JEE Advance - Mathematics (2005)

• 1

  Find the area bounded by the curves $${x^2} = y,{x^2} = - y$$ and $${y^2} = 4x - 3.$$

  回答

  (D)

  1/3 sq. units
• 1

  $$\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx} $$ is equal to

  回答

  (C)

  $$4$$
• 2

  If the incident ray on a surface is along the unit vector $$\widehat v\,\,,$$ the reflected ray is along the unit vector $$\widehat w\,\,$$ and the normal is along unit vector $$\widehat a\,\,$$ outwards. Express $$\widehat w\,\,$$ in terms of $$\widehat a\,\,$$ and $$\widehat v\,\,.$$

  回答

  (C)

  $$\widehat w = \widehat v - 2(\widehat a \cdot \widehat v)\widehat a$$
• 2

  If $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ are three non-zero, non-coplanar vectors and
  $$\overrightarrow {{b_1}} = \overrightarrow b - {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,\overrightarrow {{b_2}} = \overrightarrow b + {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a ,$$
  $$\overrightarrow {{c_1}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_2}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow a } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$$
  $$\overrightarrow {{c_3}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a + {{\overrightarrow b .\,\overrightarrow c } \over {{{\left| c \right|}^2}}}{\overrightarrow b _1},\,\,\overrightarrow {{c_4}} = \overrightarrow c - {{\overrightarrow c .\,\overrightarrow a } \over {{{\left| {\overrightarrow c } \right|}^2}}}\overrightarrow a - {{\overrightarrow b \,.\,\overrightarrow c } \over {{{\left| {{{\overrightarrow b }_1}} \right|}^2}}}{\overrightarrow b _1},$$
  then the set of orthogonal vectors is

  回答

  (B)

  $$\left( {\overrightarrow a ,\overrightarrow {{b_1}} ,\overrightarrow {{c_2}} } \right)$$
• 3

  Find the equation of the plane containing the line $$2x-y+z-3=0,3x+y+z=5$$ and at a distance of $${1 \over {\sqrt 6 }}$$ from the point $$(2, 1, -1).$$

  回答

  (C)

  $$62x+29y+19z-105=0$$ and $$50x+31y+21z-135=0$$
• 3

  A variable plane at a distance of the one unit from the origin cuts the coordinates axes at $$A,$$ $$B$$ and $$C.$$ If the centroid $$D$$ $$(x, y, z)$$ of triangle $$ABC$$ satisfies the relation $${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = k,$$ then the value $$k$$ is

  回答

  (D)

  $$9$$
• 4

  A person goes to office either by car, scooter, bus or train, the probability of which being $${1 \over 7},{3 \over 7},{2 \over 7}$$ and $${1 \over 7}$$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $${2 \over 9},{1 \over 9},{4 \over 9}$$ and $${1 \over 9}$$ respectively. Given that he reached office in time, then what is the probability that he travelled by a car.

  回答

  (A)

  1/7
• 4

  A six faced fair dice is thrown until $$1$$ comes, then the probability that $$1$$ comes in even no. of trials is

  回答

  (A)

  $$5/11$$
• 5

  If length of tangent at any point on the curve $$y=f(x)$$ intecepted between the point and the $$x$$-axis is length $$1.$$ Find the equation of the curve.

  回答

  (C)

  log |(1 - √(1 - y^2)) / y| + √(1 - y^2) = x + c
• 5

  The differential equation $${{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}$$ determines a family of circles with

  回答

  (C)

  fixed radius $$1$$ and variable centres along the $$x$$-axis.
• 6

  $$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable
  function such that $$\left| {f\left( x \right)} \right| \le 1$$ and $$f'(x)=g(x).$$
  If $${f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.$$ Prove that there exists some $$c \in \left( { - 3,3} \right)$$
  such that $$g(c).g''(c)<0.$$

  回答

  (A)

  Rolle's Theorem
• 6

  The solution of primitive integral equation $$\left( {{x^2} + {y^2}} \right)dy = xy$$
  $$dx$$ is $$y=y(x),$$ If $$y(1)=1$$ and $$\left( {{x_0}} \right) = e$$, then $${{x_0}}$$ is equal to

  回答

  (C)

  $$\sqrt 3 \,e$$
• 7

  If $$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\left[ {\matrix{ {f\left( { - 1} \right)} \cr {f\left( 1 \right)} \cr {f\left( 2 \right)} \cr } } \right] = \left[ {\matrix{ {3{a^2} + 3a} \cr {3{b^2} + 3b} \cr {3{c^2} + 3c} \cr } } \right],\,\,f\left( x \right)$$ is a quadratic
  function and its maximum value occurs at a point $$V$$. $$A$$ is a point of intersection of $$y=f(x)$$ with $$x$$-axis and point $$B$$ is such that chord $$AB$$ subtends a right angle at $$V$$. Find the area enclosed by $$f(x)$$ and chord $$AB$$.

