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1501.

Evaluate (i) int((2x+5))/((x^(2)+5x+9))dx (ii) int((6x-7))/((3x^(2)-7x+5))dx (iii) int((cosx-sinx))/((cosx+sinx))dx (iv) int(sec x)/(log(secx+tanx))dx

Answer»

Solution :(i) PUT `(x^(2)+5x+9)` =t so that (2x+5)dx=DT.
`:.int((2x+5))/((x^(2)+5x+9))dx=int(1)/(t)dt=log|t|+C`
`=log|{:x^(2)+5c+9:}|+C`.
(ii) Put (cosx+sinx)=t so that (cosx-sinx)dx=dt.
`:.int((cosx-sinx))/((cosx+sinx))dx=int(1)/(t)dt`
`=log|t|+C=log|{:(cosx+sinx):}|+C`.
(iii) Put (cosx-sinx) =t so that (cosx-sinx) dx =dt.
`:.int((cosx-sinx))/((cosx+sinx))dx=int(1)/(t)dt`
`=log|t|+C=log|{:(cosx+sinx):}|+C`
(iv) Put log(SECX+tanx)=t.
Then, on DIFFERENTIATION, we get
`(1)/((secx+tanx))*(secxtanx+sec^(2)x)dx=dt`
or sec x dx = dt.
`:.int(secx)/(log(secx+tanx))dx=int(1)/(t)dt`
`=log|t|+C=log|{:(secx+tanx):}|+C`.
Discussion
1502.

If the parabols y^(2) = 4kx (k gt 0) and y^(2) = 4 (x-1) do not have a common normal other than the axis of parabola, then k in

Answer»

`(0,1)`
`(2,OO)`
`(3,oo)`
`(0,oo)`

Solution :If the parabolas have a common normal of slope `m(m ne 0)` then it is given by
`y = mx - 2KM -KM^(3)`
and `y = m(x-1) -2m -m^(3) = mx -3m -m^(3)`
`rArr 2km + km^(3) = 3m + m^(3)`
`rArr m = 0, m^(2) =(3-2k)/(k-1)`.
If `m^(2) lt 0` then the only common normal is the axis.
`rArr (3-2k)/(k-1) lt 0`
`rArr (k-1) (2k-3) GT 0`
`k gt (3)/(2)` or `k lt 1` and `k gt 0`
Discussion
1503.

Match the conditions/expression in Column I With statement in Column II. Normals at P,Q,R are drawn to y^(2) = 4x which intersect at (3,0). Then, {:("column I","column II"),(A."Area of" Delta PQR,p.2),(B. "Radius of circumcircle of" Delta PQR,q.(5)/(2)),(C. "Centroid of" Delta PQR,r. ((5)/(2),0)),(D. "Circumcentre of" Delta PQR,s. ((2)/(3),0)):}

Answer»


Solution :since, EQUATION of normal to the parabola `y^(2) = 4ax` is `y + xt = 2at + at^(3)` passes through (3,0)
`implies 3t = 2t + t^(2)`
`implies t = 0, 1 -1`
`:. ` Coordinates of the NORMALS are `P(1,2), Q(0,0) R(1,-2)`
Thus,
A. AREA of `Delta PQR = (1)/(2) XX 1 xx 4 = 2`
C. Centriod of `Delta PQR = ((2)/(3), 0)`
Equation of circle passing through P,Q,R is
(x - 1)(x -1) + (y - 2) (y + 2) + `lambda` (x - 1) = 0
`implies 1 - 4 - lambda = 0`
`implies lambda = - 3`
`:.` Required equation of circle is
`x^(2) + y^(2) - 5x = 0`
`:.` Centre `((5)/(2) , 0)` and radius `(5)/(2)`.
Discussion
1504.

The real number k for which the equation, 2x^(3)+3x+k=0 has two distinct real roots in [0,1]

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LIES between 1 and 2
liesbetween 2 and 3
lies between -1 and 3
doesnot exist

ANSWER :D
Discussion
1505.

int_((pi)/(6))^((pi)/(3))(sec^(4)x)/("cosec"^(4) x + sec^(4)x) dx

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ANSWER :`(PI)/(12)`
Discussion
1506.

In Delta ABC, if r r_1=r_2 r_3, then the triangle is

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only I is true
only II is true
both I, II are true
neither I nor II is true

Answer :D
Discussion
1507.

Integration of some particular functions : int(dx)/(sqrt(25-9x^(2)))=....+c

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`SIN^(-1)((3x)/(5))`
`(1)/(5)sin^(-1)((3x)/(5))`
`(1)/(3)sin^(-1)((3x)/(5))`
`log|25-9x^(2)|`

ANSWER :C
Discussion
1508.

