28278 + Interview Questions in Maths in Class 12 Page 187 InterviewSolution

9301.	Evaluate the following integrals. int(e^(logx))/(x)dx
Answer» ANSWER :`E^(LOGX)+C=x+C`

Discussion

9302.	The two successive terms in the expansion (1+x)^24 whose coeff's are in the ratio 4:1 are
Answer» 18TH , 19th 19th, 20th 20th, 21th 21th, 22nd Answer :C

Discussion

9303.	Let M be a 2xx2 symmetric matrix with integer entries. Then , M is invertible, if
Answer» the FIRST column of M is the transpose of the second row of M The second row of M is the transpose of THEFIRST column of M m is a diagonal MATRIX with non- zero entries in the main diagonal the product of entries in the main diagonal of M is not the square of an integer Solution :Let `M= [[a,b],[c,d]]`, where `a, b, c, in I` M is invertible if`abs((a,b),(b,c)) ne 0 rArr AC- b^(2) ne 0 ` (a) `[[a],[b]]=[[b],[c]]rArr a = b =c rArr ac-b^(2)=0` `therefore` Option (a) is incorrect (b) `[(b,c)]= [(a,b)] rArr a = b = c rArr ac - b^(2) = 0` `therefore` Option (b) is incorrect (c) `M= [[a,0],[0,c]], ` then` abs(M) = ac ne 0` `therefore` M is invertible `therefore` Potion ( c) is correct. (d) As `acne"Integre """^(2)rArrac ne b^(2)` `therefore ` Option (d)is correct.

Discussion

9304.	10 g of hydrogen and 64 g of oxygen were filled in a steed vessel and exploded. Amount of water produced in this reaction will be :-
Answer» 2 MOLE 3 mole 4 mole 1 mole Answer :A

Discussion

9305.	A conic has latus rectum length 1, focus at (2,3)and the corresponding directrix is x+y -3=0 . Then the conic is
Answer» a parabola an ELLIPSE a HYPERBOLA a RECTANGULAR hyperbola ANSWER :C

Discussion

9306.	If A = {1,2,3,4} , define relations on A which have properties of being : Reflexive , symmetric and transitive .
Answer» SOLUTION :N/A

Discussion

9307.	Using the proprties of determinants in Exercise 7 to 9, prove that \|{:(y+z,z+y),(z,z+x,x),(y,x,x+y):}\|=4xyz
Answer» ANSWER :4XYZ

Discussion

9308.	If A=[[1,2],[-2,3]]B=[[3,2],[1,4]],C=[[2,2],[1,3]]Calculate CA.
Answer» SOLUTION :`CA=[[2,2],[1,3]][[1,2],[3,4]]` `=[[2.1+2.3""2.2+2.4],[1.1+3.3" "1.2+3.4]]=[[8,12],[10,14]]`

Discussion

9309.	Coefficient of x^(m) in the expansion of S = (1 + x)^(2m) + x(1 +·x)^(2m-1) + x^(2)(1 + x)^(2m- 2) + .... +x^(2m) is
Answer» `""^(2M)C_(m)+""^(2m+1)C_(m-1)` `""^(2m+1)C_(m)` `0` `2m^(2)+2m+1` Answer :B

