28278 + Interview Questions in Maths in Class 12 Page 146 InterviewSolution

7251.	Find the equation of the circle passing through the intersection of the circles x^2 + y^2 = 2ax and x^2 + y^2 = byand having its center on the line x/a - y/b = 2.
Answer» ANSWER :`RARR x^2 + y^2 - 3AX + by = 0`

Discussion

7252.	Examine whether the function f given by f(x)= x^2 is continuous at x= 0.
Answer» ANSWER :HENCE, F is CONTINUOUS at x=0.

Discussion

7253.	A group of students and teachers takes a field trip, such that the student to teacher ratio is 8 to 1. The total number of people on the field trip is between 60 and 70. {:("Quantity A","Quantity B"),("The number of teachers on the field trip",6):}
Answer» ANSWER :QUANTITY A is GREATER.

Discussion

7254.	Let \|(x,2,x),(x^(2),x,6),(x,x,6)\|=Ax^(4)+Bx^(3)+Cx^(2)+Dx+E. Then the value of 5A+4B+3C+2D+E is equal to
Answer» zero `-16` 16 `-11` Solution :LET the given DETERMINANT be equal to `Delta(x)`. Then, `5A+4B+3C+2D+E=Delta(1)+Delta'(1)` Now, `Delta(1)=0` as `R_(2) and R_(3)` are IDENTICAL. `Delta'(x)=\|(1,0,1),(x^(2),x,6),(x,x,6)\|+\|(x,2,x),(2x,1,0),(x,x,6)\|+\|(x,2,x),(x^(2),x,6),(1,1,0)\|` `Delta'(1)=\|(1,2,1),(2,1,0),(1,1,6)\|+\|(1,2,1),(1,1,6),(1,1,0)\|` `=-17+(12+1-1-6)=-11`

Discussion

7255.	If the third term in the expansion of (x + x log_10 x)^5 is 10^6 then x is
Answer» 1 100 `(1)/(SQRT10)` 10 Answer :D

Discussion

7256.	A sequence a _(0) , a_(1), a _(2), a_(3)………..a _(n) …. is defined such that a _(0) =a _(1) =a and a _(n-1) = ( a _(n-1) a _(n)) + 1for n ge 1. Which of the following is true ?
Answer» ANSWER :`4A _( 2004)`

Discussion

7257.	int (sin x cos x)/(1 +cos^(4) x )dx =
Answer» `(1)/(2) cot^(-1) (COS^(2)x)+ c` `(1)/(2) TAN^(-1) (cos^(2)x ) + C ` `2 cot^(-1) (cos^(2) x ) + C ` `2 tan^(-1) (cos^(2) x )+ C ` ANSWER :A

Discussion

7258.	What is the solution set to the equation2/(7-m)=4/m-(5-m)/(7-m)?
Answer» {4,7} {4,5} {1,7} {4} ANSWER :D

Discussion

7259.	Statement-I :If x_n=cos(pi//2^n)+isin (pi//2^n) then x_1.x_2.x_3…..=-1Statement-II : x_n=cos(pi//4^n)+isin(pi//4^n) then x_1.x_2.x_3…..infty=-1
Answer» Only I is true Only II is true Both I and II are true Neither I nor II are true Answer :A

Discussion

7260.	Consider the two complex numbers z and w, such that w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1). Which of the following is the value of -(b)/(a), whenever it exists?
Answer» `3tan(theta/2)` `1/3tan(theta/2)` `-(1)/(3)cot theta` `3cot""theta/2` Answer :D

Discussion

7261.	Consider the two complex numbers z and w, such that w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1). Which of the following equals to abs(z)?
Answer» `ABS(W)` `(a+1)^(2)+B^(2)` `a^(2)+(b+2)^(2)` `(a+1)^(2)+(b+1)^(2)` ANSWER :B

Discussion

7262.	If f'(x)=x^(2)+5 and f(0)=-1 then f(x)=...
Answer» `X^(3)+5x-1` `x^(3)+5x+1` `(1)/(3)x^(3)+5x-1` `(1)/(3)x^(3)+5x+1` ANSWER :C

Discussion

7263.	Chord of constanct of tangent to (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 from any point on its director circle insterset the director circle at C and D. Find the locus of point of intersection of tangents of the circle at C and D.
Answer» ANSWER :`(a^(2)+B^(2))^(3)`

