PART B MATHEMATICS (2) (4) = +

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Geoffrey Wilkinson
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1 JEE (MAIN)--CMP - PAR B MAHEMAICS. he circle passing through (, ) and touching the axis of x at (, ) also passes through the point () (, ) () (, ) () (, ) (4) (, ) Sol. () (x ) + y + λy = he circle passes through (, ) λ = λ = 4 (x ) + y + 4y = Clearly (, ) satisfies.. ABCD is a trapezium such that AB and CD are parallel and BC CD. If ADB = θ, BC = p and CD = q, then AB is equal to p + q cos θ p + q () () p cos θ + q sin θ p cos θ + q sin θ () p + q sin θ ( p cos θ + q sin θ) p + q sin θ ( ) (4) p cos θ + q sin θ Using sine rule in triangle ABD AB BD = sin θ sin θ + α AB = AB = p + q sin θ p + q sin θ = sin θ cos α + cos θ sin α sin θ q cos θ p + p + q p + q p + q sin θ θ + θ. ( p cos q sin ) A π (θ + α) p + q θ α D q α B C p. Given : A circle, x + y = and a parabola, y = 4 x. Sol. () Statement I : An equation of a common tangent to these curves is y = x +. Statement II : If the line, y = mx + (m ) is their common tangent, then m satisfies m 4 m + = m. () Statement - I is rue; Statement -II is true; Statement-II is not a correct explanation for Statement-I () Statement -I is rue; Statement -II is alse. () Statement -I is alse; Statement -II is rue (4) Statement -I is rue; Statement -II is rue; Statement-II is a correct explanation for Statement-I Let the tangent to the parabola be y = mx + (m ). m Now, its distance from the centre of the circle must be equal to the radius of the circle. So, m = + ( + m ) m = m 4 + m =. m (m ) (m + ) = m = ± So, the common tangents are y = x + and y = x. 4. A ray of light along x + y = gets reflected upon reaching x-axis, the equation of the reflected rays is IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

2 JEE (MAIN)--CMP- () y = x () y = x () y = x (4) y = x + Sol. () Slope of the incident ray is. So, the slope of the reflected ray must be. he point of incidence is (, ). So, the equation of reflected ray is y ( x ) =.. All the students of a class performed poorly in Mathematics. he teacher decided to give grace marks of to each of the students. Which of the following statistical measures will not change even after the grace marks were given? () median () mode () variance (4) mean Sol. () Variance is not changed by the change of origin. Alternate Solution: x x σ = for y = x + y = x + n y + y y y. σ = = = σ n n 6. If x, y, z are in A.P. and tan x, tan y and tan z are also in A.P., then () x = y = 6z () 6x = y = z () 6x = 4y = z (4) x = y = z If x, y, z are in A.P. y = x + z and tan x, tan y, tan z are in A.P. tan y = tan x + tan z x = y = z. Note: If y =, then none of the options is appropriate., then x f ( x ) 7. If f ( x) = Ψ ( x) x x x x C x Ψ Ψ + C is equal to x Ψ x x Ψ x + C x x x x C Ψ Ψ + () Ψ Ψ + () () (4) Sol. () f x = ψ ( x) Let x = t x = dt = tf ( t ) dt t f t dt { f t dt } dt then x f ( x ) = = x ψ x x ψ x + C. IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

3 JEE (MAIN)--CMP - 8. he equation of the circle passing through the foci of the ellipse () () x + y 6y + 7 = () x + y 6y + = (4) foci (± ae, ) We have a e = a b = 7 Equation of circle (x ) + (y ) = ( 7 ) + ( ) x + y 6y 7 =. x y + =, and having centre at (, ) is 6 9 x + y 6y = x + y 6y 7 = 9. he x-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as (, ) (, ) and (, ) is () () + () (4) + Sol. () x-coordinate = ax + bx + cx a + b + c = = = =. Alternate Solution: x-coordinate = r = (s a) tan A/ 4 + π = tan =. 4 C(, ) (, ) (, ) A(, ) (, ) B(, ) 4. he intercepts on x-axis made by tangents to the curve, y = x, are equal to () ± () ± () ± 4 (4) ± dy x = = x = ± y = t dt = for x = and y = t dt = for x = tangents are y = (x ) y = x and y + = (x + ) y = x + Putting y =, we get x = and. 4. he sum of first terms of the sequence.7,.77,.777,.., is 7 () ( 99 7 ) () ( 99 7 ) Sol. () t r = r times () + (4) x y = t dt, x R, which are parallel to the line IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

4 JEE (MAIN)--CMP- = 7 ( r ) 7 = ( r ) 9 7 r 7 S = tr = 9 r= r= 9 9 = ( ) 7 79 = ( + ) 8 4. Consider : Statement I : (p ~ q) (~ p q) is a fallacy. Statement II : (p q) (~ q ~ p) is a tautology. () Statement - I is rue; Statement -II is true; Statement-II is not a correct explanation for Statement-I () Statement -I is rue; Statement -II is alse. () Statement -I is alse; Statement -II is rue (4) Statement -I is rue; Statement -II is rue; Statement-II is a correct explanation for Statement-I Sol. () S: p q ~p ~q p^~q ~p^q (p^~q)^(~p^q) allacy S: p q ~ p ~ q p q ~ q ~ p (p q) (~ q ~ p) S is not an explanation of S autology 4. he area (in square units) bounded by the curves y = x, y x + =, x-axis, and lying in the first quadrant is () 6 () 8 () 7 4 (4) 9 x = x 4x = x 6x + 9 x x + 9 x = 9, x = y + y dy y y + y = = tan A cot A he expression + cot A tan A can be written as () seca coseca + () tana + cota () seca + coseca (4) sina cosa + Sol. (), x (, ) 9 IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

