28278 + Interview Questions in Maths in Class 12 Page 60 InterviewSolution

2951.	Find Lt_(n rarr oo){(sqrt(n))/(n^(3//2)) + (sqrt(n))/((n+3)^(3//2)) + …+ (sqrt(n))/([n+3(n-1)]^(3//2))}
Answer» ANSWER :`(1)/(3)`

Discussion

2952.	For what values of k , f(x)={:{((1-coskx)/(xsinx)," if " x ne 0),(1/2," if " x =0):} is continuous at x = 0 ?
Answer» ANSWER :`k=pm1`

Discussion

2953.	Find the all values of a for which every real root f the equation cos 3x = a cos x + (4-2\|a\|) cos ^(2) x is a root of the equation cos 3x + cos 2 x =2 cos x cos 2 x-1and vice veirs.
Answer» ANSWER :`=> a =- 3, a =-4, a LT -5-1 lt a lt 0.`

Discussion

2954.	Assertion (A) : If x+(x^(2))/(2)+(x^(3))/(3)+………oo=log((7)/(6)) then x=(1)/(7) Reason (R ) : If \|x\| lt 1, then x+(x^(2))/(2)+(x^(3))/(3)+…..oo=log((1)/(1-x))
Answer» A is TRUE, R is true and R is correct explanation of A A is true, R is true and R is not correct explanation of A A is true, R is FALSE A is false, R is true Answer :A

Discussion

2955.	A is a 3xx4 matrix .A matrix B is such that A'B and BA' are defined . Then the order of B is ……..
Answer» `3xx4` `3xx3` `4xx4` `4xx3` ANSWER :A

Discussion

2956.	A lightshines fromthe top of apole50 ft. high A ball isdropped from thesame height from a point30 ft. away fromthe light . If the shadow of theball movingat therateof 100lambda ft//sec alongthe ground 1//2sec. later [Assume the ballfalls a distance s= 16t ^(2) ft. in 't' sec ] then \| lambda\| is :
Answer» ANSWER :15

Discussion

2957.	Let veca = hati- hatk, vecb = x hati + hatj + (1-x) hatk and vec c = y hati + x hatj + (1+ x -y) hatk. Then [veca vecb vec c] dependson:
Answer» only X only y NEITHER x or y both x or y. ANSWER :C

Discussion

2958.	Ifthe pair of lines given by (x^(2) + y^(2)) cos^(2) theta = (x cos theta + y sin theta)^(2) are pendicular to each other , then theta is equal to
Answer» 0 `(PI)/(4)` `(pi)/(3)` `(3PI)/(4)` ANSWER :B

Discussion

2959.	A cylindrical vessel of volume 25(1)/(7) cu metres, open at the top is to be manufactured from a sheet of metal. If r and h are the radius and height of the vessel so that amount of metal I sused in the least possible then rh is equal to
Answer» ANSWER :4

Discussion

2960.	If x is so small such that its square and higher powers may be neglected, then find the value of ((1-2x)^(1//3) + (1+ 5x)^(-3//2))/((9+x)^(1//2))
Answer» ANSWER :`1-(149)/(36)X`

Discussion

2961.	Compute the are length of the involute of a circle x=a (cos t + t sin t), y= a(sin t-t cos t) from t=0 to t= 2pi.
Answer» ANSWER :`=2 a PI^(2)`

Discussion

2962.	For each positive integer n, consider the highest common factor hn of the two numbers n! + 1 and (n + 1)!. For n lt 100, find the largest value of h_n
Answer» ANSWER :97

Discussion

2963.	Two parabolas y^(2) = 4a(x- lambda_(1) ) and x^(2) = 4a(y- lambda_2) always touch each other, lambda_1 and lambda_2 being varaible parameters. Then show that the locus of their point of contact is a hyperbola.
Answer» ANSWER :the locus of `(x_(1) , y_(1) )` is `xy=4a^(2)` which is a HYPERBOLA.

