28278 + Interview Questions in Maths in Class 12

1.	If (7x-1)/(1-5x+6x^(2)) is expanded in ascending powers of x when abs(x) lt 1/3 then the coefficient of x^(n) is
Answer» `(-5)2^(N)+43^(n)` `52^(n)+(-4)3^(n)` `-5.2^(n)+4*3^(n)` `5.2^(n)+4.3^(n)` Answer :A

Discussion

2.	Show that the relation R in the set A of all the books in a library of a college , given by R = {(x,y) : x and y have same number of pages} is an equivalence relation.
Answer» SOLUTION :N/A

Discussion

3.	Find the sum of all odd numbers between 1 and 100 which are divisible by 3
Answer» ANSWER :`=83667`

Discussion

4.	If vec(a).vec(a)=0 and vec(a).vec(b)=0 then what can be concluded about the vector vec(b) ?
Answer» ANSWER :`VEC(B)` be any VECTOR.

Discussion

5.	If \|z-1\| = 2 then the locus of z is
Answer» X-axis y-axis x= y `x^(2) + y^(2) = 1` ANSWER :A

Discussion

6.	If alpha, beta, gammaare roots of the equation3x^(3) + 6x^(2) - 9x + 2 = 0then sum (alpha//beta) =
Answer» 12 `-12` 3 `-3` ANSWER :2

Discussion

7.	If a curve y=asqrtx+bx passes through the point (1,2) and the area bounded by the curve, line x=4 and x-axis is 8 square units, then
Answer» `a=3,b=-1` `a=3,b=1` `a=-3,b=1` `a=-3,b=-1` Solution :It is given that `y=asqrtx+bx` passes through the point (1,2). `therefore2=a+b""...(i)` It is given that `UNDERSET(0)overset(4)(INT)(asqrtx+bx)dx=8` `IMPLIES[(2)/(3)ax^(3//2)+b/2x^(2)]_(0)^(4)=8` `implies16/3a+8b=8implies2/3a+b=1""...(ii)` Solving (i)and (ii), we get `a=3, b=-1.`

Discussion

8.	If \|z - (2)/(z)\| = 1 then find the maximum value of \|z\|.
Answer» ANSWER :B

Discussion

9.	Find the value of k if the line 2y=5x+k is a tangent to the parabola y^(2)=6x
Answer» ANSWER :`K= (6)/(5)`

Discussion

10.	A small manufacures has employed 5 skilled men and 10 semiskilled men and makes an aritcle in two qualities, a de luxe model and an ordinary model. The making of a de luxe model required 2 hours' work by a skilled man and 2 hours' work by a semiskilld man. The ordinary model requires 1 hour by a skilled man and 3 hours by a semiskilled man. By union reles, no man can work more than 8 hours per day. The manufacturer gains Rs. 15 on the de luxe model and Rs 10 on the ordinary model. How many of each type should be made in order to maximize his total daily profit? Also, find the maximum daily profit.
Answer» SOLUTION :Suppose x de luxe models and y ordinary models of arti9cles be MADE each day. Then, MAXIMIZE `Z=15x+10y,` SUBJECT to the constraints `xge0y ge0,2y+yle40 and2x+3yle80.`

Discussion

11.	Differentiate the following with respect to x: x log x
Answer» ANSWER :`1+ LOG X`

Discussion

12.	Let n(A)=m" and "n(B)=n. Then the total number of non-empty relations that can be defined from A to B is :
Answer» `m^N` `n^(m)-1` `MN-1` `2^(mn)-1` ANSWER :D

Discussion

13.	The slope of one of the common tangent to circle x^(2)+y^(2)=1 and ellipse x^(2)/4+2y^(2)=1 is sqrt(a/b) where gcd(a, b)=1 then (a+b)/2 is equal to
Answer» ANSWER :`12.0`

Discussion

14.	matrix A=({:(-i,0),(0,-i):}) then A^2 =
Answer» 2I `-I` I None of these Answer :B

