| AIEEE Home page | 2007 AIEEE PAPER - 2 (Mathematics) |
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1) In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals | |||||||||||
1 )
1/2 (1 -  )
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2 )
1/2 ()
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3 )
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4 )
1/2( - 1)
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| see the answer see the solution | |||||||||||
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2) If sin-1(x/5) + cosec-1(5/4) = π/2 then a value of x is | |||||||||||
| 1 ) 1 | |||||||||||
| 2 ) 3 | |||||||||||
| 3 ) 4 | |||||||||||
| 4 ) 5 | |||||||||||
| see the answer see the solution | |||||||||||
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3) In the binomial expansion of (a - b)n, n ≥ 5, the sum of 5th and 6th terms is zero, then a/b equals | |||||||||||
| 1 ) 5/(n - 4) | |||||||||||
| 2 ) 6/(n - 5) | |||||||||||
| 3 ) (n - 5)/6 | |||||||||||
| 4 ) (n - 4)/5 | |||||||||||
| see the answer see the solution | |||||||||||
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4) The set S = {1, 2, 3, ..., 12) is to be partitioned into three sets A, B, C of equal size. Thus, A ∪ B ∪ C = S , A ∩ B = B ∩ C = A ∩ C = Φ . The number of ways to partition S is | |||||||||||
1 )
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2 )
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3 )
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4 )
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| see the answer see the solution | |||||||||||
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5) The largest interval lying in ( -π/2 , π/2 ) for which the function [ f(x) = 4x-x2 + cos-1(x/2 - 1) + log(cos x) ] is defined, is | |||||||||||
| 1 ) [ 0 , π ] | |||||||||||
| 2 ) ( -π/2 , π/2 ) | |||||||||||
| 3 ) [ -π/2 , π/2 ) | |||||||||||
| 4 ) [ 0 , π/2 ) | |||||||||||
| see the answer see the solution | |||||||||||
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6) A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends being fastened to the extremities of a rod 13 m long. If the rod be so held that the body hangs immediately below the middle point. The tensions in the strings are | |||||||||||
| 1 ) 12 kg and 13 kg | |||||||||||
| 2 ) 5 kg and 5 kg | |||||||||||
| 3 ) 5 kg and 12 kg | |||||||||||
| 4 ) 5 kg and 13 kg | |||||||||||
| see the answer see the solution | |||||||||||
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7) A pair of fair dice is thrown independently three times. The probability of getting a score of exactly 9 twice is | |||||||||||
| 1 ) 1/729 | |||||||||||
| 2 ) 8/9 | |||||||||||
| 3 ) 8/729 | |||||||||||
| 4 ) 8/243 | |||||||||||
| see the answer see the solution | |||||||||||
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8) Consider a family of circles which are passing through the point (-1, 1) and are tangent to x-axis. If (h, k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the interval | |||||||||||
| 1 ) 0 < k < 1/2 | |||||||||||
| 2 ) k ≥ 1/2 | |||||||||||
| 3 ) -1/2 ≤ k ≤ 1/2 | |||||||||||
| 4 ) k ≤ 1/2 | |||||||||||
| see the answer see the solution | |||||||||||
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9) Let L be the line of intersection of the planes 2x + 3y + z = 1 and x + 3y + 2z = 2. If L makes an angles α with the positive x-axis, then cos α equals | |||||||||||
1 )
1/
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| 2 ) 1/2 | |||||||||||
| 3 ) 1 | |||||||||||
4 )
1/
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| see the answer see the solution | |||||||||||
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10) The differential equation of all circles passing through the origin and having their centres on the x-axis is | |||||||||||
1 )
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2 )
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3 )
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4 )
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| see the answer see the solution | |||||||||||
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11) If p and q are positive real numbers such that p2 + q2 = 1, then the maximum value of (p + q) is | |||||||||||
