JEE Advance - Mathematics (1998)

• 1

  A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals

  回答

  (A)

  $$1/2$$
• 2

  For any two vectors $$u$$ and $$v,$$ prove that
  (a) $${\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}$$ and
  (b) $$\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v \right|}^2}} \right) = {\left( {1 - u.v} \right)^2} + {\left| {u + v + \left( {u \times v} \right)} \right|^2}.$$

  回答

  B

  C

  D
• 3

  Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

  回答

  (B)

  Vector methods can easily prove this by expressing the position vectors of the intersection point and midpoints and showing collinearity.
• 4

  Which of the following expressions are meaningful?

  回答

  A

  C
• 5

  For three vectors $$u,v,w$$ which of the following expression is not equal to any of the remaining three?

  回答

  (C)

  $$\,v \bullet \left( {u \times w} \right)$$
• 6

  If $$a = i + j + k,\overrightarrow b = 4i + 3j + 4k$$ and $$c = i + \alpha j + \beta k$$ are linearly dependent vectors and $$\left| c \right| = \sqrt 3 ,$$ then

  回答

  (D)

  $$\alpha = \pm 1,\,\,\beta = 1$$
• 7

  Let $${C_1}$$ and $${C_2}$$ be the graphs of the functions $$y = {x^2}$$ and $$y = 2x,$$ $$0 \le x \le 1$$ respectively. Let $${C_3}$$ be the graph of a function $$y=f(x),$$ $$0 \le x \le 1,$$ $$f(0)=0.$$ For a point $$P$$ on $${C_1},$$ let the lines through $$P,$$ parallel to the axes, meet $${C_2}$$ and $${C_3}$$ at $$Q$$ and $$R$$ respectively (see figure.) If for every position of $$P$$ (on $${C_1}$$ ), the areas of the shaded regions $$OPQ$$ and $$ORP$$ are equal, determine the function$$f(x).$$

  回答

  (C)

  f(x) = x^3 - x^2
• 8

  Three players, $$A,B$$ and $$C,$$ toss a coin cyclically in that order (that is $$A, B, C, A, B, C, A, B,...$$) till a head shows. Let $$p$$ be the probability that the coin shows a head. Let $$\alpha ,\,\,\,\beta $$ and $$\gamma $$ be, respectively, the probabilities that $$A, B$$ and $$C$$ gets the first head. Prove that $$\beta = \left( {1 - p} \right)\alpha $$ Determine $$\alpha ,\beta $$ and $$\gamma $$ (in terms of $$p$$).

  回答

  (A)

  $$\alpha = \frac{p}{1 - (1-p)^3}, \beta = \frac{(1-p)p}{1 - (1-p)^3}, \gamma = \frac{p(1-p)^2}{1 - (1-p)^3}$$
• 9

  Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

  回答

  (B)

  $$7/15$$
• 10

  There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

  回答

  (B)

  $$1/6$$
• 11

  If $$E$$ and $$F$$ are events with $$P\left( E \right) \le P\left( F \right)$$ and $$P\left( {E \cap F} \right) > 0,$$ then

  回答

  (D)

  none of the above implications holds
• 12

  If $$\overline E $$ and $$\overline F $$ are the complementary events of events $$E$$ and $$F$$ respectively and if $$0 < P\left( F \right) < 1,$$ then

  回答

  D

  A
• 13

  If from each of the three boxes containing $$3$$ white and $$1$$ black, $$2$$ white and $$2$$ black, $$1$$ white and $$3$$ black balls, one ball is drawn at random, then the probability that $$2$$ white and $$1$$ black ball will be drawn is

  回答

  (A)

  $$13/32$$
• 14

  The order of the differential equation whose general solution is given by
  $$y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},$$ where
  $${C_1},{C_2},{C_3},{C_4},{C_5},$$ are arbitrary constants, is

  回答

  (C)

  $$3$$
• 15

  Prove that $$\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.$$
  Hence or otherwise, evaluate the integral
  $$\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.} $$

  回答

  (A)

  log 2
• 16

  Let $$f\left( x \right) = x - \left[ x \right],$$ for every real number $$x$$, where $$\left[ x \right]$$ is the integral part of $$x$$. Then $$\int_{ - 1}^1 {f\left( x \right)\,dx} $$ is

  回答

  (A)

  $$1$$
• 17

  An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

  回答

  (B)

  7
• 18

  Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

  回答

  (C)

  The concurrence point is called the orthocenter of the triangle.
• 19

  If the vertices $$P, Q, R$$ of a triangle $$PQR$$ are rational points, which of the following points of the triangle $$PQR$$ is (are) always rational point(s)?

