1) Let R = { (1,3),(4,2),(2,4),(2,3),(3,1) } be th erelation on the set A = {1,2,3,4}. The relation R is

a )   a function b )   transitive c )   not symmetric d )   reflexive

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2) The range of the function f(x) = 7-xPx-3 is

a )   {1,2,3} b )   {1,2,3,4,5,6} c )   {1,2,3,4} d )   {1,2,3,4,5}

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3) Let z, w be complex numbers such that z + iw = 0 and zw = &928;

a )   π/4 b )   π/2 c )   3π/4 d )   5π/4

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4) If z = x - iy and z1/3 = p + iq, then ((x/p + y/q)/(p2 + q2) is equal to

a )   1 b )   -1 c )   2 d )   -2

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5) If | z2 - 1 | = | z2 + 1 |, then z lies on

a )   the real axis b )   the imaginary axis c )   a circle d )   an ellipse

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6)

a )   A is a zero matrix b )   A = (-1) I, where I is a unix matrix c )   A-1 d )   A2 = I

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7)

a )   -2 b )   -1 c )   2 d )   5

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8) If a1, a2, a3, ...... a1, ......are in G.P, then the value of the determinant : , is

a )   b )   1 c )   2 d )   -2

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9) Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

a )   x2 + 18x + 16 = 0 b )   x2 - 18x + 16 = 0 c )   x2 + 18x - 16 = 0 d )   x2 - 18x - 16 = 0

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10) If (1 - p) is a root of quadratic equation x2 + px + (1 - p) = 0 then its roots are

a )   0,1 b )   -1,1 c )   0,-1 d )   -1,2

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11) Let S(K) = 1 + 3 + 5 + .... + (2K - 1) = 3 + K2. Then which of the folowing is true ?

a )   S(1) is correct b )   S(K) ⇒ S(K + 1) c )   S(K) ≠> S(K + 1) d )   Principle of mathematical induction can be used to prove the formula

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12) How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order ?

a )   120 b )   240 c )   360 d )   480

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13) The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the box is empty

a )   5 b )   21 c )   38 d )   8C3

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14) If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of ' q ' is

a )   49/4 b )   12 c )   3 d )   4

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15) The coefficient of the middle term in the binomial expansion in powers of x of (1 + α x)4 and of (1 - α x)6 is the same if α equals

a )   -5/3 b )   10/3 c )   -3/10 d )   3/5

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16) The coefficient of xn in expansion of (1 + x)(1 - x)n is

a )   (n -1) b )   (-1)n(n - 1) c )   (-1)n-1(n - 1)2 d )   (-1)n-1n

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17)

a )   (1/2)n b )   (1/2)n - 1 c )   n - 1 d )   (2n - 1)/2

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18) Let Tr be the rth term of an A.P. whose first term is a and common difference is d. If for same positive integers m, n, m ≠ n, Tn = 1/m, then a - d equals

a )   b )   1 c )   1/mn d )   1/m + 1/n

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19) The sum of first n terms of the series 12 + 2 . 22 + 32 + 2 . 42 + 52 + 2 . 62 + ...... is n(n+1)2/2 when n is even. When n is odd the sum is

a )   3n(n + 1)/2 b )   n2(n + 1)/2 c )   n(n + 1)2/4 d )   [ n(n + 1)/2 ]2

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20) The sum of series 1/2! + 1/4! + 1/6! + ..... is

a )   (e2 - 1)/2 b )   (e - 1)2/(2e) c )   (e2 - 1)/(2e) d )   (e2 - 2)/e

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21) Let α, β be such that π < α - β < 3π. If sinα + sinβ = -21/65 and cosα + cosβ = -27/65 , then the value of cos((α - β)/2) is

a )   - 3/(√130) b )   3/(√130) c )   6/65 d )   - 6/65

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22) then the difference between the maximum and minimum values of u2 is given by

a )   2(a2 + b2) b )   2 √(a2 + b2) c )   (a + b)2 d )   (a - b)2

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23) The sides of a triangle are sinα , cosα and for some 0 < α < π/2. Then the greatest angle of the triangle is

a )   60° b )   90° c )   120° d )   150°

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24) A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of river is 60° and when he retires 40 meters away from the tree the angle of elevation becomes 30°. The breadth of the river is

a )   20 m b )   30 m c )   40 m d )   60 m

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25) if f : R → S, defined by f(x) = sinx - cosx + 1, is onto, then the interval if S is

a )   [ 0, 3 ] b )   [ -1, 1 ] c )   [ 0, 1 ] d )   [ -1, 3 ]