Discussion

7264.	Evaluate the following integrals int x cot^(2) x dx
Answer» Answer :`log\|sinx\|-X cot x - (1)/(2) x^(2) + C`

Discussion

7265.	There are four balls of different colours and four boxes of colours same as of the balls the number of ways in which the balls, one in each box, could be placed such that a ball does not go to a box of its own colour is
Answer» `5/8` `3/8` `1/8` `7/8` ANSWER :B

Discussion

7266.	IF a_0,a_1….,a_11 are the arithmetic progression with common difference d, then their mean deviation from their arithmetic mean is
Answer» `30/11\|d\|` `2\|d\|` `3\|d\|` `12\|d\|` ANSWER :C

Discussion

7267.	Let A,B, C be three mutually independent events. Consider the two statements S_(1)and S_(2). {:(S_(1):A and B nnC "are independent.",),(S_(2):A and B nnC "are independent.",):} Then
Answer» <P>both `S_(1) and S_(2)` are true only `S_(1)` is true only `S_(2)` is true NEITHER `S_(1)` nor `S_(2)` is true Solution :We are given that `P(AnnB)=P(A)P(B)` `P(BnnC)=P(B)P(C)` `P(CnnA)=P(C)P(A)` `P(AnnBnnC)=P(A)(B)P(C)` We have, `P(Avv(BnnC)=P(A)(B)P(C)=P(A)P(B)P(C)=P(A)P(BnnC)` `impliesA and BnnC` are independent Therefore, `S_(2)` is true. Also, `P[(Ann(BUUC)]=P[(AnnB)uu(AnnB)nn(AnnC)]` `=P(AnnB)+P(AnnC)-P(AnnBnnC)` `=P(A)P(B)+P(A)P(C)-P(A)P(B)P(C)` `P(A)[P(B)+P(C)-P(B)P(C)]` `P(A)[P(B)+P(C)-P(BnnC)]` `=P(A)P(BnnC)`

Discussion

7268.	Let R = {(3,1),(1,3),(3,3) } be a relation defined on the set A = {1,2,3}. Then , R is symmetric , transitive but not reflexive.
Answer» SOLUTION :N/A

Discussion

7269.	int sin^(2) x cos^(2) x dx =
Answer» `(1)/(2)( x -(sin 4x)/(4)) + c ` `(1)/(8)( x -(sin 4x)/(4)) + c ` `(1)/(8)( x +(sin 4x)/(4)) + c ` `(1)/(2)( x +(sin 4x)/(4)) + c ` ANSWER :B

Discussion

7270.	Let (dy)/(dx) + y = f(x) where y is a continuous function of x with y(0) = 1 and f(x) = {{:(e^(-x), if o le x le 2),(e^(-2),if x gt 2):} Which of the following hold(s) good ?
Answer» `y(1) = 2e^(-1)` `y'(1) = -e^(-1)` `y(3) = -2e^(-3)` `y'(3) = -2e^(-3)` Solution :`(dy)/(dx)+y = f(x)` is linear differential equation. `I.F. = e^(x)` `therefore"""Solution is ye"^(x) = INT e^(x) f(x) dx + C` now if `0 LE x le 2` then `ye^(x) = int e^(x) e^(-x) dx + C` `rArr""ye^(x) = x + C` `y(0) = 1, rArr C = 1` `therefore""ye^(x) = x + 1` `therefore""y=(x+1)/(e^(x))`, `therefore""y(1) = (2)/(e)` Also y' = `y'=(e^(x)-(x+1)e^(x))/(e^(2x))` `rArr""y'(1) = (e-2e)/(e^(2))=(-e)/(e^(2))=-(1)/(e)` If x `gt` 2 `ye^(x) = int e^(x-2) dx` `therefore""ye^(x) = e^(x-2) + C` `therefore""y = e^(-2) + Ce^(-x)` As y is continuous `therefore""underset(x rarr 2)("lim")(x+1)/(e^(x))=underset(x rarr 2)("lim")(e^(-2)+ Ce^(-x))` `therefore""3e^(-2) = e^(-2) + Ce^(-2) rArr C = 2` `therefore"""for x" gt 2` `y=e^(-2) + 2e^(-x)` `rArr""y(3) = 2e^(-3)+e^(-2) = e^(-2) (2e^(-1)+1)` `rArr""y'= -2e^(-x)` `rArr""y'(3) = - 2e^(-3)`

