2) 3 π. EAMCET Maths Practice Questions Examples with hints and short cuts from few important chapters

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1 EAMCET Maths Practice Questios Examples with hits ad short cuts from few importat chapters. If the vectors pi j + 5k, i qj + 5k are colliear the (p,q) ) 0 ) 3) 4) Hit : p 5 p, q q 5.If the vectors i j + k, i + j - 3k ad 3i + λ j + 5k are coplaar the λ ) 0 ) 3) 4) 4 Hit : 3 0 λ 4 3 λ 5 b + c a ( b c) 3. If a, b, c are o coplaar uit vector such that ad b is) π ) 3 π π 3) the agle betwee a 4) 6 π Hit : (. ) (. ) b c a b c a c b a b c + Comparig a. c ; a. b 3π ( a, b) 4 b c c a a b p, q 4.If a,b,c are three o coplaar vectors ad, ( abc) r ( abc), ( abc ) ) 3 ) 3) 4) a b p + b + c q + c + a r Hit:.. the

2 b c a. b c + b. b c abc + 0 Σ + Σ +. 3 abc abc abc ( a b) p ( a b) 5.ABC is equilateral triagle of side a the AB. BC + BC. CA + CA. AB is ) a a ) -3 3) a 4) Hit: AB + BC + CA 0 AB + BC + CA + AB BC + BC CA + CA AB... 0 a a a AB BC a AB. BC 6.Forces of5,3 uits actig alog 6i + j + 3k ad 3i - j + 6k respectively o a particle displaced from the poit (,,-) to the poit (4,3,). The total work doe. ) 48 7 ) ) 4) Hit: force 6i + j + 3k 3i j + 6k F i + 4 j + 33k 7 d 4i + 3 j + k i + j k i + j + k 48 w F. d 7

3 7. a,b,c are uit vectors such that the cosα + cos β a + b + c ad a b c makes α, β agle with a, b ) 3 ) 3) 4) Hit: a + b + c 8. ( a b) ( c d ) 5c 6d a b c a b b c c a [0+.. cosα +.. cos β ] cosα + cos β + a. b ( a + c + d ) the the value of ) 0 ) 3) 4) 4 a b c d a b. d c a b. c d 5c + 6d Comparig o either sides (abd) 5 (abc) -6 a. b a + c + d ( a. b a ) + ( a. b c ) + ( a. b d ) (5) If the sum of the squares of the perpedicular distaces of p from coordiate axes is the locus of p is ) 6 ) 3 3) 4)

4 y z z x x y Hit: x + y + z 6 0The ratio i which of plae divides the lie segmet joiig (-3,4,) (,,3) is ) : ) :3 3) 3: 4) : Hit: x : x 3:. If the extremities of a diagoal of a square are [,-, 3] [-3, 5] the legth of side ) 6 ) 3 3) 3 4) Hit: diagoal Side d π If a lie makes a agle of 4 with positive directio of each x axis ad y axis the the agle made by the lie with z axis is ) π ) 3 π 4 3) 3 π 4) 6 π Hit: cos α + cos β + cos γ cos + + γ γ 90

5 3. If the plae 7x + y + 3z 3003 meets the coordiate axes at A,B,C the the cetroid of the ABC is--- ) ( 43,9, 77) ) 0, 73, 0 3) (,,) 4)(3,3,3) Hit : itercepts form 7x y 3z A[ 49,0,0] B[ 0, 73,0] C [ 0,0, 3] cetroid of triagle ,, ( 43,9, 77) 4..If P(,,0), Q(,0,) the the projectio of PQ o the plae x + y + z 3 is ) 6 ) 3 3) 4) Hit : PQ OQ OP (0,-,) Plae drs (,,) Agle 0 + cosθ 0 θ 90 3 Projectio of PQ o the plae 5. Let a iˆ ˆj + 3 kˆ, b iˆ + 3 ˆj kˆ plae cotaiig a, b, the λ is equal to PQsiθ si 90 c λi ˆ + ˆ j + λ k. If c is parallel to the ad ˆ ) 0 ) 3) 4) Sol. Give a iˆ ˆj + 3 kˆ, b iˆ + 3 ˆj kˆ c λi ˆ + ˆ j + λ k ad ˆ