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26) The graph of the function y=f(x) is symmetrical about the line x = 2, then

a )   f(x + 2) = f(x - 2) b )   f(x + 2) = f(2 - x) c )   f(x) = f(-x) d )   f(x) = - f(-x)

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27) The domain of the function is

a )   [ 2, 3 ] b )   [ 2, 3 ) c )   [ 1, 2 ] d )   [ 1, 2 )

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28) then the value of a and b are

a )   a ∈ R , b ∈ R b )   a = 1 , b ∈ R c )   a ∈ R , b = 2 d )   a = 1 , b = 2

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29) Let f(x) = (1 - tanx)/(4x - π) , x ≠ π/4 , x ∈ [0 , π/2] . If f(x) is continuous in [0 , π/2], then f( π/4 ) is

a )   1 b )   1/2 c )   - 1/2 d )   -1

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30) If x = ey+ey+... to ∞, x > 0, then dy/dx is

a )   x/(1 + x) b )   1/x c )   (1 - x)/x d )   (1 + x)/x

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31) A point on the parabola y2 = 18x at which the ordinate increases at twice the rate of the abscissa is

a )   (2, 4) b )   (2, -4) c )   ( -9/8, 9/2 ) d )   ( 9/8, 9/2 )

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32) A function y =f(x) has a second order derivative f''(x) = 6(x - 1). If its graph passes through the point (2, 1) and at that point the tangent to the graph is y = 3x - 5, then the function is

a )   (x - 1)2 b )   (x - 1)3 c )   (x + 1)3 d )   (x + 1)2

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33) The normal to the curve x = a(l + cos θ), y = a sin θ at 'θ' always passes through the fixed point

a )   (a , 0) b )   (0 , a) c )   (0 , 0) d )   (a , a)

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34) If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval

a )   (0 , 1) b )   (1 , 2) c )   (2 , 3) d )   (1 , 3)

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35)

a )   e b )   e - 1 c )   1 - e d )   e + 1

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36) then value of (A , B) is

a )   ( sin α , cos α ) b )   ( cos α , sin α ) c )   ( - sin α , cos α ) d )   ( - cos α , sin α )

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37) is equal to

a )   b )   c )   d )

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38) The value of is

a )   28/3 b )   14/3 c )   7/3 d )   1/3

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39) The value of is

a )     0 b )   1 c )   2 d )   3

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40) then A is

a )     0 b )   π c )   π/4 d )   2 π

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41) If and then the value of I1/I2 is

a )   2 b )   -3 c )   -1 d )   1

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42) The area of the region bounded by the curves y= | x � 2 |, x = 1, x = 3 and the x-axis is

a )   1 b )   2 c )   3 d )   4

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43) The differential equation for the family of curves x2 + y2 - 2ay = 0, where a is an arbitrary constant is

a )   2(x2 - y2)y' = xy b )   2(x2 + y2)y' = xy c )   (x2 - y2)y' = 2xy d )   (x2 + y2)y' = 2xy

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44) The solution of the differential equation y.dx + (x + x2y).dy = 0 is

a )   - (1/xy) = C b )   - (1/xy) + log y= C c )   (1/xy) + log y= C d )   log y = Cx