Discussion

9310.	For non-negative integers n, let int(n)=(sum_(k=0)^(n)((k+1)/(n+2)pi)sin((k+2)/(n+2)pi))/(sum_(k=0)^(n)sin^2((k+1)/(n+2)pi)) Assuming cos^(-1)x takes values in [0.pi], which of the following options is/are correct?
Answer» If `mu=tan (cos^(-1)//(6),then" "ALPHA^2+2alpha-1=0` `f(4)=(sqrt(3))/(2)` `sin(7cos^(-1)f(5))=0` lim_(ntooo)f(n)=(1)/(2)` Solution : it is GIVEN that for non-negative integers 'n'. `int (n)=(sum _(k=0)^(n)sin((k+1)/(n+2))sin((k+2)/(n+2)pi))/(sum_(k=0)^(n)sin^2((k+1)/(n+2)pi))` `int (n)=(sum _(k=0)^(n)(cos.(pi)/(n+2)-cos((2k+3)/(n+2)pi)))/(sum_(k=0)^(n)(1-cos((2k+2)/(n+2)pi)))` `[because 2 sin B=cos(A-B)-cos (A+B)and 2sin^2A=1-cos 2A]` = ((cos(pi/(n+2)))sum_(k=0)^(n)1-{{:(cos'(3pi)/(n+2)+,cos'(5pi)/(n+2)+cos'(7pi)/(n+2)),(,+"........"+cos'((2n+3)/(n+2)pi)):}})/(sum_(k=0)^(n)1-{{:(cos'(2pi)/(n+2)+,cos'(4pi)/(n+2)+cos'(6pi)/(n+2)),(,+"........"+cos'((2n+2)/(n+2)pi)):}})` `[because cos (alpha)+cos(alpha+beta)+ cos (alpha+2beta)+.......` `+cos (alpha)+cos (alpha+beta)+cos(alpha+beta)+...` `+cosalpha+(n-1)(beta)=(sin((nbeta)/2))/(sin((beta)/2))cos((2alpha+(n-1)beta)/(2))]` `=((n+1)cos((pi)/(n+2))-(sin(pi-(pi)/(n+2)))/(sin((pi)/(n+2)))cos(pi+(pi)/(n+2)))/((n+1)-(sin((pi)/(n+2)))/(sin((pi)/(n+2)))cos(pi))` `=((n+1)cos((pi)/(n+2))+(sin((pi)/(n+2)))/(sin((pi)/(n+2)))cos((pi)/(n+2)))/((n+1)+(sin((pi)/(n+2)))/(sin((pi)/(n+2))))` `=((n+2)cos((pi)/(n+2)))/((n+2))=cos ((pi)/(n+2))` `rArr f(n)=cos ((pi)/(n+2))` Now, `f(6)=cos(pi)/(8)` `because alpha =tan cos^-1f(((6))=tan(pi)/(8)""{{:(because cos^-1cosx =X),(if x in (0,(pi)/(2))):}}` `=sqrt(2)-1` `rArr (alpha+1)=sqrt(2)rArr(alpha+1)^2=2rArr alpha^2+2alpha+1=2` `rArr alpha^2+2alpha-1=0` Now, `f(4)=cos ((pi)/(4+2))=cos (pi)/(6)=(sqrt(3))/(2)`, Now, `sin (7cos ^-1f(5))=sin(7cos^-1((pi)/(5+2)))` `sin(7(pi)/(7))=sinpi=0` and Now, `LIM(NTO oo)f(x)=lim(n to oo) cos (pi)/(n+2)=cos=1` Hence, options (a),(b) and (c) are correct.

Discussion

9311.	If z is a complex number such that (z-1)/(z+1) is purely imaginary then \|z\|=
Answer» `1` `2` `3` `5` ANSWER :A

Discussion

9312.	Decrypt the received encoded message [2,-3][20,4] with the encryption matrix [{:(-1,-1),(2,1):}] and the decryption matrix as its inverse, where the system of codes are described by the numbers 1-26 to the letters A-Z respectively, and the number 0 to a blank space.
Answer» ANSWER :`=["1216"]`

Discussion

9313.	If A, B and C are three independent events such that P(A) = P(B) = P(C )= p,then P ( Atleasttwo of A, B, C occur) = 3p^(2) - 2p^(3).
Answer» ANSWER :1

Discussion

9314.	If a circle passes through the point (a,b) and cuts the circle x^2+y^2=k^2 orthogonally then the locus of its centre is
Answer» `2ax+2by=a^2+b^2+k^2` `ax+by=a^2+b^2+k^2` `x^2+y^2+2ax+2by+k^2=0` `x^2+y^2-2ax+2by+k^2=0` ANSWER :A

Discussion

9315.	If t=e^athen d/dx x^t=_______
Answer» `TX^(t-1)` `x^t` `x^t `In a `tx^1` ANSWER :A

Discussion

9316.	Write the distance of the point (2,3,6) from zx-plane.
Answer» Solution :The DISTANCE of p (x,y,Z) from orgin = 4 `rArrsqrt(x^2+y^2+z^2)=4rArrx^2+y^2+z^2=4` is the REQUIRED locus.