| 1 ) 2 | |||||||||||
| 2 ) 1/2 | |||||||||||
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3 )
1/
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4 )
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| see the answer see the solution | |||||||||||
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12) A tower stands at the centre of a circular park. A and B are two points on the boundary of the park such that AB (= a) subtends an angle of 60° at the foot of the tower, and the angle of elevation of the top of the tower from A or B is 30° . The height of the tower is | |||||||||||
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1 )
2a/
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2 )
2a
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3 )
a/
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4 )
a
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| see the answer see the solution | |||||||||||
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13) The sum of the series 20C0 - 20C1 + 20C2 - 20C3 + ... - ... + 20C10 is | |||||||||||
| 1 ) - 20C10 | |||||||||||
| 2 ) (1/2)(20C10) | |||||||||||
| 3 ) 0 | |||||||||||
| 4 ) 20C10 | |||||||||||
| see the answer see the solution | |||||||||||
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14) The normal to a curve at P(x, y) meets the x-axis at G. If the distance of G from the origin is twice the abscissa of P, then the curve is a | |||||||||||
| 1 ) ellipse | |||||||||||
| 2 ) parabola | |||||||||||
| 3 ) circle | |||||||||||
| 4 ) hyperbola | |||||||||||
| see the answer see the solution | |||||||||||
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15) If |z + 4| ≤ 3, then the maximum value of |z + 1| is | |||||||||||
| 1 ) 4 | |||||||||||
| 2 ) 10 | |||||||||||
| 3 ) 6 | |||||||||||
| 4 ) 0 | |||||||||||
| see the answer see the solution | |||||||||||
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16) The resultant of two forces P N and 3 N is a force of 7 N. If the direction of 3 N force were reversed, the resultant would be  .
The value of P is
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| 1 ) 5 N | |||||||||||
| 2 ) 6 N | |||||||||||
| 3 ) 3 N | |||||||||||
| 4 ) 4 N | |||||||||||
| see the answer see the solution | |||||||||||
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17) Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is | |||||||||||
| 1 ) 0.06 | |||||||||||
| 2 ) 0.14 | |||||||||||
| 3 ) 0.2 | |||||||||||
| 4 ) 0.7 | |||||||||||
| see the answer see the solution | |||||||||||
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18) If  for x ≠ 0 , y ≠ 0 then D is | |||||||||||
| 1 ) divisible by neither x nor y | |||||||||||
| 2 ) divisible by both x and y | |||||||||||
| 3 ) divisible by x but not y | |||||||||||
| 4 ) divisible by y but not x | |||||||||||
| see the answer see the solution | |||||||||||
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19) For the hyperbola
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| 1 ) Eccentricity | |||||||||||
| 2 ) Directrix | |||||||||||
| 3 ) Abscissae of vertices | |||||||||||
| 4 ) Abscissae of foci | |||||||||||
| see the answer see the solution | |||||||||||
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20) If a line makes an angle of π/2 with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is | |||||||||||
| 1 ) π/6 | |||||||||||
| 2 ) π/3 | |||||||||||
| 3 ) π/4 | |||||||||||
| 4 ) π/2 | |||||||||||
| see the answer see the solution | |||||||||||
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21) A value of C for which the conclusion of Mean Value Theorem holds for the function f(x) = logex on the interval [1, 3] is | |||||||||||
| 1 ) 2 log3e | |||||||||||
| 2 ) (1/2)loge3 | |||||||||||
| 3 ) log3e | |||||||||||
| 4 ) loge3 | |||||||||||
| see the answer see the solution | |||||||||||
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22) The function f(x) = tan-1(sinx + cosx) is an increasing function in | |||||||||||
| 1 ) ( π/4 , π/2 ) | |||||||||||
| 2 ) ( - π/2 , π/4 ) | |||||||||||
| 3 ) ( 0 , π/2 ) | |||||||||||
| 4 ) ( - π/2 , π/2 ) | |||||||||||
| see the answer see the solution | |||||||||||
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23) Let  .