5 JEE (MAIN)--CMP -4 cot A cot A cos ec A + cot A = = cot A cot A cot A cot A cot A cot A ( ) ( ) ( ) = + sec A cosec A 4. he real number k for which the equation, x + x + k = has two distinct real roots in [, ] () lies between and () lies between and () does not exist (4) lies between and Sol. () If x + x + k = has distinct real roots in [, ], then f (x) will change sign but f (x) = 6x + > So no value of k exists. 46. lim x ( cos x) ( + cos x) x tan 4x is equal to () () () (4) 4 Sol. () ( cos x) lim + cos x x x tan 4x sin x 4x lim + cos x x x 4 tan 4x + =. 4 = 47. Let n be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If n+ n =, then the value of n is () () () 8 (4) 7 Sol. () n+ n C C = n C = n =. 48. At present, a firm is manufacturing items. It is estimated that the rate of change of production P w.r.t. Sol. () additional number of workers x is given by dp = x. If the firm employs more workers, then the new level of production of items is () () () 4 (4) P dp = x (P ) = / P =. 49. Statement I : he value of the integral b π/ π is equal to. tan x 6 π /6 + Statement II : f ( x) = f ( a + b x). a b a () Statement - I is rue; Statement -II is true; Statement-II is not a correct explanation for Statement-I () Statement -I is rue; Statement -II is alse. IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

6 JEE (MAIN)--CMP- Sol. () () Statement -I is alse; Statement -II is rue (4) Statement -I is rue; Statement -II is rue; Statement-II is a correct explanation for Statement-I I = π/ π/ 6 π/ π / 6 + tan x tan x I = + tan x π I = 6 π I =. α. If P = is the adjoint of a matrix A and A = 4, then α is equal to 4 4 () () () (4) 4 Sol. () α P = 4 4 Adj A = A Adj A = 6 ( ) α (4 6) + (4 6) = 6. α 6 = 6. α =. α =.. he number of values of k, for which the system of equations (k + )x + 8y = 4k kx + (k + )y = k has no solution, is () () () (4) infinite Sol. () or no solution k + 8 4k = () k k + k (k + ) (k + ) 8k = or k 4k + = k =, But for k =, equation () is not satisfied Hence k =.. If y = sec(tan x), then dy at x = is equal to () () () (4) IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

7 JEE (MAIN)--CMP -6 y = sec (tan x) dy = sec ( tan x) tan ( tan x) + x dy = =. x=. x y z 4 x y 4 z If the lines = = and = = are coplanar, then k can have k k () exactly one value () exactly two values () exactly three values (4) any value Sol. () k = k ( + k) + ( + k ) ( k) = k + + k + + k = k + k = (k) (k + ) = values of k. 4. Let A and B be two sets containing elements and 4 elements respectively. he number of subsets of A B having or more elements is () () 9 () (4) 6 Sol. () A B will have 8 elements. 8 8 C 8 C 8 C = = 9.. uuur uuur If the vectors AB = i ˆ + 4kˆ and AC = i ˆ j ˆ + 4kˆ are the sides of a triangle ABC, then the length of the median through A is () 7 () () 4 (4) 8 Sol. () uuur uuur uuuur AB + AC AM = uuuur AM = 4i ˆ ˆj + 4kˆ uuuur AM = = A C M B 6. A multiple choice examination has questions. Each question has three alternative answers of which exactly one is correct. he probability that a student will get 4 or more correct answers just by guessing is () () 7 () (4) Sol. () P (correct answer) = / IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

8 JEE (MAIN)--CMP-7 4 C4 + C + =. 7. If z is a complex number of unit modulus and argument θ, then + z arg + z equals π () θ () θ () π θ (4) θ Sol. () z = zz = + z + z = = z. + z + z 8. If the equations x + x + = and ax + bx + c =, a, b, c R, have a common root, then a : b : c is () : : () : : () : : (4) : : or equation x + x + = both roots are imaginary. Since a, b, c R. If one root is common then both roots are common Hence, a = b = c a : b : c = : :. 9. Distance between two parallel planes x + y + z = 8 and 4x + y + 4z + = is () () 7 () 9 (4) Sol. () 4x + y + 4z = 6 4x + y + 4z = 7 d min = 6 = 6 =. 6. x + x he term independent of x in expansion of / / / x x + x x () () () (4) 4 is Sol. () / / / ( x + ) ( x x + ) ( x + ) ( x ) / / x x + x ( x ) r+ = ( ) r C r C 4 =. r 6 x r = 4 = (x / x / ) IIJEE Ltd., IIJEE House, 9-A, Kalu Sarai, Sarvapriya Vihar, New Delhi -6, Ph 466, 66949, ax 694

JEE MAIN 2013 Mathematics

JEE MAIN 2013 Mathematics JEE MAIN 01 Mathematics 1. The circle passing through (1, ) and touching the axis of x at (, 0) also passes through the point (1) (, 5) () (5, ) () (, 5) (4) ( 5, ) The equation of the circle due to point