28278 + Interview Questions in Maths in Class 12

1.	Which of the following is not correct combination
Answer» (IV) (iii) (Q) (i) (II) (R) (III) (i) (S) (II) (iv) (P) Answer :D

Discussion

2.	Findquotientand theremainderwhen2x^5 -3x^4 +5x^3 -3x^2 +7x-9is divided by x^2 -x--3
Answer» `33x +4` `41 x+3` `47x +21` `33x-4` ANSWER :B

Discussion

3.	An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.
Answer» ANSWER :`(31)/(9)((2)/(3))^(4)`

Discussion

4.	Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha]. Which of the following statements can be concluded about (f(x))?
Answer» F(X) is discontinuous in `[1/alpha, alpha]` f(x) is INCREASING in `[1/alpha, alpha]` f(x) is decreasing in `[1/alpha, alpha]` NONE of the above Answer :B

Discussion

5.	Let f:R rarrR be definedas f(x) = x^4 Choose the correct answer
Answer» f is one-one onto f is many one f is one-one but not onto f is neither one-one nor onto Solution :f(1) = `1^4` = 1 f(-1) = `(-1)^4` = 1 f(1) = f(-1) = 1 `therefore` f is not one-one The co-domain of f is R. the range of I is `[0, infty)`. Since range of f `NE` co-domain of f, f is not onto Hence f is neither one-one nor onto.

Discussion

6.	Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha]. Which of the following statements can be concluded about f(f(x))?
Answer» F(f(X)) is DISCONTINUOUS in `[1/ALPHA, alpha]` f(f(x)) is increasing in `[1/alpha, alpha]` f(f(x)) is decreasing in `[1/alpha, alpha]` NONE of the above Answer :b

Discussion

7.	Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha]. Then f(1) is equal to
Answer» 1 0 -1 does'nt ATTAIN a UNIQUE VALUE ANSWER :a

Discussion

8.	Find the area of the region lying in the first quadrant and bounded by y = 4x^(2), x = 0, y = 1 and y = 4.
Answer» ANSWER :`7/3`

Discussion

9.	Iron filing and water were placed in a 5 litre tank and sealed.The tank was heated to 1237 k. Upon anaiysis the tank was found to contain1.10gram of hydropen and 42.5 gm of water vapour. If the reaction in the tank is represented by 3Fe(s) + 4H_(2)O(g) hArr Fe_(3)O(g) + 4H_(2)(g) the equilibrium constant will be -
Answer» `2.949 XX 10^(3)` `6.490 xx 10^(3)` `4.90 xx 10^(3)` `3.200 xx 10^(3)` ANSWER :A

Discussion

10.	Integrate the following functions : intsin^(2)xdx
Answer» ANSWER :`(X)/(2)-(SIN2X)/(4)+c`

Discussion

11.	int_(0)^(1) cot^(-1) (1-x+x^(2))dx=
Answer» `PI - LN 2` `pi/2 - ln 2` `pi + ln 2` `pi/2 + ln 2` ANSWER :B

Discussion

12.	A=[{:(4,3),(2,5):}],A^(2)-xA+yI=0.Find real numbers x and y. where I is a 2xx2 identity matrix.
Answer» ANSWER :`x=9,y=14`

Discussion

13.	If A be a matrix of order 4 x 4 and \|A\| =20, then \|A^-1\| =
Answer» 20 `1/20` 5 `1/5` ANSWER :B

Discussion

14.	If (-2, 7) is the highest point on the graph of y =-2x ^(2) -4ax+k, then k equals :
Answer» 31 11 `-1` `-1//3` ANSWER :C

Discussion

15.	If the line joining the incentreto the centroid of a triangle ABC is parallel to the side BC. Which of the following are correct ?
Answer» `2b=a+C` `2a=B+c` `"COT"(A)/(2)"cot"(C )/(2)=3 ` `"cot"(B)/(2) "cot"(C )/(2)=3 ` Answer :B::D