6 so vector ( a b ) also perpedicular to the vector c, i.e, ( θ 90 ) So, ( a b ). c should be equal to zero or ( a b ). c 0 iˆ ˆj kˆ a b 3 9 i j kˆ 3 ˆ ˆ 7iˆ + 7 ˆj + 7kˆ The from Equatio (i) ( iˆ ˆj kˆ ) ( λiˆ ˆj ( λ ) kˆ ) λ 0 Hece, the value of λ is If three uit vectors a, b, c satisfy a + b + c 0, the the agle betwee a ad b ) π 3 ) 5 π 6 3) 3 π 4) 6 π Sol. Give coditio is a + b + c 0 ad a, b, c are uit vectors the a b c Let the agle betwee a ad b is θ a + b c Now, from Equatio a + b c c c Squarig o both sides a + b + a b c. a. b + a. b c

7 { θ}..cos θ 7. ( a + b c ).( a b ) ( a b c ) π 3 is equal to ) a b c ) a b c 3) 3 a b c 4) 0 Sol. V a b c 0 3 a b c 8. If u a b, v a + b, a b, the u v is equal to 6 a. b ) 4 a. b 3) 6 a. b ) 4 a. b 4) Sol. We have, u a b, v a + b u v a b a + b 0 b a + a b 0 a b u v a b a b a b si θ ˆ { ˆ uit vector ˆ } 4.4si θ.

8 . 6 6 a b a b 6 ( a. b ) 9. If the agle θ betwee the vectors a x iˆ + 4xj ˆ + kˆ ad b 7iˆ ˆj + xkˆ is such that 90 < θ < 80, the x lies i the iterval Sol. ) 0, ), Givea x iˆ + 4 xj ˆ + kˆ, b 7i ˆj + xkˆ 3 3),, also 90 < θ < 80 4) 3, We kow that, cosθ a. b a b cosθ ( x iˆ + 4 xj ˆ + kˆ ).( 7iˆ ˆj + xkˆ ) 4 4x + 6x x θ lies betwee ( 90, 80 ) i.e, cosθ is egative i IId quadrat So, RHS is also egative i.e, 7x( x ) < 0 ( x ) 7x 4 4x + 6x x < 0 So, x 0,

9 r r r. Crx.y 0. If x + y, the r 0 equal to a) xy b) x (x+y) c) x (x+y) d) Noe of these Hit: r r r. Crx.y r 0 r r [ r(r ) + r] Crx.y r 0 x ( x - x+) x (x+y). Coefficiet of t 4 i (+t ) (+t )(+t 4 )is a) + C 6 b) C 6 c) + C 6 d) 3 + C 6 Hit: Write geeral term is (+t ) ad observe t 4 coefficiet i multiplicatio Sum of the coefficiets of the terms of degree m i the expasio of (+x) (+y) (+z) is a) ( C m ) 3 b) 3( C m ) c) C 3m d) 3 C m Hit: r + s+ t m ad required is ( + + ) C m 3 C m 3. Fid the term idepedet i x x 6 ) 5 ) 3) 4) 4

10 Hit: p 6. th 6 5 T5 C4 5 p + q ( I )4 Number of Ratioal terms are ) 3 ) 3) 4) Hit: 4 + LCM 5, 7 5 Fid the umber of terms i ( x + y + z )0 is ) 33 ) 3 3) 66 4) 6 Hit: + r 0+ 3 Cr C3 C 66 6 d, 3 rd, 4 th terms of ( ) + x are i AP the is ) 3 ) 7 3) 4) Hit: ( r ) +, substitute r gives 7 7.If the coefficiet of x i the expasio of k 5 x + x is 70 the k

11 ) 3 ) 3) 4) Hit: p s + p + q th + term k For atural umbers m, if y y a y a y m, m a ad a 0 the ) 40 ) 45 3) 0 4) 35 ( m m m ) ( C ) o C Hit : y + C y +... Co + C y + C y m 35 ; 45 Collect y, y terms ad comparig equatio 9. Coefficiet of 9 x i ( x + ) ( x + )...( x + 0) is ) 30 ) 0 3) 5 4) 55 Hit : ( + 30.The itegral part of )6 is