If |A2| = 25 , then |α| equals
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| 1 ) 52 | |||||||||||
| 2 ) 1 | |||||||||||
| 3 ) 1/5 | |||||||||||
| 4 ) 5 | |||||||||||
| see the answer see the solution | |||||||||||
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24) The sum of the series
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| 1 ) e-2 | |||||||||||
| 2 ) e-1 | |||||||||||
| 3 ) e-1/2 | |||||||||||
| 4 ) e1/2 | |||||||||||
| see the answer see the solution | |||||||||||
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25) If û and v are unit vectors and θ is the acute angle between them, then 2û X 3v is a unit vector for | |||||||||||
| 1 ) exactly two values of θ | |||||||||||
| 2 ) more than two values of θ | |||||||||||
| 3 ) no value of θ | |||||||||||
| 4 ) exactly one value of θ | |||||||||||
| see the answer see the solution | |||||||||||
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26) A particle just clears a wall of height b at distance a and strikes the ground at a distance c from the point of projection. The angle of projection is | |||||||||||
1 )
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| 2 ) 45° | |||||||||||
3 )
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4 )
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| see the answer see the solution | |||||||||||
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27) The average marks of boys in a class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is | |||||||||||
| 1 ) 40 | |||||||||||
| 2 ) 20 | |||||||||||
| 3 ) 80 | |||||||||||
| 4 ) 60 | |||||||||||
| see the answer see the solution | |||||||||||
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28) The equation of a tangent to the parabola y2 = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is | |||||||||||
| 1 ) (- 1, 1) | |||||||||||
| 2 ) (0, 2) | |||||||||||
| 3 ) (2, 4) | |||||||||||
| 4 ) (-2, 0) | |||||||||||
| see the answer see the solution | |||||||||||
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29) If (2, 3, 5) is one end of a diameter of the sphere x2 + y2 + z2 - 6x - 12y - 2z + 20 = 0, then the coordinates of the other end of the diameter are | |||||||||||
| 1 ) (4, 9, -3) | |||||||||||
| 2 ) (4, -3, 3) | |||||||||||
| 3 ) (4, 3, 5) | |||||||||||
| 4 ) (4, 3, -3) | |||||||||||
| see the answer see the solution | |||||||||||
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30) Let a = i + j + k , b = i - j + 2 k and c = x i + (x - 2)j - k . If the vector c lies in the plane of a and b then x equals | |||||||||||
| 1 ) 0 | |||||||||||
| 2 ) 1 | |||||||||||
| 3 ) -4 | |||||||||||
| 4 ) -2 | |||||||||||
| see the answer see the solution | |||||||||||
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31) Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of the triangle is 1, then the set of values which 'k' can take is given by | |||||||||||
| 1 ) {1, 3} | |||||||||||
| 2 ) {0, 2} | |||||||||||
| 3 ) {-1, 3} | |||||||||||
| 4 ) {-3, -2} | |||||||||||
| see the answer see the solution | |||||||||||
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32) Let P = (-1, 0), Q = (0, 0) and R = (3, 3 )
be three points. The equation of the bisector of
the angle PQR
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1 )
x + y = 0
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2 )
