FIITJEE SOLUTION TO AIEEE-2005 MATHEMATICS

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1 FIITJEE SOLUTION TO AIEEE-5 MATHEMATICS. If A A + I =, then the inverse of A is () A + I () A () A I () I A. () Given A A + I = A A A A + A I = A (Multiplying A on both sides) A - I + A - = or A = I A.. If the cube roots of unity are, ω, ω then the roots of the equation ( ) + 8 =, are () -, - + ω, - - ω () -, -, - () -, - ω, - ω () -, + ω, + ω. () ( ) + 8 = ( ) = (-) () / = - or -ω or -ω or n = - or ω or ω.. Let R = {(, ), (6, 6), (9, 9), (, ), (6, ), (, 9), (, ), (, 6)} be a relation on the set A = {, 6, 9, } be a relation on the set A = {, 6, 9, }. The relation is () refleive and transitive only () refleive only () an equivalence relation () refleive and symmetric only. () Refleive and transitive only. e.g. (, ), (6, 6), (9, 9), (, ) [Refleive] (, 6), (6, ), (, ) [Transitive].. Area of the greatest rectangle that can be inscribed in the ellipse () ab () ab a y + = is b () ab () a b. () Area of rectangle ABCD = (acosθ) (bsinθ) = absinθ Area of greatest rectangle is equal to ab when sinθ =. (-acosθ, bsinθ) B Y A(acosθ, bsinθ) X (-acosθ, -bsinθ)c D(acosθ, -bsinθ) 5. The differential equation representing the family of curves y = c ( c ) >, is a parameter, is of order and degree as follows: () order, degree () order, degree +, where c FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

2 () order, degree () order, degree 5. () y = c( + c) (i) yy = c or yy = c (ii) y = yy ( + yy ) [on putting value of c from (ii) in (i)] On simplifying, we get (y y ) = yy (iii) Hence equation (iii) is of order and degree. 6. lim sec sec... sec n n n n n n equals () sec () cos ec () tan () tan 6. () 9 lim sec sec sec... sec n n n n n n n n is equal to r r r r lim sec = lim sec n n n n n n n Given limit is equal to value of integral sec d or sec d sec tdt = [put = t] = tant = tan. 7. ABC is a triangle. Forces P, Q, R acting along IA, IB and IC respectively are in equilibrium, where I is the incentre of ABC. Then P : Q : R is A B C () sina : sin B : sinc () sin : sin : sin () A B C cos : cos : cos 7. () Using Lami s Theorem A B C P : Q : R = cos : cos : cos. () cosa : cosb : cosc Q I R B C 8. If in a frequently distribution, the mean and median are and respectively, then its mode is approimately (). ().5 () 5.5 (). 8. () Mode + Mean = Median Mode = = 66 =. A P FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

3 9. Let P be the point (, ) and Q a point on the locus y = 8. The locus of mid point of PQ is () y + = () y + + = () + y + = () y + = 9. () P = (, ) Q = (h, k) such that k = 8h Let (α, β) be the midpoint of PQ h+ k + α=, β= α - = h β = k. (β) = 8 (α - ) β = α - y + =.. If C is the mid point of AB and P is any point outside AB, then () PA + PB = PC () PA + PB = PC () PA + PB + PC = () PA + PB + PC =. () PA + AC + CP = PB + BC + CP = Adding, we get PA + PB + AC + BC + CP = Since AC = BC & CP = PC PA + PB PC =. A C P B. If the coefficients of rth, (r+ )th and (r + )th terms in the binomial epansion of ( + y) m are in A.P., then m and r satisfy the equation () m m(r ) + r = () m m(r+) + r + = () m m(r + ) + r = () m m(r ) + r + =. () Given m C m m r, C r, Cr+ are in A.P. m m m Cr = Cr + Cr+ m m Cr Cr+ = + m m Cr Cr r m r = + m r + r + m m (r + ) + r =. π P. In a triangle PQR, R =. If tan and tan Q are the roots of a + b + c =, a then () a = b + c () c = a + b () b = c () b = a + c. () P Q tan, tan are the roots of a + b + c = P Q b tan tan + = a FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