Discussion

16.	Which of the following functions is NOT one-one ?1. f:R to R defined by f(x)=6x-12. f: R to R defined by f(x)=x^(2)+73. f: R to R defined by f(x)=x^(3)4. f: R-{7}toR defined by f(x)=(2x+1)/(x-7)
Answer» `f:R to R` defined by `f(x)=6x-1` `f: R to R` defined by `f(x)=x^(2)+7` `f: R to R` defined by `f(x)=x^(3)` `f: R-{7}toR` defined by `f(x)=(2x+1)/(x-7)` Answer :B

Discussion

17.	Write down and simplify 10^("th")" term in "((3p)/(4)-5q)^(14)
Answer» ANSWER :`(-(2002)3^(5) 5^9)/(4^5) p^5.q^9`

Discussion

18.	The points of intersection of the line r = 2a + t (b - c) with the plane r = a + p (b + c) + q (a + 2b - c) is
Answer» 4A + 3B - 3c 4a - 3b + 3c `(1)/(2) (4a - 3b + 3c)` `(1)/(2) (4a + 3b - 3c)` Answer :D

Discussion

19.	Which of the following pair of functions have the same graph ?
Answer» `f(x) ln [1+{x}]` and `g(x) = ln(1+[{x}])`, `f(x) = 1/(1+tan^(2)x) + (1)(1+cot^(2)x)` and `g(x) = 2 SIN^(2)x + COS2X` `f(x) = 2^(sgn)(x^(2)+3x+3)` and `g(x) = 2sgn(x^(2)+3x+3)` `f(x) = (sgnx)/(sgnx)` and `g(x) = (\|x\|)/(\|x\|)` Answer :A::C::D

Discussion

20.	Integrating factor of (x+2y^(3))(dy)/(dx)=y^(2) is
Answer» `E^((1)/(y))` `e^-((1)/(y))` `y` `(-1)/(y)` ANSWER :A

Discussion

21.	Assertion (A) : The area of the functiony=sin^(2)x from 0 to pi will be more than that of curve y=sinx from 0 to pi. Reason (R ) : t^(2) lt t, if 0 lt t lt 1
Answer» Both A and R are individually true and R is the correct EXPLANATION of A Both A and R are individually true and R is not the correct explanation of A A is true R is FALSE A is false but R is true. Answer :4

Discussion

22.	Find the sum to n terms of the series 5,7,13,31,85+…….
Answer» ANSWER :`1/2 [ 3 ^(N) + 8N -1]`

Discussion

23.	Evaluate. I = int _(0) ^(1) 2 sin (pt) sin (qt) dt. If : p & q are equal and either is root of the equation tan x = x.
Answer» ANSWER :`F (x) =2e^(2X) + 3 &g(x) =3 - 2e^(2x)`

Discussion

24.	Evaluate the following : (i) int(x+2)/((x+3)(x+4)^((3)/(2))dx (ii) int((x+3)/(2x+5))^((1)/(2))*(1)/(x)dx
Answer» Solution :`(i)` LET us substitute `x+4=t^(2)` so that `dx=2tdt` and `x+2=t^(2)-2`, `x+3=t^(3)-1`. THUS `int((x+2))/((x+3)(x+4)^(3//2))dx=int((t^(2)-2))/((t^(2)-2).t^(3))2tdt=int(2(t^(2)-1-1))/((t^(2)-1)t^(2))DT` `=int(2)/(t^(2))dt-2int(1)/((t^(2)-1)t^(2))dt=2int(1)/(t^(2))dt+int(2)/(t^(2))dt-2int(1)/(t^(2)-1)dt` `=4intt^(-2)dt+int(1)/(t+1)dt-int(1)/(t-1)dt` `=4(t^(-1))/(-1)+ln\|(t+1)/(t-1)\|+C` `=-(4)/(sqrt(x+4))+ln\|(sqrt(x+4)+1)/(sqrt(x+4)-1)\|+C`, `C` is a constant of INTEGRATION `(II)` Let us substitute `(x+3)/(2x+5)=t^(2)` so that `x=(5t^(2)-3)/(1-2t^(2))` and `dx=(-2t)/((1-2t^(2))^(2))dt` Thus, `int((x+3)/(2x+5))^((1)/(2))*(1)/(x)dx=intt(-(1-2t^(2))/(5t^(2)-3))((-2t)/((1-2t^(2))^(2)))dt` `=int(-2t^(2))/((5t^(2)-3)(1-2t^(2)))dt` `=int(2t^(2))/((2t^(2)-1)(5t^(2)-3))dt` `=6int(1)/(5t^(2)-3)dt-2int(1)/(2t^(2)-1)dt` `=(3)/(sqrt(3).sqrt(5))ln\|(sqrt(5t)-sqrt(3))/(sqrt(5t)+sqrt(3))\|-(2)/(2.(1)/(sqrt(2)))ln\|(sqrt(2t)-1)/(sqrt(2t)+1)\|+C` `=sqrt((3)/(5))ln\|(sqrt(5)sqrt(x+3)-sqrt(2(2x+5)))/(sqrt(5(x+3))+sqrt(3(2x+5)))\|-sqrt(2)ln((sqrt(2(x+3))-sqrt(2x+5))/(sqrt(2(x+3))+sqrt(2x+5)))+C`