Discussion

7271.	Using differentials, find the approximate value ofeach of the following: (a)(17/81)^(1/4)""(b) (33)^(-1/5)
Answer» Solution :(a) Let `f(x)=x^(1/4)rArrf'(x) = 1/4 x ^((-3)/4)` Now, let`x = 16/81andDelta x = 1/81` `:.f(x+Delta x)~= f(x)+Delta xf'(x)` ` RARR(x+Delta x)^(1/4) ~= x^(1/4 + 1/(4x^(3/4)) xx Delta x` `rArr(16/81+1/81)^(1/4) ~=(16/81)^(1/4)+(1/81)/(4(16/81)^(3/4))` `rArr (17/81)^(1/4) ~= 2/3 + 1/(81 xx 4 (2/3)^(3))` `=2/3 + 1/96 = 65/96` `rArr(17/81)^(1/4) ~=0.677` (b) Let` f(x)=x^(-1/5) rArr f'(x)=(-1)/5 x ^(-6)/5)` Now, let x = 32AND`Delta x = 1` ` :.f(x+Delta x) ~= f(x) + Delta x f'(x)` ` rArr(x+ Delta x)^(-1//5) ~= x ^(-1//5) - 1/(5X^(6/5)) Delta x` `(33)^(-1/5) ~= (32)^((-1)/5) - 1/(5(32)^(6/5))` `= 1/2 - 1/(5 xx 2^(6)) = 0.5 - 1/320` ` = 0.5 - 0.003` ` rArr (33)^(-1/5) ~= 0.497`

Discussion

7272.	Let f(z) is of the form alpha z +beta , wherealpha, beta,z are complex numbers such that \|alpha\| ne \|beta\|.f(z) satisfies followingproperties : (i)If imaginary part of z is non zero, then f(z)+bar(f(z)) = f(barz)+bar(f(z)) (ii)If real part of of z is zero , then f(z)+bar(f(z)) =0 (iii)If z is real , then bar(f(z)) f(z) gt (z+1)^2 AA z in R (4x^2)/((f(1)-f(-1))^2)+y^2/((f(0))^2)=1 , x , y in R, in (x,y) plane will represent :
Answer» HYPERBOLA CIRCLE ellipse pair of line Answer :A

Discussion

7273.	Let f(x) is of the form alpha z _beta , where alpha, beta are constants and alpha, beta,z are complex numbers such that \|alpha\| ne \|beta\|.f(x) satisfies followingproperties : (i)If imaginary part of z is non zero, then f(x)+bar(f(z)) = f(barz)+bar(f(z)) (ii)If real part of of z is zero , then f(z)+bar(f(x)) =0 (iii)If z is real , then bar(f(x)) f(x) gt (x+1)^2 AA z in R Consider ellipse S:x^2/((Re(alpha))^2) + y^2/(Im(beta))^2) =1 , x, y in R in (x,y) plane , then point (1,1) will lie :
Answer» OUTSIDE the ELLIPSE S INSIDE the ellipse S on the ellipse S none of these Answer :B

Discussion

7274.	Assertion (A) : Orthocentre of the triangle formed by any three tangents to the parabola lies on the directrix of the parabola Reason ® : The orthocentre of the triangle formed by the tangents at t_1, t_2 ,t_3 to the parabola y^(2) =4ax is (-a ( t_1+t_2+t_3+t_1t_2t_3) )
Answer» Both A and R are true and R `RARR A ` Both A and R are true and` R CANCEL rArr A ` A is true but R is false A is false but R is true. ANSWER :A

Discussion

7275.	Express the matrix B=[{:(2,-2,-4),(-1,3,4),(1,-2,-3):}]as the sum of a symmetric and a skew symmetric matrix.
Answer» Answer :`[{:(2,-(3)/(2),-(3)/(2)),(-(3)/(2),3,1),(-(3)/(2),1,3):}]+[{:(0,-(1)/(2),-(5)/(2)),((1)/(2),0,3),((3)/(2),-3,0):}]`

Discussion

7276.	The shortest distance between the straight line passing through the point A = (6, 2,2) and parallel to the vector (1,-2,2) and the straight line passing through A (-4, 0, -1) and parallel to the vector (3,-2,-2) is
Answer» 9 8 5 2 Answer :A

