28278 + Interview Questions in Maths in Class 12 Page 20 InterviewSolution

951.	Evaluate the following determinants. [[-18,17,19],[3,0,0],[-14,5,2]]
Answer» Solution :`[[-18,17,19],[3,0,0],[-14,5,2]]` `-3[[17,19],[5,2]]` (Expanding along 2ND ROW) =-3(34-95) (-3)(-61)=183

Discussion

952.	Solve the following differential equations. x(x-1)(dy)/(dx)-(x-2)y=x^(3)(2x-1)
Answer» `y((X-1))/(x^(2)) = x^(2) - x + C` `y((x-1))/(x^(2)) = x^(2) + x + c` `y((x+1))/(x^(2)) = x^(2) - x + c` `y((x-1))/(x^(2)) = x^(2) + x + c` ANSWER :A

Discussion

953.	Let h(x) =f(x)-a(f(x))^(3) for every real number x h(x) increase as f(x) increses for all real values of x if
Answer» `a in (0,3)` `a in (-2,2)` `[3,oo)` none of these Solution :`HH(X) =-F(X)=af(x)^(2)+a(f(x))^(3)` or `h(X)=f(x)-2af(x)f(x)+3a(f(x))^(2)f(x)` `=f(x)[3a(f(x))^(2)-2af(x)+1]` Now h(x) increase if f(x) increase and `3a(f(x))^(2)-2af(x)+1gt0for all x in R` or `3agt0 and 4A^(2)-12ale0` or `AGT0 and a in [0,3]` or `a in [0,3]`

Discussion

954.	Find intsqrt(3-2x-x^(2))dx
Answer» Answer :`(1)/(2)(x+1) sqrt(3-2x-x^(2))+2 SIN^(-1)((x+1)/(2))+c`

Discussion

955.	If alpha(theta) epsilon R & beta(theta),theta epsilon R-{2n pi-(pi)/2, n epsilin I} are functions satistying (1+x)sin^(2)theta-(1+x^(2))sintheta +(x-x^(2))=0 then which of the following is/are correct?
Answer» `lim_(theta to 0^(+)){(alpha(theta))^(1/(SINTHETA))+(beta(theta))^(1/(sintheta))}=1/(e^(2))` `In (beta(theta))` is AODD `fn` `lim_(theta to 0) (sum_(r=1)^(n) r^(1/(alpha^(2)(theta))))^(alpha^(2)(theta))=n, n epsilon N, h GE2` `lim_(theta to pi//2)(alpha(theta)-(alpha(theta))^(alpha(theta)))/(1-alpha(theta)+In(alpha(theta)))=2` Solution :`x^(2)-((1+sin^(2)theta)/(1+sintheta))+((sintheta-sin^(2)theta)/(1+sintheta))=0` `x^(2)-x(sintheta+(1-sintheta)/(1+sintheta))+sintheta((1-sintheta)/(1+sintheta))=0` `impliesx=sintheta , (1-sintheta)/(1+sintheta)` `impliesalpha(theta)=sintheta, beta(theta)=(1-sintheta)/(1+sintheta)` Hence the RESULTS follows.

Discussion

956.	Assertion (A): The number of ways in which 5 boys and 5 girls can sit in a row so that all the girls sit together is 86400. Reason (R) : The number of ways in which m (first type of different) things and n (second type of different) things can be arranged in a row so that all the second type of things come together is n!""^((n+1))P_(m) The correct answer is
Answer» Both A and R are true and R is the correct explanation of A Both A and R are true but R is not correct explanation of A A is true but R is FALSE A is false but R is true Answer :C

Discussion

957.	Evaluate the following : [[-1,3,2],[1,3,2],[1,-3,-1]]
Answer» SOLUTION :`[[-1,3,2],[1,3,2],[1,-3,-1]]` `[[-2,0,0],[1,3,2],[1,-3,-1]] (R_1~~R_1-R_2)` =`-2[[3,2],[-3,-1]]=-2(-3+6)=-6`

