JEE Advance - Mathematics (1996)

• 1

  For $$n>0,$$ $$\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx = $$

  Răspuns

  (C)

  $\pi^2$
• 2

  Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the
  lines $$x=0,$$ $$y=0,$$ and $$x = {\pi \over 4}.$$ Prove that for $$n > 2,$$
  $${A_n} + {A_{n - 2}} = {1 \over {n - 1}}$$ and deduce $${1 \over {2n + 2}} < {A_n} < {1 \over {2n - 2}}.$$

  Răspuns

  A

  C

  E
• 3

  Determine the equation of the curve passing through the origin, in the form $$y=f(x),$$ which satisfies the differential equation $${{dy} \over {dx}} = \sin \left( {10x + 6y} \right).\,$$

  Răspuns

  (A)

  y = (1/3)[arctan((5tan 4x)/(4 - 3tan 4x)) - 5x]
• 4

  For the three events $$A, B,$$ and $$C,P$$ (exactly one of the events $$A$$ or $$B$$ occurs) $$=P$$ (exactly one of the two events $$B$$ or $$C$$ occurs)$$=P$$ (exactly one of the events $$C$$ or $$A$$ occurs)$$=p$$ and $$P$$ (all the three events occur simultaneously) $$ = {p^2},$$ where $$0 < p < 1/2.$$ Then the probability of at least one of the three events $$A,B$$ and $$C$$ occurring is

  Răspuns

  (A)

  $${{3p + 2{p^2}} \over 2}$$
• 5

  In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, $$3$$ in the front and $$4$$ at the back? How many seating arrangements are possible if $$3$$ girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of $$3$$ girls sitting together in a back row on adjacent seats?

  Răspuns

  (A)

  The total number of seating arrangements is $$7 \times 13!$$, the number of seating arrangements with the three girls sitting together in the back row on adjacent seats is $$12!$$ and the probability is $$1/9!$$.
• 6

  A nonzero vector $$\overrightarrow a $$ is parallel to the line of intersection of the plane determined by the vectors $$\widehat i,\widehat i + \widehat j$$ and the plane determined by the vectors $$\widehat i - \widehat j,\widehat i + \widehat k.$$ The angle between $$\overrightarrow a $$ and the vector $$\widehat i - 2\widehat j + 2\widehat k$$ is ................

  Răspuns

  C

  D
• 7

  If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$ is any vector, then $$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow c + {{\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)} \over {\left| {\overrightarrow b \times \overrightarrow c } \right|}}\left( {\overrightarrow b \times \overrightarrow c } \right) = $$ ..............

  Răspuns

  (C)

  $$\overrightarrow a$$
• 8

  The position vectors of the vertices $$A, B$$ and $$C$$ of a tetrahedron $$ABCD$$ are $$\widehat i + \widehat j + \widehat k,\,\widehat i$$ and $$3\widehat i\,,$$ respectively. The altitude from vertex $$D$$ to the opposite face $$ABC$$ meets the median line through $$A$$ of the triangle $$ABC$$ at a point $$E.$$ If the length of the side $$AD$$ is $$4$$ and the volume of the tetrahedron is $${{2\sqrt 2 } \over 3},$$ find the position vector of the point $$E$$ for all its possible positions.

  Răspuns

  (C)

  $$-\hat i + 3\hat j + 3\hat k$$
• 9

  If for nonzero $$x$$, $$af(x)+$$ $$bf\left( {{1 \over x}} \right) = {1 \over x} - 5$$ where $$a \ne b,$$ then
  $$\int_1^2 {f\left( x \right)dx} = .......$$

  Răspuns

  (A)

  $$\frac{1}{a^2 - b^2} \left[ a(\log 2 - 5) + \frac{7b}{2} \right]$$
• 10

  The angle between a pair of tangents drawn from a point P to the circle $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0$$ is $$2\,\alpha $$.
  The equation of the locus of the point P is

  Răspuns

  (D)

  $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, + \,9\,\, = \,0$$
• 11

  $${\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,$$ is true if and only if

  Răspuns

  (B)

  $$x = y,\,x \ne 0$$
• 12

  The value of the expression
  $$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\left( {n - {\omega ^2}} \right),$$

  where $$\omega $$ is an imaginary cube root of unity, is..........

  Răspuns

  (A)

  $$\frac{1}{4}n(n-1)(n^2 + 3n + 4)$$
• 13

  For positive integers $${n_1},\,{n_2}$$ the value of the expression $${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$$
  where $$i = \sqrt { - 1} $$ is real number if and only if
  

  Răspuns

  (D)

  $${n_1} > 0,\,{n_2} > 0$$
• 14

  Find all non-zero complex numbers Z satisfying $$\overline Z = i{Z^2}$$.