Discussion

9317.	If two tangents drawn from a point P to the parabola y^(2)=4x are at right angles then the locus of P is
Answer» ` 2X+ 1=0` ` x=-1` ` 2x-1=0 ` ` x=1` ANSWER :B

Discussion

9318.	Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt. y=f(x) is
Answer» injective but not surjective surjective but not injective bijective neither injective nor surjective Solution :`f(x)=x^(2)+int_(0)^(x)e^(-t)(x-t)dt`………….1 `x^(2)+int_(0)^(x)e^(-x-t)f(x-(x-t))dt` [Using `int_(a)^(b)f(x)DX=int_(a)^(b)f(a+b-x)dx`] `=x^(2)+e^(-x)int_(0)^(x)e^(t)f(t)dt`……………..2 Differentiating w.R.t.`x` we get `f'(x)=2x-e^(-x)int_(0)^(x)e^(t)f(t)dt+e^(-x)e^(x)f(x)` `=2x-e^(-x)int_(0)^(x)e^(t)f(t)dt+f(x)` `=2x+x^(2)` [using equation 2] `:. f(x)=(x^(3))/3+x^(2)+c` Also `f(0)=0` [from equation 1] or `f(x)=(x^(3))/3+x^(2)` or `f'(x)=x^(2)+2x` Thus `f'(x)=` has real roots. Hence `f(x)` is non monotonic. Hence `f(x)` is many one but range is `R` and hence, is surjective `int_(0)^(1)f(x)dx=int_(0)^(1)((x^(3))/3+x^(2))dx` `=[(x^(4))/12+(x^(3))/3]_(0)^(1)` `=1/12+1/3` `=5/12`

Discussion

9319.	The solution of (dy)/(dx) = 2xy - 3y + 2x - 3 is
Answer» `E^(x^(2)) + 3X = C(y+1)` `e^(x^(2)) - 3x = c(2y+1)` `e^(x^(2)) - 3x = c(y-1)` `e^(x^(2)-3x) = c(y+1)` Answer :D

Discussion

9320.	A coin tossed twice. Find the probability of getting at most one head
Answer» Answer :A coin is TOSS twice. `THEREFORE S={HH,HT,TH,T T}, \|S\|=4` Let A be the event of GETTING exactly one HEAD. `thereforeA={HT,TH,T T}implies\|A\|=3` `P(A)=\|A\|/\|S\|=3/4`

Discussion

9321.	Two numbers are selected from the numbers 1 to 11. If there sum is even find the probability of an event that both numbers are odd.
Answer» ANSWER :`(3)/(5)`

Discussion

9322.	Coefficient of x^n in e^(e^x)is 1/(n!) k then k =
Answer» `1+1/(1!) +1/(2!) +.......OO` `1+(2^n)/(2!) +(3N)/(3!) +.......oo` `1/(2!) -1/(3!) +1/(4!) -1/(5!)+.......oo` `1-(2^n)/(2!) +(3^n)/(3!) + ....oo` ANSWER :B

Discussion

9323.	Themaximum or minimum of the objective funtion occurs only at the corner points of the feasible region. This theorem is known as fundamental theorem of
Answer» Algebra Arithmetic Calculus Extreme point Answer :D

Discussion

9324.	The image of the point (4, -13) with respect to the line 5x+y+6=0 is :
Answer» (-1, -14) (3, 4) (1, 2) (-4, 13) ANSWER :A

Discussion

9325.	If the chords of contacts of the tangents from the points (x y,) and (x_(2), y_(2)) to the hyperbola 2x^(2) - 3y^(2) = 6 are at right angle, then 4x_(1)x_(2) + 9y_(1)y_(2) is equal to
Answer» `-1` 0 6 `-12` ANSWER :B

Discussion

9326.	Sketch theregion{(x,0) //y= sqrt( 4-x^2)} andX- axis. Findthe areaof theregionusingintegration.
Answer» ANSWER :`2 PI` SEQ. UNIT

Discussion

9327.	Lt_(ntooo)sum_(r=0)^(n-1)(1)/(n+r)=
Answer» `PI` 0 log 2 `-pi` ANSWER :C

Discussion

9328.	Ratio of seventh term from beginning and seventh term from end in the expansion of (root(3)(2)+(1)/3_sqrt(3))^(n)"is"1 : 6 .Find value of n .
Answer» ANSWER :N = 9

Discussion

9329.	Complementary events of E and Fare E' and F' respectively if 0 lt P(F) lt 1 then …………
Answer» <P>`P(E//F) + P(E'//F) = 1` `P(E//F) + P(E'//F') = 1` `P(E'//F) + P(E//F') = 1` `P(E//F) + P(E'//F') = 1` ANSWER :A