x + (/2) y = 0
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3 )
(/2) x + y = 0
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4 )
x + y = 0
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| see the answer see the solution | |||||||||||
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33) If one of the lines of my2 + (1 - m2)xy - mx2 = 0 is a bisector of the angle between the lines xy = 0, then m is | |||||||||||
| 1 ) - 1/2 | |||||||||||
| 2 ) -2 | |||||||||||
| 3 ) 1 | |||||||||||
| 4 ) 2 | |||||||||||
| see the answer see the solution | |||||||||||
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34) Let F(x) = f(x) + f(1/x), where  Then F(e) equals | |||||||||||
| 1 ) 1/2 | |||||||||||
| 2 ) 0 | |||||||||||
| 3 ) 1 | |||||||||||
| 4 ) 2 | |||||||||||
| see the answer see the solution | |||||||||||
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35) Let f : R → R be a function defined by f(x) = Min { x + 1 , |x| , 1 }. Then which of the following is true? | |||||||||||
| 1 ) f(x) ≥ for all x ∈ R | |||||||||||
| 2 ) f(x) is not differentiable at x = 1 | |||||||||||
| 3 ) f(x) is differentiable everywhere | |||||||||||
| 4 ) f(x) is not differentiable at x = 0 | |||||||||||
| see the answer see the solution | |||||||||||
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36) The function f: R ~ {0} → R given by
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| 1 ) 2 | |||||||||||
| 2 ) -1 | |||||||||||
| 3 ) 0 | |||||||||||
| 4 ) 1 | |||||||||||
| see the answer see the solution | |||||||||||
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37) The solution for x of the equation  is | |||||||||||
| 1 ) 2 | |||||||||||
| 2 ) π | |||||||||||
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3 )
/2
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4 )
2
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| see the answer see the solution | |||||||||||
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38)  is | |||||||||||
1 )
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2 )
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3 )
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4 )
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| see the answer see the solution | |||||||||||
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39) The area enclosed between the curves y2 = x and y = |x| is | |||||||||||
| 1 ) 2/3 | |||||||||||
| 2 ) 1 | |||||||||||
| 3 ) 1/6 | |||||||||||
| 4 ) 1/3 | |||||||||||
| see the answer see the solution | |||||||||||
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40) If the difference between the roots of the equation x2 + ax + 1 = 0 is less than ,
then the set of possible values of a is
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| 1 ) ( - 3, 3 ) | |||||||||||
| 2 ) ( - 3, ∞ ) | |||||||||||
| 3 ) ( 3, ∞ ) | |||||||||||
| 4 ) (- ∞ , - 3) | |||||||||||
| see the answer see the solution |
| 1) 4 | 2) 2 | 3) 4 | 4) 3 |
| 5) 4 | 6) 3 | 7) 4 | 8) 2 |
| 9) 1 | 10) 3 | 11) 4 | 12) 3 |
| 13) 2 | 14) 1,4 | 15) 3 | 16) 1 |
| 17) 2 | 18) 2 | 19) 4 | 20) 4 |
| 21) 1 | 22) 2 | 23) 3 | 24) 2 |
| 25) 4 | 26) 3 | 27) 3 | 28) 4 |
| 29) 1 | 30) 4 | 31) 3 | 32) 1 |
| 33) 3 | 34) 1 | 35) 3 | 36) 4 |
| 37) - | 38) 1 | 39) 3 | 40) 3 |
Solution
| 1 back to 1 nth term of the Geometric progression = an = arn-1 So, arn-1 = arn + arn+1 => i = r + r2 => r = 1/2( - 1)
