JEE Advance - Mathematics (1995)

• 1

  Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $$P$$ and $$Q$$. Find the minimum area of the triangle $$OPQ$$, $$O$$ being the origin.

  Jawab

  (B)

  2hk
• 1

  If $$f\left( x \right)\,\,\, = \,\,\,A\sin \left( {{{\pi x} \over 2}} \right)\,\,\, + \,\,\,B,\,\,\,f'\left( {{1 \over 2}} \right) = \sqrt 2 $$ and
  $$\int\limits_0^1 {f\left( x \right)dx = {{2A} \over \pi },} $$ then constants $$A$$ and $$B$$ are

  Jawab

  (D)

  $${4 \over \pi }$$ and $$0$$
• 2

  Let $$y=f(x)$$ be a curve passing through $$(1,1)$$ such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area $$2.$$ From the differential equation and determine all such possible curves.

  Jawab

  A

  B
• 2

  If $$\overrightarrow a ,$$ $$\overrightarrow b $$ and $$\overrightarrow c $$ are three non coplanar vectors, then
  $$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right).\left[ {\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow c } \right)} \right]$$ equals

  Jawab

  (D)

  $$-\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$$
• 3

  Consider a square with vertices at $$(1,1), (-1,1), (-1,-1)$$ and $$(1, -1)$$. Let $$S$$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region $$S$$ and find its area.

  Jawab

  (D)

  The region S consists of all points inside the square which are nearer to the origin than to any edge. The area of S is (16√2 - 20)/3.
• 3

  Let $$\overrightarrow u ,\overrightarrow v $$ and $$\overrightarrow w $$ be vectors such that $$\overrightarrow u + \overrightarrow v + \overrightarrow w = 0.$$ If $$\left| {\overrightarrow u } \right| = 3,\left| {\overrightarrow v } \right| = 4$$ and $$\left| {\overrightarrow w } \right| = 5,$$ then $$\overrightarrow u .\overrightarrow v + \overrightarrow v .\overrightarrow w + \overrightarrow w .\overrightarrow u $$ is

  Jawab

  (B)

  $$-25$$
• 4

  Evaluate the definite integral : $$$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx$$$

  Jawab

  (A)

  $$\frac{\pi}{12} \left[ \pi + 3 \log_e(2 + \sqrt{3}) - 4 \sqrt{3} \right]$$
• 4

  If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ are non coplanar unit vectors such that $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\left( {\overrightarrow b + \overrightarrow c } \right)} \over {\sqrt 2 }},\,\,$$ then the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is

  Jawab

  (A)

  $${{3\pi } \over 4}$$
• 5

  Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${I_m} = m\,\pi ,m = 0,1,2,........$$

  Jawab

  (D)

  This type of question is not suitable for a multiple-choice format.
• 5

  Let $$\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.$$ If $$\overrightarrow d $$ is a unit vector such that $$\overrightarrow a .\overrightarrow d = 0 = \left[ {\overrightarrow b \overrightarrow c \overrightarrow d } \right],$$ then $$\overrightarrow d $$ equals

  Jawab

  (A)

  $$ \pm {{\widehat i + \widehat j - 2k} \over {\sqrt 6 }}$$
• 6

  The minimum value of the expression $$\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,$$ where $$\alpha ,\,\beta ,\,\gamma $$ are real numbers satisfying $$\alpha + \beta + \gamma = \pi $$ is

  Jawab

  (C)

  negative
• 6

  Let $$0 < P\left( A \right) < 1,0 < P\left( B \right) < 1$$ and
  $$P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( A \right)P\left( B \right)$$ then

  Jawab

  C

  D
• 7

  Let '$$d$$' be the perpendicular distance from the centre of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ to the tangent drawn at a point $$P$$ on the ellipse. If $${F_1}$$ and $${F_2}$$ are the two foci of the ellipse, then show that $${\left( {P{F_1} - P{F_2}} \right)^2} = 4{a^2}\left( {1 - {{{b^2}} \over {{d^2}}}} \right)$$.

  Jawab

  (E)

  All of the above.
• 7

  The probability of India winning a test match against West Indies is $$1/2$$. Assuming independence from match to match the probability that in a $$5$$ match series India's second win occurs at third test is

  Jawab

  (B)

  $$1/4$$
• 8

  Show that the locus of a point that divides a chord of slope $$2$$ of the parabola $${y^2} = 4x$$ internally in the ratio $$1:2$$ is a parabola. Find the vertex of this parabola.

