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 :

  پاسخ دهید

  (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 :

  پاسخ دهید

  (C)

  $${\pi \over 2}$$
• 3

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

  پاسخ دهید

  (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 :

  پاسخ دهید

  (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 :

  پاسخ دهید

  (A)

  $$-$$$${{{32} \over 4}}$$
• 6

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

  پاسخ دهید

  (A)

  2 + $${2 \over 5}\sqrt {30} $$
• 7

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

  پاسخ دهید

  (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 :

  پاسخ دهید

  (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 :

  پاسخ دهید

  (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 :

  پاسخ دهید

  (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?
  
  

  پاسخ دهید

  (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 :

  پاسخ دهید

  (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 :

  پاسخ دهید

  (C)

  4
• 14

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

  پاسخ دهید

  (B)

  $${1 \over 2}$$| z |²
• 15

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

  پاسخ دهید

  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 _____________.

  پاسخ دهید

  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 __________.

  پاسخ دهید

  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 ____________.

  پاسخ دهید

  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 ____________ .

  پاسخ دهید

  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 ____________.

  پاسخ دهید

  6