4 P Q c tan tan = a P Q tan tan + P Q = tan P Q + = tan tan b a b a c = = b = a c c a a a a c = a + b.. The system of equations α + y + z = α -, + αy + z = α -, + y + αz = α - has no solution, if α is () - () either or () not - (). () α + y + z = α - + αy + z = α - + y + zα = α - α = α α = α(α ) (α - ) + ( - α) = α (α - ) (α + ) (α - ) (α - ) (α - )[α + α - ] = (α - )[α + α - ] = [α + α - α - ] = (α - ) [α(α + ) (α + )] = (α - ) =, α + = α =, ; but α.. The value of α for which the sum of the squares of the roots of the equation (a ) a = assume the least value is () () () (). () (a ) a = α + β = a α β = (a + ) α + β = (α + β) - αβ = a a + 6 = (a ) + 5 a =. 5. If roots of the equation b + c = be two consectutive integers, then b c equals () () FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

5 5 () () 5. () Let α, α + be roots α + α + = b α(α + ) = c b c = (α + ) - α(α + ) =. 6. If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number () 6 () 6 () 6 () 6 6. () Alphabetical order is A, C, H, I, N, S No. of words starting with A 5! No. of words starting with C 5! No. of words starting with H 5! No. of words starting with I 5! No. of words starting with N 5! SACHIN The value of 5 C r C is r= () 55 C () 55 C () 56 C () 56 C 7. () 6 56 r C r= C C C C C C C C C C C C C C 5 C = C + C + C + C + C + C C + 55 C = 56 C. 8. If A = and I =, then which one of the following holds for all n, by the principle of mathematical indunction () A n = na (n )I () A n = n- A (n )I () A n = na + (n )I () A n = n- A + (n )I 8. () By the principle of mathematical induction () is true. 9. If the coefficient of 7 in a + b equals the coefficient of -7 in a b, then a and b satisfy the relation () a b = () a + b = 9. () () a = () ab = b FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

6 6 r r + = b b T r + in the epansion a Cr ( a ) r r = C r (a) r (b) -r () r = 7 r = 5 coefficient of 7 = C 5 (a) 6 (b) -5 () r r r = b b Again T r + in the epansion a C ( a) = C r a r (-) r (b) -r () -r () - r Now r = -7 r = 8 r = 6 coefficient of -7 = C 6 a 5 (b) -6 5 C ( a) ( b) = C a ( b) ab = Let f : (-, ) B, be a function defined by f() = tan and onto when B is the interval π π (), (), π π π π (), (),. () Given f() = tan for (-, ) π π clearly range of f() =, π π co-domain of function = B =,., then f is both one-one. If z and z are two non-zero comple numbers such that z + z = z + z then argz argz is equal to π () () - π π () () -. () z + z = z + z z and z are collinear and are to the same side of origin; hence arg z arg z =. z. If ω = and ω =, then z lies on z i () an ellipse () a circle () a straight line () a parabola.. () FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

7 7 z z As given w = w = = distance of z from origin and point z i z i, is same hence z lies on bisector of the line joining points (, ) and (, /). Hence z lies on a straight line.. If a + b + c = - and f() = + ( + ) ( + ) ( + ) + ( + ) ( + a ) ( + b ) + c a b c a b c polynomial of degree () () () (). () + ( ) ( + ) ( + ) ( ) ( ) + ( ) ( + ) + ( + ) ( + ) + ( + ) ( + ) + a b c b c then f() is a f = + a + b + c + + b + c, Applying C C + C + C = a b c b c b c b c b c a + b + c + = f() = ; Applying R R R, R R R + b + c f() = ( ) Hence degree =.. The normal to the curve = a(cosθ + θ sinθ), y = a( sinθ - θ cosθ) at any point θ is such that () it passes through the origin () it makes angle π + θ with the -ais π () it passes through a, a () it is at a constant distance from the origin. () Clearly dy = tan θ slope of normal = - cot θ d Equation of normal at θ is y a(sin θ - θ cos θ) = - cot θ( a(cos θ + θ sin θ) y sin θ - a sin θ + a θ cos θ sin θ = - cos θ + a cos θ + a θ sin θ cos θ cos θ + y sin θ = a Clearly this is an equation of straight line which is at a constant distance a from origin. FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