Integration of some particular functions : int(dx)/(sqrt(25+9x^(2)))=......+c

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`sin^(-1)((3x)/(5))`
`(1)/(3)sin^(-1)((3x)/(5))`
`(1)/(3)log|3x- SQRT(25+9x^(2))|+c`
`(1)/(3)log|3x- sqrt(25+9x^(2)+25)|+c`

Answer :D
Discussion
1509.

If 2x-3y=14 and 5x+3y=21, what is the value of x ?

Answer»

`-1`
0
`7/3`
5

Solution :This system is already set up perfectly to solve using combination because the y terms `(-3 y and 3Y)` are OPPOSITES. Add the two equation to CANCEL `-3yand 3y.` Then solve the resulting equation for x:
`2x -3y=14`
`(+5x +3y =21)/({:(7x, =53),(x, =5):})`
Choice (D) is correct. The equation asks only for the value of x, so you don't need to substitute x back into EITHER equation to find the balue of y.
Discussion
1510.

In a Binomial disribution, the probability of getting success is 1/4 and the standard deviation is 3 . Then its mean is

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6
8
10
12

Answer :D
Discussion
1511.

Evalute the following integrals int (" tan x")/(1 + cos^(2) x)dx

Answer»


ANSWER :` log | (sqrt(1+cos^2 X))/cos x | +C`
Discussion
1512.

sin(xy)+x/y=x^2-y

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ANSWER :`(2xy^2-y-y^3cos(XY))/(xy^2cos(xy)-x+y^2)`
Discussion
1513.

[d b c] a + [a d c] b + [a b c] c =

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[a B C]
[a b c] d
d
0

Answer :B
Discussion
1514.

The numberof waysofdistributing8 indenticalballsin3distinctboxesso thatnoneof theboxesemptyis

Answer»

5
`""^(8)C_(3)`
`3^(8)`
21

Answer :D
Discussion
1515.

The odds against an event are 5 to 2 and the odds in favour of another disjoint event are 3 to 5. Then the probability that one atleast of the event will happen is

Answer»

`(29)/(30)`
`(49)/(50)`
`(17)/(50)`
`(37)/(56)`

ANSWER :D
Discussion
1516.

If the average of the first n numbers in the sequence 148, 146, 144,….., is 125, then n =

Answer»

18
24
30
36

Answer :B
Discussion
1517.

If n is a positive integer, then |(n!,(n+1)!,(n+2)!)),((n+1)!,(n+2)!,(n+3)!)),((n+2)!,(n+3)!,(n+4)!))|=

Answer»

`2N!(n+1)!`
`2n!(n+1!(n+2)!`
`2n!(n+3)!`
`2(n+1)!(n+2)!(n+3)!`

ANSWER :B
Discussion
1518.

The foci of the ellips(x^(2))/( 16) +(y^(2))/( b^(2) ) =1and the hyperbola(x^(2))/( 144) -( y^(2))/( 81) =(1)/(25)coincide ,then the value ofb^(2)is

Answer»

only I is TRUE
only II is true
both I and II are ture
NEITHER I nor II true

ANSWER :C
Discussion
1519.

int (1)/(1+cosx)dx=

Answer»


ANSWER :`-cotx+cosecx+c`
Discussion
1520.

Find the area of the region bounded by the curves y= e^(x),y= e^(-x) and the line x=1

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ANSWER :`E+ (1)/(e )-2`
Discussion
1521.

Find the area of the parallelogram whose adjacent sides are determined by the vectors. veca = hati + hatj + 3hatk and vecb = 2hati + 7hatj + hatk

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ANSWER :`=3sqrt(50)` SQ. UNITS.
Discussion
1522.

The solution of (dx)/(dy) - (2)/(3)xy = x^(4)y^(3) is

Answer»

`(1)/(x^(3)) = (3)/(2)(1-y^(2)) + C e^(-y^(2))`
`(1)/(x^(3)) = (3)/(2)(1+y^(2)) + c e^(-y^(2))`
`(1)/(x^(4)) = (3)/(2)(1-y^(2)) + c e^(y^(2))`
`(1)/(x^(4)) = (3)/(2)(1+y^(2)) + c e^(y^(2))`

ANSWER :A
Discussion
1523.

The position vectors of A and B are 3hati - hatj +7hatk and 4hati-3hatj-hatk. Find the magnitude and direction cosines of vec(AB).

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SOLUTION :N/A
Discussion
1524.

A company produces 10000 items per day. On a particular day 2500 items were produced on machine A, 3500 on machine B and 4000 on machine C. The probability that an item produced by the machines A, B, C to be defective is respectively 2%,3% and 5%. If one item is selected at random from the output and is found to be defective, then the probability that it was produced machine C, is

Answer»

`(10)/(71)`
`(16)/(71)`
`(40)/(71)`
`(21)/(71)`

Answer :C
Discussion
1525.

Let P(n) be the statement n^3+n is 3m such that m is a positive integer, then which of the following is true?