  回答

  A

  C

  D
• 20

  If $$\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)$$ and $$S\left( {a,b} \right)$$ are the vertices of a parrallelogram $$PQRS,$$ then

  回答

  (C)

  $$a = 2,\,b = 3$$
• 21

  The diagonals of a parralleogram $$PQRS$$ are along the lines $$x + 3y = 4$$ and $$6x - 2y = 7$$. Then $$PQRS$$ must be a.

  回答

  (D)

  rhombus.
• 22

  If $$x > 1,y > 1,z > 1$$ are in G.P., then $${1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}$$ are in

  回答

  (B)

  H.P.
• 23

  Let $${T_r}$$ be the $${r^{th}}$$ term of an A.P., for $$r=1, 2, 3, ....$$ If for some positive integers $$m$$, $$n$$ we have
  $${T_m} = {1 \over n}$$ and $${T_n} = {1 \over m},$$ then $${T_n} = {1 \over m},$$ equals

  回答

  (C)

  $$1$$
• 24

  Let $$n$$ be an odd integer. If $$\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $$ for every value of $$\theta ,$$ then

  回答

  (B)

  $${b_0} = 0,\,{b_1} = n$$
• 25

  [Hint: You may use the fact that $${\left( {1 + x} \right)^{\left( {m + 1} \right)p}} = {\left( {1 + x} \right)^p}{\left( {1 + x} \right)^{mp}}$$]

  回答

  (A)

  The problem asks to prove a divisibility property of binomial coefficients involving a prime number p. The hint suggests using the binomial theorem and induction.
• 26

  If the circle $${x^2}\, + \,{y^2} = \,{a^2}$$ intersects the hyperbola $$xy = {c^2}$$ in four points $$P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R\,({x_3},\,{y_3}),\,S\,({x_4},\,{y_4}),$$ then

  回答

  B

  A

  C

  D
• 27

  If $${a_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}},\,\,\,then\,\,\,\sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}}} } $$ equals

  回答

  (C)

  $${1 \over 2}n{a_n}$$
• 28

  Number of divisor of the form 4$$n$$$$ + 2\left( {n \ge 0} \right)$$ of the integer 240 is

  回答

  (A)

  4
• 29

  Prove that $$\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha $$

  回答

  (E)

  Use the identity $\tan x - \cot x = -2\cot 2x$ repeatedly.
• 30

  Which of the following number(s) is /are rational?

  回答

  (C)

  $$\sin {\mkern 1mu} {15^ \circ }{\mkern 1mu} \cos {\mkern 1mu} {15^ \circ }$$
• 31

  The number of values of $$x\,\,$$ in the interval $$\left[ {0,\,5\pi } \right]$$ satisfying the equation $$3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0$$ is

  回答

  (C)

  6
• 32

  If $$\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy$$ , then

  回答

  (D)

  x = 0, y = 0
• 33

  The value of the sum $$\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})$$ , where i = $$\sqrt { - 1} $$, equals

  回答

  (B)

  i - 1
• 34

  Let $${A_0}{A_1}{A_2}{A_3}{A_4}{A_5}$$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $${A_0}{A_1},{A_0}{A_2}$$ and $${A_0}{A_4}$$ is

  回答

  (C)

  $$3$$
• 35

  If $$\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} } $$ then the value of $$f(1)$$ is

  回答

  (A)

  $$1/2$$
• 36

  Suppose $$f(x)$$ is a function satisfying the following conditions
  (a) $$f(0)=2,f(1)=1$$,
  (b) $$f$$has a minimum value at $$x=5/2$$, and
  (c) for all $$x$$, $$$f'\left( x \right) = \matrix{ {2ax} & {2ax - 1} & {2ax + b + 1} \cr b & {b + 1} & { - 1} \cr {2\left( {ax + b} \right)} & {2ax + 2b + 1} & {2ax + b} \cr } $$$
  where $$a,b$$ are some constants. Determine the constants $$a, b$$ and the function $$f(x)$$.