Discussion

958.	The term independent of x in the expansion of (1 + x)^n (1 + 1/x)^n is
Answer» `C_0^2 + C_1^2 + C_2^2 + ……….+C_n^2` `""^(2N)C_n` `(1.3.5…..(2n - 1))/(N!) 2^n` All of the above Answer :B

Discussion

959.	Gross calorific value and physiological value of carbohydrate and liqid is :-
Answer» 4.1 KCAL, 4.0 Kcal and 9.45 Kcal, 9.0 Kcal 4.1 Kcal, 4.0 Kcal and 5.65 Kcal, 5.00 Kcal 4.1 Kcal, 4.0 Kcal and 5.65 Kcal, 4.0 Kcal 9.65 Kcal, 5.65 Kcal and 4 Kcal, 4.1 Kcal Answer :A

Discussion

960.	Solve graphically y gt 5
Answer» SOLUTION :

Discussion

961.	IF f(x) = (cosx)(cos2x)… (cosnx), then f(x)+sum_(t prop1)^(1)(rtanrx)f(x) is equal to
Answer» F(x) 0 `-f(x)` `2F(x)` Answer :B

Discussion

962.	Find the angle between the lines, whose direction cosines are given by the equation 3l+m+5n=0 and 6mn-2nl+5lm=0.
Answer» ANSWER :`COS^(-1)(1)/(6)`

Discussion

963.	Let f(x)=(3)/(x^(4)+3x^(2)+9) and g(x) =(x^(2))/(x^(4)+3x^(2)+9)
Answer» `(1)/(2sqrt(3))LOG \|(x^(2)-sqrt(3)x-3)/(x^(2)+sqrt(3)x-3)\|+c` `(1)/(3) tan^(-1) ((x^(2)-3x)/(3x))+c` `(1)/(2sqrt(3)) log \|(x^(2)-sqrt(3)x+3)/(x^(2)+sqrt(3)x+3)\|+c` `(1)/(3) tan^(-1)((x^(2)+3x)/(3x))+c` ANSWER :B

Discussion

964.	Let alpha" and "beta be the roots of the equation px^(2)+qx+r=0. If p,q, r in A.P. and alpha+beta=4, then alpha beta is equal to
Answer» `-9` `9` `-5` `5` ANSWER :A

Discussion

965.	Eco Wildlife preserve contains 5x zebras and 2x lions, where x is a positive integer. If the lions succeed in killing z of t he zebras, is the new ratio of zebras to lions less than 2 to 1 ? (1) z > x (2) z = 4
Answer» ANSWER :A

Discussion

966.	lim_(x rarr oo) (sqrt(x^(2) + 1) - root(3)(x^(3) + 1))/(root(4)(x^(4) + 1) - root(5) (x^(4) + 1)) equals :
Answer» -1 0 1 None of these ANSWER :B

Discussion

967.	bar(a)=2bar(i)-3bar(j)+6bar(k) and bar(b)=-2bar(i)+2bar(j)-bar(k) then ("Proj"_(bar(b))bar(a))/("Proj"_(bar(a))bar(b))=.......
Answer» `(3)/(7)` `(7)/(3)` 3 7 Answer :B