Discussion

25.	The value of the integral int(cos7x-cos8x)/(1+2cos5x)dx can be
Answer» `(SIN2X)/(2)-(sin3x)/(3)+c` `(SIN5X)/(5)-(cos5x)/(4)+c` `(cosx)/(3)-(sin4x)/(2)+c` none of these Answer :A

Discussion

26.	Let z_(1) and z_(2) be two non-zero complex numbers such that (z_(1))/(z_(2)) + (z_(2))/(z_(1)) = 1 , then the origin and points represented by z_(1) and z_(2)
Answer» LIE on a STRAIGHT line Form a right TRIANGLE Form an EQUILATERAL triangle form an isosceles triangle Answer :C

Discussion

27.	Let : P_(1):3y+z+1=0 and P_(2):2x-y+3z-7=0 and the equation of line AB is (x-1)/(2)=(y-3)/(-1)=(z-4)/(3) in 3D space. Shortest distance between the line of intersection of planes P_(1) and P_(2) and the line AB is equal to
Answer» `(7)/(sqrt(10))" UNITS"` `7sqrt((2)/(5))" units"` `(6)/(sqrt(10))" units"` `2sqrt((2)/(5))" units"` ANSWER :B

Discussion

28.	Determine whether a*b ="LCM" {a,b} "on" Noperations as defined by are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.
Answer» SOLUTION :for all `a,B in N , LCM{a,b } in N` `IMPLIES ab in N` ` implies ` is a bimary operation on N.

Discussion

29.	Let f(x) = [{:((pi/2.sin^(-1)(1-x^(2))sin^(-1(1-{x})))/(sqrt(2((x)-(x)^(3)))), "for " x ne 0),(""pi/2, x=0):} where {x} is the fractional part of x. Consider another function g(x), such that g(x) = {{:(f(x), for, x ge 0),(2sqrt(2)f(x), for , x lt 0):} Discuss the continuity of the function f(x) & g(x) at x=0.
Answer» ANSWER :A::B::C::D

Discussion

30.	If a, b, c in Rand the equations ax^(2) + bx + c = 0 and x^(3) + 3x + 2 = 0 have two common roots, then
Answer» a = B = - C a = - b = c a = b = c NONE of these Answer :C

Discussion

31.	x=(1)/(3)+(1)/(3.3^(3))+(1)/(5.3^(5))+….. y=(1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+…. z=1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+….. Then descending order of x, y, z
Answer» Z, y, X z, x, y x, y, z y, z, x ANSWER :B

Discussion

32.	Derive the equation of a plane perpendicular to a given vector and passing through a given point in both vector form and Cartesian form.
Answer» ANSWER :`A (x-x_(1))+B(y-y_(1))+C(z-z_(1))=0`

Discussion

33.	Integration using rigonometric identities : int (sin x cos x)/(sqrt(3+5 sin ^(2)x))dx=...+c
Answer» `(1)/(5) sqrt(3+5 SIN^(2)X)` `(1)/(10) sqrt(3+5 sin^(2)x)` `(1)/(20) sqrt(3+5 sin^(2)x)` `(1)/(2sqrt(3)) sqrt(3+5 sin^(2)x)` Answer :A

