JEE MAIN - Mathematics (2002)

• 1

  f(x) and g(x) are two differentiable functions on [0, 2] such that
  
  f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, g(2) = 9
  
  then f(x) - g(x) at x = $${3 \over 2}$$ is

  回答

  (D)

  -5
• 2

  In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

  回答

  (B)

  65
• 3

  $$\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}$$ is

  回答

  (D)

  does not exist
• 4

  The domain of $${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]$$ is

  回答

  (A)

  [1, 9]
• 5

  Let $$f(2) = 4$$ and $$f'(x) = 4.$$
  
  Then $$\mathop {\lim }\limits_{x \to 2} {{xf\left( 2 \right) - 2f\left( x \right)} \over {x - 2}}$$ is given by

  回答

  (C)

  $$- 4$$
• 6

  $$\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}$$

  回答

  (A)

  $${e^4}$$
• 7

  $$\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}$$, $$n \in N$$, ( [x] denotes the greatest integer less than or equal to x )

  回答

  (D)

  does not exist
• 8

  If $$f\left( 1 \right) = 1,{f'}\left( 1 \right) = 2,$$ then
  $$\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}}$$ is

  回答

  (A)

  $$2$$
• 9

  $$f$$ is defined in $$\left[ { - 5,5} \right]$$ as
  
  $$f\left( x \right) = x$$ if $$x$$ is rational
  
  $$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = - x$$ if $$x$$ is irrational. Then

  回答

  (B)

  $$f(x)$$ is discontinuous at every $$x,$$ except $$x = 0$$
• 10

  If f(x + y) = f(x).f(y) $$\forall $$ x, y and f(5) = 2, f'(0) = 3, then
  f'(5) is

  回答

  (C)

  6
• 11

  A triangle with vertices $$\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)$$ is :

  回答

  (A)

  isosceles and right angled
• 12

  Locus of mid point of the portion between the axes of
  
  $$x$$ $$cos$$ $$\alpha + y\,\sin \alpha = p$$ where $$p$$ is constant is :

  回答

  (D)

  $${1 \over {{x^2}}} + {1 \over {{y^2}}} = {4 \over {{p^2}}}$$
• 13

  If $$y=f(x)$$ makes +$$ve$$ intercept of $$2$$ and $$0$$ unit on $$x$$ and $$y$$ axes and encloses an area of $$3/4$$ square unit with the axes then $$\int\limits_0^2 {xf'\left( x \right)dx} $$ is

  回答

  (D)

  $$-3/4$$
• 14

  If $$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},$$ then $$\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$$ is

  回答

  (A)

  $${n^2}y$$
• 15

  $${\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x,$$ then sin x is equal to :

  回答

  (A)

  $${\tan ^2}\left( {{\alpha \over 2}} \right)$$
• 16

  The maximum distance from origin of a point on the curve
  $$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$
  $$y = a\cos t - b\cos \left( {{{at} \over b}} \right),$$ both $$a,b > 0$$ is

  回答

  (B)

  $$a+b$$
• 17

  If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then
  $$\left| {\matrix{ a & b & {ax + b} \cr b & c & {bx + c} \cr {ax + b} & {bx + c} & 0 \cr } } \right|$$ is equal to

  回答

  (C)

  $$-ve$$
• 18

  $${I_n} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx} $$ then $$\,\mathop {\lim }\limits_{n \to \infty } \,n\left[ {{I_n} + {I_{n + 2}}} \right]$$ equals

  回答

  (B)

  $$1$$
• 19

  $$\int\limits_0^{10\pi } {\left| {\sin x} \right|dx} $$ is

  回答

  (A)

  $$20$$
• 20

  $$\int\limits_0^2 {\left[ {{x^2}} \right]dx} $$ is

  回答

  (D)

  $$ - \sqrt 2 - \sqrt 3 + 5$$
• 21

  $$\int_{ - \pi }^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} dx$$ is

  回答

  (B)

  $${{\pi ^2}}$$
• 22

  The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length $$3$$$$a$$ is :

  回答

  (C)

  $${x^2}\, + \,{y^2} = 4{a^2}$$
• 23

  The area bounded by the curves $$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$ and $$y = \left| {\ln \left| x \right|} \right|$$ is :

  回答

  (A)

  $$4$$sq. units
• 24

  The order and degree of the differential equation
  $$\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$$ are

  回答

  (C)

  $$(3,3)$$
• 25

  The order and degree of the differential equation
  $$\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$$ are

  回答

  (C)

  $$(3,3)$$
• 26

  A problem in mathematics is given to three students $$A,B,C$$ and their respective probability of solving the problem is $${1 \over 2},{1 \over 3}$$ and $${1 \over 4}.$$ Probability that the problem is solved is :

  回答

  (A)

  $${3 \over 4}$$
• 27

  $$A$$ and $$B$$ are events such that $$P\left( {A \cup B} \right) = 3/4$$,$$P\left( {A \cap B} \right) = 1/4,$$
  $$P\left( {\overline A } \right) = 2/3$$ then $$P\left( {\overline A \cap B} \right)$$ is :