12 ) 98 ) 5 3) 0 4) C o + C Hit : f 97 + F Where F ad f are fractios I Number of terms i ( x + a ) + ( x a ) are ) 0 ) 3) 4) 5 Hit : terms No zero terms i ( + x ) + ( x ) + ( + ix ) + ( ix ) Hit : terms ) ) 3) 0 4) r 5 5 a + x ao + ax + + a5 x r r a r... ) 0 ) 5 3) 0 4) 35 Hit : It is i the form of (5)th syopsis 5 ( 5 )

13 34 x < the the coefficiet of + x r x i ( x ) is ) r. r ) ( r ). r 3) + r. r 4) ( x + ) r Hit : + x x + x + x + 3 x... + x + 4x + x +... Put r Collect coefficiet x 8x + x 0 r (. + ) 0 (4) optio 35 o C C C C ) +. Hit: Put ) +. + ( + ). 3) 4) C +. C + 3. C o Optio () If x is positive, the first egative term i the expasio of ( + x ) 7 5 is ) 7 th term ) 5 th term 3) 8 th term 4) 6 th term Hit: [7/5] th term 37. If, α β are roots of the equatio x + 6x + b 0( b < 0) the α β + β α is less tha

14 a) b) c) 8 d) Noe of these Hit: α + β 3, αβ b / ( ) D 36 4b > 0 b < 0 ( α + β) α β α + β αβ + β α αβ αβ ( α+β ) 8 < αβ b 39.If α, β are the roots of ax + bx + c 0; α + h, β + h are the roots of px + qx + r 0; D, D the respective discrimiats of these equatios, the D : D ad a a) p b) these b q c) c r d) Noe of Hit: Let A α + h, B β + h A B α β ( A B) ( α β) D a D P 40.The values of m; for which oe of the roots of roots of x x + m 0 is (are) x 3x + m 0 is double of oe of the a) 0, b) 0, c), d) Noe of these Hit: be the root of x x + m 0ad α be the root of x 3x + m 0 x α + m 0ad 4α 6α + m 0 α α m m m m

15 m 0 m ta x si x 4. lim x 0 x is equal to ) 0 ) 3) 4) ta x si x 0 Sol. lim form x 0 x 0 By L Hospital Rule lim x 0 sec x cos x x Agai, by L Hospital Rule lim x 0 sec x.sec x.ta x + si x If f : R R the k is equal to defied by + 3x cos x, for x 0 f x x k, for x 0 is cotiuous at x 0, ) ) 5 3) 6 4) 0 Sol. f ( x) + x x for x 3 cos, 0 k, for x 0 RHL f ( h) h 0 ( h ) ( h) ( 0 + h ) cos lim

16 + 3h cos h lim h 0 h + 3h si h lim h 0 h + 3h + si lim h 0 h h si lim 3 + h 0 h h si h 3 +.lim h 0 h LHL f ( h) h 0 ( h) ( h) ( 0 h) cos 0 0 lim If f ( x) ( cos x)( cos x)...( cos x), the f ( x) ( r ta rx) f ( x) + is equal to ) f ( x) ) 0 3) f ( x) 4) f ( x) Sol. f ( x) ( cos x)( cos x)...( cos x) d f ' x si x.cos x...cos x cos x cos x.cos3 x...cos x dx r + { } f ' x si x.cos x...cos x cos x.si x...cos3x ( 3cos x.cos x.si3 x...cos x) ( cos x.cos x...si x) So, f '( x) ( r ta rx) f ( x) + r f '( x) + { ta x + ta x + 3ta3 x ta x} f ( x)

17 f ' x f x ta x f x ta x... f x ta x f '( x) ( si x.cos x...cos x) ( cos x.si x...cos x)... ( cos x.cos x...si x) f ( x) f ( x) ' ' 0 Hece, f ( x) ( r rx) f ( x) ' + ta 0 r 44. If a x ax y cos si +, the dy a + x a + x dx is equal to a ) x + a a ) x + a 4a 3) x + a 4) 4a x + a Sol. a x ax y cos si + a + x a + x Put x a ta θ x θ ta y cos ( cos θ) + si ( si θ) y 4θ a x dy 4a y 4 ta a dx a + x 45. If f ( x) si x + cos x the f π ( iv) π f 4 4 is equal to ) ) 3) 3 4) 4 Sol. f ( x) si x + cos x, f ' x cos x si x f '' x si x cos x So, f π π π π f '''' si + cos