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<P>`P(1)`
`p(2)`
`P(3)`
`P(4)`

Answer :C
Discussion
1526.

The value of ""^(n)P_(1)+(""^(n)P_(2))/(2!)+(""^(n)P_(3))/(3!)+......+(""^(n)P_(n))/(n!)

Answer»

`2^(N)`
`2^(n)+1`
`2^(n)-1`
`2^(n-1)`

ANSWER :C
Discussion
1527.

There are 4 machines out of which 2 are defective. They are tested one by one at random till both the defective machines are identified. The probability that only two tests are needed for this is

Answer»

`(1)/(2)`
`(1)/(3)`
`(1)/(4)`
`(1)/(6)`

ANSWER :B
Discussion
1528.

Let veca = 2hati+hatj, vecb = -hati+3hatj+hatk and vecc = hati+2hatj+5hatk be three vectors. Find (veca-2vecb)xxvecc

Answer»

SOLUTION :
= `HATI(-25+4)-HATJ(20+2)+HATK(8+5)`
= `-21hati-22hatj+13hatk`.
Discussion
1529.

The value ofint _(0)^(pi//2) sin ^(8) x dxis

Answer»

`(105pi)/(32(4!))`
` (105pi)/(14(4!))`
` (105)/(16pi(4!))`
None of these

Solution :Let `l=int_(0)^(PI//2) SIN^(8) x dx = (7.5.3.1)/(8.6.4.2.2).(pi)/(2)`[By Walli's FORMULA]
`=(105 pi)/(32(4.3.2.1))=(105 pi)/(32.4!)`
Discussion
1530.

If |x|lt 1, tnen the coefficient of x^(5) in the expansion of (1-x)log_(e )(1-x) is

Answer»

`(1)/(2)`
`(1)/(4)`
`(1)/(20)`
`(1)/(10)`

Answer :C
Discussion
1531.

If the coordinates of four concyclic point on the rec­tangular hyperbola xy =c^(2) are i=1,2,3,4 then

Answer»

`t_(1)t_(2)t_(3)t_(4)=-1`
`t_(1)t_(2)t_(3)t_(4)=1`
`t_(1)t_(3)=t_(2)t_(4)`
`t_(1)t_(2)t_(3)t_(4)=C^(2)`

ANSWER :b
Discussion
1532.

The standard deviations of two sets containing 10 and 20 members are 2 and 3 respectively measured from their common means 5. The S.D. for the whole set of 30 members is

Answer»

`(2)/(SQRT(3))`
`sqrt(6)`
`sqrt(((22)/(3)))`
`sqrt(3)`

Answer :C
Discussion
1533.

""^((2n + 1))C_0 - ""^((2n+ 1))C_1 + ""^((2n + 1))C_2 - ……+""^((2n + 1))C_(2n) =

Answer»

`-1`
1
3
`-2`

ANSWER :B
Discussion
1534.

The area common to the circle x^2+y^2=64 and the parabola y^2=12x is equal to

Answer»


ANSWER :16/3 (squre3-2pi)
Discussion
1535.

Let C _(1) and C _(2) be the centres of the circles x ^(2) + y ^(2) - 2x - 2y -2 =0 and x ^(2) +y ^(2) - 6x + 14=0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC _(1) QC_(2) is:

Answer»

8
6
9
4

Answer :A
Discussion
1536.

The solution of the differential equation (tan y)(dy)/(dx) = sin (x+y) + sin (x-y) is

Answer»

SECY + 2cos x = C
SEC y - 2 cos x = C
cosy - 2 sin x = C
tan y - 2 sec x= C

Answer :A
Discussion
1537.

Solve (dy)/(dx) + y cot x = 5e^(cos x). Find the x = (pi)/(2), y = -4.

Answer»


Answer :`y SIN x = -5e^(COS x) + c` and `y sin x+5 e^(COSX) = 1`
Discussion
1538.

Consider an equation z^(197)=1(z is a complex number). If alpha, beta are its two randomly chosen roots. 'p' denotes the probability that sqrt(2+sqrt(3)) le|alpha+beta|, then

Answer»

`p=1/1996,aepsilonI`
`[1/p]` lies in `[5,15]`
`[1/pxx83/50]` is 8 is divisible by 2
`[1/p]+2` is 6

Solution :Take `beta=1`
`implies|1+alpha|^(2)=|(1+costheta)+isintheta|^(2)=2+2costhetage2+SQRT(3)`
`impliescosthetage(sqrt(3))/2` so, we GET 332 angles
`:.p=332/1996`
Discussion
1539.

If A and B are independent events such that 0ltP(A)ltq, 0ltP(B)lt1 then

Answer»

<P>A, B are mutually EXCLUSIVE
A and B' are INDEPENDENT
A' and B' are independent
`P(A|B)_P(A'|B)=1`

Answer :B::C::D
Discussion
1540.