Discussion

968.	Let sec ^(2)((pi)/(9))+sec^(2)((2pi)/(9))+sec_(2)((4pi)/(9))=S and Sigma_(k-1)^(89) cos^(6)(k^(o))=(a)/(b) ,where a,b are coprime, then which of the following is/are correct?
Answer» Number of positive divisors of a+B + S is 4 Number of positive divisors of a+b + S is 8 S is a perfect square a+b is a prime number Solution :`sec^(2)((PI)/(9))+sec^(2)((2pi)/(9))+sec^(2)((4pi)/(9))` `=((cos.(2pi)/(9))^(2)(cos.(4pi)/(9))^(2) +(cos.(4pi)/(9))^(2)(cos.(pi)/(9))^(2)+cos^(2)((pi)/(9))cos^(2)((2pi)/(9)))/(cos.(pi)/(9)cos.(2pi)/(9)cos.(4pi)/(9))^(2)` Using CD formulas `=(1)/(4)([(cos.(2pi)/(3)+cos.(2pi)/(9))^(2)+(cos.(5pi)/(9)+cos.(pi)/(3))^(2)+(cos.(pi)/(3)+cos.(pi)/(9))^(2)])/(((1)/(64)))` `= 16 ((3)/(4)+cos^(2).(pi)/(9)+cos^(2).(2pi)/(9)+cos^(2).(4pi)/(9)+cos.(pi)/(9)-cos^(2).(2pi)/(9)-cos.(4pi)/(9))` `=16((3)/(9)+(3)/(2))` `therefore cos .(pi)/(9)-cos^(2).(2pi)/(9)-cos.(4pi)/(9)=0 =36` `underset(k=1)overset(89)Sigmacos^(6)(k^(@))= underset(k=1)overset(89)Sigmasin^(6)(k^(@))=(1)/(2)underset(k=1)overset(89)sum(SIN^(6)(k^(@))+cos^(6)(k^(@)))` `(1)/(2)(underset(k=1)overset(89)Sigma(1-(3)/(4)sin^(2)(2k^(@))))=(89)/(2)-(3)/(8)underset(k=1)overset(89)Sigma(2k)^(@)` `=(89)/(2)-(3)/(8)xx45` `(a)/(b)=(221)/(8)` `a221,b=8`

Discussion

969.	If a pair of dice is thrown 5 times then find the probability of getting three doublets.
Answer» Solution :In a single through of a pair of dice p(a DOUBLET) `=6/36=1/6` p (a non doublet) = 1-p9a doublet)`=1-1/6=5/6` CLEARLY the given experiment is a binomial experiment with n=5 `p=1/6` and `q=5/6` p(3 DOUBLETS in 5 THROW) = `^5C_3p^3q^2=10 cdot(1/6)^3(5/6)^2=250/6^5=125/3888`

Discussion

970.	Compute the integral I_(n) = int_(0)^(a) (a^(2) - x^(2))^(n)dx , where n is a natural number.
Answer» ANSWER :`(1)/(2)`

Discussion

971.	The third term in the expansion of (1/x + x ^[log_10 x])^5 is 10^6 then x =
Answer» `10^(-1)` 10 `10^2` `10^3` ANSWER :B

Discussion

972.	If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is
Answer» `(x^(2)+y^(2))^(2)=4R^(2)x^(2)y^(2)` `(x^(2)+y^(2))^(3)=4R^(2)x^(2)y^(2)` `(x^(2)+y^(2))(x+y) =R^(2)XY` `(x^(2)+y^(2))^(2)=4Rx^(2)y^(2)` Solution :Let the foot of perpendicular be P(h,k). Then, the SLOPE of line ` OP = (k)/(h)` `therefore` Line AB is perpendicular to line OP, so slope of line `AB = -(h)/(k)`[`therefore` product of slope of two perpendicular line s is (-1)] Now, the equation of line AB is `y-k=-(h)/(k)(x-h)rArrhx+ky=h^(2)+k^(2)` or`(x)/(((h^(2)+k^(2))/(h)))+(y)/(((h^(2)+k^(2))/(k)))=1` So, point `A((h^(2)+k^(2))/(h),0) and B(0,(h^(2)+k^(2))/(h))` `therefore Delta AOB` is a RIGHT angled triangle, so AB is ONE of the diameter of the circle having radius R (given). `rArr AB = 2R` `rArrsqrt(((h^(2)+k^(2))/(k))^(2)+((h^(2)+k^(2))/(k)))=2R` `rArr (h^(2)+k^(2))^(2)((1)/(h^(2))+(1)/(k^(2)))=4R^(2)` `rArr(h^(2)+k^(2)) =4R^(2) h^(2)k^(2)` On replacing h by x and k by y, we get `(x^(2)+y^(2))^(3)=4R^(2)x^(2)y^(2)`, which is teh REQUIRED locus.