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| 2 back to 2 sin-1(x/5) + cosec-1(5/4) = π/2 => sin-1(x/5) = π/2 - cosec-1(5/4) = π/2 - sin-1(4/5) => sin-1(x/5) = cos-1(4/5) = sin-1(3/5) => x = 3 | |||||||||||||||||||||||
| 3 back to 3 nC4 an-4(-b)4 + nC5 an-5(-b)5 = 0 => a/b = (n-5+1)/5 = (n-4)/5 | |||||||||||||||||||||||
| 4 back to 4 Number of ways to partition : 12C4 x 8C4 x 4C4 =
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| 5 back to 5 x/2 - 1 must be between -1 and 1 , so -1 ≤ x/2 - 1 ≤ 1 .......... (i) and, cosx must be greater than 0, i.e cosx > 0 ..................(ii) By solving (i), 0 ≤ x ≤ 4 By solving (ii), π/2 < x < -π/2 So, x ∈ [ 0 , π/2 ) | |||||||||||||||||||||||
6 back to 6 T2cosθ +T1sinθ = mg .....................(i) T2sinθ = T1cosθ .....................(ii) T2 = mgcosθ .....................(iii) By solving (ii) and (iii) T1 = mg sinθ tanθ = 5/12 So, T1 = 13kg x (5/13) = 5 Kg And, T2 = 13kg x (12/13) = 12 Kg | |||||||||||||||||||||||
| 7 back to 7 Probability of getting score 9 in a single throw = 4/36 = 1/9 Probability of getting sum nine exactly two times out of three draws = 3C2 (1/9)2(8/9) = 8/243 | |||||||||||||||||||||||
| 8 back to 8 Equation of circle is: (x-h)2 + (y-k)2 = k2 The circle passes thru (-1,1), so (-1-h)2 + (1-k)2 = k2 h2 + 2h -2k + 2 = 0 D ≥ 0 2k-1 ≥ 0 ⇒ k ≥ 1/2 | |||||||||||||||||||||||
| 9 back to 9 If direction cosines of L be l, m, n, then 2l + 3m + n = 0 ............(i) l + 3m + 2n = 0 ............(ii) By solving (i) and (ii) l/3 = -m/3 = n/3 So, l:m:n = 1/ : -1/ : 1/ So, cosα = 1/
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| 10 back to 10 Equation of circle passing through origin and having their centres on x-axis is : x2 + y2 + 2gx = 0 ............ (1)
Replacing value of g from equation (i)
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| 11 back to 11 We know that: Arithmetic Mean ≥ Geometric Mean. So
=> 1/2 ≥ pq => 1 ≥ 2pq We know that p2 + q2 + 2pq = (p+q)2 1 + 1 ≥ (p+q)2 => ≥ (p+q)
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12 back to 12 ∠ACB = 60° ΔABC is an equilateral triangle Radius of the circle = a DC/AC = tan30° DC = a/
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| 13 back to 13 (1+x)20 = 20C0 + 20C1x + 20C2x2 + .... + 20C20x20 Let x = -1, then 0 = 20C0 - 20C1 + 20C2 + .... + 20C20 0 = 2(20C0 - 20C1 + 20C2 + .... - 20C9) + 20C10 20C0 - 20C1 + 20C2 + .... - 20C9 + 20C10 = (1/2)20C10 | |||||||||||||||||||||||
| 14 back to 14 Equation of normal at P(x,y) is : Y-y = (dy/dx) (X-x) Co-ordinate of point G is (x+y(dy/dx) , 0) |x+y(dy/dx)| = 2x ................ (i) => y(dy/dx) = x OR y(dy/dx) = -3x => y dy = x dx OR ydy= -3xdx => y2/2 = x2/2 + c OR y2/2 = -3x2/2 + c => x2 - y2 = -2c OR 3x2 + y2 = 2c | |||||||||||||||||||||||
15 back to 15 |z + 4| ≤ 3 => z can be on the circle or inside the circle of radius=3 and center at: (-4,0). => So maximum value of |z+1| will be 6 | |||||||||||||||||||||||
16 back to 16 72 = P2 + 9 + 6Pcosθ => 6Pcosθ = 40 - P2 ............................(i) 19 = P2 + 9 + 6Pcos(π - θ) => 19 = P2 + 9 - 6Pcosθ ............................(ii) Solving (i) and (ii) 19 = P2 + 9 - 40 + P2 So, P = 5N | |||||||||||||||||||||||
| 17 back to 17 Required probability = 0.7 x 0.2 + (0.7) (0.8) (0.7) (0.2) + (0.7) (0.8) (0.7) (0.8) (0.7) (0.2) + ... = 0.14 [ 1 + (0.56) + (0.56)2 + (0.56)3 + (0.56)4..... ] = 0.14 (1/(1-0.56)) = 0.14/0.44 = 7/22 | |||||||||||||||||||||||