18 + The, π π f f '''' f If y si ( msi x), the x y xy is equal to (Here, y deotes d y dx ) m y ) Sol. y si ( msi x) m y 3) m y 4) m y y cos msi x. m. x where y dy dx y x + y.( x) y x m cos m si x m d y msi msi x. y + x dx m si ( msi x) + x ( ) x y x xy m y 47. If 4 4 x y u si +, the x + y u u x + y is equal to δ x y ) 3u ) 4u 3) 3siu 4) 3tau Sol. 4 4 x y u si + x + y Let v si u 4 4 x + y x + y, here degree is homogeeous, so 4 3 By Euler s theorem,

19 v v v v x ( si u) + y ( si u) 3si u, xcosu + y cosu 3si u x y x y v v x + y 3ta u x y 48. If f ( x) logloglog...log x (log is repeated times), the ( xf ) x f x... f x f ( x) f+ x + c ) ) + c f + ( x) + + c 3) f ( x) + c 4) f x x (upto terms) Sol. log.log.log...log f x log x log log log...log (upto ( ) f x x times)... xf x f x f x dx Now, dx xf ( x) f ( x)... f ( x)... xf x f x f x dt dt xf x f x... f x. t t If ( cos x ) cos ec x dx f ( x ) + c, the f x is equal to ) ta x ) cot x 3) ta x 4) x ta cos x cosec x dx Sol.

20 x x si. dx cos x si si x x x sec dx ta. + c x x ta + c f x + c f ( x) ta 50. If I π 4 ta x dx, the I + I 4, I 3 + I 5, I 4 + I , are i o progressio ) Arithmetic progressio ) geometric progressio 3) Harmoic progressio 4) arithmetic geometric Sol. I π 4 o ta r + dx We have, Ir + Ir+ r + i.e, I + I4 3 I I + I I Which are clearly i HP 5. If a straight lie L is perpedicular to the lie 4x y ad forms a triagle of area 4 square uits with the coordiate axes, the the equatio of the lie L is ) x + 4y ) x 4y ) x + 4y ) 4x y 8 0 As. 3

21 c λ Sol: Perpedicular to the lie 4x y is x + y + λ 0 ad area is 4 ab 4 λ 4 x + y + 4 x + 4y The image of the poit ( 4, 3) with respect to the lie 5x + y is ) (, 4) ) ( 3,4) 3) (, ) 4) ( 4,3) As. h x k y Sol: Usig formula a b ( ) + ( ) a h x b k y a + b 53. The image of the lie x + y 0 i the y axis is ) x y + 0 ) y x + 0 3) x + y + 0 4) x + y 0 As. Sol :Itercepts of give lie are A(,0),B(0,-) thus w.r.t y-axes A (-,0) ad B(0,-) Thus by usig two poit formula image is x y The distace betwee the two lies represeted by is 8x 4xy + 8y 6x + 9y 5 0 ) 0 ) ) 6 3 4) 7 3 As. 4 g - ac Sol: distace betwee the two lies a a + b

22 55. A pair of perpedicular lies passes through the origi ad also through the poits of itersectio of the curve x + y 4 with x + y a, where a > 0. The a is equal to As. ) ) 3 3) 4 4) 5 Sol: Solvig two equatios x + ( a x) 4 a ± 8 a Gives x as poit of itersectio is real umber 8 a 0 a thus a 56. If the agle θ is acute, the the acute agle betwee the pair of straight lies x (cosθ si θ ) + xy cosθ + y (cosθ + siθ ) 0 is ) θ ) 3θ 3) 0 4) θ As. 4 Sol : If α is the agle betwee the lies the Cos α cosθ siθ + cosθ + siθ (cos si cos si ) + 4 cos θ θ θ θ θ cos θ cos θ α θ. 57. If the slope of oe of the lies is twice the slope of the other i the pair of straight lies + hxy + by 0 the 8h ) ab ) 3ab 3) 4ab 4) 9ab ax As. 4