Discussion

9330.	Special fluid helps in transportation of substances in complex animals is/are :-
Answer» BLOOD and lymph Only blood Only lymph RBC, WBC and platelets Answer :A

Discussion

9331.	(i) y=e^(x)sin^(3)xcos^(4)x (ii) y=x*e^(xsinx)
Answer» Answer :(i) `e^(X)sin^(3)XCOS^(4)x[1+3cotx-4tanx]` (ii) `e^(XSINX)(xcosx+sinx)`

Discussion

9332.	(1)/(3sqrt(8.16))~~....
Answer» `0.4` `0.48` `0.4997` `0.488` ANSWER :C

Discussion

9333.	Use induction to prove that, int _(0) ^(pi//2) cos ^(n-1) x sin nx dx = AA n ge 2, n in N
Answer» ANSWER :`p(k) + int _(0)^(pi//2) cos ^(k -2) sin x. cos (k +1) x DX`

Discussion

9334.	Let f :R to R is not identically zero, differentiable function and satisfy the equals d (xy)= f(x) f(y) and f (x+z) + f(x) + f (a), then f (5)=
Answer» 3 5 10 15 Answer :B

Discussion

9335.	Find the area bounded by the the curve y=cos x between x=0 and x=2pi.
Answer» ANSWER :4

Discussion

9336.	A chemist wishes to provide his consumers, at least cost, the minimum daily requirements of two vitamins A, B by using a mixture of two products M and N. The amount of each vitamin in one gram of each product, cost per gram of each product and the minimum daily requirement are given below: Find the least expensive combination which provides the minimum requirement of the two vitamins.
Answer» Answer :1 GM of product M and 3 GMS of product N.

Discussion

9337.	A car moving with a speed of 40 km/hr can be stopped by applying brakes after at least 2 m. If the same car is moving with a speed of 80 km/hr. What is the stopping distance ?
Answer» 2 m 4 m 6 m 8 m Answer :A

Discussion

9338.	Matrices of any order can be added .
Answer» ANSWER :FALSE

Discussion

9339.	The value of int_(0)^(a)(sqrt(a^(2)-x^(2)))^(3) dx is……. .
Answer» `(pia^(3))/(16)` `(3pia^(4))/(16)` `(3pia^(2))/(8)` `(3pia^(4))/(8)` ANSWER :B

Discussion

9340.	If f(x) = sin^(-1) x. cos^(-1) x. tan^(-1) x . cot^(-1) x. sec^(-1) x. cosec^(-1) x, then which of the following statement (s) hold(s) good?
Answer» The graph of y = F (x) does not lie above x axis The non - NEGATIVE difference between maximumminimum value of the function ` y = f (x) " is " (3 PI^(6))/64` The function y = f (x) is not injective. Number of non-negative INTEGERS in the domain of f (x) is two . Answer :A::B

Discussion

9341.	Find the number of ways of arranging the letters of the word SINGING so that the two G's come together
Answer» ANSWER :180

Discussion

9342.	If I= int_(0)^(pi//4) ( sqrt(sin x) + sqrt(cos x) )^(-4) dx then I equals
Answer» ANSWER :`0.08`

Discussion

9343.	Let Z be the set of integers and o be a bonary operation of z defined as aob = a + b - ab for all a, b in Z. The inverse of an element a (ne 1 ) in Z is
Answer» `a//(a - 1)` `a//(1 - a)` `(a - 1)//a` NONE of these Answer :A

Discussion

9344.	If alpha, beta , gamma are the roots of x^(3) + 3px + q = 0then the equation whose roots are(alpha + 1)/( beta + gamma - alpha), (beta + 1)/(gamma + alpha - beta) and (gamma + 1)/(alpha + beta - gamma) is
Answer» `8Y^(3) + 12y^(2) + (6 + 6P) y + 1 + 3p - q = 0 ` `8qy^(3) - 12y (q + p) y^(2) + 6(q - 2p)y + (q + 3p - 1 ) = 0` `8qy^(3) + 12 (q + p) y^(2) + 6(q - 2p)y + (q + 3p - 1) = 0 ` `8qy^(3) - 12(q - p) y^(2) - 6(q - 2p) y + (q - 3p + 1) = 0` Answer :4

