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5351.

Integrate the rational functions in exercise. (1)/(x^(4)-1)

Answer»


ANSWER :`(1)/(4)LOG|(x-1)/(x+1)|-(1)/(2)tan^(-1)x+c`
Discussion
5352.

Express the matrix A=[{:(2,3,4),(5,6,-2),(1,4,5):}]as the sum of a symmetric and a skew symmetric matrix .

Answer»


ANSWER :`A=[{:(2,4,(5)/(2)),(4,6,1),((5)/(2),1,5):}]+[{:(0,-1,(3)/(2)),(1,0,-3),(-(3)/(2),3,0):}]`
Discussion
5353.

Integrate the function is Exercise. (x^(2)+x+1)/((x+1)^(2)+(x+2))

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ANSWER :`-2log|x+1|-(1)/(x+1)+3log|x+2|+c`
Discussion
5354.

Evalute the following integrals int (1)/(2 + 3" sin x")dx

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Answer :`(1)/(SQRT(5)) " log "|(2 TAN((x)/(2)) + 3 - sqrt(5))/(2 tan ((x)/(2)) + 3 + sqrt(5))| + C `
Discussion
5355.

A random variable X has its range {0,1,2,3,……..}. If P(X=r)=(c(r+1))/(3^(r)) for r=0,1,2,………. Then c =

Answer»

2
`1//2`
`4//9`
`1//4`

ANSWER :C
Discussion
5356.

Let Delta(x)=|{:(cos^(2)x,cosxsinx,-sinx,),(cosx sinx,sin^(2)x,cosx,),(sinx,-cosx,0,):}|"then" int_(0)^(pi//2)[Delta(x)+Delta'(x)]dx equals

Answer»

`(pi)/(3)`
`(pi)/(2)`
`2PI`
`(3pi)/(2)`

Solution :f(n) = integer closest to `sqrt(n)`.
Let `f(n)=K,""kinN`
`implies k-(1)/(2)ltsqrt(n)LTK+(1)/(2)impliesk^(2)-k+(1)/(4)ltnltk^(2)+k+(1)/(4)impliesk^(2)-k+1lenlek^(2)+k""(therefore nin N)`
`therefore` Number of integers for which `f(n)=k" is "(k^(2)+k)-(k^(2)-k)=2k`
`{:(therefore,,f(n)=1" for",,n=1.2),(,,f(n)=2" for",,n=3.4.5.6),(,,f(n)=3" for",,n=7.8.9.10.11.12):}`
`therefore underset(2)ubrace((1)/(f(1))+(1)/(f(2)))+underset(2)ubrace((1)/(f(3))+(1)/(f(4))+(1)/(f(5))+(1)/(f(6)))+.......+(1)/(f(2015))+(1)/(f(2016))`
Sum of first 2 terms = 2
Sum of next 4 terms =2
Sum of next 6 terms = 2
Discussion
5357.

If n is a multiple of 3, then coefficient of x^(n) in log(1+x+x^(2)) is

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`-(2)/(N)`
`(2)/(n)`
`(1)/(n)`
`-(1)/(n)`

ANSWER :A
Discussion
5358.

Writethe set builder form A= {-1,1}

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A= {X: x is a root of the EQUATION `x^(2) + 1=0`}
A= {x: x is a root of the equation `x^(2)-1=0`}
A= {x : x is an integer}
A = {x : x is a REAL NUMBER}

Answer :B
Discussion
5359.

If lim_(xrarroo) xlog_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)=-5. where alpha, beta, gamma are finite real numbers, then

Answer»

`alpha=2, BETA=1, gamma in R`
`alpha=2, beta=2, gamma=5`
`alpha in R, beta =1, gamma in R`
`alpha in R, beta = 1, gamma=5`

Solution :Let, `L=UNDERSET(xrarroo)(lim)X.log_(e)(|(alpha//x,1,gamma),(0,1//x,beta),(1,0,1//x)|)`
`=underset(xrarroo)(lim)x.log_(e)((alpha)/(x^(3))+beta-(gamma)/(x))`
For LIMIT to EXIST, `beta=1`
`therefore""L=underset(xrarroo)(lim)x.log_(e)(2+(alpha)/(x^(3))-(gamma)/(x))`
`=underset(xrarroo)(lim)x((alpha)/(x^(3))-(gamma)/(x))`
`=underset(xrarroo)(lim)((alpha)/(x^(2))-gamma)=-gamma=-5 and alpha in R`
Discussion
5360.