Discussion

973.	If 1,alpha,alpha^2,…...,alpha^(n-1) are the n^(th) roots of unity, then the value of (3-alpha)(3-alpha^2)…...(3-alpha^(n-1)) is
Answer» n 0 `(3^n-1)//2` `(3^n+1)//2` ANSWER :C

Discussion

974.	Using problems are of the Inequality Type. Examples of this type are as follows:
Answer» Solution :Let `P(n): tan n alpha gt n tan alpha` Step I For `n=2, tan 2 alpha gt 2 tan alpha ` `rArr (2 tan alpha)/(1-tan^2alpha)-2tan alpha gt 0` `rArr 2 tan alpha ((1-(1-tan^2alpha)/(1-tan^2alpha))gt0` `rArr tan^2alpha.tan 2alpha gt 0 [because 0 lt alpha lt (pi)/(4) "for"n=2]` `rArr tan 2 alpha gt 0 [because 0 lt 2 alpha lt (pi)/(2)]` Which is true (`because` in first quadrant , `tan2alpha ` is POSITIVE) Therefore , P(2)is true. Step II ASSUME that P(k) is true , then `P(k): tan k alpha gt tan alpha` Step III For `n=k+1`, we shall prove that `tan(k+1)alpha gt (k+1)tan alpha` `becausetan (k+1)alpha=(tan k alpha +tan alpha)/(1-tank alpha tan alpha )` .......(i) When `0 lt alpha lt (pi)/(4k)or 0 lt kalpha lt (pi)/(4)` i.e., `0 lt tan k alpha lt1`, also `0 lt tan alpha lt 1` `therefore tan k alpha tan alpha lt1` `1-tan k alpha tan alpha gt 0 and 1-tan k alpha tan alpha lt 1`......(ii) From Eqs. (i) and (ii) , we GET `tan(k+1alpha gt (tan kalpha +tanalpha)/(1) gt tan kalpha +tanalpha gt k tan alpha +tan alpha ` [ by assumption step ] `therefore tan (k+1)alpha gt (k+1)tan alpha` Therefore , `P(k+1)` is true , Hence by the PRINCIPLE of mathematical induction P(n) is true for all `n in N`.

Discussion

975.	By using the properties of definite integrals evaluate the integrals in exercise. overset(5)underset(-5)int\|x+2\|dx
Answer» ANSWER :29

Discussion

976.	The sumof the numbers formedfromthe digits 2,3,4,5 is
Answer» 1,03,124 93324 78456 1,15,576 Answer :A

Discussion

977.	The plane x-2y + 3z= 2 makes an angle ... With Y-axis.
Answer» `cos^(-1)""(2)/(sqrt(14))` `SIN^(-1)""(2)/(14)` `TAN^(-1)""(2)/(sqrt(14))` `sin^(-1)""(2)/(sqrt(10))` Answer :B

Discussion

978.	Evaluate int " x cot"^(2) x dx
Answer» ANSWER :`-` x cot x + LOG \|SIN x\| - `(x^(2))/(2) + `

Discussion

979.	If A =[(1,2,1),(0,1,-1),(3,-1,1)] and AA' =I , then x+y is equal to
Answer» O I A `A^(2)` ANSWER :A

Discussion

980.	If P, Q, R lie on xy=c^(2), then the orthocentre of DeltaPQR lies on
Answer» `x+y=0` `2x+3y=C` `xy=c^(2)` NONE ANSWER :C

Discussion

981.	Choose the correct answer from the bracket. The antiderivative of (sqrtx + 1/sqrtx) equals
Answer» `1/3 X^(1/3) +2X^(1/2) + C` `2/3 x^(2/3) + 1/2 x^2 +c` `2/3 x^(3/2) + 2x^(1/2) +c` `3/2 x^(3/2) +1/2 x^(1/2) +c` ANSWER :C