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45) Let A(2, - 3) and B(- 2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1, then the locus of the vertex C is the line

a )   2x + 3y = 9 b )   2x - 3y = 7 c )   3x + 2y = 5 d )   3x - 2y = 3

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46) The equation of the straight line passing through the point (4, 3) and making intercepts on the co-ordinate axes whose sum is - 1 is

a )   x/2 + y/3 = -1 and x/(-2) + y/1 = -1 b )   x/2 - y/3 = -1 and x/(-2) + y/1 = -1 c )   x/2 + y/3 = 1 and x/2 + y/1 = 1 d )   x/2 - y/3 = 1 and x/(-2) + y/1 = 1

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47) If the sum of the slopes of the lines given by x2 - 2cxy - 7y2 = 0 is four times their product, then c has the value

a )   1 b )   -1 c )   2 d )   -2

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48) If one of the lines given by 6x2 - xy + 4cy2 = 0 is 3x + 4y = 0, then c equals:

a )   1 b )   -1 c )   3 d )   -3

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49) If a circle passes through the point (a, b) and cuts the circle x2 + y2 = 4 orthogonally, then the locus of its centre is:

a )   2ax + 2by + ( a2 + b2 + 4 ) = 0 b )   2ax + 2by - ( a2 + b2 + 4 ) = 0 c )   2ax - 2by + ( a2 + b2 + 4 ) = 0 d )   2ax - 2by - ( a2 + b2 + 4 ) = 0

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50) A variable circle passes through the fixed point A(p, q) and touches x-axis. The locus of the other end of the diameter through A is:

a )   (x - p)2 = 4qy b )   (x - q)2 = 4py c )   (y - p)2 = 4qx d )   (y - q)2 = 4px

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51) If the lines 2x + 3y + 1 = 0 and 3x - y - 4 = 0 lie along diameters of a circle of circumference 10 π , then the equation of the circle is:

a )   x2 + y2 - 2x + 2y - 23 = 0 b )   x2 + y2 - 2x - 2y - 23 = 0 c )   x2 + y2 + 2x + 2y - 23 = 0 d )   x2 + y2 + 2x - 2y - 23 = 0

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52) The intercept on the line y = x by the circle x2 + y2 - 2x = 0 is AB. Equation of the circle on AB as a diameter is:

a )   x2 + y2 - x - y = 0 b )   x2 + y2 - x + y = 0 c )   x2 + y2 + x + y = 0 d )   x2 + y2 + x - y = 0

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53) If a ≠ 0 and the line 2bx + 3cy + 4d = 0 passes through the points of intersection of the parabolas y2 = 4ax and x2 = 4ay, then:

a )   d2 + (2b + 3c)2 = 0 b )   d2 + (3b + 2c)2 = 0 c )   d2 + (2b - 3c)2 = 0 d )   d2 + (3b - 2c)2 = 0

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54) The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x = 4, then the equation of the ellipse is:

a )   3x2 + 4y2 = 1 b )   3x2 + 4y2 = 12 c )   4x2 + 3y2 = 12 d )   4x2 + 3y2 = 1

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55) A line makes the same angle θ , with each of the x and z axis. If the angle β, which it makes with y-axis, is such that sin2β = 3sin2θ then cos2θ equals:

a )   2/3 b )   1/5 c )   3/5 d )   2/5

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56) Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is:

a )   3/2 b )   5/2 c )   7/2 d )   9/2

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57) A line with direction cosines proportional to 2, 1, 2 meets each of the line x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the points of intersection are given by:

a )   (3a, 3a, 3a,), (a, a, a) b )   (3a, 2a, 3a), (a, a, a) c )   (3a, 2a, 3a,), (a, a, 2a) d )   (2a, 3a, 3a), (2a, a, a)

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58) If the straight lines x = 1 + s, y = -3 - λs , z = 1 + λs and x = t/2 , y = 1 + t , z = 2 - t, with parameter s and t respectively, are co-planar, then λ equals:

a )   -2 b )   -1 c )   - 1/2 d )     0

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59) The intersection of the spheres x2 + y2 + z2 + 7x - 2y - z = 13 and x2 + y2 + z2 - 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane:

a )   x - y - z = 1 b )   x - 2y - z = 1 c )   x - y - 2z = 1 d )   2x - y - z = 1