By approximatly what percent did the number of questions decrease in CET 1994 over the previous year?

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0.16
0.19
0.35
0.4

Answer :A
Discussion
5361.

The locus of the point z=x+iy satisfying |(z-2i)/(z+2i)|=1 is

Answer»

X-axis
y-axis
y = 2
x = 2

ANSWER :A
Discussion
5362.

Two unbiased dice are thrown. The probability that the sum of the numbers appearing on the top face of two dice is greater than 7, if 4 appear on the top face of first dice is ………..

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`(1)/(3)`
`(1)/(2)`
`(1)/(12)`
`(1)/(6)`

ANSWER :B
Discussion
5363.

int_(0)^(1) ((tan^(-1)x)^(3))/(1+x^(2))dx=

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`1/4 (pi/12)^(6)`
`1/4 (pi/12)^(4)`
`1/4 ((pi/4)^(4) - (pi/6)^(4))`
`(pi/8)^(4)`

ANSWER :C
Discussion
5364.

{:(,"Hyperbola point ",,"Tangent"),(I,3x^(2) -4y ^(2) =12","3x+2y +6=0, ,(a) ("2,-3")),(II,x^(2) -3y^(2) =3","2x+ y-1 =0,,(b) ("-1,-3")),(III,3x^(2) -4y^(2) =8","(2","-1) ,,(c) x+2y-4=0),(IV,9x^(2) -16y^(2) =144","3x-16y + 48 =0,,(d) ("6,-1")):}

Answer»

a,B,C
b,c,a
c,b,a
a,c,b

Answer :B
Discussion
5365.

(i) A manufacturer produces nuts and bolts for industrial machinery.It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts while it takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts.He earns a profit og ₹ 17.50 per package of nuts and ₹ 7.00 per package of bolts.How many packages of each should be produced each day so as to maiximise his profit, if he operate his machine for at the most 12 hous a day?Form an LPP for the problem and solve it graphically. (ii) A manufacturer produces nuts and bolts.It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on amchine A and 1 hour on machine B to produce a package of bolts.He earns a profit ₹35 per package of nuts and ₹14 per package of bolts.How many packages of each should be produced each day so as to maximise his profit, if he operates each machine for atmost 12 hours a day? Convert it into an LPP and solve graphically.

Answer»


ANSWER :(i) -(ii) 3 PACKAGES each of NUTS and BOLTS daily.
Discussion
5366.

ABCD is a square of side 'a'. If AB and AD are taken as co-ordinate axes, prove that the equation of the circle circumscribing the square is x^2+y^2=a(x+y).

Answer»

SOLUTION :`THEREFORE` CENTRE of the circle is at (a/2, a/2) and radius is `(asqrt2)/2=a/sqrt2`
`therefore` Equation of the circle is
`(x-a/2)^2+(y-a/2)^2=a^2/2`
or, `x^2+a^2/4-ax+y^2+a^2/4-ay=a^2/2`
or, `x^2+y^2-ax-ay=0`
or, `x^2+y^2=a(x+y)`
Discussion
5367.

After beating Donkey Kong, Mario was informed by his friend Toad the Mushroom Guy that Bowser, the evil dragon who has kidnapped Princess Peach, uses a different lan- guage to communicate which is based on the English alphabets in matrices, and that the princess was not in the castle he was in! So, he decided to use the warp tunnel and left for the same. On reaching the warp zone, he saw the below pattern on the tunnel and real- ized that Bowser had put it there to prevent him from entering the warp zone. Figure the one important word which the sentence implied by this pattern on the Warp tunnel says to help Mario get out of the castle through the warp.Noticing his dilemma, Toad hinted that it is very well versed with the word.eodC dorW +*-!Write down the 2nd letter of the word.