  回答

  (B)

  125/3 sq. units
• 7

  For the primitive integral equation $$ydx + {y^2}dy = x\,dy;$$
  $$x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,$$ then $$y(-3)$$ is

  回答

  (C)

  $$1$$
• 8

  If one the vertices of the square circumscribing the circle $$\left| {z - 1} \right| = \sqrt 2 \,is\,2 + \sqrt {3\,} \,i$$. Find the other vertices of the square.

  回答

  A

  B

  C
• 8

  If $$y=y(x)$$ and it follows the relation $$x\cos \,y + y\,cos\,x = \pi $$ then $$y''(0)=$$

  回答

  (C)

  $${\pi}$$
• 9

  Evaluate $$\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$$

  回答

  (A)

  $$\frac{24}{5}\left[e\cos \left(\frac{1}{2}\right) + \frac{1}{2}e\sin \left(\frac{1}{2}\right) - 1\right]$$
• 9

  The area bounded by the parabola $$y = {\left( {x + 1} \right)^2}$$ and
  $$y = {\left( {x - 1} \right)^2}$$ and the line $$y=1/4$$ is

  回答

  (D)

  $$1/3$$ sq. units
• 10

  If $$p(x)$$ be a polynomial of degree $$3$$ satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maxima at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. Find the distance between the local maxima and local minima of the curve.

  回答

  (C)

  $$4\sqrt{65}$$
• 10

  $$a,\,b,\,c$$ are integers, not all simultaneously equal and $$\omega $$ is cube root of unity $$\left( {\omega \ne 1} \right),$$ then minimum value of $$\left| {a + b\omega + c{\omega ^2}} \right|$$ is

  回答

  (B)

  1
• 11

  If $$\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},$$ for all $${x_1},{x_2} \in R$$. Find the equation of tangent to the cuve $$y = f\left( x \right)$$ at the point $$(1, 2)$$.

  回答

  (C)

  y = 2
• 11

  If $$\int\limits_{\sin x}^1 {{t^2}f\left( t \right)dt = 1 - \sin x,} $$ then f$$\left( {{1 \over {\sqrt 3 }}} \right)$$ is

  回答

  (C)

  $$3$$
• 12

  In an equilateral triangle, $$3$$ coins of radii $$1$$ unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

  回答

  (B)

  $$6 + 4\sqrt 3 $$
• 12

  If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$$ then

  回答

  (B)

  $$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,2} \right)$$
• 13

  Find the equation of the common tangent in $${1^{st}}$$ quadrant to the circle $${x^2} + {y^2} = 16$$ and the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over 4} = 1$$. Also find the length of the intercept of the tangent between the coordinate axes.

  回答

  (A)

  $$y = -\frac{2}{\sqrt{3}}x + 4\sqrt{\frac{7}{3}}, \,\,\,\,\frac{14}{\sqrt{3}}$$
• 13

  In a triangle $$ABC$$, $$a,b,c$$ are the lengths of its sides and $$A,B,C$$ are the angles of triangle $$ABC$$. The correct relation is given by

  回答

  (B)

  $$\left( {b - c} \right)cos\left( {{A \over 2}} \right) = a\,sin{{B - C} \over 2}$$
• 14

  Tangents are drawn from any point on the hyperbola $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ to the circle $${x^2} + {y^2} = 9$$.Find the locus of mid-point of the chord of contact.

  回答

  (C)

  ${{{x^2}} \over 9} - {{{y^2}} \over 4} = {\left( {{{{x^2} + {y^2}} \over 9}} \right)^2}$
• 14

  If $$f(x)$$ is a twice differentiable function and given that $$f\left( 1 \right) = 1;f\left( 2 \right) = 4,f\left( 3 \right) = 9$$, then

  回答

  (D)

  $$f''\left( x \right) = 2$$ for some $$x \in \left( {1,3} \right)$$
• 15

  Circles with radii 3, 4 and 5 touch each other externally. It P is the point of intersection of tangents to these circles at their points of contact, find the distance of P from the points of contact.

