Aieee Maths 2004

aieEE–Past papers,[object Object],MATHEMATICS- UNSOLVED PAPER - 2004,[object Object]

SECTION – A,[object Object],Single Correct Answer Type,[object Object],There are five parts in this question. Four choices are given for each part and one of them is correct. Indicate you choice of the correct answer for each part in your answer-book by writing the letter (a), (b), (c) or (d) whichever is appropriate,[object Object]

01,[object Object],Problem,[object Object],Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is,[object Object],a function,[object Object],reflexive,[object Object],not symmetric ,[object Object],transitive,[object Object]

Problem,[object Object],02,[object Object],The range of the function f(x)= 7-xPx-3 is,[object Object],{1, 2, 3},[object Object],{1, 2, 3, 4, 5},[object Object],{1, 2, 3, 4} ,[object Object],{1, 2, 3, 4, 5, 6},[object Object]

Problem,[object Object],03,[object Object],Let z, w be complex numbers such that and argzw = π. Then arg z equals,[object Object],4/π,[object Object],5π/4,[object Object],3π/4,[object Object],2/π,[object Object]

Problem,[object Object],04,[object Object],If z = x – i y and Z1/33 = p + iq , then is equal to,[object Object],1,[object Object],-2,[object Object],2 ,[object Object],-1,[object Object]

Problem,[object Object],05,[object Object], then z lies on,[object Object],the real axis,[object Object],an ellipse,[object Object],a circle ,[object Object],the imaginary axis.,[object Object]

Problem,[object Object],06,[object Object],Let A The only correct statement about the matrix A is,[object Object],A is a zero matrix ,[object Object],A2 = I,[object Object],A−1does not exist ,[object Object],A = (−1)I, where I is a unit matrix,[object Object]

Problem,[object Object],07,[object Object],Let A= (10)B= . If B is the inverse of matrix A, ,[object Object],then α is,[object Object],-2 ,[object Object],5,[object Object],2,[object Object],-1,[object Object]

Problem,[object Object],08,[object Object],If a1, a2, a3 , ....,an , .... are in G.P., then the value of the determinant,[object Object], 0 ,[object Object], -2,[object Object], 2 ,[object Object], 1,[object Object]

Problem,[object Object],09,[object Object],Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation,[object Object],x2 + 18x + 16 = 0 ,[object Object],x2 −18x −16 = 0,[object Object],x2 + 18x −16 = 0 ,[object Object],x2 −18x + 16 = 0,[object Object]

Problem,[object Object],10,[object Object],If (1 – p) is a root of quadratic equation x2 + px + (1− p) = 0 , then its roots are,[object Object],0, 1 ,[object Object], -1, 2,[object Object],0, -1 ,[object Object],-1, 1,[object Object]

Problem,[object Object],11,[object Object],Let S(K) = 1+ 3 + 5 + ... + (2K −1) = 3 + K2 . Then which of the following is true?,[object Object],S(1) is correct,[object Object],Principle of mathematical induction can be used to prove the formula,[object Object],S(K) ⇒ S(K + 1),[object Object],S(K)⇒ S(K + 1),[object Object]

Problem,[object Object],12,[object Object],How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?,[object Object],120 ,[object Object],480,[object Object],360,[object Object],240,[object Object]

Problem,[object Object],13,[object Object],The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is,[object Object],5 ,[object Object],8C3,[object Object],38,[object Object],21,[object Object]

Problem,[object Object],14,[object Object],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,[object Object],49/4,[object Object],4,[object Object],3 ,[object Object],12,[object Object]

Problem,[object Object],15,[object Object],The coefficient of the middle term in the binomial expansion in powers of x of (1+ αx)4 and of (1− αx)6 is the same if α equals,[object Object],- 5/3,[object Object],3/5,[object Object],-3/10,[object Object],10/3,[object Object]

Problem,[object Object],16,[object Object],The coefficient of xn in expansion of (1+ x) (1− x)n is,[object Object],(n – 1) ,[object Object],(-1)n(1− n) ,[object Object],(-1)n-1(1− n)2,[object Object],(-1)n-1n ,[object Object]

Problem,[object Object],17,[object Object],If = and , then n,[object Object],is equal to,[object Object],a.,[object Object],b.,[object Object],c. n-1,[object Object],d.,[object Object]

Problem,[object Object],18,[object Object],Let Tr be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, m ≠ n , and , then a – d equals,[object Object],a. 0,[object Object],b. 1,[object Object],c. 1/mn,[object Object],d.,[object Object]