Discussion

9345.	Let f(x) = [{:((g(x) - g(a))/(x-a),x ne 0),(g'(a),x = a):}, where g is a function derivable at x = a, then at x = a :
Answer» F is continuous only if G'(a) = 0 f is continuous f is DISCONTINUOUS None of these Answer :B

Discussion

9346.	If the orthocentre of the triangle formedby the lines 2x+3y-1=0, x+2y-1=0, ax+by-1=0 is at the origin, then (a, b) is given by
Answer» (6,4) (-3, 3) (-8, 8) (0, 7) ANSWER :C

Discussion

9347.	Equation of a line passing through (-1, 2, -3) and perpendicular to the plane 2x + 3y + z + 5=0 is
Answer» `(x-1)/(-1)=(y+2)/(1)=(z-3)/(-1)` `(x+1)/(-1)=(y-2)/(1)=(z+3)/(1)` `(x+1)/(2)=(y-2)/(3)=(z+3)/(1)` none Answer :C

Discussion

9348.	Evaluate the following integrals int (x^3-x^2 +x -1)/(x-1) dx
Answer» SOLUTION :`INT (x^3-x^2+x-1)/(x-1) DX = int(x^2(x-1)+x-1)/(x-1) dx = int (x^2+1)dx = x^3/3 +x+c`

Discussion

9349.	Consider the parabola x ^(2) + 4x - 2y + 6 = 0. If its chord AB 2mx - 2y + (4m + 3) =0 intersect its axis at K and directrix at M, then AM, KM and BM are in
Answer» A.P G.P H.P none of these ANSWER :C

Discussion

9350.	You are running out of time but have no idea about the route from Nepal to IIT Guwahati. The various cut-out parts of the map that can help you reach IIT Guwahati are hidden in the houses shown in the figure below. You will get the parts of the map only on delivering the right article(one per house) to all the houses. (i)An ‘X’ coloured ARTICLE should be delivered to an ‘X’ coloured HOUSE only by using an ‘X’ colouredTRUCK. (ii)A truck can also carry other coloured articles so that it can place them at any of the CHECKPOINTS from where other truck can later carry it to the destination. (iii) Trucks can carry any number of articles at a time. (iv) The main objective is to start all the trucks at the same time, with same speed and deliver the articles without collision of trucks. Also, a truck picks up every article that comes on its way. (v)Last and the most importantly, the PATHS of the trucks should not overlap at any point other thanjunctions.At junctions, the paths can though cross each other (but both trucks should not reach that junction at same time which leads to collision). The trucks can move only forward. Give the total number of articles carried by the red truck on it’s way.
Answer» 1 2 3 4 Solution :B From the above FIGURE, No. of turns taken by BLUE truck = 6. No. of articles CARRIED by RED truck = 2 ( 1red + 1blue ).

Discussion

Explore topic-wise InterviewSolutions in .

Evaluate the following integrals. int(e^(logx))/(x)dx

The two successive terms in the expansion (1+x)^24 whose coeff's are in the ratio 4:1 are

Let M be a 2xx2 symmetric matrix with integer entries. Then , M is invertible, if

10 g of hydrogen and 64 g of oxygen were filled in a steed vessel and exploded. Amount of water produced in this reaction will be :-

A conic has latus rectum length 1, focus at (2,3)and the corresponding directrix is x+y -3=0 . Then the conic is

If A = {1,2,3,4} , define relations on A which have properties of being : Reflexive , symmetric and transitive .

Using the proprties of determinants in Exercise 7 to 9, prove that |{:(y+z,z+y),(z,z+x,x),(y,x,x+y):}|=4xyz

If A=[[1,2],[-2,3]]B=[[3,2],[1,4]],C=[[2,2],[1,3]]Calculate CA.

Coefficient of x^(m) in the expansion of S = (1 + x)^(2m) + x(1 +·x)^(2m-1) + x^(2)(1 + x)^(2m- 2) + .... +x^(2m) is

For non-negative integers n, let int(n)=(sum_(k=0)^(n)((k+1)/(n+2)pi)sin((k+2)/(n+2)pi))/(sum_(k=0)^(n)sin^2((k+1)/(n+2)pi)) Assuming cos^(-1)x takes values in [0.pi], which of the following options is/are correct?