Discussion

7277.	Find the number of garlands that can be made using 6 different coloured flowers taken 4 at a time
Answer» ANSWER :45

Discussion

7278.	If the line joining the points A (1, 0, 0) andB(0, 0, 1) is a normal to the plane (pi)which passes through the point A, them the angle between the planes (pi) and x+y+z=6 is
Answer» `(PI)/(6)` `(pi)/(4)` `(pi)/(3)` `(pi)/(2)` ANSWER :A

Discussion

7279.	If 1, 2, 3 and 4 are the roots of the equation x^(4)+ax^(3)+bx^(2)+cx+d=0 , then a+2b+c is equal to
Answer» -25 0 10 24 Answer :C

Discussion

7280.	The value of cosec 20^(@).cosec40^(@).cosec60^(@).cosec80^(@) is
Answer» `3//16` `16//3` `5//16` `16//5` ANSWER :B

Discussion

7281.	If the 4th, 10th and 16th terms of a G.P. are x,y and z respectively,then x,y,z are in
Answer» G.P. A.P. H.P. A.G.P. Answer :A

Discussion

7282.	Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY-plane.
Answer» ANSWER :`((13)/(5),(23)/(5),0)`

Discussion

7283.	A car accelerates from rest at a constant rate of 2 m//s^(2) and comes to rest. If it remains in motion for 3 seconds, then the maximum speed attained by the car is :
Answer» 2 m/s 3 m/s 4 m/s 6 m/s Answer :A

Discussion

7284.	The value of m for which straight line 3x-2y+z+3=0=4x-3y+4z+1 is parallel to the plane 2x-y+mz-2=0 is ___
Answer» `-2` 8 -18 11 Answer :A

Discussion

7285.	A factory produces lazor blades and 1 in 500 blades is estimated to be defective. The blades are supplied in packetsof 10. IN a consignment of 10,000 packets, using poisson distribution, find approximately the number of packets which contain no defective blades. (Given e^(-0.02) = 0.9802)
Answer» ANSWER :9802

Discussion

7286.	At what point, the slope of the curve y=-x^(3)+3x^(2)+9x-27 is maximum ? Also find the maximum slope.
Answer» ANSWER :`=-16`

Discussion

7287.	Find the value of a so that the points (1,4),(2,7),(3,a) are collinear.
Answer» SOLUTION :As the point (1,4),(2,7),(3,a) are COLLINEAR we have the area of the TRIANGLE with vertices (1,4),(2,7),(3,a) is zero. `THEREFORE` `1/2{1(7-a)+2(a-4)+3(4-7)}=0` or, 7-a+2a-8+12-21=0. `implies` a=10.

Discussion

7288.	Resolve (2x^(3)+3x^(2)+5x+7)/((x+1)^(5)) into partial fractions.
Answer» ANSWER :`:. (2x^(3)+3X^(2)+5x+7)/((x+1)^(5)) = - (2)/((x+1)^(2)) - (3)/((x+1))+(5)/((x+1)^(4)) + (3)/((x+1)^(5))`

Discussion

7289.	If the foot of perpendicular drawn from the point (1, 0, 3) on a line passing through ( alpha, 7, 1) is((5)/(3), (7)/(3), (17)/(3)), then alpha is equal to …………
Answer» 3 6 5 4 Answer :D