23 Sol: Let the slopes be m, m Their ratio is : Required coditio is 8h 9ab ab( m) + 4h lm ab + 4h () () 58 The coditio that oe of the lies coordiate axes i ax + hxy + by 0 will bisect the agle betwee the ) 4h ) 3 4h 3) 4h 4) 4h As. 3 Sol : Equatio of the agle bisector of the coordiate axes are y ± x. ax + hxy + a + b ± h by 0 ( a b) ax + hx ( ± x) + b(±x) 0 (a + b) + 4h. x ( ± h) x 59 The area (i square uits) of the triagle formed by the lies x 0, y 0 ad 3x + y 7 is As. 4 Sol : Area of the t) ) 3 3) 4 4) 49 riagle (7) 3 49 square uit. 60 The area of the triagle formed by the axes ad the lie (coshα sih α ) x + (coshα sih α ) y i sq. uit, is ) ) 3 3) 4 4) 5 As.

24 ( ) Sol : Area of the triagle (coshα sih α)(coshα + sih α) ( ) cosh α sih α 6. If A ad B are square matrices of the same order ad A is o-sigular the for a positive - iteger, ( A BA) a) is equal to - A B A b) - A B A c) - - A B A d) ( A BA) HINT: If Geerally Hece 3rd optios is correct 6. If x x x y y y z z z ( x y)( y z)( z x) + + x y z the value of is a) - b) - c) d) HINT: Order of determiat is Order of R.H.S +- Hece st optios is correct

25 63. If f (x) ta x ad A, B, C are the agles of ABC, the f ( A) f ( π / 4) f ( π / 4) f ( π / 4) f ( B) f ( π / 4) f ( π / 4) f ( π / 4) f ( C) a) 0 b) - c) d) HINT: Hece 3rd optios is correct If α, β, γ are the roots of equals + + 0, where q 0, the 3 x px q / α / β / γ / β / γ / α / γ / α / β a) -p/q b) /q c) p /q d) 0 HINT:. R + R + R3 α β γ β γ α γ α β β γ α γ α β

26 Hece 4th optios is correct The characteristic roots of the matrix A is/are give by a), 4 b), 4,7 c), 4, 7 d) Noe of these HINT: Characteristic equatio of A is A λi 0. Hece d optios is correct 66. If a upto terms b upto terms c upto terms The a b 4c 3 5 a) ( 30) b) 0 c) 0 d) HINT: 3 5 R3 R two rows are equal 0 3 5

27 Hece 3rd optios is correct 67 Let a, b, c be positive real umbers. The followig system of equatios i x, y, z x y z x y z x y z has a b c a b c a b c ) No solutio ) uique solutio 3) Ifiitely may solutio 4) fiitely may solutio Sol. Let x y z X, Y, Z a b c X + Y Z, X Y Z, - X + Y + Z Determiat of coefficiet 0 Hece 4th optios is correct 68 If A is o-sigular ad (A-I) (A-4I) 0, the 4 A + A 6 3 a) I b) 0 c) I d) 6I HINT: Takig o both the sides Hece st optios is correct 69. The mea ad the variace of a biomial distributio are 4 ad respectively. The the probability of successes is ) 8 56 ) 56 3) 4 4) 8

28 Sol: I a B.D. : mea p 4. (), variace pq () () () : q 4 p 8 Probability of successes (B.D) 8 8 C A pair of fair dice is throw idepedetly three times. The probability of gettig a score of exactly 9 twice is ) 8/79 ) 8/43 3) /79 4) 8/9 Sol. Probability of gettig a score 9 whe a pair of dice is throw 4 P q 9 9 Required probability 3 C P A 7.It is give that the evets A ad B are such that B P A 3 The P(B), 4 P A B ad ) 6 ) 3 3) 3 4) A P( A B) P P( A B) P( B) B P Sol. ( B) () ( B) P( A) B P A P P( A B) P( A) A () P( B) From () ad () : P(B) Four umber are chose at radom from {,,3,..40}. The probability that they are ot cosecutive is]

