JEE MAIN - Mathematics (2007)

• 1

  The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

  回答

  (A)

  80
• 2

  The function $$f:R/\left\{ 0 \right\} \to R$$ given by
  
  $$f\left( x \right) = {1 \over x} - {2 \over {{e^{2x}} - 1}}$$
  
  can be made continuous at $$x$$ = 0 by defining $$f$$(0) as

  回答

  (B)

  1
• 3

  Let $$f:R \to R$$ be a function defined by
  
  $$f(x) = \min \left\{ {x + 1,\left| x \right| + 1} \right\}$$, then which of the following is true?

  回答

  (A)

  $$f(x)$$ is differentiale everywhere
• 4

  The largest interval lying in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ for which the function
  
  $$f\left( x \right) = {4^{ - {x^2}}} + {\cos ^{ - 1}}\left( {{x \over 2} - 1} \right)$$$$ + \log \left( {\cos x} \right)$$,
  
  is defined, is

  回答

  (B)

  $$\left[ {0,{\pi \over 2}} \right)$$
• 5

  If sin^-1$$\left( {{x \over 5}} \right)$$ + cosec^-1$$\left( {{5 \over 4}} \right)$$ = $${\pi \over 2}$$, then the value of x is :

  回答

  (D)

  3
• 6

  If a line makes an angle of $$\pi /4$$ with the positive directions of each of $$x$$-axis and $$y$$-axis, then the angle that the line makes with the positive direction of the $$z$$-axis is :

  回答

  (B)

  $${\pi \over 2}$$
• 7

  Two aeroplanes $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ bomb a target in succession. The probabilities of $${\rm I}$$ and $${\rm I}$$$${\rm I}$$ scoring a hit correctly are $$0.3$$ and $$0.2,$$ respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is :

  回答

  (D)

  0.32
• 8

  The area enclosed between the curves $${y^2} = x$$ and $$y = \left| x \right|$$ is :

  回答

  (A)

  $$1/6$$
• 9

  Let $$I = \int\limits_0^1 {{{\sin x} \over {\sqrt x }}dx} $$ and $$J = \int\limits_0^1 {{{\cos x} \over {\sqrt x }}dx} .$$ Then which one of the following is true?

  回答

  (B)

  $$1 < {2 \over 3}$$ and $$J < 2$$
• 10

  The solution for $$x$$ of the equation $$\int\limits_{\sqrt 2 }^x {{{dt} \over {t\sqrt {{t^2} - 1} }} = {\pi \over 2}} $$ is

  回答

  (D)

  None
• 11

  Let $$F\left( x \right) = f\left( x \right) + f\left( {{1 \over x}} \right),$$ where $$f\left( x \right) = \int\limits_l^x {{{\log t} \over {1 + t}}dt,} $$ Then $$F(e)$$ equals

  回答

  (C)

  $$1/2$$
• 12

  $$\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}} $$ equals

  回答

  (C)

  $$\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C$$
• 13

  If $$D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|$$ for $$x \ne 0,y \ne 0,$$ then $$D$$ is :

  回答

  (D)

  divisible by both $$x$$ and $$y$$
• 14

  If $$p$$ and $$q$$ are positive real numbers such that $${p^2} + {q^2} = 1$$, then the maximum value of $$(p+q)$$ is

  回答

  (C)

  $${\sqrt 2 }$$
• 15

  The function $$f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$ is an incresing function in

  回答

  (D)

  $$\left( { - {\pi \over 2},{\pi \over 4}} \right)$$
• 16

  If $$\widehat u$$ and $$\widehat v$$ are unit vectors and $$\theta $$ is the acute angle between them, then $$2\widehat u \times 3\widehat v$$ is a unit vector for :

  回答

  (B)

  exactly one value of $$\theta $$
• 17

  For the Hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$ , which of the following remains constant when $$\alpha $$ varies$$=$$?

  回答

  (B)

  abscissae of foci
• 18

  Let A $$\left( {h,k} \right)$$, B$$\left( {1,1} \right)$$ and C $$(2, 1)$$ be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is $$1$$ square unit, then the set of values which $$'k'$$ can take is given by :

  回答

  (A)

  $$\left\{ { - 1,3} \right\}$$
• 19

  In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals

  回答

  (B)

  $$\,{1 \over 2}\left( {\sqrt 5 - 1} \right)$$
• 20

  In the binomial expansion of $${\left( {a - b} \right)^n},\,\,\,n \ge 5,$$ the sum of $${5^{th}}$$ and $${6^{th}}$$ terms is zero, then $$a/b$$ equals

  回答

  (B)

  $${{n - 4} \over 5}$$
• 21

  The set S = {1, 2, 3, ........., 12} is to be partitioned into three sets A, B, C of equal size. Thus $$A \cup B \cup C = S,\,A \cap B = B \cap C = A \cap C = \phi $$. The number of ways to partition S is

  回答

  (A)

  $${{12!} \over {{{(4!)}^3}}}\,\,$$
• 22

  If the difference between the roots of the equation $${x^2} + ax + 1 = 0$$ is less than $$\sqrt 5 ,$$ then the set of possible values of $$a$$ is

  回答

  (C)

  $$\left( { - 3,3} \right)$$
• 23

  If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is :

  回答

  (A)

  6
• 24

  Let $$L$$ be the line of intersection of the planes $$2x+3y+z=1$$ and $$x+3y+2z=2.$$ If $$L$$ makes an angle $$\alpha $$ with the positive $$x$$-axis, then cos $$\alpha $$ equals

  回答

  (C)

  $${1 \over {\sqrt 3 }}$$