Discussion

2964.	If I=int2/x(x^("lnx"))(nx)^(3)dx=Ax^("lnx"^(2))-Bx^("lnx")+C, then A/B is equal to:
Answer» 1 `-1` 2 `-2` ANSWER :a

Discussion

2965.	Which of the following statements is tautology ?
Answer» <P>`(~~Q^^p)^^q` `(~~q^^p)^^(p ^^ ~~q)` `(~~q^^p)VV(PV ~~p)` `(p^^q) ^(~~(p ^^q))` SOLUTION :

Discussion

2966.	int_(0)^(a) \|x-1\|dx
Answer» Solution :`" Let I "= int_(0)^(4) \|X-1\|dx` `:. I= int_(0)^(1) \|x-1\|dx+INT _(1)^(4) \|x-1\|dx` `=int_(0)^(1) (1-x)dx+ int_(1)^(4) (x-1) dx` `=[x-(x^(2))/(2)]_(0)^(1) +[(x^(2))/(2)-x]_(1)^(4)` `=(1-(1)/(2))-0+((4^(2))/(2)-x)-((1)/(2)-1)` `=(1)/(2)+4+(1)/(2)=5`

Discussion

2967.	Find the area lying above x-axis and included between the circle x^2+y^2=8x and inside of the parabola y^2=4x.
Answer» ANSWER :`4/3(8+3pi)`

Discussion

2968.	If the tangents drawn from a point P to the ellipse 4x^(2) + 9 y^(2) - 24x + 36 y = 0 are perpendicular, then the locus of P is
Answer» `x^(2) +y^(2) - 6x +4y +13= 0` `x^(2) +y^(2) - 6x +4y -13= 0` `x^(2) +y^(2)+6x -4y -13= 0` `x^(2) + y^(2) = 26` Answer :B

Discussion

2969.	If 1,alpha,alpha^2,…,alpha^(n-1) are the n^(th) roots of unity show that sum_(r=1)^(n-1)r(alpha_r+alpha_(n-r))=-n.
Answer» ANSWER :`=N{alpha_1+alpha_2+alpha_3+…....+alpha_(n-1)=n{-1}=-n`

Discussion

2970.	Show that the following vector are co-planar. hati+2hatj+3hatk, -2hati-4hatj+5hatk, 3hati+6hatj+hatk
Answer» SOLUTION :LET `veca = hati+2hatj+3hatk` `vecb = -2hati-4hatj+5hatk` `vecc = 3hati+6hatj+hatk` = 1(-4-30)-2(-2-15)+3(-12+12) = -34+34 = 0 As we know scalar TRIPLE product`(VECAXXVECB).vecc = 0`, it follows that the given vectors `veca.vecb.vecc` are coplanar.

Discussion

2971.	Integrate the functions sin^(-1)((2x)/(1+x^(2)))
Answer» ANSWER :`2xtan^(-1)x-log(1+x^(2))+C`

Discussion

2972.	Find the cartesian equation of the plane through the point (2,-1, 1) and perpendicular to the vector 4hati+2hatj-3hatk.
Answer» ANSWER :`4x+2y-3=3`

Discussion

2973.	Find the equation of a curve passing through the point (-2, 3), given that the slopw of the tangent to the curve at any point (x, y) is (2x)/y^(2).
Answer» ANSWER :`y^(3)/3=x^(2)+5`

Discussion

2974.	A straight line passes through the points (5, 0) and (0, 3). The length of perpendicular from the point (4, 4) on the line is :
Answer» `(15)/(SQRT(34))` `(sqrt(17))/(2)` `(17)/(2)` `sqrt((17)/(2))` ANSWER :D

Discussion

2975.	If therootsof theequationx^3 +3px^2+ 3qx-8=0areina geometricprogression, then(q^3)/( p^3)=
Answer» 1 `-2` `4` `-8` ANSWER :D

Discussion

2976.	Let y be an implicit fuction of x defined by : x^(2x)-2x^(x) cot y-1=0. Then f'(1) equals :
Answer» `-1` 1 `LOG2` `-log2`. ANSWER :A

