JEE Advance - Mathematics (1999)

• 1

  For which of the following values of $$m$$, is the area of the region bounded by the curve $$y = x - {x^2}$$ and the line $$y=mx$$ equals $$9/2$$?

  Odpověď

  B

  D
• 2

  Integrate $$\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.} $$

  Odpověď

  (C)

  $$\frac{\pi}{2}$$
• 3

  Find the area of the region in the third quadrant bounded by the curves $$x = - 2{y^2}$$ and $$y=f(x)$$ lying on the left of the line $$8x+1=0.$$

  Odpověď

  (A)

  257/192 sq. units
• 4

  A solution of the differential equation
  $${\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0$$ is

  Odpověď

  (C)

  $$y=2x-4$$
• 5

  The differential equation representing the family of curves
  $${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c$$ is a positive parameter, is of

  Odpověď

  A

  C
• 6

  If the integers $$m$$ and $$n$$ are chosen at random from $$1$$ to $$100$$, then the probability that a number of the form $${7^m} + {7^n}$$ is divisible by $$5$$ equals

  Odpověď

  (A)

  $$1/4$$
• 7

  The probabilities that a student passes in Mathematics, Physics and Chemistry are $$m, p$$ and $$c,$$ respectively. Of these subjects, the student has a $$75%$$ chance of passing in at least one, a $$50$$% chance of passing in at least two, and a $$40$$% chance of passing in exactly two. Which of the following relations are true?

  Odpověď

  C

  B
• 8

  Eight players $${P_1},{P_2},.....{P_8}$$ play a knock-out tournament. It is known that whenever the players $${P_i}$$ and $${P_j}$$ play, the player $${P_i}$$ will win if $$i < j.$$ Assuming that the players are paired at random in each round, what is the probability that the player $${P_4}$$ reaches the final?

  Odpověď

  (D)

  4/35
• 9

  Let $$a=2i+j-2k$$ and $$b=i+j.$$ If $$c$$ is a vector such that $$a.$$ $$c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2 $$ and the angle between $$\left( {a \times b} \right)$$ and $$c$$ is $${30^ \circ },$$ then $$\left| {\left( {a \times b} \right) \times c} \right| = $$

  Odpověď

  (B)

  $$3/2$$
• 10

  Let $$a=2i+j+k, b=i+2j-k$$ and a unit vector $$c$$ be coplanar. If $$c$$ is perpendicular to $$a,$$ then $$c =$$

  Odpověď

  (A)

  $${1 \over {\sqrt 2 }}\left( { - j + k} \right)$$
• 11

  Let $$a$$ and $$b$$ two non-collinear unit vectors. If $$u = a - \left( {a\,.\,b} \right)\,b$$ and $$v = a \times b,$$ then $$\left| v \right|$$ is

  Odpověď

  A

  C
• 12

  Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove that $$\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2$$ and that the equality holds if and only if $$u$$ is perpendicular to $$v .$$

  Odpověď

  (D)

  The magnitude of the scalar triple product (u x v) . w is always less than or equal to 1/2 and equality holds if and only if u is perpendicular to v.
• 13

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

  Odpověď

  (A)

  $$2$$
• 14

  If two distinct chords, drawn from the point (p, q) on the circle $${x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)$$ are bisected by the x - axis, then

  Odpověď

  (D)

  $${p^2}\, > \,\,8\,{q^2}$$.
• 15

  $$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$$ is equal to

  Odpověď

  (C)

  $$i\sqrt 3 $$
• 16

  For a positive integer $$\,n$$, let
  $${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {2^n}\theta } \right).$$ Then

  Odpověď

  B

  A

  C

  D
• 17

  For complex numbers z and w, prove that $${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$ if and only if $$ z = w\,or\,z\overline {\,w} = 1$$.

  Odpověď

  (A)

  The statement is true and the provided condition is both necessary and sufficient.
• 18

  If the roots of the equation $${x^2} - 2ax + {a^2} + a - 3 = 0$$ are real and less than 3, then

  Odpověď

  (A)

  $$a < 2$$
• 19

  If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$ the coefficients of $$x$$ and $${x^2}$$ are $$3$$ and $$-6$$ respectively, then $$m$$ is

  Odpověď

  (C)

  12
• 20

  for each non-be gatuve integer $$m \le n.$$ $$\,\left( {Here\left( {\matrix{ p \cr q \cr } } \right) = {}^p{C_q}} \right).$$

  Odpověď

  C

  E
• 21

  Let $${a_1},{a_2},......{a_{10}}$$ be in $$A,\,P,$$ and $${h_1},{h_2},......{h_{10}}$$ be in H.P. If $${a_1} = {h_1} = 2$$ and $${a_{10}} = {h_{10}} = 3,$$ then $${a_4}{h_7}$$ is

  Odpověď

  (D)

  6
• 22

  The harmonic mean of the roots of the equation $$\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8 + 2\sqrt 5 = 0$$ is

  Odpověď

  (B)

  4
• 23

  Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations
  u + 2v + 3w = 6
  4u + 5v + 6w = 12
  6u + 9v = 4

  then show that the roots of the equation $$\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}$$
  $$ + [{(b - c)^2} + {(c - a)^2} + {(d - b)^2}]x + u + v + w = 0$$ and $$20{x^2} + 10{(a - d)^2}x - 9 = 0$$ are reciprocals of each other.