8 8 5. A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval Function () (-, ) + + () [, ) + 6 (), + () (-, -] () Clearly function f() = + is increasing when f () = 6 [/, ) Hence () is incorrect. 6. Let α and β be the distinct roots of a + b + c =, then equal to a () ( α β ) () a () ( α β ) () α β 6. () ( α)( β) sin a cosa( α)( β) Given limit = lim = lim α α α α ( α)( β) sin a = lim α ( α) a ( α) ( β) a ( α β) =. a ( α) ( β) α cos a + b + c lim ( α) 7. Suppose f() is differentiable = and lim f ( + h) = 5, then f () equals h h () () () 5 () 6 7. () f( + h) f f () = lim ; As function is differentiable so it is continuous as it is given h h f( + h) that lim = 5 and hence f() = h h f( + h) Hence f () = lim = 5 h h Hence () is the correct answer. 8. Let f be differentiable for all. If f() = - and f () for [, 6], then () f(6) 8 () f(6) < 8 () f(6) < 5 () f(6) = 5 is FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

9 9 8. () As f() = - & f () [, 6] Applying Lagrange s mean value theorem f( 6) f = f ( c) 5 f(6) + f() f(6) f(6) If f is a real-valued differentiable function satisfying f() f(y) ( y),, y R and f() =, then f() equals () - () () () 9. () f( + h) f f () = lim h h ( + ) f h f h f ( ) = lim lim h h h h f () f () = f() = constant As f() = f() =.. If is so small that and higher powers of may be neglected, then / + + ( ) / may be approimated as () () () () 8 8. () ( ) / + + = ( ) / 8 = If = n n n a, y= b, z= c where a, b, c are in A.P. and a <, b <, c <, n= n= n= then, y, z are in () G.P. () A.P. () Arithmetic Geometric Progression () H.P.. () n = a = n= a a = n y = b = b b = n= y FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

10 n z = c = c = n= c z a, b, c are in A.P. b = a + c = + y y = + y z, y, z are in H.P.. In a triangle ABC, let C = π. If r is the inradius and R is the circumradius of the the triangle ABC, then (r + R) equals () b + c () a + b () a + b + c () c + a. () ab r + R = c + a + b + c a + = b = a+ b ( a+ b+ c ) a+ b+ c ( since c = a + b ).. If cos cos y = α, then y cos α + y is equal to () sin α () () sin α () sin α. () cos - cos - y = α y y cos + =α y y y cos + + = α y + y = cos α + y y cosα + y y cosα = sin α.. If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sin B, sin C are in () G.P. () A.P. () Arithmetic Geometric Progression () H.P.. () = pa = pb = pb p, p, p are in H.P.,, are in H.P. a b c,, are in H.P a b c a, b, c are in A.P. sina, sinb, sinc are in A.P. FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

11 = 5. If I = d, I d, I = d and I = d then () I > I () I > I () I = I () I > I 5. () I = d, I = d, I = d, I = d < <, > d > d I > I. 6. The area enclosed between the curve y = log e ( + e) and the coordinate aes is () () () () 6. () Required area (OAB) = ln( + ) = e ln + e d + e =. e d 7. The parabolas y = and = y divide the square region bounded by the lines =, y = and the coordinate aes. If S, S, S are respectively the areas of these parts numbered from top to bottom; then S : S : S is () : : () : : () : : () : : 7. () y = and = y are symmetric about line y = area bounded between y = and y = is A s = 6 and As A s A s : 8. If dy d A s : = = 6 A s :: : :. 8 d = = y (log y log + ), then the solution of the equation is () y log = c y () log y = c 8. () dy = y (log y log + ) d dy y y = log + d Put y = v () log y = cy () log = cy y FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