Discussion

2977.	Find the sum .^(n)C_(0) + 2 xx .^(n)C_(1) + xx .^(n)C_(2) + "….." + (n+1) xx .^(n)C_(n).
Answer» Solution :Method I : `.^(n)C_(0)+2xx.^(n)C_(1)+3xx.^(n)C_(2)+"...."+(n+1)xx .^(n)C_(n)` `= UNDERSET(r=0)overset(n)sum(r+1).^(n)C_(r)` `=underset(r=0)overset(n)sum[r.^(n)C_(r)+.^(n)C_(r)]` `=n underset(r=0)overset(n)sum.^(n-1)C_(r-1)+underset(r=0)overset(n)sum.^(n)C_(r)` `= n(.^(n-1)C_(0) + .^(n-1)C_(1) + .^(n-1)C_(2)+"..."+.^(n-1)C_(n-1)) + (.^(n)C_(0)+.^(n)C_(1)+.^(n)C_(2)+"....."+.^(n)C_(n))` `= n2^(n-1) + 2^(n)` `= (n+2)2^(n-1)` Method II : We have `(1+x)^(n) = .^(n)C_(0)+.^(n)C_(1)x+.^(n)C_(2)x^(2)+"....."+.^(n)C_(n)x^(n)` `:. x(1+x)^(n) = .^(n)C_(0)x+.^(n)C_(1)x^(2)+.^(n)C_(2)x^(3) + "....." + .^(n)C_(n)x^(n+1)` Differentiating w.r.t. x, we get `n(n1+x)^(n-1)x+(1+x)^(n)=.^(n)C_(0)+2xx.^(n)C_(1)x+3xx.^(n)C_(2)x^(2)+"..."+(n+1)xx.^(n)C_(n)x^(n)` Putting `x = 1`, we get `n2^(n-1)+2^(n)=.^(n)C_(0) +2xx.^(n)C_(1)+3xx.^(n)C_(2)+"....."+(n+1)xx.^(n)C_(n)`

Discussion

2978.	If x,y,z in R satisfies the system of equations x +(y) + (s) =12.7, [x]+{y}+z=4.1 and {x} +y+[z] =2 where {.} and [.] denotes the fractional and integral parts respectively) then match the following
Answer» ANSWER :`A toR; BTO S,; C to P; D to Q`

Discussion

2979.	The p^(th), q^(th) and r^(th)terms of an A.P. are a, b, c, respectively. Show that (q-r) a+(r-p) b+(q-p)c = 0
Answer» ANSWER :` = A(0) + B(0) = 0`

Discussion

2980.	Write the negation of the following statements There exists x in N , x + 3 =10
Answer» SOLUTION :For all`X in N, x+3 != 10`

Discussion

2981.	Prove that (3 !)/(2(n + 3)) = sum_(r=0)^(n)(-1)^(r ) ((""^(r )C_(r ))/(""^(r + 3)C_(r )))
Answer» Solution :`sum_(R=0)^(n) (-1) (""^(n)C_(r))/(""^(r+3)C_(r))` `= sum_(r=0)^(n) (-1)^(r)(n!.3!)/((n-r)!(r+3))=3!sum_(r=0)^(n) (-1)^(r)(n!)/((n-r)!(r+3)!)` `= (3!)/((n+1)(n+2)(n+3)).sum_(r=0)^(n) ((-1)^(r).(n+3)!)/((n-r)!(r+3)!)` `=(3!)/((n+1)(n+2)(n+3)) =.sum_(r=0)^(n) (-1)^(r).""^(n+3)C_(r+3)` `= (3!(-1)^(3))/((n+1)(n+2)(n+3))sum_(s=0)^(n) (-1)^(s).""^(n+3)C_(3)` `=(-3!)/((n+1)(n+2)(n+3))(sum_(s=0)^(n+3) (-1)^(s).""^(n+3)C_(s))_(""^(n+3)C_(0)+""^(n+3)C_(1) -""^(n+3)C_(2))` `=(-3!)/((n+1)(n+2)(n+3)){0-1+(n+3)-((n+3)(n+2))/(2!)}` `=(-3!)/((n+1)(n+2)(n+3)).((n+2)(2-n-3))/(2) = (3!)/(2(n+3))`

Discussion

2982.	(C_1)/(2) + (C_3)/(4) + …….+(C_15)/(16)=
Answer» `(2^15 - 1)/(16)` `(2^15 - 1)/(16)` `(2^14 - 1)/(16)` `(2^40+ 1)/(16)` ANSWER :A

Discussion

2983.	Find (dy)/(dx) of y=e^(-x)
Answer» SOLUTION :`d/dx(E^(-X))=e^(-x)xxd/dx(-x)=-e^(-x)`

Discussion

2984.	intxsin^(2)xdx
Answer» ANSWER :`(x^(2))/(2)-(x)/(2)SIN2X-(1)/(8)cos2x+c`

Discussion

2985.	An equation of the ellipse whose length of the major axis is 10 and foci are (pm 2, 0)is
Answer» `x^2/25+y^2/21=1` `x^2/25+y^2/4=1` `x^2/25+y^2/16=1` `x^2/29+y^2/25=1` ANSWER :A

