт

  6