Discussion

7290.	Evaluate the following integrals : (i) int(1)/((x+1)sqrt(2x-3))dx (ii) int(1)/((x+1)sqrt(x^(2)+x-1))dx (iii) int(1)/((x^(2)+3x+3)sqrt(x+1))dx (iv) int(1)/((x^(2)-3x+2)sqrt(x^(2)-2))dx
Answer» Solution :`(i)` We have, `int(1)/((x+1)sqrt(2x-3))dx` Let us put `2x-3=t^(2)` so that `dx=tdt` and `x=(t^(2)+3)/(2)impliesx+1=(t^(2)+5)/(2)` `=int(tdt)/((t^(2)+5)/(2).t)=2int(1)/(t^(2)+5)dt=(2)/(sqrt(5))tan^(-1)((t)/(sqrt(5)))+C` `==(2)/(sqrt(5))tan^(-1)((sqrt(2x-3))/(sqrt(5)))+C` `(ii)` We have, `(1)/((x+1)sqrt(x^(2)+x-1))dx` Let us put `(1)/(x+1)=t` so that `x=(1)/(t)-1` and `dx=-(1)/(t^(2))dt` `=int(t)/(sqrt((1)/(t^(2))-(2)/(t)+1+(1)/(t)-1-1))xx-(1)/(t^(2))dt` `=-int(1)/(sqrt(1-t-t^(2)))dt=-int(1)/(sqrt((5)/(4)-(t+(1)/(2))^(2)))dt` `=cos^(-1)((t+(1)/(2))/((sqrt(5))/(2)))+C` `=cos^(-1)((2T+1)/(sqrt(5)))+C=cos^(-1)((x+3)/(sqrt(5)(x+1)))+C` `(iii) int(1)/((x^(2)+3x+3)sqrt(x+1))dx` Let us put `x+1=t^(2)` so that `dx=2tdt` and `x^(2)+3x+3=(t^(2)-1)^(2)+3(t^(2)-1)+3` `=t^(4)-2t^(2)+1+3t^(2)-3+3` `=t^(4)+t^(2)+1` `=int(1)/((t^(4)+t^(2)+1)t)2tdt` `=int(2)/(t^(4)+t^(2)+1)dt=int((2)/(t^(2)))/(t^(2)+(1)/(t^(2))+1)dt` `=int((1+(1)/(t^(2)))-(1-(1)/(t^(2))))/(t^(2)+(1)/(t^(2))+1)` `=int(1+(1)/(t^(2)))/((t-(1)/(t))^(2)+3)dt-int(1-(1)/(t^(2)))/((t+(1)/(t))^(2)-1)dt` `=int(d(t-(1)/(t)))/((t-(1)/(t))^(2)+3)-int(d(t+(1)/(t)))/((t+(1)/(t))^(2)-1)dt` `=(1)/(sqrt(3))tan^(-1)((t-(1)/(t))/(sqrt(3)))-(1)/(2)ln\|(t+(1)/(t)-1)/(t+(1)/(t)+1)\|+C` `=(1)/(sqrt(3))tan^(-1)((x)/(sqrt(3(x+1))))-(1)/(2)ln\|((x+2)-sqrt(x+1))/((x+2)+sqrt(x+1))\|+C`, where `C` is a constant of INTEGRATION. `(iv)` We have, `int(1)/((x^(2)-3x+2)sqrt(x^(2)-2))dx=int(1)/((x-1)(x-2))*(1)/(sqrt(x^(2)-2))dx` `=int((1)/(x-2)-(1)/(x-1))(1)/(sqrt(x^(2)-2))dx` `=int(1)/((x-2)sqrt(x^(2)-2))dx-int(1)/((x-1)sqrt(x^(2)-2))dx` `=l_(1)-l_(2)`,SAY Now `l_(1)=int(1)/((x-2)sqrt(x^(2)-2))dx` `=int(txxt)/(sqrt(2t^(2)+4t+1))xx-(1)/(t^(2))dt` Let us put `x-2=(1)/(t)` `=(-1)/(sqrt(2))int(1)/(sqrt(t^(2)+2t+(1)/(2)))dt` `dx=-(1)/(t^(2))dt` `=(-1)/(sqrt(2))int(1)/(sqrt((t+1)^(2)-(1)/(2)))dt` `x=(1)/(t)+2=(1+2t)/(t)` `=-(1)/(sqrt(2))ln\|(t+1)+sqrt(t^(2)+2t+(1)/(2))\|+C_(1)` `x^(2)-2=(1+4t+4t^(2)-2t^(2))/(t^(2))` `=-(1)/(sqrt(2))ln\|(x-1)/(x-2)+(sqrt(x^(2)-2))/(sqrt(2)(x-2))\|+C_(1)`,`C_(1)` is a constant of integration. and `l_(2)=int(1)/((x-1)sqrt(x^(2)-2))dx` Let us put `x-1=(1)/(u)` `=int(uxxu)/(sqrt(1+2u-u^(2))xx-(1)/(u^(2))du` `dx=-(1)/(u^(2))du` `=-int(1)/(sqrt(2-(u+1)^(2))du` `x^(2)-2=((1)/(u)+1)^(2)-2` `=cos^(-1)((u-1)/(sqrt(2)))+C_(2)=cos^(-1)((x-2)/(sqrt(2)(x-1)))+C_(2)` `=(1+2u-u^(2))/(u^(2))` Hence, `l=int(1)/((x^(2)-3x+2)sqrt((x^(2)-2)))dx=-(1)/(sqrt(2))ln\|(x-1)/(x-2)+(sqrt(x^(2)-2)/(sqrt(2)(x-2))\|-cos^(-1)((x-2)/(sqrt(2)(x-1)))+k` where `k` is a constant.