Discussion

34.	If int(e^(4x)-1)/(e^(2)x)log((e^(2x)+1)/(e^(2x)-1))dx (t^(2))/(2)logt-(t^(2))/(4)+(u^(2))/(4)+C them
Answer» `t=E^(-X)-e^(x)u=e^(x)+e^(-x)` `t=e^(x)-e^(-x),u=e^(x)+e^(-x)` `t=e^(x)+e^(-x),u=e^(x)-e^(-x)` NONE of these Answer :C

Discussion

35.	{:("Quantity A","Quantity B"),(abc,c-d):}
Answer» QUANTITY A is greater. Quantity B is greater. The TWO quantities are equal. The relationship cannot be DETERMINED from the INFROMATION given. Answer :A

Discussion

36.	If I is the incentre of a triangle whose is radius and circumradius are r and R respepectively : I_1,I_2,I_3 is its ex-centre triangle, then I I_1. I I_2. I I_3 is equal to
Answer» `R^2r` `16R^2 r` `R r^2` `16Rr^2` ANSWER :B

Discussion

37.	If plane 2x+3y+6z+k=0 is tangent to the sphere x^(2)+y^(2)+z^(2)+2x-2y+2z-6=0, then a value of k is
Answer» 26 16 `-26` NONE of these Solution :Center and radius of the sphere are `(-1,1,-1)` and 3, respectively. Distance of `(-1,1,-1)` from the plane is `\|(-2+3-6+k)/SQRT(4+9+36)\|`. Since, the plane is TANGENT to the plane. `therefore \|(k-5)/7\|=3`. `therefore k=-16,26`

Discussion

38.	Sum to n terms of the series (1)/(1 cdot 2 cdot 3 cdot 4)+(1)/(2 cdot 3 cdot 4 cdot 5)+(1)/(3 cdot 4 cdot 5 cdot 6)+… is
Answer» `(1)/(24)-(1)/(n)` `(1)/(18)-(1)/(3(n+1)(n+2)(n+3))` `(1)/(2^(n))-(1)/(n^(2))` `(1)/(18)-(1)/(3(n+3)(n+4))` Answer :B

Discussion

39.	If y^(2)= p(x) the is polynomial of order 3, then 2(d)/(dx) [y^(3) (d^(2)y)/(dx^(2))]=……….
Answer» <P>`p'''(X) + p'(x)` `p''(x).p'''(x)` `p(x).p'''(x)` `p(x) +p'''(x)` ANSWER :C

Discussion

40.	Assertion(A): x^(2)+x+1 gt 0 for all positive real values of x only. Reason (R) :When b^(2)-4ac lt 0, a , ax^(2)+bx+x have same sign for all real values of x.
Answer» Both A, R are TRUE and R explain Assertion Both A, R are true but R does't explain A A is true R is FALSE A is false R is true Answer :D

Discussion

41.	Minimise Z = 6x + 3y, subject to the constraints 4x + y ge 80, x + 5y ge 115, 3x + 2y le 150, x ge 0, y ge 0.
Answer» Answer :MINIMUM value of Z = 150 at X = 15 and y = 20

Discussion

42.	If ……….., then f(x)=x^(2)-kx+20, [0,3] is strictly increasing.
Answer» `k lt 0` `0 lt k lt 1` `1 lt k lt 2` `2 lt k lt 3` ANSWER :A

Discussion

43.	PQ is a double ordinate of the parabola y^(2)+4x . The locus of its point of trisection is
Answer» `9y^(2)=4AX` `9y^(2)=16ax` `3y^(2)=8AX` `9y^(2)=8ax` ANSWER :A

Discussion

44.	The periodof the functionf(x)= \| sin2x\| + \| cos8x \|is
Answer» `2PI` `PI` `(2pi)/(3)` `pi/2` ANSWER :D

Discussion

45.	For x ne 1, f is defined by f(x)=1/(log x) - 1/(x-1) The value of f(1), so that f is a continous function is
Answer» 1 `1/2` 0 2 Answer :B