Answer»

S
H
I
V

Solution : The key is that there are 26 COLUMNS which can be considered as the 26 alphabets,where, each word FITS in each ROW and each column corresponds to an alphabet in theorder.
eodC dorW `+*­!` , when descrambled gives us, Code Word `!*­+`So, when one follows the pattern of !*­+ in each row, it literally translates to:Wise is the code word. 
HENCE, the Answer is option: C)I
Discussion
5368.

Matrix A=[(a,-b,0),(c,0,b),(0,-c,-a)] then :

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`A^9=(a^2-2bc)^3A`
`A^9=(a^2-2bc)^4A`
`|A^2+(2bc-a^2)I_3|=0`
`|A^2+(a^2-2bc)I_3|=0`

ANSWER :B::C
Discussion
5369.

If 24 cos^(2)x+8sin 2x cos x +18 cos x+6 sin 2x=15 +10 sin x then general solutionof x is

Answer»

`60^(@)`
`90^(@)`
`120^(@)`
`135^(@)`

Answer :b
Discussion
5370.

Three vertices of a convex n sided polygon are selected. If the number of triangles that can be constructed such that none of the sides of the triangles is also the side of the polygon is 30, then the polygon is a

Answer»

Heptagon
Octagon
Nonagon
Decagon

Answer :C
Discussion
5371.

If w(ne 1) is a cube root of unity and (1+w^2)^n = (1+w^4)^n then find the least positive integral value of n

Answer»


ANSWER :`implies` N=…....-6,-3,0,3,6,…..
LEAST POSITIVE integral value of n is 3
Discussion
5372.

If y= sin x^(o)," find "(dy)/(dx).

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ANSWER :`(PI)/(180^(@))COS X^(@)`.
Discussion
5373.

int_(-1)^(1)(x^(27)Cos x + e^(x))dx=

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0
1
`E +1/e`
`e -1/e`

ANSWER :D
Discussion
5374.

if f(x)=(x^(3)-3x^(2)+3x-x^(2)l n(e^((1)/(x^(2)))(x^(2)-x+2)))/(l n(x^(2)-x+2)) then f^('')(1) is equal to

Answer»

`(2)/((l N2)^(2))`
`-2`
`(-2)/((l n2)^(2))`
`(1-2l n2)/((l n2)^(4))`

SOLUTION :`F(x) (x^(3)-3x^(2)+3x-1-x^(2)l N (x^(2)-x+2))/(l n (x^(2)-x+2))`
`=((x-1)^(3))/(ln(x^92)-x+2)`
`thereforef^('')(1)=-2`
Discussion
5375.

Find the angle between the lines vecr = (hati+hatj)+lambda(3hati+2hatj+6hatk) and vecr = (hati-hatk) +mu(hati+2hatj+2hatk)

Answer»

Solution :Here `VECM = 3hati+2hatj+6hatk` and `vecn=hati+2hatj+2hatk`
Let ` theta` be the ANGLE between TWOLINES .
`:. cos theta = (vecm.vecn)/(|vecm||vecn|) = ((3hati+2hatj+6hatk).(hati+2hatj+2hatk))/(|3hati+2hatj+6hatk|| hati+2hatj+2hatk|)`
`= (3+4+12)/(sqrt(9+4+36)sqrt(1+4+4)) = (19)/(7xx3) = (19)/(21)`
`theta = cos^(-1) (19/21)`
Discussion
5376.

Fill in the blank choosing correct answer from the brackets cot^(-1)[(sqrt(1-sin x ) +sqrt(1+sinx))/(sqrt(1-sin x)- (sqrt 1+ sin x))]=_______(2pi -x/2,x/2,pi- x/2)

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SOLUTION :`pi-x/2`
Discussion
5377.

If the lines of regreasion of y on x and x on y makes angle 30 ^(@) and 60^(@)respectively with positive direction of x - axis the correlationcoefficient between x on y is

Answer»

` +-(1)/(3)`
` SQRT3 `
` +-(1)/(sqrt3) `
` ( 1)/(sqrt3)`

ANSWER :D
Discussion
5378.