  Jawab

  (D)

  The locus is a parabola with vertex (2/9, 8/9).
• 8

  Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral, equals

  Jawab

  (C)

  $$1/10$$
• 9

  The orthocentre of the triangle formed by the lines $$xy=0$$ and $$x+y=1$$ is

  Jawab

  (C)

  $$\left( {0,\,0} \right)$$
• 9

  $$\,3{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x} \right) = $$

  Jawab

  (C)

  13
• 10

  Let $$a,\,b,\,c$$ be real. If $$a{x^2} + bx + c = 0$$ has two real roots $$\alpha $$ and $$\beta ,$$ where $$\alpha < - 1$$ and $$\beta > 1,$$ then show that $$1 + {c \over a} + \left| {{b \over a}} \right| < 0.$$

  Jawab

  (E)

  $$1 + {c \over a} + \left| {{b \over a}} \right| < 0$$
• 10

  The value of $$\int\limits_\pi ^{2\pi } {\left[ {2\,\sin x} \right]\,dx} $$ where [ . ] represents the greatest integer function is

  Jawab

  (A)

  $${{ - 5\pi } \over 3}$$
• 11

  Find the smallest positive number $$p$$ for which the equation $$\cos \left( {p\,\sin x} \right) = \sin \left( {p\cos x} \right)$$ has a solution $$x\, \in \,\left[ {0,2\pi } \right]$$.

  Jawab

  (C)

  $$\frac{\pi \sqrt{2}}{4}$$
• 11

  The value of the integral $$\int {{{{{\cos }^3}x + {{\cos }^5}x} \over {{{\sin }^2}x + {{\sin }^4}x}}} \,dx\,$$ is

  Jawab

  (C)

  $$\sin x - 2{\left( {\sin x} \right)^{ - 1}} - 6{\tan ^{ - 1}}\left( {\sin x} \right) + c$$
• 12

  If $$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that $${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}$$

  Jawab

  (D)

  The inequality holds with the given conditions.
• 12

  The slope of the tangent to a curve $$y = f\left( x \right)$$ at $$\left[ {x,\,f\left( x \right)} \right]$$ is $$2x+1$$. If the curve passes through the point $$\left( {1,2} \right)$$, then the area bounded by the curve, the $$x$$-axis and the line $$x=1$$ is

  Jawab

  (A)

  $${5 \over 6}$$
• 13

  If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.

  Jawab

  (B)

  |z| = 1
• 13

  On the interval $$\left[ {0,1} \right]$$ the function $${x^{25}}{\left( {1 - x} \right)^{75}}$$ takes its maximum value at the point

  Jawab

  (B)

  $${1 \over 4}$$
• 14

  The function $$f\left( x \right) = {{in\,\left( {\pi + x} \right)} \over {in\,\left( {e + x} \right)}}$$ is

  Jawab

  (B)

  decreasing on $$\left( {0,\infty } \right)$$
• 15

  In a triangle $$ABC$$, $$\angle B = {\pi \over 3}$$ and $$\angle C = {\pi \over 4}$$. Let $$D$$ divide $$BC$$ internally in the ratio $$1:3$$ then $${{\sin \angle BAD} \over {\sin \angle CAD}}$$ is equal to

  Jawab

  (A)

  $${1 \over {\sqrt 6 }}$$
• 16

  The radius of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and having its centre at $$(0, 3)$$ is

  Jawab

  (A)

  $$4$$
• 17

  Consider a circle with its centre lying on the focus of the parabola $${y^2} = 2px$$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is

  Jawab

  (A)

  $$\left( {{p \over 2},p} \right)$$ or $$\left( {{p \over 2},- p} \right)$$
• 18

  Let $$z$$ and $$\omega $$ be two non zero complex numbers such that
  $$\left| z \right| = \left| \omega \right|$$ and $${\rm A}rg\,z + {\rm A}rg\,\omega = \pi ,$$ then $$z$$ equals

  Jawab

  (D)

  $$ - \overline \omega $$
• 19

  If $$\omega \,\left( { \ne 1} \right)$$ is a cube root of unity and $${\left( {1 + \omega } \right)^7} = A + B\,\omega $$ then $$A$$ and $$B$$ are respectively

  Jawab

  (B)

  1, 1
• 20

  Let $$z$$ and $$\omega $$ be two complex numbers such that
  $$\left| z \right| \le 1,$$ $$\left| \omega \right| \le 1$$ and $$\left| {z + i\omega } \right| = \left| {z - i\overline \omega } \right| = 2$$ then $$z$$ equals

  Jawab

  (C)

  $$1$$ or $$ - 1$$
• 21

  The general values of $$\theta $$ satisfying the equation $$2{\sin ^2}\theta - 3\sin \theta - 2 = 0$$ is

  Jawab

  (D)

  $$n\pi + {\left( { - 1} \right)^n}7\pi /6$$