If z is a complex number such that (z-1)/(z+1) is purely imaginary then |z|=

Decrypt the received encoded message [2,-3][20,4] with the encryption matrix [{:(-1,-1),(2,1):}] and the decryption matrix as its inverse, where the system of codes are described by the numbers 1-26 to the letters A-Z respectively, and the number 0 to a blank space.

If A, B and C are three independent events such that P(A) = P(B) = P(C )= p,then P ( Atleasttwo of A, B, C occur) = 3p^(2) - 2p^(3).

If a circle passes through the point (a,b) and cuts the circle x^2+y^2=k^2 orthogonally then the locus of its centre is

If t=e^athen d/dx x^t=_______

Write the distance of the point (2,3,6) from zx-plane.

If two tangents drawn from a point P to the parabola y^(2)=4x are at right angles then the locus of P is

Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt. y=f(x) is

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

A coin tossed twice. Find the probability of getting at most one head

Two numbers are selected from the numbers 1 to 11. If there sum is even find the probability of an event that both numbers are odd.

Coefficient of x^n in e^(e^x)is 1/(n!) k then k =

Themaximum or minimum of the objective funtion occurs only at the corner points of the feasible region. This theorem is known as fundamental theorem of

The image of the point (4, -13) with respect to the line 5x+y+6=0 is :

If the chords of contacts of the tangents from the points (x y,) and (x_(2), y_(2)) to the hyperbola 2x^(2) - 3y^(2) = 6 are at right angle, then 4x_(1)x_(2) + 9y_(1)y_(2) is equal to

Sketch theregion{(x,0) //y= sqrt( 4-x^2)} andX- axis. Findthe areaof theregionusingintegration.

Lt_(ntooo)sum_(r=0)^(n-1)(1)/(n+r)=

Ratio of seventh term from beginning and seventh term from end in the expansion of (root(3)(2)+(1)/3_sqrt(3))^(n)"is"1 : 6 .Find value of n .

Complementary events of E and Fare E' and F' respectively if 0 lt P(F) lt 1 then …………

Special fluid helps in transportation of substances in complex animals is/are :-

(i) y=e^(x)sin^(3)xcos^(4)x (ii) y=x*e^(xsinx)

(1)/(3sqrt(8.16))~~....

Use induction to prove that, int _(0) ^(pi//2) cos ^(n-1) x sin nx dx = AA n ge 2, n in N

Let f :R to R is not identically zero, differentiable function and satisfy the equals d (xy)= f(x) f(y) and f (x+z) + f(x) + f (a), then f (5)=

Find the area bounded by the the curve y=cos x between x=0 and x=2pi.

A car moving with a speed of 40 km/hr can be stopped by applying brakes after at least 2 m. If the same car is moving with a speed of 80 km/hr. What is the stopping distance ?

Matrices of any order can be added .

The value of int_(0)^(a)(sqrt(a^(2)-x^(2)))^(3) dx is……. .

If f(x) = sin^(-1) x. cos^(-1) x. tan^(-1) x . cot^(-1) x. sec^(-1) x. cosec^(-1) x, then which of the following statement (s) hold(s) good?

Find the number of ways of arranging the letters of the word SINGING so that the two G's come together

If I= int_(0)^(pi//4) ( sqrt(sin x) + sqrt(cos x) )^(-4) dx then I equals

Let Z be the set of integers and o be a bonary operation of z defined as aob = a + b - ab for all a, b in Z. The inverse of an element a (ne 1 ) in Z is

If alpha, beta , gamma are the roots of x^(3) + 3px + q = 0then the equation whose roots are(alpha + 1)/( beta + gamma - alpha), (beta + 1)/(gamma + alpha - beta) and (gamma + 1)/(alpha + beta - gamma) is

Let f(x) = [{:((g(x) - g(a))/(x-a),x ne 0),(g'(a),x = a):}, where g is a function derivable at x = a, then at x = a :

If the orthocentre of the triangle formedby the lines 2x+3y-1=0, x+2y-1=0, ax+by-1=0 is at the origin, then (a, b) is given by

Equation of a line passing through (-1, 2, -3) and perpendicular to the plane 2x + 3y + z + 5=0 is

Evaluate the following integrals int (x^3-x^2 +x -1)/(x-1) dx

Consider the parabola x ^(2) + 4x - 2y + 6 = 0. If its chord AB 2mx - 2y + (4m + 3) =0 intersect its axis at K and directrix at M, then AM, KM and BM are in