Given sum of first n terms of an A.P. is 2n + 3n^(2). rel then Another A.P. is formed with the same first term and double the common difference. Let Sn) denote the 98. sum to n terms of new A.P., then S(10)=

Answer»


ANSWER :590
Discussion
5379.

Let A_(n) = a_(1) + a_(2) + "……" + a_(n), B_(n) = b_(1) + b_(2) + b_(3) + "…." + b_(n), D_(n) = c_(1) + c_(2) + "….." + c_(n) and c_(n) = a_(1)b_(n) + a_(2)b_(n-1) + "……." + a_(n)b_(1)Aan in N. Using mathematical induction , prove that (a) D_(n) = a_(1)B_(n) + a_(2)B_(n-1) + "....."+a_(n)B_(1) = b_(1)A_(n) + b_(2)A_(n-1) + "......"+b_(n)A_(1) AA n in N (b) D_(1) + D_(2) + "......"+ D_(n) = A_(1)B_(n) + A_(2)B_(n-1) + "....." + A_(n)B_(1) AA n in N

Answer»

Solution :Let ` P(n) : D_(n) a_(n)B_(n) + a_(n)B_(n-1)+"……."+a_(n)B_(1)`
For `n = 1,P(1) = D_(1) = a_(1)B_(1) = a_(2)b_(1) = c_(1)`
Thus, ` P(1)` is true.
Let`P(n)`be true for `n = K`
i.e., `P(k) = D_(k) = a_(1)B_(k)= a_(1)B_(k) + a_(2)B_(k-1) + "....."+ a_(k)B_(1)`
Now, `D_(k+1) = D_(k) + c_(k+1)`
`= a_(1)B_(k) + a_(2)B_(k-1) + "......"+a_(k)B_(1) +a_(1)b_(k+1) + a_(2)b_(k) + "....."+a_(k+1)b_(1)`
`=a_(1)(B_(k)+b_(k+1))+ a_(2)(B_(k-1)+b_(k)) + "......" + a_(k)(B_(1) + b_(2)) + a_(k+1)b_(1)` ltbrlt `= a_(1)B_(k+1)+a_(2)B_(2)+"......."+a_(k)B_(2) +a_(k+1)B_(1)`.
Thus, `P(k+1)` is ALSO truewhenever`P(k)` is true.
So, by the principle of mathematical induction `P(n)` is true
for any NATURAL number n.
Let `P_(1)(n) : D_(n)= b_(1)A_(n) + b_(2)A_(n-1)+"......."+b_(n)A_(1)`
For `n = 1,P_(1)(1) = D_(1) = b_(1)A_(1) = b_(1)a_(1) = c_(1)`
Thus, `P_(1)(1)` is true.
Let, `P_(1)(n)` be true for `n= k`.
i.e, `P_(1)(k) = b_(1)A_(k) + b_(2)A_(k-1) + "....." + b_(k)A_(1)`
Now, `D_(k+1) = D_(k) +c_(k+1)`
`= b_(1)A_(k) + b_(2)a_(k-1) + "......." + b_(k)A_(1) + a_(1)b_(k+1) + a_(2)b_(k) + "....." + a_(k+1)b_(1)`
`= b_(1)(A_(k) + a_(k+1))b_(2)(A_(k-1)+a_(k))+"......"+b_(k)(A_(1) + a_(2)) + b_(k+1)a_(1)`
`= b_(1)A_(k+1) + b_(2)A_(k) + "......"+b_(k)A_(2) + b_(k+1)A_(1)`
Thus, `P_(1)(k+1)` is alsotrue.
So, by the principle of mathematical induction `P_(1)(n)` is TRUEFOR any natural number n.
(b) Say,`P_(2)(n) = D_(1) + D_(2) + "......"+D_(n) = A_(1)B_(n) + A_(2)B_(n-1) + "......" + A_(n)B_(2)`
For `n = 1, P_(2)(1) = D_(1) = A_(1)B_(1) = a_(1)b_(1) = c_(1)`
Thus, `P_(2)(1)` is true .
Let `P_(2)(n)` be true for `n = k`
i.e, ` P_(2)(k) = A_(1)B_(k) + A_(2)B_(k+1) + "......" + A_(k)B_(1)`
Now, `P_(2)(k+1)`
`= P_(2)(k) + D_(k+1)`
`= A_(1)B_(k) + A_(2)B_(k-1) +"........"+)A_(k)B_(1) +b_(1)A_(k+1) + b_(2)A_(k) + "......" + b_(k)A_(2) + b_(k+1)A_(1)`
`= A_(1)(B_(k) + b_(k+1)) + A_(2)(B_(k-1)+b_(k)) +"....." + A_(k)(B_(1) + b_(2)) + A_(k+1)b_(1)`
` = A_(1)B_(k+1) + A_(2)B_(k) + "....." + A_(k)B_(2) + A_(k+1)B_(1)`
Thus, `P_(2)(k+1)` is also true.
So, by the principle of MATHEMATICALINDUCTION, `P_(2)(n)` is true for any natural number n.
Discussion
5380.

