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1 AIEEE//Math S. No Questions Solutions Q. Lets cos (α + β) = and let sin (α + β) = 5, where α, β π, then tan α = 5 (a) 56 (b) 9 (c) 7 (d) 5 6 Sol: (a) cos (α + β) = 5 tan (α + β) = tan α = than (α + β + α β) = = 56 sin(α β) = 5 tan (α β) = 5 Q. Let S be a non-epty subset of R. Consider the following stateent: P: There is a rational nuber x S such that x >. Which of the following stateents is the negation of the stateent P? (a) There is no rational nuber x S such that x (b) Every rational nuber x S satisfies x (c) x S and x _ x is not rational (d) There is a rational nuber x S such that x Q. Let a = j k and c = j k. Then vector b satisfying a b + c = and a. b = is (a) i - j + k (b) i - j- k (c) i + j- k (d) -i + j-k Q. The equation of the tangent to the curve y = x +, that is parallel to the x- axis, is (a) y = (b) y = (c) y = (d) y = X Sol: (b) P: there is a rational nuber x Î S such that x > ~P: Every rational nuber x Î S satisfies x Sol: (d) c = b a b. c = (b i + b j + b k). (i j k) b -b b = and a. b = b b = b = b +b = +b Sol: (c) b = (+b ) i + (+b ) j +b k Parallel to x-axis dy dx - 8 = x x = y = Equation of tangent is y = (x ) y = Q.5 Solution of the differential equation cos x dy = y(sin x y) dx, < x < π is (a) y sec x = tan x + c (b) y tan x = sec x + c (c) tan x = (sec x + c)y (d) sec x = (tan x + c)y Sol: 5 (d) cos x dy = y (sin x y ) dx dy = y tanx = dx y sec x dy tanx = - sec x y dx y Let y = t - y dy dx = dt dx - dy dt - t tan x = - sec x + (tan x) t = sec x. dx dx I.F = e tan x dx = sec x Solution is t (I.F) = I. F. sec x dx y sec x = tan x + c Poornia University, For any query, contact us at: , 8

2 AIEEE//Math Q.6 The area bounded by the curves y = cos x and y = sin x between the ordinates x = and x = π is (a) + (b) (c) + (d) Sol: 6 (d) π 5π (cos x sin x ) dx + (sin x cos x) dx + (cos x sin x) = π π 5π Q.7 If two tangents drawn fro a point P to the parabola y = x are at right angles, then the locus of P is (a) x + = (b) x = (c) x = (d) x = Q.8 If the vectors a = i j + k, b = i + j + k and c = λi + j + μk are utually orthoronal, then λ, μ = (a) (, ) (b) (, ) (c) (, ) (d) (, ) Q.9 Consider the following relations: R = {(x, y) x, y are real nubers and x = wy for soe rational nuber w}; S = n, p q, n, p and q are integers such that n, q and q = p. then (a) neither R nor S is an equivalence relation (b) S is an equivalence relation but R is not an equivalence relation (c) R and S both are equivalence relations (d) R is an equivalence relation but S is not an equivalence relation Q. Let f: R R be defined by f(x) = k x, if x. If f has a local inu at x = x +, if x >, then a possible value of k is (a) (b) (c) - (d) Sol: 7 (b) The locus of perpendicular tangents is directrix i.e, x = a; x = Sol: 8 (d) a.b =, b.c =, c.a = λ + + μ = λ + μ = Solving we get: λ =, μ = Sol: 9 (b) xry need not iplies y Rx S: s p q = pn n q s reflexive n n s p n q p q s n syetric s p, p s r q = pn, ps = rq s = rn transitive. n q q s S is an equivalence relation. Sol: (c) f(x) = k x if x = x + if x > Poornia University, For any query, contact us at: , 8