Discussion

982.	Two tangentsare drawfroma pointP, 40 units fromthe centreof the circleand inclinedto eachother at60^(@). Whatis the area , to the nearet integer, of theregionAXBP ?
Answer» ANSWER :72

Discussion

983.	One plane is parallel to the vectors hat i + hat j + hat k and 2 hat i Other plane is parallel to the vectroshat i + hat j andhat i - hat k . The angle between the line of intersection of both the planes and the vector 2 hat i - hat j is ................
Answer» `COS^(-1)"" (3/(sqrt(50)))` `cos^(-1)""( 2/(sqrt(30)))` `cos^(-1)""((1)/(sqrt(10)))` `cos^(-1)""(19/(sqrt(30)))` Answer :C

Discussion

984.	Express in polar form (mod-Amplitude form) -1 -i
Answer» ANSWER :B::C::D

Discussion

985.	Digits 1, 2, 3, ….. 9 are written in a random order to form a 9 digit number. If the number happens to be divisible by 4. Then probability that it will be divisible by 36 is given by
Answer» `2/9` `3/4` `7/9` 1 Answer :D

Discussion

986.	Find the number of ways of arranging 8 persons around a circle
Answer» ANSWER :`7!`

Discussion

987.	int (x^(2))/(x^(6)+2x^(3)+2)dx=
Answer» Answer :`(1)/(3)TAN^(-1)(X^(3)+1)+c`

Discussion

988.	Find least non negative integer r such that 1936xx8789 -= r"(mod4)"
Answer» SOLUTION :`1936xx8789-= R ("MOD "4) ` `1936xx8789 -= 0 "mod" 4 ` `r=0`

Discussion

989.	int(sin^(3)x)/((cos^(4)x+3cos^(2)x+1)tan^(-1)(secx+cosx))dx=
Answer» `tan^(-1)(SECX+cosx)+C` `log_(e)\|tan^(-1)(secx+cosx)\|+C` `1/((secx+cosx)^(2))+C` None of these Solution :Let `I=int_(sin^(3)x)/((COS^(4)x+3cos^(2)x+1)tan^(-1)(secx+cosx))dx` `impliesI=int((sin^(3)x)/(cos^(2)x))/((cos^(2)x+3+sec^(2)x)tan^(-1)(secx+cosx))dx` `impliesI=int1/(1+(secx+cosx)^(2))XX(sinx(1-cos^(2)x))/(cos^(2)x)xx1/(tan^(-1)(secx+cosx))dx` `impliesI=int1/(tan^(-1)(secx+cosx))xx1/(1+(secx+cosx))^(2)xx(tanx secx -sinx)dx` `impliesI=int1/(tan^(-1)(secx+cosx))d{tan^(-1)(secx+cosx)}` `impliesI=log\|tan^(-1)(secx+cosx)\|+C`

Discussion

990.	int_(-3)^(3) cot^(-1)x dx=
Answer» `6PI` `3PI` 3 0 Answer :B

Discussion

991.	Rankof a null matrixis :
Answer» 1 0 does not EXIST NONE of these ANSWER :C

Discussion

992.	Evalute : int (cos 2 x - cos2 alpha)/(cos x - cos alpha)dx.
Answer» `2(sin X + x cos theta)+C` `2(sin x - x cos theta)+C` `2(sin x + 2x cos theta)+C` `2(sin x - 2x cos theta)+C` ANSWER :A

Discussion

993.	Differntiate the following functions by proper substitution.sin^(-1)((2x)/(1+x^2))
Answer» Solution :`y=sin^(-1)frac(2x)(1+x^2)`[PUT x=`tantheta` `sin^(-1)frac(2 TAN theta)(1+tan^2theta)=sin^(-1)sin2theta` `=2theta=2 tan^(-1)x` `thereforedy/dx=2/(1+x^2)`

Discussion

994.	The random variable X has a probability distribution P(X) of the following form, where k is some number.Determine the value of k P(x)={k, if x=0 ""2k, if x=1 ""3k if x=2 ""0, otherwise}
Answer» SOLUTION :`SUMP(X)=1` `RARR k+2k+3k+0=1 rArr k=1/6`