Problem,[object Object],19,[object Object],The sum of the first n terms of the series 12 + 2 ⋅ 22 + 32 + 2 ⋅ 42 + 52 + 2 ⋅ 62 + ... iswhen n is even. When n is odd the sum is,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],20,[object Object],The sum of series is,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],21,[object Object],Let α, β be such that π < α - β < 3π. If sinα + sinβ = and cosα + cosβ = , then the value of cos,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],22,[object Object],If u = , then the difference between the maximum and minimum values of u2 is given by,[object Object],a. 2(a2 + b2 ) ,[object Object],b. ,[object Object],c. (a + b)2,[object Object],d. (a -b)2,[object Object]

Problem,[object Object],23,[object Object],The sides of a triangle are sinα, cosα and for some . Then the greatest angle of the triangle is,[object Object],60o,[object Object],90o,[object Object],120o ,[object Object],150o,[object Object]

Problem,[object Object],24,[object Object],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 the river is 60o and when he retires 40 meter away from the tree the angle of elevation becomes30o . The breadth of the river is,[object Object],20 m ,[object Object],30 m,[object Object],40 m ,[object Object],60 m,[object Object]

Problem,[object Object],25,[object Object],If f :R -> S, defined by f(x) = sin x − √3 cos x + 1, is onto, then the interval of S is,[object Object],[0, 3] ,[object Object],[-1, 1],[object Object],[0, 1] ,[object Object],[-1, 3],[object Object]

Problem,[object Object],26,[object Object],The graph of the function y = f(x) is symmetrical about the line x = 2, then,[object Object],f(x + 2)= f(x – 2) ,[object Object],f(2 + x) = f(2 – x),[object Object],f(x) = f(-x) ,[object Object],f(x) = - f(-x),[object Object]

Problem,[object Object],27,[object Object],The domain of the function is,[object Object],[2, 3] ,[object Object],[2, 3),[object Object],[1, 2] ,[object Object],[1, 2),[object Object]

Problem,[object Object],28,[object Object],If , then the values of a and b, are,[object Object],a∈ , b∈,[object Object],a = 1, b∈,[object Object],a∈ , b = 2 ,[object Object],a = 1 and b = 2,[object Object]

Problem,[object Object],29,[object Object],Let f(x) . If f(x) is continuous in , then f is,[object Object],a. 1 ,[object Object],b. 1/2,[object Object],c. -1/2,[object Object],d. -1,[object Object]

Problem,[object Object],30,[object Object],If xy+e y+...to ∞ , x > 0, then dy/dx is,[object Object],x/1+ x,[object Object],1/x,[object Object],1-x/x,[object Object],1+x/x,[object Object]

Problem,[object Object],31,[object Object],A point on the parabola y2 = 18x at which the ordinate increases at twice the rate of the abscissa is,[object Object],a.(2, 4) ,[object Object],b. (2, -4),[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],32,[object Object],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,[object Object],(x −1)2 ,[object Object],(x −1)2,[object Object],(x+1)3 ,[object Object],(x+1)2,[object Object]

Problem,[object Object],33,[object Object],The normal to the curve x = a(1 + cosθ), y = asinθ at ‘θ’ always passes through the fixed point,[object Object],(a, 0) ,[object Object],(0, a),[object Object],(0, 0) ,[object Object],(a, a),[object Object]

Problem,[object Object],34,[object Object],If 2a + 3b + 6c =0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval,[object Object],(0, 1) ,[object Object], (1, 2),[object Object],(2, 3) ,[object Object],(1, 3),[object Object]

Problem,[object Object],35,[object Object], is,[object Object],e ,[object Object],e – 1,[object Object],1 – e ,[object Object],e + 1,[object Object]

Problem,[object Object],36,[object Object],If dx=Ax+ B log sin(x ) C sin(x- α ) , then value of (A, B) is,[object Object],(sinα, cosα) ,[object Object],(cosα, sinα),[object Object],(- sinα, cosα),[object Object],(- cosα, sinα),[object Object]

Problem,[object Object],37,[object Object],is equal to,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],38,[object Object],The value of dx is,[object Object],28/3,[object Object],14/3,[object Object],7/3,[object Object],1/3,[object Object]

Problem,[object Object],39,[object Object],The value of I = dxis,[object Object],0 ,[object Object],1,[object Object],2 ,[object Object],3,[object Object]

Problem,[object Object],40,[object Object],If xf(sin x)dx=A f (sin x) then A is,[object Object],0 ,[object Object],π,[object Object],4/π,[object Object],2π,[object Object]

Problem,[object Object],41,[object Object],If f(x) = xg{x(1 x)}dx and g{x(1 x)}dx then the ,[object Object],value of is,[object Object],2 ,[object Object],–3,[object Object],–1 ,[object Object],1,[object Object]

Problem,[object Object],42,[object Object],The area of the region bounded by the curves y = |x – 2|, x = 1, x = 3 and the x-axis is,[object Object],1 ,[object Object],2,[object Object],3 ,[object Object],4,[object Object]