Discussion

2986.	Evaluation of definite integrals by subsitiution and properties of its : int_(a)^(a+1)\|a-x\|dx=.........(ainN)
Answer» `(1)/(2)` a 0 `(1)/(3)` ANSWER :A

Discussion

2987.	int_0^1x(1-x)^100dx
Answer» SOLUTION :`I=int_0^1x(1-x)^100dx` =`int_0^1(1-x)x^100dx` =`int_0^1(x^100-x^101)DX` =`[x^101/101-x^102/102]_0^1` =1/101-1/102=1/10302

Discussion

2988.	If(1+ x+ x ^ 2+ x^ 3 )^5= sum _(k = 0) ^(15)a _k x ^k, thensum _ (k= 0 ) ^(7) a _(2k)isequalto
Answer» 128 256 512 1024 Solution :Giventhat, ` (1 + x + x ^ 2+ x ^ 3 ) ^5 =sum _(k = 0) ^(15 )=a _ k x ^k ` `RARR (1 +x+x ^ 2+x ^3) ^5 =a_ 0+ a _ 1 x ^ 1+ a _2x ^ 2+… +a _ (14)x ^(14 )+a _ (15)x ^(15)`… (1) Substituting`x = 1`,in (1) ` rArr(1 +1+ 1+ 1 ) ^(5)= a_ 0+ a _ 1 + a _ 2+ ... +a _ (14) +a _ (15)` ` rArr4 ^(5)=a _ 0+a _ 1+a _ 2+...+a _(14)+a _ (15) ""... (2) ` Substituting` x =-1`in (1) ` rArr(1 - 1 + (1) ^2+ (1)^3 =a _0- a _1+ a _2-a_3 +... +a _ (14)-a _ (15) ` `0 =a _ 0- a _ 1+a _ 2 +..+ a _ (14 )+a _ (15) `...(3) Adding(2)and(3) `rArr 4 ^ 5= 2 ( a _ 0+a _ 2+a_4+... +a _ (14)) ` `RARRA _ 0+a _ 2 +... +a _ (14 )= ( 4 ^5 )/(2 ) ` `rArra_ 0+a _ 2+... +a _ (14)=2 ^9 ` ` = 512 `

Discussion

2989.	IF x=1+log_abc,y=1+log_bca,z=1+log_cab, prove that xyz=xy+yz+zx.
Answer» ANSWER :xyz=xy+yz+zx

Discussion

2990.	Evaluate (ii) int_(0)^(1) x^(5) (1-x)^(5//2)dx
Answer» ANSWER :`(512)/(153153)`

Discussion

2991.	If the sixth term in the expansion of [3log_(3sqrt(9^(x-1)+7))+1/(3^(log3(3^(x-1)+1)))]^(7) is 84, ttien sum of the possible values of x is __________.
Answer» ANSWER :3

Discussion

2992.	If A is a square matrix such that A^(2)=I then (A-I)^(3)+(A+I)^(3)-7Ais equal to …….
Answer» A `I-A` `I+A` 3A Answer :A

Discussion

2993.	A straight line 4x +y -1 = 0throughthe point A(2,-7) meets the line BC whose equation is 3x - 4y +1at the point B . Then the equation of the line AC such thatAB = AC , is
Answer» 89x - 52y - 162 = 0 52x + 89y + 519 = 0 4x - y - 15 = 0 4x +3y +13 = 0 ANSWER :B

Discussion

2994.	Find the number of ways of arranging the letters of the word SINGING so that they begin and end with I
Answer» ANSWER :30

Discussion

2995.	If [[2x , 5], [4,2]]is singular, then x =
Answer» 20 4 5 10 Answer :C

Discussion

2996.	Discuss the continuity of the functionf (x)= \|x\| " at "x=0
Answer» ANSWER :DISCONTINUOUS

Discussion

2997.	The minimum value of the expression sin theta_(1) + sin theta_(2) + sin theta_(3), where theta_(1), theta_(2), theta_(3) are real numbers satisfyingtheta_(1) + theta_(2) + theta_(3) = pi is :
Answer» negative positive zero `-4.` ANSWER :A