3 AIEEE//Math Q. The nuber of non-singular atrices, with four entries as and all other entries as, is (a) 5 (b) 6 (c) at least 7 (d) less than Sol: (c) First row with exactly one zero; total nuber of cases = 6 First row zeros we get ore cases Total we get ore than 7. Directions: Questions Nuber 7 to 76 are Assertion Reason type questions. Each of these questions contains two stateents. Stateent-: (Assertion) and Stateent-: (Reason) Each of these questions also has four alternative choices, only one of which is the correct answer. You have to select the correct choice. Q. Four nubers are chosen at rando (without replaceent) fro the set {,,,..., }. Stateent-: The probability that the chosen nubers when arranged in soe order will for an AP is 85 Stateent-: If the four chosen nubers fro an AP, then the set of all possible values of coon difference is {±, ±, ±, ±, ±5}. (a) Stateent- is true, Stateent- is true; Stateent- is not the correct explanation for Stateent- (b) Stateent- is true, Stateent- is false (c) Stateent- is false, Stateent- is true (d) Stateent- is true, Stateent- is true; Stateent- is the correct explanation for Stateent- Q. Stateent-: The point A(,, 6) is the irror iage of the point B(,, ) in the plane x y + z = 5. Stateent-: The plane x y + z = 5 bisects the line segent joining A(,, 6) and B(,, ). (a) Stateent- is true, Stateent- is true; Stateent- is not the correct explanation for Stateent- (b) Stateent- is true, Stateent- is false (c) Stateent- is false, Stateent- is true (d) Stateent- is true, Stateent- is true; Stateent- is the correct explanation for Stateent- Sol: (b) N(S) = c Stateent-: coon difference is ; total nuber of cases = 7 coon difference is ; total nuber of cases = coon difference is ; total nuber of cases = coon difference is ; total nuber of cases = 8 coon difference is 5; total nuber of cases = 5 coon difference is 6; total nuber of cases = Prob. = C = 85 Sol: (a) A(,, 6); B = (,, ) Mid-point of AB = (,, 5) lies on the plane. and d.r s of AB =,, ) d.r s Of noral to plane =,, ). AB is perpendicular bisector A is iage of B Stateent- is correct but it is not correct explanation. Q. Let S = j(j ) C j S = J J= J= c j and S = J= j C j. Stateent-: S = 55 9 Stateent-: S = 9 8 and S = 8. (a) Stateent- is true, Stateent- is true; Stateent- is not the correct explanation for Stateent- (b) Stateent- is true, Stateent- is false (c) Stateent- is false, Stateent- is true (d) Stateent- is true, Stateent- is true; Stateent- is the correct explanation for Stateent- Sol: (b) S = j= j(j ) S = S = j=! 8! j j j! j! j! 8 j!! 9! j! 8 j!! j! j! j= j = j j! 9 j! j =. 9 = 9. 8 j= j j + j = j= j(j ) C j = C j = = =. 8 = Poornia University, For any query, contact us at: , 8

4 AIEEE//Math Q.5 Let A be a atrix with non-zero entries and let A = I, where I is identity atrix. Define Tr(A) = su of diagonal eleents of A and A = deterinant of atrix A. Stateent-: Tr(A) = Stateent-: A = () Stateent- is true, Stateent- is true; Stateent- is not the correct explanation for Stateent- () Stateent- is true, Stateent- is false () Stateent- is false, Stateent- is true () Stateent- is true, Stateent- is true; Stateent- is the correct explanation for Stateent- Q.6 Let f : R R be a continuous function defined by f (x) = Stateent : f (c) =, for soe c R. Stateent : <f(x), for all x R e x +e x. () Stateent- is true, Stateent- is true; Stateent- is not the correct explanation for Stateent- () Stateent- is true, Stateent- is false () Stateent- is false, Stateent- is true () Stateent- is true, Stateent- is true; Stateent- is the correct explanation for Stateent- Q.7 For a regular polygon, let r and R be the radii of the inscribed and the circuscribed circles. A false stateent aong the following is (a) There is a regular polygon with r R = (b) There is a regular polygon with r R = (c) There is a regular polygon with r R = (d) There is a regular polygon with r = R Q.8 If a and b are the roots of the equation x x + =, then a 9 + b 9 = (a) (b) (c) (d) Sol: 5 (b) Let A = a b, abcd c d A = a b c d. a b A c d = a + bc =, bc + d = ab + bd = ac + cd = c and b a+d = Trace A = a + d = A = ad - bc = - a bc =. Sol: 6 (d) ex f(x) = = e x +e x e x + fʼ(x) = (ex +)e x e x.e x (e x + ) fʼ(x) = e x + = e x e x = e x = Maxiu f(x) = = <f(x) Since < < Sol: 7 (b) x R r = a cot π n a is side of polygon. R = a cosec π n Sol: 8 (b) for soe c R f(c) = r = cot π n R cosec π n = cos π n x + x + = x= + i x = α = + i, β = + i α = cos π + isinπ, β = cosπ isinπ a + bc ab + bd ac + cd bc + d cos π for any n N. n α 9 + β 9 = cos9 π = cos 668π + π + π = cos π + π Poornia University, For any query, contact us at: , 8 = cos π = =

