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1 AIEEE/1/MAHS 1 S. No Questios Solutios Q.1 he circle passig through (1, ) ad touchig the axis of x at (, ) also passes through the poit (a) (, ) (b) (, ) (c) (, ) (d) (, ) Q. ABCD is a trapezium such that AB ad CD are parallel ad BC CD. If ADB θ, BC p ad CD q, the AB is equal to (a) (c) p +q cos θ (b) p +q p cos θ+q si θ p cos θ+q si θ (p +q )si θ (d) (p +q )si θ (p cos θ+q si θ) p cos θ+q si θ Q. Give : A circle, x + y ad a parabola, y x. Statemet I : A equatio of a commo taget to these curves is y x +. Statemet II : If the lie,y mx (m ) is their commo taget, the m Q. satisfies m m +. (1) Statemet - I is rue; Statemet -II is true; Statemet-II is ot a correct explaatio for Statemet-I () Statemet -I is rue; Statemet -II is alse. () Statemet -I is alse; Statemet -II is rue () Statemet -I is rue; Statemet -II is rue; Statemet-II is a correct explaatio for Statemet-I m Sol: 1 (b) (x ) + y + λy he circle passes through (1, ) + λ λ (x ) + y + y Clearly (, ) satisfies. Sol: (d) Usig sie rule i triagle ABD AB BD si θ si (θ+a) p AB +q si θ p +q si θ si θ cos α+cos θ si α si θ.q + cos θ.p p +q p +q AB Sol: (a) p +q si θ. (p cos θ+q si θ) Let the taget to the parabola be y mx + (m ). Now, its distace from the cetre of the circle must be equal to the radius of the circle. So, m 1 + m (1+m )m m + m. (m 1) m + m +1 So, the commo tagets are y x + ad y - x -. Sol: (a) m A ray of light alog x + y gets reflected upo readig x axis, the equatio of the reflected rays is (a) y x - (b) y x (c) y x 1 (d) y x + Slope of the icidet ray is - 1. So, the slope of the reflected ray must be 1. he poit of icidece is (, ). So, the equatio of reflected ray is y 1 (x- ). Q. All the studets of a class performed poorly i Mathematics. he teacher decided to give grace marks of 1 to each of the studets. Which of the followig statistical measures will ot chage eve after the grace marks were give? (a) media (b) mode (c) variace (d) mea Sol: (c) Variace is ot chage by the chage of origi. Alterate Solutio: σ x x for y x + 1 y x +1 σ 1 y+1 y 1 y y σ Poorima Uiversity, or ay query, cotact us at: ,18

2 AIEEE/1/MAHS Q.6 If x, y, z are i A.P. ad ta 1 x, ta 1 y ad ta 1 z are also i A.P., the (a) x y 6z (b) 6x y z (c) 6x y z (d) x y z Q.7 If f(x) dx Ψ (x), the x f(x )dx is equal to (a) 1 x Ψ (x ) x Ψ (x ) dx+c Q.8 (b) 1 x Ψ (x ) x Ψ (x ) dx+c (c) 1 x Ψ x x Ψ (x ) dx +C (d) 1 x Ψ (x ) x Ψ (x ) dx +C he equatio of the circle passig through the foci of the ellipse x 1, ad 16 9 havig cetre (,) is (a) x + y - 6y + 7 (b) x + y - 6y (c) x + y - 6y + (d) x + y - 6y 7 Q.9 he x-coordiate of the icetre of the triagle that has the coordiates of mid poits of its sides as (, 1) (1, 1) ad (1, ) is (a) - (b) 1+ (c) 1 - (d) + x Q.1 he itercepts o x axis made by tagets to the curve, y t which are parallel to the lie y x, are equal to (a) + (b) + (c) + (d) + 1 Q.11 he sum of first terms of the sequece.7,.77,.777,.., is (a) 7 9 (99-1 ) (b) 7 81 ( ) (c) 7 9 ( ) (d) 7 81 (179-1 ) dt, x R, Sol: 6 (d) If x, y, z are i A.P. y x + z ad ta 1 x, ta 1 y, ta 1 z are i A.P. ta 1 y ta 1 x + ta 1 z x y z. Note: If y, the oe of the optios is appropriate. Sol: 7 (b) f (x)dx Ψ (x) Let x t x dx dt the x f(x )dx 1 tf(t)dt 1 t f t dt {1. f t dt}dt 1 x Ψ (x ) - x Ψ(x )dx+c. Sol: 8 (d) foci (+ae,) We have a e a b 7 Equatio of circle (x ) + (y ) ( 7 ) + ( ) x +y 6y 7. Sol: 9 (a) x coordiate ax 1+bx +cx a+b+c Alterate Solutio: x coordiate r (s a) ta A/ + + ta π -. Sol: 1 (d) x x + y t dt for x dx ad y t dt - for x - tagets are y (x ) y x ad y + (x + ) y x + Puttig y, we get x 1 ad l. Sol: 11 (b) t r.77 r times 7 ( r ) 7 ( r ) S r1 t r 7 9 r1 1 r ( (179+1 ) Poorima Uiversity, or ay query, cotact us at: ,18