Problem,[object Object],43,[object Object],The differential equation for the family of curves x2 + y2 − 2ay = 0 , where a is an arbitrary constant is,[object Object],2(x2 − y2 )y′ = xy,[object Object],2(x2 + y2 )y′ = xy,[object Object],(x2 − y2 )y′ = 2xy ,[object Object],(x2 + y2 )y′ = 2xy,[object Object]

Problem,[object Object],44,[object Object],The solution of the differential equation y dx + (x + x2y) dy = 0 is,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],45,[object Object],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,[object Object],2x + 3y = 9 ,[object Object],2x – 3y = 7,[object Object],3x + 2y = 5 ,[object Object],3x – 2y = 3,[object Object]

Problem,[object Object],46,[object Object],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,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],47,[object Object],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,[object Object],1 ,[object Object],-1,[object Object],2 ,[object Object],-2,[object Object]

Problem,[object Object],48,[object Object],If one of the lines given by 6x2 − xy + 4cy2 = 0 is 3x + 4y = 0, then c equals,[object Object],1 ,[object Object],-1,[object Object],3 ,[object Object],-3,[object Object]

Problem,[object Object],49,[object Object],If a circle passes through the point (a, b) and cuts the circle x2 + y2 = 4 orthogonally, then the locus of its centre is,[object Object],2ax + 2by + (a2 + b2 + 4) = 0 ,[object Object],2ax + 2by − (a2 + b2 + 4) = 0,[object Object],2ax − 2by + (a2 + b2 + 4) = 0 ,[object Object],2ax − 2by − (a2 + b2 + 4) = 0,[object Object]

Problem,[object Object],50,[object Object],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,[object Object],(x − p)2 = 4qy ,[object Object],(x − q)2 = 4py,[object Object],(y − p)2 = 4qx ,[object Object],(y − q)2 = 4px,[object Object]

Problem,[object Object],51,[object Object],If the lines 2x + 3y + 1 = 0 and 3x – y – 4 = 0 lie along diameters of a circle of circumference 10π, then the equation of the circle is,[object Object],x2 + y2 − 2x + 2y − 23 = 0,[object Object],x2 + y2 − 2x − 2y − 23 = 0,[object Object],x2 + y2 + 2x + 2y − 23 = 0,[object Object],x2 + y2 + 2x − 2y − 23 = 0,[object Object]

Problem,[object Object],52,[object Object],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,[object Object],x2 + y2 − x − y = 0 ,[object Object], x2 + y2 − x + y = 0,[object Object],x2 + y2 + x + y = 0 ,[object Object],x2 + y2 + x − y = 0,[object Object]

Problem,[object Object],53,[object Object],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,[object Object],d2 + (2b + 3c)2 = 0 ,[object Object],d2 + (3b + 2c)2 = 0,[object Object],d2 + (2b − 3c)2 = 0,[object Object],d2 + (3b − 2c)2 = 0,[object Object]

Problem,[object Object],54,[object Object],The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directives is x =4, then the equation of the ellipse is,[object Object],3x2 + 4y2 = 1 ,[object Object],3x2 + 4y2 = 12,[object Object],4x2 + 3y2 = 12 ,[object Object],4x2 + 3y2 = 1,[object Object]

Problem,[object Object],55,[object Object],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 sin2 β = 3sin2 θ , then cos2 θ equals,[object Object],2/3,[object Object],1/5,[object Object],3/5,[object Object],2/5,[object Object]

Problem,[object Object],56,[object Object],Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is,[object Object],3/2,[object Object],5/2,[object Object],7/2,[object Object],9/2,[object Object]

Problem,[object Object],57,[object Object],A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the point of intersection are given by,[object Object],(3a, 3a, 3a), (a, a, a) ,[object Object],(3a, 2a, 3a), (a, a, a),[object Object],(3a, 2a, 3a), (a, a, 2a) ,[object Object],(2a, 3a, 3a), (2a, a, a),[object Object]

Problem,[object Object],58,[object Object],If the straight lines x = 1 + s, y = –3 – λs, z = 1 + λs and x = t/2, y = 1 + t, z = 2 – t withparameters s and t respectively, are co-planar then λ equals,[object Object],–2 ,[object Object],–1,[object Object], – 1/2,[object Object], 0,[object Object]

Problem,[object Object],59,[object Object],The intersection of the spheres x2 + y2 + z2 + 7x − 2y − z = 13 and,[object Object],x2 + y2 + z2 − 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane,[object Object],x – y – z = 1 ,[object Object],x – 2y – z = 1,[object Object],x – y – 2z = 1,[object Object],2x – y – z = 1,[object Object]