| 18 back to 18 Do following: Column3 = Column3 - Column1 Column2 = Column2 - Column1 So,  So, D is divisible by x and y. | |||||||||||||||||||||||
| 19 back to 19 Eccentricity: Eccentricity measures as how much the conic section deviates from being circular. a2 = cos2α ................(i) b2 = sin2α ................(ii) b2 = a2(e2 - 1) ................(iii) By solving (i), (ii) and (iii) e = secα coordinates of focii : (±ae , 0) = (±1 , 0) Hence abscissae of foci remain constant when α varies. | |||||||||||||||||||||||
| 20 back to 20 cos2α + cos2β + cos2γ = 1 => cos2&pi/4 + cos2&pi/4 + cos2γ = 1 => 1/2 + 1/2 + cos2γ = 1 => cos2γ = 0 => γ = π/2 | |||||||||||||||||||||||
| 21 back to 21 f'(C) = (f(3)-f(1))/(3-1) => 1/c = (loge3)/2 => c = 2log3e | |||||||||||||||||||||||
| 22 back to 22 f(x) = tan-1(sinx + cosx) f'(x) = (cosx - sinx)/( 1+(sinx + cosx)2 ) f'(x) = ( cos(x+π/4))/( 1+(sinx + cosx)2 ) So, f(x) increases if -π/2 < x + π/4 < π/2 => So, f(x) increases if -3π/4 < x < π/4 => So f(x) increases in x ∈ (-π/2 , π/4) | |||||||||||||||||||||||
| 23 back to 23 |A . A| = |A|.|A| = (25α). (25α) = 25 So, α2 = 1/25 => α = ± 1/5 | |||||||||||||||||||||||
24 back to 24
Repace x with 1, then
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| 25 back to 25 |2u × 3v| = 1 6|u × v| = 1 |u × v| = 1/6 sinθ = ±1/6 As θ is accute angle so θ can have only value | |||||||||||||||||||||||
26 back to 26 a = u cosα t ................... (i) b = u sinα t - 1/2 gt2 .........................(ii) c = (u2 sin2α)/2 ..............................(iii) Using (i) and (ii) b = a tanα - (1/2)g (a2)/(u2cos2α) .........(iv) Repacing value of u2 from (iii) in (iv) b = (a tanα) - (a2g sin2α sec2α)/(2cg) Use, sin2α = 2sinα cosα b = a tanα - (a2 2tanα)/(2c) tanα = (bc)/(a(c-a)) | |||||||||||||||||||||||
| 27 back to 27 Let number of boys = x number of girls - y Total marks = 52x + 42y = 50 (x+y) => 2x = 8y => x = 4y So percent of boys = 100x/(x+y) = 400y/(5y) = 80% | |||||||||||||||||||||||
| 28 back to 28 Point of intersection of two perpendicular tangents to the parabola must be on directrix of the parabola. Equation of directrix is x + 2 = 0 Hence the point is (-2, 0). | |||||||||||||||||||||||
| 29 back to 29 Coordinates of centre = (3,6,1) Let coordinates of other end of the diameter is (α, β, γ) So, (α+2)/2 = 3 (β+3)/2 = 6 (γ+5)/2 = 1 So, α = 4 β = 9 γ = -3 | |||||||||||||||||||||||
30 back to 30 By solving above , 2x + 4 = 0 x = -2 | |||||||||||||||||||||||
31 back to 31 A = (1/2).1.|k-1| = 1 => k-1 = 2 OR k-1= -2 =>k = 3 OR k = -1 | |||||||||||||||||||||||
32 back to 32 Slope of QR = So ∠PQR = 120° Slope of QS = tan120° = - So equation of QS will be y=- x
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| 33 back to 33 my2 + (1 - m2)xy - mx2 = 0 => my2 + m2xy + xy - mx2 = 0 => my(y-mx)+x(y-mx) = 0 => (my+x)(y-mx) = 0 => y=mx and y=(-1/m)x So, m=±1 | |||||||||||||||||||||||
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35 back to 35 f(x) = Min { x + 1 , |x| , 1 } (under progress) | |||||||||||||||||||||||
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| 40 back to 40 α+β=-a and α.β=1 |α-β| < => (α-β)2 < 5 => (α+β)2 - 4α.&beta < 5 => a2-4 < 5 => a ∈ (-3,3) |
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