Discussion

995.	Let A=[[2,4],[3,2]] , B=[[1,3],[-2,5]] , C=[[-2,5],[3,4]] Find each of the folowing AB
Answer» SOLUTION :`AB=[[2,4],[3,2]],[[1,3],[-2,5]]=[[2xx1+4xx-2, 2xx3+4xx5],[3xx1+2xx-2, 3xx3+2xx5]]` `=[[2-8, 6+20],[3-4, 9+10]]=[[-6,26],[-1,19]]`

Discussion

996.	Solve the following Linear Programming Problems graphically : Minimise Z = 3x + 5y such that x+3y ge 3, x+y ge 2, x, y ge 0.
Answer» <P> ANSWER :The MINIMUM value of Z = 3x + 5Y is 7 at the POINT `P(3/2, 1/2)`

Discussion

997.	Find the area of the region bounded by the curves y = x^(2) + 2, y = x, x = 0 and x = 3.
Answer» ANSWER :`(21)/(2)`

Discussion

998.	Read of the following two statements I: sqrt(3)x-y+4=0 is tangent to the circle x^(2)+y^(2)=4 II: y=(sqrt(m^(2)-1))x+-mr is tangent to the circle x^(2)+y^(2)=r^(2)
Answer» I is true, II is true, II is CORRECT explanation of I. I is true,II is true, II is not correct explanation of I. I is FALSE, II is false I is true , II is true Answer :B

Discussion

999.	For what value of k, the function f(x) ={:{(2x+1", "x gt2),(""k ", " x=2),(3x-1 ", "x lt2):}, is continuous at x=2.
Answer» ANSWER :5

Discussion

1000.	Evaluate the following integrals inte^(x)((x-1))/((x+1)^(3))dx
Answer» ANSWER :`(E^(X))/((x+1)^(2))+C`

Discussion

Explore topic-wise InterviewSolutions in .

Evaluate the following determinants. [[-18,17,19],[3,0,0],[-14,5,2]]

Solve the following differential equations. x(x-1)(dy)/(dx)-(x-2)y=x^(3)(2x-1)

Let h(x) =f(x)-a(f(x))^(3) for every real number x h(x) increase as f(x) increses for all real values of x if

Find intsqrt(3-2x-x^(2))dx

If alpha(theta) epsilon R & beta(theta),theta epsilon R-{2n pi-(pi)/2, n epsilin I} are functions satistying (1+x)sin^(2)theta-(1+x^(2))sintheta +(x-x^(2))=0 then which of the following is/are correct?

Evaluate the following : [[-1,3,2],[1,3,2],[1,-3,-1]]

The term independent of x in the expansion of (1 + x)^n (1 + 1/x)^n is

Gross calorific value and physiological value of carbohydrate and liqid is :-

Solve graphically y gt 5

IF f(x) = (cosx)(cos2x)… (cosnx), then f(x)+sum_(t prop1)^(1)(rtanrx)f(x) is equal to

Find the angle between the lines, whose direction cosines are given by the equation 3l+m+5n=0 and 6mn-2nl+5lm=0.

Let f(x)=(3)/(x^(4)+3x^(2)+9) and g(x) =(x^(2))/(x^(4)+3x^(2)+9)

Let alpha" and "beta be the roots of the equation px^(2)+qx+r=0. If p,q, r in A.P. and alpha+beta=4, then alpha beta is equal to

Eco Wildlife preserve contains 5x zebras and 2x lions, where x is a positive integer. If the lions succeed in killing z of t he zebras, is the new ratio of zebras to lions less than 2 to 1 ? (1) z > x (2) z = 4

lim_(x rarr oo) (sqrt(x^(2) + 1) - root(3)(x^(3) + 1))/(root(4)(x^(4) + 1) - root(5) (x^(4) + 1)) equals :

bar(a)=2bar(i)-3bar(j)+6bar(k) and bar(b)=-2bar(i)+2bar(j)-bar(k) then ("Proj"_(bar(b))bar(a))/("Proj"_(bar(a))bar(b))=.......