3 AIEEE/1/MAHS Q.1 Cosider : Statemet I : (p ~ q) (~ p q) is a fallacy. Statemet II : (p q) (~ q ~ p) is a tautology. (1) Statemet - I is rue; Statemet -II is true; Statemet-II is ot a correct explaatio for Statemet-I () Statemet -I is rue; Statemet -II is alse. () Statemet -I is alse; Statemet -II is rue () Statemet -I is rue; Statemet -II is rue; Statemet-II is a correct explaatio for Statemet-I Q.1 he area (i square uits) bouded by the curves y x,y x +, x axis, ad lyig i the first quadrat is (a) 6 (b) 18 (c) 7 (d) 9 Q.1 he expressio ta A ca be writte as 1 cot A 1 ta A (a) seca coseca + 1 (b) taa + cota (c) seca + coseca (d) sia cosa + 1 Q.1 he real umber k for which the equatio x + x + k has two distict real roots i [,1] (a) lies betwee ad (b) lies betwee -1 ad (c) does ot exist (d) lies betwee 1 ad Sol: 1 (a) S1: p q ~ p ~ q p ~ q ~ p q (p ~ q) (~ p q S: p q ~ p ~ q p q ~ p ~ q (p q) (~ p ~ q S is ot a explaatio of S1 Sol: 1 (d) x x x x - 6x + 9 x 1x + 9 x 9, x 1 y + y ] y + y y Sol: 1 (a) 1 cota (1 cota ) - co t A 1 co t A cos ec A+cot A (1 cota ) cota (1 cota ) cot A allacy 1 + sec A cosec A autology Sol: 1 (c) If x + x + k has distict real roots i [,1], the f (x) will chage sig but f (x) 6x + > So o value of k exists. Q.16 Sol: 16 (c) Q.17 Let be the umber of all possible triagles formed by joiig vertices of a sided regular polygo. If +1 1, the the value of is (a) (b) 1 (c) 8 (d) 7 Sol: 17 (a) +1 C C 1 C 1. Poorima Uiversity, or ay query, cotact us at: ,18