29 ) 470 ) ) ) Sol. 4 cosecutive umbers are,,3,4;,3,4,5;.; 37,38,39, Probability for the 4 umbers to be cosecutive C If three six faced fair dice are throw together the the probability that the sum of the umber appearig o the dice is K K ( 3 K 8) is ( K )( K ) K ( K ) K K ) 43 ) 43 3) 43 4) 6 K Hit: Coefficiet of X k i (X+X.+X 6 ) 3 C 74. If four people are chose at Radom. The the probability that o two of them were ) fore o the same day of the week is ) Hit : ) ) A ma throw a die util he gets a umber bigger tha 3. The probability that he gets a 5 i the last throw 3 ) ) 3 3) 3 4) 5 P ( 5) 3 Hit: 6 P ( or or 3) O a toss a two dice. A throw a total of 5 the the probability that he will throw aother 5 before he throws 7 is 5 ) 9 ) 6 3) 5 4) P 5 36 P( 5 or 7) Hit:

30 P ( 7) a r I a Poisso o distributio variace is m. The sum of the term i odd places i this distributio m ) e m m m ) e 3) e cosh m 4) e si m Hit : P(x0)+P (x).. e m cosh m 78. If the mea of Biomial distributio with 9 Trials is 6 the its variace. ) ) 3 3) 4 4) Hit: p6; σ pq 79. I a book of 500 pages, it is foud that there are 50 typig errors. Assume that poisso law holds for the umber of errors per page. The the probability that a radom sample of pages will cotai o error is : ) Sol: 0.3 e ) 0 λ λ 0! e e 50 λ Here e 3) e 4) The probability that a radom sample of pages will cotai o error is e 80.I a polygo of sides has 75 diagoals the ) 0 ) 30 3) 4 4) 5 Hit: ( 3) 75 5

31 8. Number of ways of distictly 8 idetical balls m 3 distict boxes. So that oe of boxes is empty ) 0 ) 3) 4 4) 5 Hit: cr 8 c3 7c 8. The umber of ways of selectig 0 balls out of ulimited umber of white, red, gree ad blue balls. ) 00 ) 309 3) 40 4) 86 + r c Hit: r c4 3c The umber of positive divisor of 3 7 Hit: Positive divisor (3+)(6+)(+) 84 Proper divisor (3+)(6+)(+)- 8 Odd divisor (6+)(+) ) 0 ) 30 3) 4) The umber of three digit umbers havig oly two cosecutive digits idetical is a) 53 b) 6 c) 68 d) 63 Hit:9 x 9 x + 9 x x Give five lie segmets of legth,3,4,5,6 uits. The the umber of triagles that ca be formed by joiig these lies is a) 5 C3 3 b) 5 C3 c) 5 C3 d) 5 C3 Hit: 5c 3. 3 sice,3,5;,4,6;,3,6 does ot form a triagle 86. Let T deote the umber of triagles which ca be formed usig the vertices of a regular polygo of sides. If T + T, the equals a) 5 b) 7 c) 6 d) 4 Hit: c3 + thus 7 c3

32 87.I a - storey house te people eter a lift cabi. It is kow that they will leave i groups of, 3 ad 5 people at differet stories. The umber of ways they ca do so if the lift does ot stop upto the secod storey is a) 78 b) c) 70 d) 3. Hit: 0 p The sum of itegers from to 00 that are divisible by or 5 is a) 3000 b) 3050 c) 3600 d) 350 Hit: Required sum ( ) + ( ) ( ) The umber of distict ratioal umbers such that p,q {,,3,4,5,6} is p 0 < < ad, where q a) 5 b) 3 c) d) Hit: From 6 digits digits ca be selectio 6 C way. Out of these ways,, ways & they ca be arraged i oly oe 4 represet same umber, 3 6, represet same umber No. of umbers represet same umber 90. I a chess touramet, where the participats were to play oe game with aother. Two chess players fell ill, havig played 3 games each. If the total umber of games played is 84, the umber of participats at the begiig was a) 5 b) 6 c) 0 d) Hit:( ) the 5 C

3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B

3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B 1. If A ad B are acute positive agles satisfyig the equatio 3si A si B 1 ad 3si A si B 0, the A B (a) (b) (c) (d) 6. 3 si A + si B = 1 3si A 1 si B 3 si A = cosb Also 3 si A si B = 0 si B = 3 si A Now,