Discussion

7291.	F: R to Rdefinedbyf (x)= (1)/(2x^2 +5)therangeof F is
Answer» A. `(5,OO)` B. `[0,1/5]` C. `[1/5 ,5]` D. noneof these Answer :D

Discussion

7292.	The probabilities of solving a problem by three students A,B,C independently are 1/3,1/4,1/5. The probability that the problems will be solved is………
Answer» `(1)/(60)` `(3)/(5)` `(48)/(60)` `(57)/(60)` ANSWER :B

Discussion

7293.	Find the scalar and vector projection of veca on vecb. veca = hati,vecb = hatj
Answer» SOLUTION :Scalar PROJECTION of `VECA` on `vecb` = `(veca.vecb)/\|vecb\| =(hati.hatj)/\|hatj\| = 0[becausehati.hatj = 0]` VECTOR projection of `veca` on `vecb = VEC0`

Discussion

7294.	From the two system of lines x - h = 0 where 0lehle5 and y - k = 0 where 0lekle3,how many squares can be formed.
Answer» ` ANSWER :26

Discussion

7295.	If A(3,4,5),B(4,6,3),C(-1,2,4) and D(-1,2,4) are suchthat the angle betweenthe lines DC and AB is theta, then costheta is equal is
Answer» `7/9` `2/9` `4/9` `5/9` ANSWER :C

Discussion

7296.	Determine whether or not each of the definition of given below gives a binary operation. In the even that is not a binary operation , give justification for this . On Z^(+) , define ** by a" * " b = a - b
Answer» SOLUTION :N/A

Discussion

7297.	Determine whether or not each of the definition of given below gives a binary operation. In the even that is not a binary operation , give justification for this . On Z^(+) , define *by a" " b = ab
Answer» SOLUTION :N/A

Discussion

7298.	The modulus-amplitude form of ((1-i)^(3) (2-i))/((2+i) (1 +i)) is
Answer» `2 CIS (pi "tan"^(-1) (4)/(3))` `2 cis (-"tan"^(-1) (4)/(3))` `2 cis (-pi + "tan"^(-1) (4)/(3))` `2 cis ("tan"^(-1) (4)/(3))` Answer :A

Discussion

7299.	3. ^n C_0-8. ^n C_1 + 13. ^n C_1 - 18 , ^n C_3 + ….. (n+1) terms =
Answer» 0 `-1` 1 2 Answer :A

Discussion

7300.	Find the centre and radius of each of the circles whose equations are given below. 3x^(2) + 3y^(2) - 5x - 6y + 4 =0
Answer» ANSWER :` ((5)/( 6), 1) , ( SQRT(13))/(6) `

Discussion

Explore topic-wise InterviewSolutions in .

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Consider the two complex numbers z and w, such that w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1). Which of the following is the value of -(b)/(a), whenever it exists?

Consider the two complex numbers z and w, such that w=(z-1)/(z+2)=a+ib, " where " a,b in R " and " i=sqrt(-1). Which of the following equals to abs(z)?

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There are four balls of different colours and four boxes of colours same as of the balls the number of ways in which the balls, one in each box, could be placed such that a ball does not go to a box of its own colour is

IF a_0,a_1….,a_11 are the arithmetic progression with common difference d, then their mean deviation from their arithmetic mean is

Let A,B, C be three mutually independent events. Consider the two statements S_(1)and S_(2). {:(S_(1):A and B nnC "are independent.",),(S_(2):A and B nnC "are independent.",):} Then

Let R = {(3,1),(1,3),(3,3) } be a relation defined on the set A = {1,2,3}. Then , R is symmetric , transitive but not reflexive.

int sin^(2) x cos^(2) x dx =

Let (dy)/(dx) + y = f(x) where y is a continuous function of x with y(0) = 1 and f(x) = {{:(e^(-x), if o le x le 2),(e^(-2),if x gt 2):} Which of the following hold(s) good ?

