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.

  Trả lời

  (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 $$\overrightarrow b \, = $$.........

  Trả lời

  (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.

  Trả lời

  (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 $${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$$

  Trả lời

  (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 ...............

  Trả lời

  (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 ______.

  Trả lời

  (B)

  1/64
• 7

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

  Trả lời

  (D)

  never less than $$n$$
• 8

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

  Trả lời

  (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.

  Trả lời

  (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 $$\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$$.

  Trả lời

  (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 $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$

  Trả lời

  (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 ...............

  Trả lời

  (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.

  Trả lời

  (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 $$\lambda $$ = .........

  Trả lời

  (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 $$y + 2x = 5$$.

  Trả lời

  (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.

  Trả lời

  (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.

  Trả lời

  B

  D
• 18

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

  Trả lời

  (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.

  Trả lời

  (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.

  Trả lời

  (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 $${{ar} \over {1 - {r^2}}}$$

  Trả lời

  A

  B

  D
• 22

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

  Trả lời

  (D)

  $$\frac{6 + \pi}{6}$$
• 23

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

  Trả lời

  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.

  Trả lời

  (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 $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$

  Trả lời

  (C)

  intersects the curve at at least one point
• 26

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

  Trả lời

  A

  C