Discussion

2998.	If A = [{:(0,-1,2),(2,-2,0):}], B = [{:(0,1),(1,0),(1,1):}] and M = AB, then thevalue of M^(-1) is
Answer» `[{:((2)/(3),(-1)/(3)),((1)/(3),(4)/(5)):}]` `[{:((1)/(3),(-1)/(3)),((1)/(3),(1)/(6)):}]` `[{:(2,(-1)/(3)),((2)/(3),0):}]` NONE OT these Answer :B

Discussion

2999.	Integrate the functions (e^(x)-e^(-x))/(e^(x)+e^(-x))
Answer»

Discussion

3000.	AB and CD are two equal and parallel chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1. Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ
Answer» passes through the origin is bisected at the origin cannot pass through the origin is not bisected at the origin Solution :LET P and Q be the points `(ALPHA, beta)` and `(alpha_(1),beta_(1))` `rArr` Equations of Ab and CD are `(X)/(a) alpha + (y)/(b) beta =1` and `(x)/(a) alpha_(1) +(y)/(b) beta_(1) =1` (Chord of CONTACT) These lines are parallel `rArr (alpha)/(alpha_(1)) = (beta)/(beta_(1)) =k` ALSO `(alpha^(2))/(alpha^(2)) +(beta^(2))/(b^(2)) = (alpha_(1)^(2))/(a^(2)) + (beta_(1)^(2))/(b^(2))` `rArr (alpha)/(alpha_(1)) = (beta)/(beta_(1)) =-1` `rArr PQ` passes through origin and is bisected at the origin.

Discussion

Explore topic-wise InterviewSolutions in .

Find Lt_(n rarr oo){(sqrt(n))/(n^(3//2)) + (sqrt(n))/((n+3)^(3//2)) + …+ (sqrt(n))/([n+3(n-1)]^(3//2))}

For what values of k , f(x)={:{((1-coskx)/(xsinx)," if " x ne 0),(1/2," if " x =0):} is continuous at x = 0 ?

Find the all values of a for which every real root f the equation cos 3x = a cos x + (4-2|a|) cos ^(2) x is a root of the equation cos 3x + cos 2 x =2 cos x cos 2 x-1and vice veirs.

Assertion (A) : If x+(x^(2))/(2)+(x^(3))/(3)+………oo=log((7)/(6)) then x=(1)/(7) Reason (R ) : If |x| lt 1, then x+(x^(2))/(2)+(x^(3))/(3)+…..oo=log((1)/(1-x))

A is a 3xx4 matrix .A matrix B is such that A'B and BA' are defined . Then the order of B is ……..

A lightshines fromthe top of apole50 ft. high A ball isdropped from thesame height from a point30 ft. away fromthe light . If the shadow of theball movingat therateof 100lambda ft//sec alongthe ground 1//2sec. later [Assume the ballfalls a distance s= 16t ^(2) ft. in 't' sec ] then | lambda| is :

Let veca = hati- hatk, vecb = x hati + hatj + (1-x) hatk and vec c = y hati + x hatj + (1+ x -y) hatk. Then [veca vecb vec c] dependson:

Ifthe pair of lines given by (x^(2) + y^(2)) cos^(2) theta = (x cos theta + y sin theta)^(2) are pendicular to each other , then theta is equal to

A cylindrical vessel of volume 25(1)/(7) cu metres, open at the top is to be manufactured from a sheet of metal. If r and h are the radius and height of the vessel so that amount of metal I sused in the least possible then rh is equal to

If x is so small such that its square and higher powers may be neglected, then find the value of ((1-2x)^(1//3) + (1+ 5x)^(-3//2))/((9+x)^(1//2))

Compute the are length of the involute of a circle x=a (cos t + t sin t), y= a(sin t-t cos t) from t=0 to t= 2pi.

For each positive integer n, consider the highest common factor hn of the two numbers n! + 1 and (n + 1)!. For n lt 100, find the largest value of h_n

Two parabolas y^(2) = 4a(x- lambda_(1) ) and x^(2) = 4a(y- lambda_2) always touch each other, lambda_1 and lambda_2 being varaible parameters. Then show that the locus of their point of contact is a hyperbola.