  回答

  (A)

  $$a = 1/4, b = -5/4, f(x) = (1/4)x^2 - (5/4)x + 2$$
• 37

  A curve $$C$$ has the property that if the tangent drawn at any point $$P$$ on $$C$$ meets the co-ordinate axes at $$A$$ and $$B$$, then $$P$$ is the mid-point of $$AB$$. The curve passes through the point $$(1, 1)$$. Determine the equation of the curve.

  回答

  (C)

  xy = 1
• 38

  Let $$h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$$ for every real number $$x$$. Then

  回答

  C

  A
• 39

  If $$f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},$$ for every real number $$x$$, then the minimum value of $$f$$

  回答

  (D)

  is equal to -1
• 40

  The number of values of $$x$$ where the function
  $$f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)$$ attains its maximum is

  回答

  (B)

  $$1$$
• 41

  Prove that a triangle $$ABC$$ is equilateral if and only if $$\tan A + \tan B + \tan C = 3\sqrt 3 $$.

  回答

  A

  B

  C

  D

  E
• 42

  A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $${60^ \circ }$$ and $${30^ \circ }$$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $$P$$ and $$Q$$ respectively on its path. Let $$\theta $$ be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between $$P$$ and $$Q$$. Find the numerical value of $${\tan ^2}\theta $$. (Assume that the observer is not inside the vertical projection of the path of the bird.)

  回答

  (C)

  3/5
• 43

  If $${\omega}$$ is an imaginary cube root of unity, then $${(1\, + \omega \, - {\omega ^2})^7}$$ equals

  回答

  (D)

  $$ - 128{\omega ^2}$$
• 44

  If in a triangle $$PQR$$, $$\sin P,\sin Q,\sin R$$ are in $$A.P.,$$ then

  回答

  (B)

  the altitudes are in $$H.P.$$
• 45

  If$$\,\,\,$$ $$y = {{a{x^2}} \over {\left( {x - a} \right)\left( {x - b} \right)\left( {x - c} \right)}} + {{bx} \over {\left( {x - b} \right)\left( {x - c} \right)}} + {c \over {x - c}} + 1$$,
  prove that $${{y'} \over y} = {1 \over x}\left( {{a \over {a - x}} + {b \over {b - x}} + {c \over {c - x}}} \right)$$.

  回答

  (D)

  Take the natural logarithm of both sides and then differentiate.
• 46

  The angle between a pair of tangents drawn from a point $$P$$ to the parabola $${y^2} = 4ax$$ is $${45^ \circ }$$. Show that the locus of the point $$P$$ is a hyperbola.

  回答

  A

  B

  C

  D

  E
• 47

  If $$P=(x, y)$$, $${F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)$$ and $$16{x^2} + 25{y^2} = 400,$$ then $$P{F_1} + P{F_2}$$ equals

  回答

  (C)

  $$10$$
• 48

  The number of values of $$c$$ such that the straight line $$y=4x + c$$ touches the curve $$\left( {{x^2}/4} \right) + {y^2} = 1$$ is

  回答

  (C)

  $$2$$
• 49

  $$C_1$$ and $$C_2$$ are two concentric circles, the radius of $$C_2$$ being twice that of $$C_1$$. From a point P on $$C_2$$, tangents PA and PB are drawn to $$C_1$$. Prove that the centroid of the triangle PAB lies on $$C_1$$.

  回答

  A

  B

  C

  D
• 50

  The number of common tangents to the circles $${x^2}\, + \,{y^2} = 4$$ and $${x^2}\, + \,{y^2}\, - 6x\, - 8y = 24$$ is

  回答

  (B)

  1