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60) If a , b and c be three non-zero vectors such that no two of these are collinear. If the vector a + 2b is collinear with c and b + 3c is is collinear with a ( λ being some non-zero scalar ) then a + 2b + 6c equals

a )   λ a b )   λ b c )   λ c d )     0

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61) A particle is acted upon by constant forces 4 i + j - 3k and 3i + j - k which displace it from a point i + 2j +3k to the point 5i + 4j + k . The work done is standard units by the forces is given by:

a )   40 b )   30 c )   25 d )   15

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62) If a , b ,c are non-coplanar vectors and λ is a real number, then the vectors a + 2b + 3c, λb + 4c and (2λ - 1) c are non-coplanar for:

a )   all values of λ b )   all except one value of λ c )   all except two values of λ d )   no value of λ

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63) Let u , v ,w be such that | u | = 1 , | v | = 2 , | w | = 3 , If the projection v along u is equal to that of w along u and v , w are perpendicular to each other then | u - v + w | equals:

a )   2 b )   c )   d )   14

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64) Let a , b and c be non-zero vectors such that If θ is the acute angle between the vectors b and c , then sin θ equals:

a )   1/3 b )   /3 c )   2/3 d )   2/3

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65) Consider the following statements: (i) Mode can be computed from histogram (ii) Median is not independent of change of scale (iii) Variance is independent of change of origin and scale Which of these is/are correct?

a )   only (i) b )   only (ii) c )   only (i) and (ii) d )   (i), (ii) and (iii)

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66) In a series of 2n observations, half of them equal a and remaining half equal -a. If the standard deviation of the observations is 2, then a equals:

a )   1/n b )   c )   2 d )   /n

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67) The probability that A speaks truth 4/5 is while this probability for B is 3/4. The probability that they contradict each other when asked to speak on a fact is:

a )   3/20 b )   1/5 c )   7/20 d )   4/5

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68) A random variable X has the probability distribution: X : 1 2 3 4 5 6 7 8 P(X): 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

a )   0.87 b )   0.77 c )   0.35 d )   0.50

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69) The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is:

a )   37/256 b )   219/256 c )   128/256 d )   28/256

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70) With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then the forces are

a )   (2+)N and (2-)N b )   (2 + )N and (2-)N c )   (2 + (1/2))N and (2 - (1/2))N d )   (2 + (1/2))N and (2 - (1/2))N

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71) In a right angle Δ ABC , ∠A = 90° and sides a, b, c are respectively, 5cm, 4 cm and 3 cm. If a force F has moments 0, 9 and 16 in N cm unit respectively about vertices A, B and C, the magnitude of F is:

a )   3 b )   4 c )   5 d )   9

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72) Three forces P , Q and R acting along IA, IB and IC, where I is the incentre of a Δ ABC, are in equilibrium. Then P : Q : R is:

a )   cos A/2 : cos B/2 : cos C/2 b )   sin A/2 : sin B/2 : sin C/2 c )   sec A/2 : sec B/2 : sec C/2 d )   cosec A/2 : cosec B/2 : cosec C/2

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73) A particle moves towards east from a point A to a point B at the rate of 4 km/h and then towards north from B to C at rate of 5 km/h. If AB = 12 km and BC = 5 km, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively:

a )   17/4 km/h and 13/4 km/h b )   13/4 km/h and 17/4 km/h c )   17/9 km/h and 13/9 km/h d )   13/9 km/h and 17/9 km/h

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74) A velocity 1/4 m/s is resolved into two components along OA and OB making angles 30° and 45° respectively with the given velocity. Then the component along OB is:

a )   1/8 m/s b )   1/4 ( - 1 ) m/s c )   1/4 m/s d )   1/8 ( - ) m/s

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75) If t1 and t2 are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then t12 + r22 is equal to:

a )   u2/g b )   4 u2/g2 c )   u2/2g2 d )   1

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