Discussion

46.	Show that the tangent to the ellipse x^2/a^2+y^2/b^2=1 at points whose eccentric angles differ by pi/2 intersect on the ellipse x^2/a^2+y^2/b^2=2
Answer» ANSWER :`(1+sin^(2) THETA)^(-1//2)`

Discussion

47.	A tangential force 'F' is applied at the topmost point of a spherical shell of mass 'm' kept on a rough horizontal surface. If it rolls without slipping, it's acceleration is (6F)/(xm)" then "'x'
Answer» SOLUTION :`F+f=ma` `FR="f"R=Ia//R` `:.F=(5)/(6)"ma"` `:.a=(6F)/(5M)`

Discussion

48.	Evaluate int 2 x sin(x^(2) + 1)dx
Answer» Answer :`-COS(X^(2) +1) +c`

Discussion

49.	If A is 2xx2 matrix such that A[{:(" "1),(-1):}]=[{:(-1),(2):}]and A^2[{:(" "1),(-1):}]=[{:(1),(0):}], then trace of A is (wherethe trace of the matrix is the sum of all principal diagonal elements of the matrix )
Answer» 1 0 2 5 Answer :A

Discussion

50.	Integrate the functions in exercise. (1)/(sqrt(7-6x-x^(2)))
Answer» ANSWER :`SIN^(-1)((x+3)/(4))+C`

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Which of the following is not correct combination

Findquotientand theremainderwhen2x^5 -3x^4 +5x^3 -3x^2 +7x-9is divided by x^2 -x--3

An experiment succeeds twice as often as it fails. Find the probability that in the﻿ next six trials, there will be atleast 4 successes.﻿

Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha]. Which of the following statements can be concluded about (f(x))?

Let f:R rarrR be definedas f(x) = x^4 Choose the correct answer

Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha]. Which of the following statements can be concluded about f(f(x))?

Consider alpha gt 1 and f:[1/alpha,alpha] rarr [1/alpha,alpha] be bijective function. Suppose that f^(-1)(x)=1/f(x), " for all " in [1/alpha,alpha]. Then f(1) is equal to

Find the area of the region lying in the first quadrant and bounded by y = 4x^(2), x = 0, y = 1 and y = 4.

Integrate the following functions : intsin^(2)xdx

int_(0)^(1) cot^(-1) (1-x+x^(2))dx=

A=[{:(4,3),(2,5):}],A^(2)-xA+yI=0.Find real numbers x and y. where I is a 2xx2 identity matrix.

If A be a matrix of order 4 x 4 and |A| =20, then |A^-1| =

If (-2, 7) is the highest point on the graph of y =-2x ^(2) -4ax+k, then k equals :

If the line joining the incentreto the centroid of a triangle ABC is parallel to the side BC. Which of the following are correct ?

Which of the following functions is NOT one-one ?1. f:R to R defined by f(x)=6x-12. f: R to R defined by f(x)=x^(2)+73. f: R to R defined by f(x)=x^(3)4. f: R-{7}toR defined by f(x)=(2x+1)/(x-7)

Write down and simplify 10^("th")" term in "((3p)/(4)-5q)^(14)

The points of intersection of the line r = 2a + t (b - c) with the plane r = a + p (b + c) + q (a + 2b - c) is

Which of the following pair of functions have the same graph ?

Integrating factor of (x+2y^(3))(dy)/(dx)=y^(2) is

Assertion (A) : The area of the functiony=sin^(2)x from 0 to pi will be more than that of curve y=sinx from 0 to pi. Reason (R ) : t^(2) lt t, if 0 lt t lt 1

Find the sum to n terms of the series 5,7,13,31,85+…….

Evaluate. I = int _(0) ^(1) 2 sin (pt) sin (qt) dt. If : p & q are equal and either is root of the equation tan x = x.