If I=int2/x(x^("lnx"))(nx)^(3)dx=Ax^("lnx"^(2))-Bx^("lnx")+C, then A/B is equal to:

Which of the following statements is tautology ?

int_(0)^(a) |x-1|dx

Find the area lying above x-axis and included between the circle x^2+y^2=8x and inside of the parabola y^2=4x.

If the tangents drawn from a point P to the ellipse 4x^(2) + 9 y^(2) - 24x + 36 y = 0 are perpendicular, then the locus of P is

If 1,alpha,alpha^2,…,alpha^(n-1) are the n^(th) roots of unity show that sum_(r=1)^(n-1)r(alpha_r+alpha_(n-r))=-n.

Show that the following vector are co-planar. hati+2hatj+3hatk, -2hati-4hatj+5hatk, 3hati+6hatj+hatk

Integrate the functions sin^(-1)((2x)/(1+x^(2)))

Find the cartesian equation of the plane through the point (2,-1, 1) and perpendicular to the vector 4hati+2hatj-3hatk.

Find the equation of a curve passing through the point (-2, 3), given that the slopw of the tangent to the curve at any point (x, y) is (2x)/y^(2).

A straight line passes through the points (5, 0) and (0, 3). The length of perpendicular from the point (4, 4) on the line is :

If therootsof theequationx^3 +3px^2+ 3qx-8=0areina geometricprogression, then(q^3)/( p^3)=

Let y be an implicit fuction of x defined by : x^(2x)-2x^(x) cot y-1=0. Then f'(1) equals :

Find the sum .^(n)C_(0) + 2 xx .^(n)C_(1) + xx .^(n)C_(2) + "….." + (n+1) xx .^(n)C_(n).

If x,y,z in R satisfies the system of equations x +(y) + (s) =12.7, [x]+{y}+z=4.1 and {x} +y+[z] =2 where {.} and [.] denotes the fractional and integral parts respectively) then match the following

The p^(th), q^(th) and r^(th)terms of an A.P. are a, b, c, respectively. Show that (q-r) a+(r-p) b+(q-p)c = 0

Write the negation of the following statements There exists x in N , x + 3 =10

Prove that (3 !)/(2(n + 3)) = sum_(r=0)^(n)(-1)^(r ) ((""^(r )C_(r ))/(""^(r + 3)C_(r )))

(C_1)/(2) + (C_3)/(4) + …….+(C_15)/(16)=

Find (dy)/(dx) of y=e^(-x)

intxsin^(2)xdx

An equation of the ellipse whose length of the major axis is 10 and foci are (pm 2, 0)is

Evaluation of definite integrals by subsitiution and properties of its : int_(a)^(a+1)|a-x|dx=.........(ainN)

int_0^1x(1-x)^100dx

If(1+ x+ x ^ 2+ x^ 3 )^5= sum _(k = 0) ^(15)a _k x ^k, thensum _ (k= 0 ) ^(7) a _(2k)isequalto

IF x=1+log_abc,y=1+log_bca,z=1+log_cab, prove that xyz=xy+yz+zx.

Evaluate (ii) int_(0)^(1) x^(5) (1-x)^(5//2)dx

If the sixth term in the expansion of [3log_(3sqrt(9^(x-1)+7))+1/(3^(log3(3^(x-1)+1)))]^(7) is 84, ttien sum of the possible values of x is __________.

If A is a square matrix such that A^(2)=I then (A-I)^(3)+(A+I)^(3)-7Ais equal to …….

A straight line 4x +y -1 = 0throughthe point A(2,-7) meets the line BC whose equation is 3x - 4y +1at the point B . Then the equation of the line AC such thatAB = AC , is

Find the number of ways of arranging the letters of the word SINGING so that they begin and end with I

If [[2x , 5], [4,2]]is singular, then x =

Discuss the continuity of the functionf (x)= |x| " at "x=0

The minimum value of the expression sin theta_(1) + sin theta_(2) + sin theta_(3), where theta_(1), theta_(2), theta_(3) are real numbers satisfyingtheta_(1) + theta_(2) + theta_(3) = pi is :

If A = [{:(0,-1,2),(2,-2,0):}], B = [{:(0,1),(1,0),(1,1):}] and M = AB, then thevalue of M^(-1) is

Integrate the functions (e^(x)-e^(-x))/(e^(x)+e^(-x))

AB and CD are two equal and parallel chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1. Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