12 dy dv = v + d d dv v+ = v log v+ d dv vlogv d = dv d = vlogv put log v = z dv = dz v dz d = z ln z = ln + ln c z = c log v = c y log = c. 9. The line parallel to the ais and passing through the intersection of the lines a + by + b = and b ay a =, where (a, b) (, ) is () below the ais at a distance of () below the ais at a distance of () above the ais at a distance of () above the ais at a distance of from it from it from it from it 9. () a + by + b + λ(b ay a) = (a + bλ) + (b aλ)y + b - λa = a + bλ = λ = -a/b a + by + b - a (b ay a) = b a a a + by + b a + y+ = b b a a y b+ + b+ = b b b + a b + a y = b b ( + a) a + b y = = b y = so it is / units below -ais. FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

13 . A spherical iron ball cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of 5 cm /min. When the thickness of ice is 5 cm, then the rate at which the thickness of ice decreases, is () 6π cm/min () 8π cm/min (). () dv 5π cm/min () 5 6π cm/min 5 dt = πr dr 5 dt = dr 5 = where r = 5 dt π 5 = 6π. (log ). d is equal to ( + (log ) log () (log ) + + C e () + + C. () ( log ) ( + ( log) ) d log = d ( log ) ( ( log) ) + + t t = e t e dt put log = t d = e t dt + t ( + t ) t t e dt + t ( + t ) t e = + c = + t + log + c () + + C () (log ) + + C. Let f : R R be a differentiable function having f () = 6, f () = 8. Then f() t lim 6 dt equals () () 6 () () 8 FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

14 . () f t lim dt Applying L Hospital rule lim f f = f() f () = 6 8 = 8.. Let f () be a non negative continuous function such that the area bounded by the π π curve y = f (), ais and the ordinates = and = β > π is βsinβ+ cosβ+ β. Then f π is π π () + () + π π () () +. () β π Given that f d =βsinβ+ cosβ+ β π / Differentiating w. r. t β f(β) = β cosβ + sinβ - π sinβ + π π π π f sin = + = +.. The locus of a point P (α, β) moving under the condition that the line y = α + β is a y tangent to the hyperbola = is a b () an ellipse () a circle () a parabola () a hyperbola. () y Tangent to the hyperbola = is a b y = m ± am b Given that y = α + β is the tangent of hyperbola m = α and a m b = β a α b = β Locus is a y = b which is hyperbola. 5. If the angle θ between the line + y z = = and the plane y + λ z + = is such that sin θ = the value of λ is () 5 () 5 FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

15 5 () () 5. () Angle between line and normal to plane is π + λ cos θ = where θ is angle between line & plane 5+λ λ sinθ = = 5+λ λ = The angle between the lines = y = z and 6 = y = z is () () 9 () 5 () 6. () Angle between the lines = y = - z & 6 = -y = -z is 9 Since a a + b b + c c =. 7. If the plane a ay + az + 6 = passes through the midpoint of the line joining the centres of the spheres + y + z + 6 8y z = and + y + z + y z = 8, then a equals () () () () 7. () Plane a ay + az + 6 = passes through the mid point of the centre of spheres + y + z + 6 8y z = and + y + z + y z = 8 respectively centre of spheres are (-,, ) & (5, -, ) Mid point of centre is (,, ) Satisfying this in the equation of plane, we get a a + a + 6 = a = The distance between the line r = i ˆ j ˆ+ k ˆ +λ(i ˆ ˆj+ k) ˆ r (i ˆ+ 5j ˆ+ k) ˆ = 5 is () () 9 () () 8. () Distance between the line r = i ˆ j ˆ+ kˆ +λ( ˆi ˆj+ kˆ ) and the plane r ( ˆ i + 5j ˆ + k ˆ ) = 5 is equation of plane is + 5y + z = 5 Distance of line from this plane = perpendicular distance of point (, -, ) from the plane + 5 i.e. = and the plane FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