Evaluate the following : (i) int(x+2)/((x+3)(x+4)^((3)/(2))dx (ii) int((x+3)/(2x+5))^((1)/(2))*(1)/(x)dx

The value of the integral int(cos7x-cos8x)/(1+2cos5x)dx can be

Let z_(1) and z_(2) be two non-zero complex numbers such that (z_(1))/(z_(2)) + (z_(2))/(z_(1)) = 1 , then the origin and points represented by z_(1) and z_(2)

Let : P_(1):3y+z+1=0 and P_(2):2x-y+3z-7=0 and the equation of line AB is (x-1)/(2)=(y-3)/(-1)=(z-4)/(3) in 3D space. Shortest distance between the line of intersection of planes P_(1) and P_(2) and the line AB is equal to

Determine whether a**b ="LCM" {a,b} "on" Noperations as defined by * are binary operations on the sets specified in each case. Give reasons if it is not a binary operation.

If a, b, c in Rand the equations ax^(2) + bx + c = 0 and x^(3) + 3x + 2 = 0 have two common roots, then

x=(1)/(3)+(1)/(3.3^(3))+(1)/(5.3^(5))+….. y=(1)/(5)+(1)/(3.5^(3))+(1)/(5.5^(5))+…. z=1+(1)/(3.2^(2))+(1)/(5.2^(4))+(1)/(7.2^(6))+….. Then descending order of x, y, z

Derive the equation of a plane perpendicular to a given vector and passing through a given point in both vector form and Cartesian form.

Integration using rigonometric identities : int (sin x cos x)/(sqrt(3+5 sin ^(2)x))dx=...+c

If int(e^(4x)-1)/(e^(2)x)log((e^(2x)+1)/(e^(2x)-1))dx (t^(2))/(2)logt-(t^(2))/(4)+(u^(2))/(4)+C them

{:("Quantity A","Quantity B"),(abc,c-d):}

If I is the incentre of a triangle whose is radius and circumradius are r and R respepectively : I_1,I_2,I_3 is its ex-centre triangle, then I I_1. I I_2. I I_3 is equal to

If plane 2x+3y+6z+k=0 is tangent to the sphere x^(2)+y^(2)+z^(2)+2x-2y+2z-6=0, then a value of k is

Sum to n terms of the series (1)/(1 cdot 2 cdot 3 cdot 4)+(1)/(2 cdot 3 cdot 4 cdot 5)+(1)/(3 cdot 4 cdot 5 cdot 6)+… is

If y^(2)= p(x) the is polynomial of order 3, then 2(d)/(dx) [y^(3) (d^(2)y)/(dx^(2))]=……….

Assertion(A): x^(2)+x+1 gt 0 for all positive real values of x only. Reason (R) :When b^(2)-4ac lt 0, a , ax^(2)+bx+x have same sign for all real values of x.

Minimise Z = 6x + 3y, subject to the constraints 4x + y ge 80, x + 5y ge 115, 3x + 2y le 150, x ge 0, y ge 0.

If ……….., then f(x)=x^(2)-kx+20, [0,3] is strictly increasing.

PQ is a double ordinate of the parabola y^(2)+4x . The locus of its point of trisection is

The periodof the functionf(x)= | sin2x| + | cos8x |is

For x ne 1, f is defined by f(x)=1/(log x) - 1/(x-1) The value of f(1), so that f is a continous function is

Show that the tangent to the ellipse x^2/a^2+y^2/b^2=1 at points whose eccentric angles differ by pi/2 intersect on the ellipse x^2/a^2+y^2/b^2=2

A tangential force 'F' is applied at the topmost point of a spherical shell of mass 'm' kept on a rough horizontal surface. If it rolls without slipping, it's acceleration is (6F)/(xm)" then "'x'

Evaluate int 2 x sin(x^(2) + 1)dx

If A is 2xx2 matrix such that A[{:(" "1),(-1):}]=[{:(-1),(2):}]and A^2[{:(" "1),(-1):}]=[{:(1),(0):}], then trace of A is (wherethe trace of the matrix is the sum of all principal diagonal elements of the matrix )

Integrate the functions in exercise. (1)/(sqrt(7-6x-x^(2)))

An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be atleast 4 successes.