5 AIEEE//Math 5 Q.9 The nuber of coplex nubers z such that z = z + = z i equals (a) (b) (c) (d) Q. A line AB in three-diensional space akes angles 5 and with the positive x-axis and the positive y-axis respectively. If AB akes an acute angle q with the positive z-axis, then q equals (a) 5 (b) 6 (c) 75 (d) Sol: 9 (a) Let z = x + iy z = z + Re z = x = z = z i x = y z + = z i y = x Only (, ) will satisfy all conditions. Nuber of coplex nuber z = Sol: (b) l = cos 5 = = cos = n = cos θ where θ is the angle which line arkes with positive z axis. Now l θ + + n = + + cos θ = Q. The line L given by x + y = passes through the point (, ). The line K is 5 b parallel to L and has th equation x + y =. Then the distance between L and K c is (a) 7 (b) 7 5 (c) (d) 7 5 cos θ = cos θ = θ = π. Sol: (c) Slope of line L = - b 5 Slope of line K = - C Line L is parallel to line k. 5 + b = b = (θ Being acute) b = - c = - Equation of K: y x = Distance between L and K = = 7 Q. A person is to count 5 currency notes. Let a n denote the nuber of notes he counts in the n t inute. If a = a =... = a = 5 and a, a,... are in A.P. with coon difference, then the tie taken by hi to count all notes is (a) inutes (c) 5 inutes (b) 5 inutes (d) inutes Sol: (a) Toll th inute nuber of counted notes = 5 = n [ 8 + (n )(- )] = n [8 n +] n 9n + = n = 5, n = 5 is not possible. Total tie = + = inutes. Poornia University, For any query, contact us at: , 8

6 AIEEE//Math 6 Q. Let f: R R be a positive increasing function with f(x) f(x) = (a) (a) 8 (a) (a) li f(x) x ω =. Then li f(x) x ω Sol: ( d) f(x) is a positive increasing function < f(x) < f(x) < f(x) < < f(x) < f(x) f(x) f(x) li x ω li x ω By sandwich theore. li f(x) x ω = f(x) f(x) f(x) li f(x) x ω f(x) Q. Let p(x) be a function defined R such that p (x) = p ( - x), for all x ω [,], p() = and p() =. Then p x dx equals (a) (b) (c) (d) Sol: (a) p'(x) = p (-x) p(x) = - p(-x)+c at x = p() = - p() + c = c now p(x) = - p(-x) + p(x)+p(-x) = I = I = x dx = p x dx dx I =. Q.5 Let f: (-,) R be a differentiable function with f() = - and f () =. Let g(x) = [f(f(x)+)] Then g () = (a) - (b) (c) - (d) Sol: 5 (a) g (x) = (f(f(x) + )) d dx (f f x + ) = f(f(x) + f (f(x)+). (f (x)) g () = f(f()+). f (f()+). (f ()=f() f () = (-) () = - Q.6 There are two urns. Urn A has distinct red balls and urn B has 9 distinct blue balls. Fro each urn two balls are taken out at rando and then transferred to the other. The nuber of ways in which this can be done is Sol: 6 (c) Total nuber of ways = C 9 C = 9 8 = 6 = 8 (a) 6 (b) 66 (c) 8 (d) Poornia University, For any query, contact us at: , 8

7 AIEEE//Math 7 Q.7 Consider the syste of linear equations: x + x + x = x + x + x = x + 5x + x = The syste has (a) exactly solutions (b) a unique solution (c) no solution (d) infinite nuber of solutions Sol: 7 ( c) D = = 5 D = = 5 Given syste, does not have any solution. No solution. Q.8 An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at rando without replaceent fro the urn. The probability that the three balls have differentcolour is (a) 7 (b) (c) (d) Q.9 For two data sets, each of size 5, the variances are given to be and 5 and the corresponding eans are given to be and, respectively. The variance of the cobined data set is Sol: 8 (a) n(s) = 9 C n(e) = C C C Probability = = 9 C Sol: 9 (a) σ x = σ x = 5 x = y = Xi 5! 9! 6 6! = = = xi = ; yi = σ x = x i - (x) = ( y 5 i ) 6 x i = x i = 5 σ z = ( x i + y i ) x+y = = 5 9 = 55 = Q. The circle x + y = x + 8y + 5 intersects the line x y = at two distinct points if (a) 5 < < 5 (b) 5 < < 65 (c) 5 < < 85 (d) 85 < < 5_ Sol: (a) Circle x + y x 8y 5 = Centre = (,), Radius = = 5 If circle is intersecting line x y = at two distinct points. length of perpendicular fro centre < radius <5 + < 5-5 < + < 5-5 < < 5. Poornia University, For any query, contact us at: , 8