16 6 9. For any vector a, the value of ˆ (a i) (a ˆ + j) + (a k) ˆ () a () a () a () a 9. () Let a= i ˆ+ yj ˆ+ zkˆ a ˆi = zj ˆ ykˆ a ˆ i = y + z a ˆ j = + z similarly a k ˆ = + y and a ˆ j = + z similarly a k ˆ = + y and a ˆ i = y + z a ˆi + a ˆj + a kˆ = ( + y + z ) = a. is equal to 5. If non-zero numbers a, b, c are in H.P., then the straight line + y + = always a b c passes through a fied point. That point is () (-, ) () (-, -) () (, -) (), 5. () a, b, c are in H.P. = b a c y + + = a b c y = = =, y = - 5. If a verte of a triangle is (, ) and the mid-points of two sides through this verte are (-, ) and (, ), then the centroid of the triangle is 7 7 (), (), 7 7 (), (), 5. () Verte of triangle is (, ) and midpoint of sides A(, ) through this verte is (-, ) and (, ) verte B and C come out to be (-, ) and (5, ) centroid is + 5, + + (-, ) (, ) (, 7/) B C FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

17 7 5. If the circles + y + a + cy + a = and + y a + dy = intersect in two distinct points P and Q then the line 5 + by a = passes through P and Q for () eactly one value of a () no value of a () infinitely many values of a () eactly two values of a 5. () S = + y + a + cy + a = S = + y a + dy = Equation of radical ais of S and S S S = 5a + (c d)y + a + = Given that 5 + by a = passes through P and Q a c d a+ = = b a a + = -a a + a + = No real value of a. 5. A circle touches the -ais and also touches the circle with centre at (, ) and radius. The locus of the centre of the circle is () an ellipse () a circle () a hyperbola () a parabola 5. () Equation of circle with centre (, ) and radius is + (y ) =. Let locus of the variable circle is (α, β) It touches -ais. It equation ( - α) + (y - β) = β Circles touch eternally α + β = +β α + (β - ) = β + + β α = (β - /) Locus is = (y /) which is parabola. (α, β) 5. If a circle passes through the point (a, b) and cuts the circle + y = p orthogonally, then the equation of the locus of its centre is () + y a by + (a + b p ) = () a + by (a b + p ) = () + y a by + (a b p ) = () a + by (a + b + p ) = 5. () Let the centre be (α, β) It cut the circle + y = p orthogonally (-α) + (-β) = c p c = p Let equation of circle is + y - α - βy + p = It pass through (a, b) a + b - αa - βb + p = Locus a + by (a + b + p ) =. 55. An ellipse has OB as semi minor ais, F and F its focii and the angle FBF is a right angle. Then the eccentricity of the ellipse is () () FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

18 8 () 55. () FBF = 9 o ( ae b ) ( ae b ) = (ae) (a e + b ) = a e e = b /a Also e = - b /a = e () B(, b) F (-ae, ) O F(ae, ) e =, e =. 56. Let a, b and c be distinct non-negative numbers. If the vectors ai ˆ+ aj ˆ+ ck, ˆ ˆi + kˆ and ci ˆ+ cj ˆ+ bkˆ lie in a plane, then c is () the Geometric Mean of a and b () the Arithmetic Mean of a and b () equal to zero () the Harmonic Mean of a and b 56. () Vector ai ˆ+ aj ˆ+ ckˆ, ˆ i + k ˆ and ci ˆ+ cj ˆ+ bkˆ are coplanar a a c = c = ab c c b a, b, c are in G.P. 57. If a, b, c are non-coplanar vectors and λ is a real number then λ ( a+ b) λ b λ c = a b+ c b for () eactly one value of λ () no value of λ () eactly three values of λ () eactly two values of λ 57. () λ a+ b λ b λ c = a b+ c b λ λ λ = λ λ = - Hence no real value of λ. ˆ ˆ a i k, b i ˆ ˆ j k ˆ 58. Let = = + + ( ) c yi ˆ j ˆ y k ˆ. Then a, b, c and = + + ( + ) depends on () only y () only () both and y () neither nor y 58. () a = ˆi kˆ, b = i ˆ + ˆ j+ ( ) k ˆ abc = a ( b c) c yi ˆ j ˆ y k ˆ and = + + ( + ) FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