Using differentials, find the approximate value ofeach of the following: (a)(17/81)^(1/4)""(b) (33)^(-1/5)

Assertion (A) : Orthocentre of the triangle formed by any three tangents to the parabola lies on the directrix of the parabola Reason ® : The orthocentre of the triangle formed by the tangents at t_1, t_2 ,t_3 to the parabola y^(2) =4ax is (-a ( t_1+t_2+t_3+t_1t_2t_3) )

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

The shortest distance between the straight line passing through the point A = (6, 2,2) and parallel to the vector (1,-2,2) and the straight line passing through A (-4, 0, -1) and parallel to the vector (3,-2,-2) is

Find the number of garlands that can be made using 6 different coloured flowers taken 4 at a time

If the line joining the points A (1, 0, 0) andB(0, 0, 1) is a normal to the plane (pi)which passes through the point A, them the angle between the planes (pi) and x+y+z=6 is

If 1, 2, 3 and 4 are the roots of the equation x^(4)+ax^(3)+bx^(2)+cx+d=0 , then a+2b+c is equal to

The value of cosec 20^(@).cosec40^(@).cosec60^(@).cosec80^(@) is

If the 4th, 10th and 16th terms of a G.P. are x,y and z respectively,then x,y,z are in

Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XY-plane.

A car accelerates from rest at a constant rate of 2 m//s^(2) and comes to rest. If it remains in motion for 3 seconds, then the maximum speed attained by the car is :

The value of m for which straight line 3x-2y+z+3=0=4x-3y+4z+1 is parallel to the plane 2x-y+mz-2=0 is ___

A factory produces lazor blades and 1 in 500 blades is estimated to be defective. The blades are supplied in packetsof 10. IN a consignment of 10,000 packets, using poisson distribution, find approximately the number of packets which contain no defective blades. (Given e^(-0.02) = 0.9802)

At what point, the slope of the curve y=-x^(3)+3x^(2)+9x-27 is maximum ? Also find the maximum slope.

Find the value of a so that the points (1,4),(2,7),(3,a) are collinear.

Resolve (2x^(3)+3x^(2)+5x+7)/((x+1)^(5)) into partial fractions.

If the foot of perpendicular drawn from the point (1, 0, 3) on a line passing through ( alpha, 7, 1) is((5)/(3), (7)/(3), (17)/(3)), then alpha is equal to …………

Evaluate the following integrals : (i) int(1)/((x+1)sqrt(2x-3))dx (ii) int(1)/((x+1)sqrt(x^(2)+x-1))dx (iii) int(1)/((x^(2)+3x+3)sqrt(x+1))dx (iv) int(1)/((x^(2)-3x+2)sqrt(x^(2)-2))dx

F: R to Rdefinedbyf (x)= (1)/(2x^2 +5)therangeof F is

The probabilities of solving a problem by three students A,B,C independently are 1/3,1/4,1/5. The probability that the problems will be solved is………

Find the scalar and vector projection of veca on vecb. veca = hati,vecb = hatj

From the two system of lines x - h = 0 where 0lehle5 and y - k = 0 where 0lekle3,how many squares can be formed.

If A(3,4,5),B(4,6,3),C(-1,2,4) and D(-1,2,4) are suchthat the angle betweenthe lines DC and AB is theta, then costheta is equal is

Determine whether or not each of the definition of ** given below gives a binary operation. In the even that ** is not a binary operation , give justification for this . On Z^(+) , define ** by a" * " b = a - b

Determine whether or not each of the definition of ** given below gives a binary operation. In the even that ** is not a binary operation , give justification for this . On Z^(+) , define **by a" * " b = ab

The modulus-amplitude form of ((1-i)^(3) (2-i))/((2+i) (1 +i)) is

3. ^n C_0-8. ^n C_1 + 13. ^n C_1 - 18 , ^n C_3 + ….. (n+1) terms =

Find the centre and radius of each of the circles whose equations are given below. 3x^(2) + 3y^(2) - 5x - 6y + 4 =0

Determine whether or not each of the definition of given below gives a binary operation. In the even that is not a binary operation , give justification for this . On Z^(+) , define ** by a" * " b = a - b

Determine whether or not each of the definition of given below gives a binary operation. In the even that is not a binary operation , give justification for this . On Z^(+) , define **by a" * " b = ab