JEE Advance - Mathematics (1978)

• 1

  Find the derivative of $$\sin \left( {{x^2} + 1} \right)$$ with respect to $$x$$ first principle.

  回答

  (C)

  $$2x,cos left( {{x^2} + 1} ight)$$
• 2

  From a point $$O$$ inside a triangle $$ABC,$$ perpendiculars $$OD$$, $$OE, OF$$ are drawn to the sides $$BC, CA, AB$$ respectively. Prove that the perpendiculars from $$A, B, C$$ to the sides $$EF, FD, DE$$ are concurrent.

  回答

  (C)

  The perpendiculars from A, B, C to EF, FD, DE are concurrent at a point related to the isogonal conjugate of O with respect to triangle ABC.
• 3

  Balls are drawn one-by-one without replacement from a box containing $$2$$ black, $$4$$ white and $$3$$ red balls till all the balls are drawn. Find the probability that the balls drawn are in the order $$2$$ black, $$4$$ white and $$3$$ red.

  回答

  (A)

  1/1260
• 4

  Evaluate $$\int {{{\sin x} \over {\sin x - \cos x}}dx} $$

  回答

  (D)

  ${1 \over 2}\log |\sin x - \cos x| + {x \over 2} + C
• 5

  A triangle $$ABC$$ has sides $$AB=AC=5$$ cm and $$BC=6$$ cm Triangle $$A'B'C'$$ is the reflection of the triangle $$ABC$$ in a line parallel to $$AB$$ placed at a distance $$2$$ cm from $$AB$$, outside the triangle $$ABC$$. Triangle $$A''B''C''$$ is the reflection of the triangle $$A'B'C'$$ in a line parallel to $$BC$$ placed at a distance of $$2$$ cm from $$B'C'$$ outside the triangle $$A'B'C'$$. Find the distance between $$A$$ and $$A''$$.

  回答

  (C)

  8\sqrt{17/5} cm
• 6

  If x = a + b, y = a$$\gamma $$ + b$$\beta $$ and z = a$$\beta $$ +b$$\gamma $$ where $$\gamma $$ and $$\beta $$ are the complex cube roots of unity, show that xyz = $${a^3} + {b^3}$$.

  回答

  (B)

  The statement is true; xyz = a^3 + b^3.
• 7

  Find the equation of the circle whose radius is 5 and which touches the circle $${x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,$$ at the point (5, 5).

  回答

  (A)

  $${x^2}, + ,{y^2}, - ,18x,, - 16y, + 120 = 0,$$
• 8

  One side of rectangle lies along the line $$4x + 7y + 5 = 0.$$ Two of its vertices are $$(-3, 1)$$ and $$(1, 1).$$ Find the equations of the other three sides.

  回答

  A

  B

  C
• 9

  A straight line segment of length $$\ell $$ moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio $$1 : 2$$

  回答

  (C)

  $$9{x^2} + 36{y^2} = 4{\ell ^2}$$
• 10

  Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.

  回答

  (A)

  26
• 11

  Sketch the solution set of the following system of inequalities: $$${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$$

  回答

  (D)

  The solution set is a bounded region.
• 12

  Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$

  回答

  (C)

  x = 3
• 13

  Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$$

  回答

  (C)

  x = 3
• 14

  Show that the square of $$\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}$$ is a rational number.

  回答

  (C)

  The square of the expression equals 1.
• 15

  Solve the following equation for $$x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0$$

  回答

  A

  D
• 16

  Solve for $$x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.$$

  回答

  (C)

  $$5/4$$
• 17

  Solve for $$x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}$$

  回答

  (C)

  3/2
• 18

  where $$m$$ and $$n$$ are positive integers $$\left( {n \le m} \right),$$ show that
  $$\left( {m,n + 1} \right) = \left( {m - 1,\,n + 1} \right) + {x^{m - n - 1}}\left( {m - 1,n} \right).$$

  回答

  A

  B

  C

  D

  E
• 19

  If $$\tan \alpha = {m \over {m + 1}}\,$$ and $$\tan \beta = {2 \over {2m + 1}},$$ find the possible values of $$\left( {\alpha + \beta } \right).$$

  回答

  (C)

  $$n\pi + \frac{\pi}{4}$$
• 20

  Express $${1 \over {1 - \cos \,\theta + 2i\sin \theta }}$$ in the form x + iy.

  回答

  (D)

  $${{rac{1}{5 + 3cos heta}}} + i{{rac{-2cot( heta/2)}{5 + 3cos heta}}}$$