19 9 ˆi ˆj kˆ b c = = î ( + ) - ĵ ( + - y y + y) + ˆk( y) y + y a. ( b c) = which does not depend on and y. 59. Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is () 9 () 9 () 8 9 () () For a particular house being selected Probability = Prob(all the persons apply for the same house) = = A random variable X has Poisson distribution with mean. Then P(X >.5) equals () () e () () e e 6. () k λ λ P( = k) = e k! P( ) = P( = ) P( = ) λ = e -λ e -λ! = - e. =, P( A B) 6 where A stands for complement of event A. Then events A and B are () equally likely and mutually eclusive () equally likely but not independent () independent but not equally likely () mutually eclusive and independent 6. Let A and B be two events such that P( A B) 6. () ( B) =, P(A B) = and P( A) P A 6 P(A B) = 5/6 P(A) = / Also P(A B) = P(A) + P(B) P(A B) P(B) = 5/6 / + / = / P(A) P(B) = / / = / = P(A B) = = and P( A) =, FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

20 Hence A and B are independent but not equally likely. 6. A lizard, at an initial distance of cm behind an insect, moves from rest with an acceleration of cm/s and pursues the insect which is crawling uniformly along a straight line at a speed of cm/s. Then the lizard will catch the insect after () s () s () s () s 6. () t = + t t =. 6. Two points A and B move from rest along a straight line with constant acceleration f and f respectively. If A takes m sec. more than B and describes n units more than B in acquiring the same speed then () (f - f )m = ff n () (f + f )m = ff n () ( f f ) m ffn + = () ( f f) n= ffm 6. () v = f(d + n) = f d v = f (t) = (m + t)f eliminate d and m we get (f - f)n = ff m. 6. A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance H H () () A B A + B H H () () A + B A B 6. () (A + B) = d = H H d = A + B. 65. The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is () : () : () : () : 65. () F = F cos θ F F = F sin θ F F = F F : F : : :. F FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

21 66. The sum of the series ad inf. is.! 6.! 6.6! () e () e + e e () e () e + e e 66. () 6 e + e = !! 6! putting = / we get e+. e 67. The value of π cos π + a d, a >, is () a π () π () a π () π 67. () π π cos π d = cos d = + a. π 68. The plane + y z = cuts the sphere + y + z + z = in a circle of radius () () () () 68. () Perpendicular distance of centre,, from + y = + = 6 radius = 5 =. 69. If the pair of lines a + (a + b)y + by = lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then () a ab + b = () a ab + b = () a + ab + b = () a + ab + b = 69. () a+ b ab a+ b = (a + b) = (a + b + ab) a + b + ab =. FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

22 7. Let,,, n be n observations such that = and i i = 8. Then a possible value of n among the following is () 5 () 8 () 9 () 7. () i i n n n A particle is projected from a point O with velocity u at an angle of 6 o with the horizontal. When it is moving in a direction at right angles to its direction at O, its velocity then is given by () u () u () u u () 7. () u cos 6 o = v cos o v =. o 6 o o 7. If both the roots of the quadratic equation k + k + k 5 = are less than 5, then k lies in the interval () (5, 6] () (6, ) () (-, ) () [, 5] 7. () b < 5 a f(5) > k (-, ). 7. If a, a, a,, a n, are in G.P., then the determinant logan logan+ logan+ = logan+ logan+ logan+ 5is equal to loga loga loga n+ 6 n+ 7 n+ 8 () () () () 7. () C C, C C two rows becomes identical Answer:. 7. A real valued function f() satisfies the functional equation f( y) = f() f(y) f(a ) f(a + y) where a is a given constant and f() =, f(a ) is equal to () f() () f() () f(a) + f(a ) () f(-) FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

23 7. () f(a ( a)) = f(a) f( a) f() f() = -f() =, y =, f = f f ( a) f ( a) = f( a) =. 75. If the equation n n an + an a =, a, n, has a positive root = α, then the n n equation nan + ( n ) an a = has a positive root, which is () greater than α () smaller than α () greater than or equal to α () equal to α 75. () f() =, f(α) = f (k) = for some k (, α). FIITJEE Ltd. ICES House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi - 6, Ph : 65599, 65699, Fa : 659

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