Problem,[object Object],60,[object Object],Let , and be three non-zero vectors such that no two of these are collinear. If the vector + 2 is collinear with and + 3 is collinear with,[object Object],(λ being some non-zero scalar) then equals,[object Object],a.,[object Object],b.,[object Object],c.,[object Object],d. 0,[object Object]

Problem,[object Object],61,[object Object],A particle is acted upon by constant forces 4î +ĵ− 3kˆ and 3ˆi + ĵ − which displace it from a point î + 2ĵ + 3 to the point5 î+ 4ĵ + . The work done in standard units by the forces is given by,[object Object],40 ,[object Object],30,[object Object],25 ,[object Object],15,[object Object]

Problem,[object Object],62,[object Object],If are non-coplanar vectors and λ is a real number, then the vectors,[object Object], and (2λ −1) are non-coplanar for,[object Object],all values of λ ,[object Object],all except one value of λ,[object Object],all except two values of λ,[object Object],no value of λ,[object Object]

Problem,[object Object],63,[object Object],Let be such that . If the projection along is equal to that of w along u and v, w are perpendicular to each other then u − v + w equals,[object Object],2,[object Object],√7,[object Object],√14,[object Object],14,[object Object]

Problem,[object Object],64,[object Object],Let a, b and c be non-zero vectors such that . If θ is the acute angle between the vectors and , then sin θ equals,[object Object],a. 1/3,[object Object],b. √2/3,[object Object],c. 2/3,[object Object],d. 2√2/3,[object Object]

Problem,[object Object],65,[object Object],Consider the following statements:,[object Object],(a) Mode can be computed from histogram,[object Object],(b) Median is not independent of change of scale,[object Object],(c) Variance is independent of change of origin and scale.,[object Object],Which of these is/are correct?,[object Object],only (a),[object Object],only (b),[object Object],only (a) and (b),[object Object],(a), (b) and (c),[object Object]

Problem,[object Object],66,[object Object],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,[object Object],a. 1/n,[object Object],b. √2,[object Object],c. 2,[object Object],d. √2/n,[object Object]

Problem,[object Object],67,[object Object],The probability that A speaks truth is 4/5, while this probability for B is 3/4,[object Object],. The probability that they contradict each other when asked to speak on a fact is,[object Object],a. 3/20,[object Object],b. 1/5,[object Object],c. 7/20,[object Object],d. 4/5,[object Object]

Problem,[object Object],68,[object Object],A random variable X has the probability distribution: For the events E = {X is a prime number} and F = {X < 4}, the probability P (E ∪ F) is,[object Object],0.87,[object Object],0.77,[object Object],0.35,[object Object],0.50,[object Object]

Problem,[object Object],69,[object Object],The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is,[object Object],37/256,[object Object],219/256,[object Object],128/256,[object Object],28/256,[object Object]

Problem,[object Object],70,[object Object],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,[object Object],a. (2 + √2)N and (2 − √2)N,[object Object],b. (2 + √3)N and (2 − √3)N,[object Object],c.,[object Object],d.,[object Object]

Problem,[object Object],71,[object Object],In a right angle ΔABC, ∠A = 90° and sides a, b, c are respectively, 5 cm, 4 cm and 3 cm. If a force has moments 0, 9 and 16 in N cm. units respectively about vertices A, B and C, then magnitude of is,[object Object],3 ,[object Object],4,[object Object],5 ,[object Object],9,[object Object]

Problem,[object Object],72,[object Object],Three forces P, Q and R rrr acting along IA, IB and IC, where I is the incentre of a ΔABC, are in equilibrium. Then P : Q :R rrr is,[object Object],Cos : cos : cos,[object Object],Sin : sin : sin,[object Object], sec : sec : sec ,[object Object],cosec : cosec : cosec ,[object Object]

Problem,[object Object],73,[object Object],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 the 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,[object Object], 17/4 km/h and 13/4 km/h ,[object Object], 13/4 km/h and 17/4 km/h,[object Object], 17/9 km/h and 13/9 km/h ,[object Object], 1/9 km/h and 17/9 km/h,[object Object]

Problem,[object Object],74,[object Object],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,[object Object],1/8 m/s ,[object Object],¼ ( √3 -1) m/s,[object Object], ¼ m/s ,[object Object],1/8 ( √6-√2) m/s,[object Object]

Problem,[object Object],75,[object Object],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 2 2 t21+ t22is equal to,[object Object],a. u2/g,[object Object],b. 4u2/g,[object Object],c. 2u2/g,[object Object],d.1,[object Object]

FOR SOLUTION VISIT WWW.VASISTA.NET,[object Object]

Aieee Maths 2004

Aieee Maths 2004

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