  回答

  (A)

  $$5/12$$
• 28

  If $$\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2$$ and the angle between $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$\pi /6$$ then $${\left( {\overrightarrow a \times \overrightarrow b } \right)^2}$$ is equal to :

  回答

  (B)

  $$16$$
• 29

  If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then :

  回答

  (B)

  $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$
• 30

  If $$\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3$$ thus what will be the value of $$\left| {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarrow c + \overrightarrow c .\overrightarrow a } \right|,$$ given that $$\overrightarrow a + \overrightarrow b + \overrightarrow c = 0$$ :

  回答

  (A)

  $$25$$
• 31

  $$\overrightarrow a = 3\widehat i - 5\widehat j$$ and $$\overrightarrow b = 6\widehat i + 3\widehat j$$ are two vectors and $$\overrightarrow c $$ is a vector such that $$\overrightarrow c = \overrightarrow a \times \overrightarrow b $$ then $$\left| {\overrightarrow a } \right|:\left| {\overrightarrow b } \right|:\left| {\overrightarrow c } \right|$$ =

  回答

  (B)

  $$\sqrt {34} :\sqrt {45} :39$$
• 32

  If the vectors $$\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k$$ and $$\widehat b = \widehat j$$ are such that $$\overrightarrow a ,\overrightarrow c $$ and $$\overrightarrow b $$ form a right handed system then $${\overrightarrow c }$$ is :

  回答

  (A)

  $$z\widehat i - x\widehat k$$
• 33

  Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :

  回答

  (D)

  720
• 34

  The period of $${\sin ^2}\theta $$ is

  回答

  (B)

  $$\pi $$
• 35

  Which one is not periodic?

  回答

  (B)

  $$\cos \sqrt x + {\cos ^2}x$$
• 36

  z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ then z equals

  回答

  (B)

  $$ - \overline \omega $$
• 37

  If $$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by :

  回答

  (C)

  $${\mathop{\rm Re}\nolimits} (z) > 3$$
• 38

  The locus of the centre of a circle which touches the circle $$\left| {z - {z_1}} \right| = a$$ and$$\left| {z - {z_2}} \right| = b\,$$ externally
  
  ($$z,\,{z_1}\,\& \,{z_2}\,$$ are complex numbers) will be :

  回答

  (B)

  a hyperbola
• 39

  If $$\alpha \ne \beta $$ but $${\alpha ^2} = 5\alpha - 3$$ and $${\beta ^2} = 5\beta - 3$$ then the equation having $$\alpha /\beta $$ and $$\beta /\alpha \,\,$$ as its roots is

  回答

  (A)

  $$3{x^2} - 19x + 3 = 0$$
• 40

  Product of real roots of equation $${t^2}{x^2} + \left| x \right| + 9 = 0$$

  回答

  (A)

  is always positive
• 41

  Difference between the corresponding roots of $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0$$ is same and $$a \ne b,$$ then

  回答

  (A)

  $$a + b + 4 = 0$$
• 42

  If $$p$$ and $$q$$ are the roots of the equation $${x^2} + px + q = 0,$$ then

  回答

  (A)

  $$p = 1,\,\,q = - 2$$
• 43

  If $$a,\,b,\,c$$ are distinct $$ + ve$$ real numbers and $${a^2} + {b^2} + {c^2} = 1$$ then $$ab + bc + ca$$ is

  回答

  (A)

  less than 1
• 44

  The coefficients of $${x^p}$$ and $${x^q}$$ in the expansion of $${\left( {1 + x} \right)^{p + q}}$$ are

  回答

  (A)

  equal
• 45

  The positive integer just greater than $${\left( {1 + 0.0001} \right)^{10000}}$$ is

  回答

  (D)

  3
• 46

  Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is :

  回答

  (C)

  374
• 47

  The sum of integers from 1 to 100 that are divisible by 2 or 5 is :

  回答

  (B)

  3050
• 48

  Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are :

  回答

  (D)

  216
• 49

  If 1, $${\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)$$ are in A.P. then x equals

  回答

  (B)

  $$1 - \,{\log _3}\,4\,$$
• 50

  l, m, n are the $${p^{th}}$$, $${q^{th}}$$ and $${r^{th}}$$ term of a G.P all positive, $$then\,\left| {\matrix{ {\log \,l} & p & 1 \cr {\log \,m} & q & 1 \cr {\log \,n} & r & 1 \cr } } \right|\,equals$$

  回答

  (D)

  0
• 51

  Fifth term of a GP is 2, then the product of its 9 terms is

  回答

  (B)

  512
• 52

  Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

  回答

  (B)

  3/5
• 53

  If the chord y = mx + 1 of the circle $${x^2}\, + \,{y^2} = 1$$ subtends an angle of measure $${45^ \circ }$$ at the major segment of the circle then value of m is :

  回答

  (C)

  $$- 1\, \pm \,\sqrt 2 \,\,$$