Let A,B,C be the vertices of a triangle ABC in which B is taken an origin of reference and position vectors of A and C bara and barc respectively. A line AR parallel to BC is drawn from A PR(P is mid point of AB) meets AC at Q and area of triangle ACR is 2 times area of triangle ABC. Positions vector of R in terms of bara and barc is z 1. 2. bara+3barc 3. bara+barc 4.bara+4barc Position vector of Q is (considering result of Q(7)). 1.(2bara+3barc)/5 2.(3bara+2barc)/5 3. (bara+2barc)/3 4. None of these (PQ/QR). (AQ/QC) is equal to 1. 1//4 2. 2//5 3. 3//5 1//6

Answer»


ANSWER :A,B,D
Discussion
1541.

The solution of 2x(dy)/(dx) + y = 2x^(3) is

Answer»

`ysqrt(x) = (2)/(5)x^(5//2)+C`
`ysqrt(x) = (2)/(7)x^(7//2)+c`
`ysqrt(x) = (2)/(9)x^(9//2)+c`
`ysqrt(x) = (2)/(3)x^(3//2)+c`

Answer :B
Discussion
1542.

If absA = m , absB =n ,what can you say about abs(A xx B)

Answer»

SOLUTION :If `absA` = m , `absB` = n then `ABS(A xx B) = MN`
Discussion
1543.

If veca = (1,1,1), vecb = (-1,3,0) andvecc = (2,0,2), find veca+2vecb-1/2vecc.

Answer»

SOLUTION :`VECA + 2vecb-1/2vecc`
(1,1,1) +2(-1,3,0) -1/2(2,02)
Discussion
1544.

Consider a family of circles which are passing through the point (-1,1) and are tangent to x-axis. If (h,k) are the co-ordinates of the centre of the circles, then the set of values of k is given by the internal.

Answer»

`-1/2lekle1`
`kle1/2`
`0ltklt1/2`
`kle1/2`

ANSWER :D
Discussion
1545.

Find the sumsum_(0leiltjlen)1.

Answer»

SOLUTION :`underset(0leiltjlen)(sumsum1)=(sum_(i=1)^(n)sum_(j=1)^(n)1-sum_(i=j)SUM1)/2`
`=((sum_(j=1)^(n)1)(sum_(j=1)^(n)1)-sum_(j=1)^(n))/2`
`=(n^(2)-n)/2`
`=(n(n-1))/2`
Discussion
1546.

int_(2)^(4) (3x^(2)+1)/((x^(2)-1)^(3))dx = (lambda)/(n^(2)) where lambda, n in N and gcd(lambda,n) = 1, then findthe value of lambda + n

Answer»


SOLUTION :N//A
Discussion
1547.

int_(0)^(pi)(x tan x)/(Sec x+ Cos x) dx =

Answer»

`PI/4`
`pi/2`
`pi^(2)/4`
`pi^(2)/(2)`

ANSWER :C
Discussion
1548.

Which of the following is the graph of the equation y = 2x − 5 in the xy-plane?

Answer»




Solution :In the xy-plane, the graph of the equation y = mx + b, where m and b are constants, is a LINE with slope m and y-intercept (0, b). Therefore, the graph of y = 2x − 5 in the xy-plane is a line with slope 2 and a y-intercept (0, −5). Having a slope of 2 means that for each INCREASE in x by 1, the value of y INCREASES by 2. Only the graph in choice D has a slope of 2 and crosses the y-axis at (0, −5). Therefore, the graph shown in choice D must be the correct answer.
Choices A, B, and C are incorrect. The graph of y = 2x − 5 in the xy-plane is a line with slope 2 and a y-intercept at (0, −5). The graph in choice A crosses the y-axis at the point (0, 2.5), not (0, −5), and it has a slope of `1/2`, not 2. The graph in choice B crosses the y-axis at (0, −5), however, the slope of this line is −2, not 2. The graph in choice C has a slope of 2, however, the graph crosses the y-axis at (0, 5), not (0, −5).
Discussion
1549.

A : If A+B+C=180^(@) " then " cos^(2) A + cos^(2) B+ cos^2 C =1-2 cos A cos B cos C. R : If A+B+C=180^(@) " then " cos 2A + cos 2B + cos 2C=-1-4 cos A cos B cos C

Answer»

A is true , R is true and R is CORRECT EXPLANATION of A
A is true , R is trueand R is not correct explanation of A
A is true , R is false
A is false , R is true

ANSWER :A
Discussion
1550.

If int(sqrt(1+3sqrt(x)))/(x^(2//3))dx=2f(x)^(3//2+C then f(x) is equa to

Answer»

`1 + X^(2//3)`
`1+x^(1//3)`
`1-x^(1//3)`
`1-x^(2//3)`

ANSWER :B
Discussion