  Răspuns

  A

  C

  D
• 15

  Find all values of $$\theta $$ in the interval $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ satisfying the equation $$\left( {1 - \tan \,\theta } \right)\left( {1 + \tan \,\theta } \right)\,\,{\sec ^2}\theta + \,\,{2^{{{\tan }^2}\theta }} = 0.$$

  Răspuns

  B

  C
• 16

  Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k$$, all integers, satisfying $${x_1} + {x_2} + \,..... + {x_k} = n,\,$$ is......................................

  Răspuns

  (A)

  $${{\left[ {k + \left( {n - {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
• 17

  Using mathematical induction prove that for every integer $$n \ge 1,\,\,\left( {{3^{2n}} - 1} \right)$$ is divisible by $${2^{n + 2}}$$ but not by $${2^{n + 3}}$$.

  Răspuns

  A

  B

  C

  D
• 18

  For any odd integer $$n$$ $$ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$$

  Răspuns

  (A)

  $$\frac{1}{4}(n+1)^2(2n-1)$$
• 19

  The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are in AP. Find the intervals in which $$\beta \,\,and\,\gamma $$ lie.

  Răspuns

  (A)

  $$\beta \, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left[ { - {1 \over {27}},\infty } \right)$$
• 20

  A rectangle $$PQRS$$ has its side $$PQ$$ parallel to the line $$y = mx$$ and vertices $$P, Q$$ and $$S$$ on the lines $$y = a, x = b$$ and $$x = -b,$$ respectively. Find the locus of the vertex $$R$$.

  Răspuns

  (A)

  x(m^2 - 1) - ym + (m^2 + 1)b + am = 0
• 21

  The intercept on the line y = x by the circle $${x^2} + {y^2} - 2x = 0$$ is AB. Equation of the circle with AB as a diameter is................................

  Răspuns

  (A)

  x² + y² - x - y = 0
• 22

  General value of $$\theta $$ satisfying the equation $${\tan ^2}\theta + \sec \,2\,\theta = 1$$ is _________.

  Răspuns

  A

  E
• 23

  A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are $$\,\alpha \,\,and\,\,\beta $$ respectively and the distance between the point A and the mid point of the line segment DC is d, prove that the area of the circle is $$${{\pi \,{d^2}\,\,{{\cos }^2}\,\,\alpha } \over {{{\cos }^2}\,\alpha \, + \,{{\cos }^2}\,\beta \, + \,\,2\,\cos \,\,\alpha \,\,\cos \,\beta \,\cos \,\,(\beta - \alpha )\,}}$$$

  Răspuns

  (A)

  $${{pi ,{d^2},,{{cos }^2},,alpha } over {{{cos }^2},alpha , + ,{{cos }^2},eta , + ,,2,cos ,,alpha ,,cos ,eta ,cos ,,(eta - alpha ),}}$$
• 24

  Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $$\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)$$ to the circle $$\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt 2 a)\,x - (1 - \sqrt 2 a)\,y = 0$$.

  Răspuns

  (A)

  a ∈ (-∞, -2) ∪ (2, ∞)
• 25

  An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one directrix is the common tangent, nearer to the point $$P$$, to the circle $${x^2} + {y^2} = 1$$ and the hyperbol;a $${x^2} - {y^2} = 1$$. The equation of the ellipse, in the standard form, is ............

  Răspuns

  (A)

  ${{{{\left( {x - {1 \over 3}} \right)}^2}} \over {{{\left( {{1 \over 3}} \right)}^2}}} + {{{{\left( {y - 1} \right)}^2}} \over {{{\left( {{1 \over {2\sqrt 3 }}} \right)}^2}}} = 1$
• 26

  Points $$A, B$$ and $$C$$ lie on the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$A, B$$ and $$C$$, taken in pairs, intersect at points $$P, Q$$ and $$R$$. Determine the ratio of the areas of the triangles $$ABC$$ and $$PQR$$.

  Răspuns

  (B)

  2:1
• 27

  From a point $$A$$ common tangents are drawn to the circle $${x^2} + {y^2} = {a^2}/2$$ and parabola $${y^2} = 4ax$$. Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of contact of the parabola.

  Răspuns

  (A)

  $$rac{15a^2}{4}$$
• 28

  If $$x{e^{xy}} = y + {\sin ^2}x,$$ then at $$x = 0,{{dy} \over {dx}} = ..............$$

  Răspuns

  (B)

  1
• 29

  In a triangle $$ABC$$, $$a:b:c=4:5:6$$. The ratio of the radius of the circumcircle to that of the incircle is ............

  Răspuns

  (D)

  16:7
• 30

  A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.

  Răspuns

  (A)

  y = e^(a(x-1))
• 31

  Determine the points of maxima and minima of the function
  $$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a constant.

  Răspuns

  A

  E
• 32

  Where a is a positive constant. Find the interval in which $$f'(x)$$ is increasing.

  Răspuns

  (C)

  (-2/a, a/3)
• 33

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

  Răspuns

  (A)

  $$\log \left( {\frac{{1 + x{e^x}}}{{x{e^x}}}} \right) - \frac{1}{{1 + x{e^x}}} + C$$