  Odpověď

  (B)

  The roots of both equations are real, and one root of each equation is the reciprocal of a root of the other.
• 24

  For a positive integer $$n$$, let
  $$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$$. Then

  Odpověď

  A

  D
• 25

  Lt $$PQR$$ be a right angled isosceles triangle, right angled at $$P(2, 1)$$. If the equation of the line $$QR$$ is $$2x + y = 3,$$ then the equation representing the pair of lines $$PQ$$ and $$PR$$ is

  Odpověď

  (B)

  $$3{x^2} - 3{y^2} + 8xy - 20x - 10y + 25 = 0$$
• 26

  If $${x_1},\,{x_2},\,{x_3}$$ as well as $${y_1},\,{y_2},\,{y_3}$$, are in G.P. with the same common ratio, then the points $$\left( {{x_1},\,{y_1}} \right),\left( {{x_2},\,{y_2}} \right)$$ and $$\left( {{x_3},\,{y_3}} \right).$$

  Odpověď

  (A)

  lie on a straight line
• 27

  Let $${L_1}$$ be a straight line passing through the origin and $${L_2}$$ be the straight line $$x + y = 1$$. If the intercepts made by the circle $${x^2} + {y^2} - x + 3y = 0$$ on $${L_1}$$ and $${L_2}$$ are equal, then which of the following equations can represent $${L_1}$$?

  Odpověď

  B

  C
• 28

  In a triangle $$PQR,\angle R = \pi /2$$. If $$\,\,\tan \left( {P/2} \right)$$ and $$\tan \left( {Q/2} \right)$$ are the roots of the equation $$a{x^2} + bx + c = 0\left( {a \ne 0} \right)$$ then.

  Odpověď

  (A)

  $$a + b = c$$
• 29

  Let $${T_1}$$, $${T_2}$$ be two tangents drawn from (- 2, 0) onto the circle $$C:{x^2}\,\, + \,{y^2} = 1$$. Determine the circles touching C and having $${T_1}$$, $${T_2}$$ as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.

  Odpověď

  A

  B

  C

  D
• 30

  If $$(h, k)$$ is the point of intersection of the normals at $$P$$ and $$Q$$, then $$k$$ is equal to

  Odpověď

  (D)

  $$ - \left( {{{{a^2} + {b^2}} \over b}} \right)$$
• 31

  The curve described parametrically by $$x = {t^2} + t + 1,$$ $$y = {t^2} - t + 1 $$ represents

  Odpověď

  (C)

  a parabola
• 32

  If $$x$$ $$=$$ $$9$$ is the chord of contact of the hyperbola $${x^2} - {y^2} = 9,$$ then the equation of the vcorresponding pair of tangents is

  Odpověď

  (B)

  $$9{x^2} - 8{y^2} - 18x + 9 = 0$$
• 33

  On the ellipse $$4{x^2} + 9{y^2} = 1,$$ the points at which the tangents are parallel to the line $$8x = 9y$$ are

  Odpověď

  B

  D
• 34

  Find the co-ordinates of all the points $$P$$ on the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, for which the area of the triangle $$PON$$ is maximum, where $$O$$ denotes the origin and $$N$$, the foot of the perpendicular from $$O$$ to the tangent at $$P$$.

  Odpověď

  A

  B

  C

  D
• 35

  Consider the family of circles $${x^2} + {y^2} = {r^2},\,\,2 < r < 5$$. If in the first quadrant, the common taingent to a circle of this family and the ellipse $$4{x^2} + 25{y^2} = 100$$ meets the co-ordinate axes at $$A$$ and $$B$$, then find the equation of the locus of vthe mid-point of $$AB$$.

  Odpověď

  (B)

  25/x^2 + 4/y^2 = 4
• 36

  Let $$ABC$$ be a triangle having $$O$$ and $$I$$ as its circumcenter and in centre respectively. If $$R$$ and $$r$$ are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\nolimits} $$. Further show that the triangle BIO is a right-angled triangle if and only if $$b$$ is arithmetic mean of $$a$$ and $$c$$.

  Odpověď

  (B)

  The distance between the incenter and circumcenter squared is equal to the circumradius squared minus twice the product of the circumradius and inradius.
• 37

  The number of real solutions of
  $${\tan ^{ - 1}}\,\,\sqrt {x\left( {x + 1} \right)} + {\sin ^{ - 1}}\,\,\sqrt {{x^2} + x + 1} = \pi /2$$ is

  Odpověď

  (C)

  two
• 38

  The function $$f(x)=$$ $${\sin ^4}x + {\cos ^4}x$$ increases if

  Odpověď

  (B)

  $$\pi /4 < x < 3\pi /8$$
• 39

  The function $$f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}} $$ $$dt$$ has a local minimum at $$x=$$

  Odpověď

  B

  D
• 40

  Integrate $$\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.} $$

  Odpověď

  (A)

  -{1/2}log|x+1| + {1/4}log(x^2+1) + {3/2}tan^{-1}x + x/(1+x^2) + C
• 41

  If for a real number $$y$$, $$\left[ y \right]$$ is the greatest integer less than or
  equal to $$y$$, then the value of the integral $$\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx} $$ is

  Odpověď

  (C)

  $$ - \pi /2$$