If the roots of the equation x^(2)+x+a=0 exceed a,then

Answer»

Option1`a GT 2`
OPTION2`a lt -2`
OPTION3`2 lt a lt 3`
Option4`-2 lt a lt -1`

ANSWER :B
Discussion
5381.

If P_(1), P_(2), P_(3) are the perimeters of the three circles x^(2) + y^(2) + 8x - 6y = 0, 4x^(2) + 4y^(2) - 4x - 12y - 186 =0 and x^(2) + y^(2) - 6x + 6y - 9 = 0 respectively, then

Answer»

`P_1 LT P_2 lt P_3`
`P_1 lt P_3 lt P_2`
`P_3 lt P_2 lt P_1`
`P_2 lt P_3 lt P_1`

ANSWER :C
Discussion
5382.

Let C be the locus of the circumcentre of a variable traingle having sides Y-axis ,y=2 and ax+by=1, where (a,b) lies on the parabola y^2=4lamdax. For lamda=2 , the product of coordinates of the vertex of the curve C is

Answer»

-8
-6
6
8

Answer :B
Discussion
5383.

If A_(j) = ( x-a_(j))/( | x- a_(j)|) , j = 1,2,"........" nand a_(1) lt a_(2) lt a_(3)lt"....."lt a_(n)underset( x rarr a_(m)) ( "lim") (A_(1).A_(2)."........" A_(n)), 1 le m le n

Answer»

is EQUAL to `( -1)^(N -m+1)`
is equal to `( -1) ^(n-m)`
is equal to `( -1)^(m)`
does not exist

Answer :D
Discussion
5384.

Match the following for the system of linear equations lambda x+ y+z= 1x + lambda y+ z=lambdax+y +lambda z=lambda ^(2) (##FIITJEE_MAT_MB_07_C02_E04_001_Q01.png" width="80%">

Answer»


ANSWER :A::B::C::D
Discussion
5385.

If n= 10, sum_(i=1)^(10) x_(i) = 60 and sum_(i=1)^(10) x_(i)^(2) = 1000 then find s.d.

Answer»


ANSWER :8
Discussion
5386.

Integrate the function (3x)/(1+2x^(4))

Answer»


Answer :`3/(2sqrt2)TAN^(-1)sqrt2x^(2)+C`
Discussion
5387.

Find the equation of a circle which passes through (4,1) (6,5) and having the centre on 4x + 3y - 24 = 0

Answer»


ANSWER :`X^(2) +y^(2) -6X -8Y +15=0 `
Discussion
5388.

Using elementry transformation, find the inverse of the matrices. A = [(4,5),(3,4)]

Answer»

SOLUTION :`A^(-1) = [(4,-5),(-3,4)]`
Discussion
5389.

Find the Cartesian equation of the line parallel to y - axis and passing through the point (1,1,1).

Answer»


ANSWER :`=(z-1)/( 0)`
Discussion
5390.

Protective components of food are :-

Answer»

MINERALS, VITAMINS, and WATER
Minerals, CARBOHYDRATE and proteins
Minerals, carbohydrate and fat
Vitamins, water and carbohydrates

Answer :A
Discussion
5391.