  回答

  (C)

  $$\sqrt{5}$$
• 15

  Tangent to the curve $$y = {x^2} + 6$$ at a point $$(1, 7)$$ touches the circle $${x^2} + {y^2} + 16x + 12y + c = 0$$ at a point $$Q$$. Then the coordinates of $$Q$$ are

  回答

  (D)

  $$(-6, -7)$$
• 16

  The area of the triangle formed by intersection of a line parallel to $$x$$-axis and passing through $$P (h, k)$$ with the lines $$y = x $$ and $$x + y = 2$$ is $$4{h^2}$$. Find the locus of the point $$P$$.

  回答

  (B)

  y = 2x + 1 or y = -2x + 1
• 16

  The minimum area of triangle formed by the tangent to the $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ and coordinate axes is

  回答

  (A)

  $$ab$$ sq. units
• 17

  If total number of runs scored in n matches is $$\left( {{{n + 1} \over 4}} \right)\,\,({2^{n + 1}} - n - 2)\,$$ where $$n > 1$$, and the runs scored in the $${k^{th}}$$ match are given by k. $$\,{2^{n + 1 - k}}$$, where $$1 \le k \le n$$. Find n.

  回答

  (C)

  7
• 17

  A circle is given by $${x^2}\, + \,{(y\, - \,1\,)^2}\, = \,1$$, another circle C touches it externally and also the x-axis, then thelocus of its centre is

  回答

  (D)

  $$\{ (x,\,y):\,\,{x^2} = \,4y\} \, \cup \,\{ (0,\,\,y):\,\,y \le \,0\,\} $$
• 18

  Find the range of values of $$\,t$$ for which $$$2\,\sin \,t = {{1 - 2x + 5{x^2}} \over {3{x^2} - 2x - 1}},\,\,\,\,\,t\, \in \,\left[ { - {\pi \over 2},\,{\pi \over 2}} \right].$$$

  回答

  (C)

  $$\left[ {{{ - \pi } \over 2},\,{{ - \pi } \over {10}}} \right]\, \cup \,\left[ {{{3\pi } \over {10}},\,{\pi \over 2}} \right]$$
• 18

  In the quadratic equation $$\,\,a{x^2} + bx + c = 0,$$ $$\Delta $$ $$ = {b^2} - 4ac$$ and $$\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},$$ are in G.P. where $$\alpha ,\beta $$ are the root of $$\,\,a{x^2} + bx + c = 0,$$ then

  回答

  (C)

  $$c\Delta = 0$$
• 19

  If the LCM of p, q is $${r^2}\,{r^4}\,{s^2}$$, where r, s, t are prime numbers and p, q are the positive integers then number of ordered pair (p, q) is

  回答

  (C)

  225
• 20

  A rectangle with sides of lenght (2m - 1) and (2n - 1) units is divided into squares of unit lenght by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is

  回答

  (C)

  $${m^2}\,{n^2}$$
• 21

  The value of $$$\left( {\matrix{ {30} \cr 0 \cr } } \right)\left( {\matrix{ {30} \cr {10} \cr } } \right) - \left( {\matrix{ {30} \cr 1 \cr } } \right)\left( {\matrix{ {30} \cr {11} \cr } } \right) + \left( {\matrix{ {30} \cr 2 \cr } } \right)\left( {\matrix{ {30} \cr {12} \cr } } \right)....... + \left( {\matrix{ {30} \cr {20} \cr } } \right)\left( {\matrix{ {30} \cr {30} \cr } } \right)$$$
  is where $$\left( {\matrix{ n \cr r \cr } } \right) = {}^n{C_r}$$

  回答

  (A)

  $$\left( {\matrix{ {30} \cr {10} \cr } } \right)$$
• 22

  $$\cos \left( {\alpha - \beta } \right) = 1$$ and $$\,\cos \left( {\alpha + \beta } \right) = 1/e$$ where $$\alpha ,\,\beta \in \left[ { - \pi ,\pi } \right].$$
  Paris of $$\alpha ,\,\beta $$ which satisfy both the equations is/are
  

  回答

  (D)

  4