Let sec ^(2)((pi)/(9))+sec^(2)((2pi)/(9))+sec_(2)((4pi)/(9))=S and Sigma_(k-1)^(89) cos^(6)(k^(o))=(a)/(b) ,where a,b are coprime, then which of the following is/are correct?

If a pair of dice is thrown 5 times then find the probability of getting three doublets.

Compute the integral I_(n) = int_(0)^(a) (a^(2) - x^(2))^(n)dx , where n is a natural number.

The third term in the expansion of (1/x + x ^[log_10 x])^5 is 10^6 then x =

If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is

If 1,alpha,alpha^2,…...,alpha^(n-1) are the n^(th) roots of unity, then the value of (3-alpha)(3-alpha^2)…...(3-alpha^(n-1)) is

Using problems are of the Inequality Type. Examples of this type are as follows:

By using the properties of definite integrals evaluate the integrals in exercise. overset(5)underset(-5)int|x+2|dx

The sumof the numbers formedfromthe digits 2,3,4,5 is

The plane x-2y + 3z= 2 makes an angle ... With Y-axis.

Evaluate int " x cot"^(2) x dx

If A =[(1,2,1),(0,1,-1),(3,-1,1)] and AA' =I , then x+y is equal to

If P, Q, R lie on xy=c^(2), then the orthocentre of DeltaPQR lies on

Choose the correct answer from the bracket. The antiderivative of (sqrtx + 1/sqrtx) equals

Two tangentsare drawfroma pointP, 40 units fromthe centreof the circleand inclinedto eachother at60^(@). Whatis the area , to the nearet integer, of theregionAXBP ?

One plane is parallel to the vectors hat i + hat j + hat k and 2 hat i Other plane is parallel to the vectroshat i + hat j andhat i - hat k . The angle between the line of intersection of both the planes and the vector 2 hat i - hat j is ................

Express in polar form (mod-Amplitude form) -1 -i

Digits 1, 2, 3, ….. 9 are written in a random order to form a 9 digit number. If the number happens to be divisible by 4. Then probability that it will be divisible by 36 is given by

Find the number of ways of arranging 8 persons around a circle

int (x^(2))/(x^(6)+2x^(3)+2)dx=

Find least non negative integer r such that 1936xx8789 -= r"(mod4)"

int(sin^(3)x)/((cos^(4)x+3cos^(2)x+1)tan^(-1)(secx+cosx))dx=

int_(-3)^(3) cot^(-1)x dx=

Rankof a null matrixis :

Evalute : int (cos 2 x - cos2 alpha)/(cos x - cos alpha)dx.

Differntiate the following functions by proper substitution.sin^(-1)((2x)/(1+x^2))

The random variable X has a probability distribution P(X) of the following form, where k is some number.Determine the value of k P(x)={k, if x=0 ""2k, if x=1 ""3k if x=2 ""0, otherwise}

Let A=[[2,4],[3,2]] , B=[[1,3],[-2,5]] , C=[[-2,5],[3,4]] Find each of the folowing AB

Solve the following Linear Programming Problems graphically : Minimise Z = 3x + 5y such that x+3y ge 3, x+y ge 2, x, y ge 0.

Find the area of the region bounded by the curves y = x^(2) + 2, y = x, x = 0 and x = 3.

Read of the following two statements I: sqrt(3)x-y+4=0 is tangent to the circle x^(2)+y^(2)=4 II: y=(sqrt(m^(2)-1))x+-mr is tangent to the circle x^(2)+y^(2)=r^(2)

For what value of k, the function f(x) ={:{(2x+1", "x gt2),(""k ", " x=2),(3x-1 ", "x lt2):}, is continuous at x=2.

Evaluate the following integrals inte^(x)((x-1))/((x+1)^(3))dx