The point on the curve y^(2)=x, where the tangent makes an angle of (pi)/(4) with x-axis is :

Answer»

`((1)/(2),(1)/(4))`
`((1)/(4),(1)/(2))`
(4, 2)
(1, 1)

ANSWER :B
Discussion
5392.

y=x is tangent to the ellipse whose foci are (1, 0) and (3, 0) then:

Answer»

MAJOR axis of ellipse is `sqrt(6)`
Major axis of ellipse is `sqrt(10)`
`((3)/(4),(3)/(4))`is the point of CONTACT of this ellipse and its TANGENT
`((1)/(2),(1)/(2))`is the point of contact of this ellipse and its tangent

Answer :B::C
Discussion
5393.

(.^(n)C_(0))^(2)+(.^(n)C_(1))^(2)+(.^(n)C_(2))^(2)+ . . .+(.^(n)C_(n))^(2) equals

Answer»

`.^(2n)C_(N)`
`((2n-1)!)/(((n-1)!)^(2))((2)/(n))`
`(n!)^(2)`
`(.^(.2n)C_(n))^(2)`

Answer :A::B
Discussion
5394.

Resolve into Partial Fractions (v) (2x^(3)+3)/(x^(2)-5x+6)

Answer»


ANSWER :`2x+10 - (19)/((x-2))+(57)/((x-3))`
Discussion
5395.

If sinx+siny=a and cosx+cosy=b then find the values of "tan"(x+y)/2,cot (x+y) and "sin"(x-y)/2 interms of a and b (given a!=0 !=b)

Answer»


ANSWER :`a//B,(b^(2)-a^(2))/(2AB)`,+-SQRT(4-a^(2)-b^(2))/2`
Discussion
5396.

Larger of 99^50+100^50 and 101^50

Answer»

`101^50`
`99^50 + 100^50`
Both are EQUAL
can not be DECIDED

ANSWER :A
Discussion
5397.

AD is a median of the Delta ABC.If AE are medians of the Delta ABD and DeltaADC respectively, and AD=m_(1), AE =m_(2),AF =m_(3), then find the value of (a^(2))/(8).

Answer»


ANSWER :`m_(2)^(2)+ m_(3)^(2) -2m_(1)^(2)`
Discussion
5398.

The value of sum_(k=0)^(7)[(((7),(k)))/(((14),(k)))sum_(r=k)^(14)((r ),(k))((14),(r ))], where ((n),(r )) denotes "^(n)C_(r ) is

Answer»

`6^(7)`
greater than `7^(6)`
`8^(7)`
greater than `7^(8)`

Solution :`(a,b)` `sum_(k=0)^(7)(('^(7)C_(k))/('^(14)C_(k))sum_(r=k)^(14)'^(r )C_(k)*^(14)C_(r ))`
`=sum_(k=0)^(7)(('^(7)C_(k))/(14!)xxk!(14-k)!sum_(r=k)^(14)(r !)/(k!(r-k)!)*(14!)/(r!(14-r)!))`
`=sum_(k=0)^(7)('^(7)C_(k)sum_(r=k)^(14)'^(14-k)C_(r-k))`
`=sum_(k=0)^(7)'^(7)C_(k)*2^(14-k)=2^(14)sum_(k=0)^(7)'^(7)C_(k)((1)/(2))^(k)`
`=2^(14)*(1+(1)/(2))^(7)=6^(7) gt 7^(6)`
Discussion
5399.

Let f(x) bea polynomialfor whichthe remainderswhendividedbyx-1, x-2 , x-3respectivelyare 3,7,13thenthe remainderof f(x)whendividedby(x-1) (x-2)(x-3)is

Answer»

`F(X)`
`x^2+X+1`
`x^2 +1`
none

Answer :B
Discussion
5400.

int sec^(5)xdx.

Answer»


ANSWER :`(SEC^(3)xtanx)/(4)+(3)/(8)secxtanx+(3)/(8)log|secx+tanx|+c`
Discussion