JEE Advance - Mathematics (1991)

• 1

  In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is $$1/3$$ and the probability that he copies the answer is $$1/6$$. The probability that his answer is correct given that he copied it, is $$1/8$$. Find the probability that he knew the answer to the questions given that he correctly answered it.

  Responder

  (C)

  24/29
• 2

  Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3$$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c ,$$ then $$\overrightarrow b \, = $$.........

  Responder

  (A)

  $$\left( {\frac{5}{3},\frac{2}{3},\frac{2}{3}} \right)$$
• 3

  Determine the value of $$'c'$$ so that for all real $$x,$$ the vector
  $$cx\widehat i - 6\widehat j - 3\widehat k$$ and $$x\widehat i + 2\widehat j + 2cx\widehat k$$ make an obtuse angle with each other.

  Responder

  (B)

  -4/3 < c < 0
• 4

  If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}$$ satiesfies the equation $${x^2} - 9x + 8 = 0,$$ find the value of $${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$$

  Responder

  (B)

  $$\frac{\sqrt{3} - 1}{2}$$
• 5

  If the mean and the variance of binomial variate $$X$$ are $$2$$ and $$1$$ respectively, then the probability that $$X$$ takes a value greater than one is equal to ...............

  Responder

  (B)

  11/16
• 6

  The value of
  $$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}$$ is equal to ______.

  Responder

  (B)

  1/64
• 7

  The product of $$n$$ positive numbers is unity. Then their sum is

  Responder

  (D)

  never less than $$n$$
• 8

  Using induction or otherwise, prove that for any non-negative integers $$m$$, $$n$$, $$r$$ and $$k$$ ,
  $$\sum\limits_{m = 0}^k {\left( {n - m} \right)} {{\left( {r + m} \right)!} \over {m!}} = {{\left( {r + k + 1} \right)!} \over {k!}}\left[ {{n \over {r + 1}} - {k \over {r + 2}}} \right]$$

  Responder

  (D)

  Mathematical induction can be directly applied with respect to $$k$$.
• 9

  Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made.

  Responder

  (A)

  ${}^{11}{C_5} \times 9! \times 9!
• 10

  Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and $$\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$$.

  Responder

  (D)

  The statement's validity depends on the specific values of the two numbers and n.
• 11

  If $${S_1}$$, $${S_2}$$, $${S_3}$$,.............,$${S_n}$$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are $${1 \over 2}$$, $${1 \over 3}$$, $${1 \over 4}$$,....................$$\,{1 \over {n + 1}}$$ respectively, then find the values of $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$

  Responder

  (C)

  ${{{}^n(2n + 1),(4n + 1) - 3} \over 3}$
• 12

  Let the algebraic sum of the perpendicular distances from the points $$\left( {2,0} \right),\,\left( {0,\,2} \right)$$ $$\left( {1,\,1} \right)$$ to a variable straight line be zero; then the line passes through a fixed point whose cordinates are ...............

  Responder

  (A)

  (1, 1)
• 13

  Show that all chords of the curve $$3{x^2} - {y^2} - 2x + 4y = 0,$$ which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

  Responder

  (B)

  The chords pass through the point (1, -2).
• 14

  If a circle passes through the points of intersection of the coordinate axes with the lines $$\lambda \,x - y + 1 = 0$$ and x - 2y + 3 = 0, then the value of $$\lambda $$ = .........

  Responder

  (B)

  2
• 15

  Find the equation of the line passing through the point $$(2, 3)$$ and making intercept of length 2 units between the lines $$y + 2x = 3$$ and $$y + 2x = 5$$.

  Responder

  (A)

  3x + 4y - 18 = 0 or x - 2 = 0
• 16

  Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.

  Responder

  (A)

  x^2 + y^2 + 6x + 2y - 15 = 0 and x^2 + y^2 - 10x - 10y + 25 = 0
• 17

  Three normals are drawn from the point $$(c, 0)$$ to the curve $${y^2} = x.$$ Show that $$c$$ must be greater than $$1/2$$. One normal is always the $$x$$-axis. Find $$c$$ for which the other two normals are perpendicular to each other.

  Responder

  B

  D
• 18

  Find $${{{dy} \over {dx}}}$$ at $$x=-1$$, when
  $${\left( {\sin y} \right)^{\sin \left( {{\pi \over 2}x} \right)}} + {{\sqrt 3 } \over 2}{\sec ^{ - 1}}\left( {2x} \right) + {2^x}\tan \left( {In\left( {x + 2} \right)} \right) = 0$$

  Responder

  (A)

  0
• 19

  The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

  Responder

  (C)

  4, 5, 6
• 20

  A man notices two objects in a straight line due west. After walking a distance $$c$$ due north he observes that the objects subtend an angle $$\alpha $$ at his eye; and, after walking a further distance $$2c$$ due north, an angle $$\beta $$. Show that the distance between the objects is $${{8c} \over {3\cot \beta - \cot \alpha }}$$; the height of the man is being ignored.

  Responder

  (A)

  The distance between the objects is $${{8c} over {3\cot \beta - \cot \alpha }}$$
• 21

  In a triangle of base a the ratio of the other two sides is $$r\left( { < 1} \right)$$. Show that the altitude of the triangle is less than of equal to $${{ar} \over {1 - {r^2}}}$$

  Responder

  A

  B

  D
• 22

  What is the ratio for the sides of the rectangle so that the window transmits the maximum light ?

  Responder

  (D)

  $$\frac{6 + \pi}{6}$$
• 23

  Evaluate $$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $$

  Responder

  B

  C

  D
• 24

  Sketch the curves and identify the region bounded by
  $$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.

  Responder

  (C)

  $$\,{{4 - \sqrt 2 } \over {\log 2}} - {5 \over 2}\log 2 + {3 \over 2}$$
• 25

  If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to \infty ,$$ then show that every line $$y=mx$$
  intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$

  Responder

  (C)

  intersects the curve at at least one point
• 26

  For any two events $$A$$ and $$B$$ in a simple space

  Responder

  A

  C