4 AIEEE/1/MAHS Q.18 At preset, a firm is maufacturig items. It is estimated that the rate of chage of productio P w.r.t. additioal umber of workers x is give by dp dx 1 1 x. If the firm employs more workers, the the ew level of productio of items is (a) (b) (c) (d) Q.19 π/ dx Statemet I: he value of the itegral is equal to π. 6 Q. b b π/6 1+ ta x Statemet II: f x dx f a + b x dx. a a (a) Statemet I is rue; Statemet II is true; Statemet II is ot a correct explaatio for statemet I (b) Statemet I is rue; Statemet II is alse. (c) Statemet I alse; Statemet II is rue (d) Statemet I rue; Statemet II is rue; Statemet II is a correct explaatio for Statemet I 1 α If P 1 is the adjoit of a matrix A ad A, the α is equal to (a) 11 (b) (c) (d) Sol: 18 (b) P dp 1 1 x dx (P ) 1 1 () / P. Sol: 19 (c) I I I π 6 I π 1. π/ dx π/6 1+ ta x π/ ta x π/6 1+ ta x dx Sol: (a) 1 a P 1 Adj A A Adj A 16 1(1 1) α ( 6) + ( 6) 16. α α 11. Q.1 he umber of values of k, for which they system of equatios (k + 1)x + 8y k kx + (k + )y k -1 has o solutio, is (a) 1 (b) (c) (d) ifiite Q. If y sec(ta 1 x), the at x 1is equal to (a) 1 Q. If the lies x dx (b) 1 (c) (d) 1 y z x 1 ad y z 1 1 k k 1 (a) exactly oe value (c) exactly three values are coplaar, the k ca have (b) exactly two values (d) ay value Q. Let A ad B be two sets cotaiig elemets ad elemets respectively. he umber of subsets of A B havig or more elemets is (a) (b) 19 (c) 11 (d) 6 Sol: 1 (a) or o solutio k+1 8 k k + (1) k k 1 (k + 1) (k + ) 8k or k k + k 1, But for k 1, equatio (1) is ot satisfied Hece k. Sol: (d) y sec (ta 1 x) sec dx (ta 1 x) ta (ta 1 1 x). 1+x dx x Sol: (b) k k 1 1(1 + k) + 1(1 + k ) 1 ( k) k k k k + k (k) (k + ) values of k. Sol: (b) A B will have 8 elemets. 8 8 C 8 C 1 8 C Poorima Uiversity, or ay query, cotact us at: ,18

5 AIEEE/1/MAHS Q. If the vectors AB i ad AC i - j + k are the sides of a triagle ABC, the the legth of the media through A is (a) 7 (b) (c) (d) 18 Q.6 A multiple choice examiatio has questios. Each questio has three alterative aswers of which exactly oe is correct. he probability that a studet will get or more correct aswers just by guessig is (b) (c) (d) (a) 1 Sol: (b) AM AB +AC AM i - j +k AM Sol: 6 (b) P (correct aswer) 1/ 1 + C 1 C () () Q.7 If z is a complex umber of uit modulus ad argumet θ, the arg 1+z equals Sol: 7 (b) 1+z (a) π θ (b) θ (c) π θ (d) θ z 1 zz 1 1+z 1+z 1+z 1+ 1 z z Q.8 If the equatios x + x + ad ax + bx + c, a, b, c R, have a commo root, the a : b : c is (a) : : 1 (b) 1 : : (c) : 1 : (d) 1 : : Sol: 8 (d) or equatio x + x + both roots are imagiary. Sice a, b, c R. If oe root is commo the both roots are commo Q.9 Distace betwee two parallel plaes x + y +z 8 ad x + y + z + is (a) (b) 7 (c) 9 (d) Q. x+1 he term idepedet of x i expasio of x 1 1 x / x 1/ +1 x x is 1/ (a) 1 (b) 1 (c) 1 (d) Hece, a b c 1 a : b : c 1 : : Sol: 9 (b) x + y + z 16 x + y +z - d mi Sol: (b) (x 1/ + 1) (x / x 1/ + +1) x / x 1/ +1 1 x. x + 1 x 1 x 1 (x 1/ x 1/ ) 1 r+1 ( - 1) r 1 C r x r r 1 C Poorima Uiversity, or ay query, cotact us at: ,18

PART B MATHEMATICS (2) (4) = +

PART B MATHEMATICS (2) (4) = + 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