Discussion

15.	By Simpson's rule, value of int_(1)^(2)(dx)/(x) dividing the interval (1,2) into four equal parts, is
Answer» `0.6932` `0.6753` `0.6692` `7.1324` SOLUTION :`h=(2-1)/(4)=(1)/(4)` Now, `x_(0)1, x_(1)=1+(1)/(4), x_(2)=1+2xx(1)/(4)` `x_(3)=1+3xx(1)/(4),x_(4)=1+4xx(1)/(4)` ie, `x_(0)=1, x_(1) =1.25, x_(2)= 1.5, x_(3)` ` =1.75, x_(4)=2` `rArr y_(0)=1, y_(1)=0.8, y_(2)=0.667, y_(3)` `=0.5571, y_(4)=0.5` `:. ` Using simpson's `(1)/(3)` RD rule `INT _(1)^(2)(dx)/(X)= (1)/(12)[(1+0.5)+4(0.8+0.571)+2(0..667)]` `=(1)/(12)[1.5+5.484+1.334]` `=(1)/(12) [8.318]=0.6932`

Discussion

16.	An oil company has two depots, A and B, with capacities of 7000 L and 4000 L respectively. The company is to supply oil to three pumps D,E,F whose requirementsare 4500 L, 3000 L and 3500 L respectively. The distance (in Km) between the depots and the petrol pumps are given in the following table : Assuming that the transportation cost of10 litres of oil is ₹ 1 per km , how should the delivery be scheduled in order that the transportation cost is minimum ?
Answer» ANSWER :A to D 500 LITRES : A to E 3000 LITRE : A to F3500 litres: B to D 4000 litres : B to E 0 litre : B to F 0 litre

Discussion

17.	(i) Let z be a non-real complex number lying on the circle \|z\|=1. Then prove that z= (1+ I tan (("arg" z)/(2)))/(1-I tan(("arg"z )/(2))) (ii) Find the modulus and the argument of the complex number z_1, where z_1 = z^2 - z and z = cos theta+I sin theta.
Answer» Answer :(II) `\|z_1\| =2\|sin THETA//2\|, "ARG" (z_1) = (pi+3 theta)/(2` or `\|z_1\|=0`, "arg" (z_1)` is not defined

Discussion

18.	Prove that the circle on a focal radius of a parabola, as diameter touches the tangent at the vertex.
Answer» ANSWER :the CIRCLE TOUCHES the TANGENT at the VERTEX

Discussion

19.	Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area ?
Answer» Answer :RADIUS `r=((50)/(pi))^((1)/(3))` and height `h=2((50)/(pi))^((1)/(3))` CM.

Discussion

20.	The scalartriple product[veca+vecb-vec c" "vecb+vec c -veca" "vec c+veca-vecb] is equal to :
Answer» 0 `[VECAVECB vec c]` `2[veca vecb vec c]` `4[veca vecb vec c]` ANSWER :D

Discussion

21.	1+ cos 2x + cos 4 x+ cos 6x- 4 cos x cos 2x cos 3x =
Answer» 1 `-1` `2` `0` ANSWER :D

Discussion

22.	Given that the 4th term in the expansion of (2+(3)/(8)x)^(10) has the maximum numerical value, find the range of x for which the statement will be true.
Answer» ANSWER :`(3)(4) lt X lt (8)/(7)`

Discussion

23.	x^(3)/((2x-1)(x+2)(x-3))=A+B/(2x-1)+C/(x+2)+D/(x-3) rArr A =
Answer» `1/2` `- 1/50` `- 8/25` `- 27/25` ANSWER :A

Discussion

24.	f:R rarrR , f(x) =x/(sqrt(1+x^2)),AAx inR.Then find (fofof) (x).
Answer» `X/(1+x^2)` `(1+x^2)/(x)` `x/(SQRT(1+2x^2))` `x/(sqrt(1+3x^2))` SOLUTION :N/A

Discussion

25.	The position vectors of the points A, B, C and D are 4hati+3hatj-hatk, 5hati+2hatj+2hatk, 2hati-2hatj-3hatk and 4hati-4hatj+3hatk respectively. Show that AB and CD are parallel.
Answer» Solution :Given that the POSITION vector of A = `4hati+3hatj-hatk` position vector of B = `5hati+2hatj+2hatk` position vector of C = `2hati-2hatj-3hatk` position vector of D = `4hati-4hatj+3hatk` Then `VEC(AB) = (5hati+2hatj+2hatk)-(4hati+3hatj-hatk)` =`hati-hatj+3hatk` `vec(CD) = (4hati-4hatj+3hatk)-(2hati-2hatj-3hatk)` =`2hati-2hatj+6hatk` =`2(hati-hatj+3hatk) = 2vec(AB)`.

Discussion

26.	The product of all the values of \|lambda\|, such that the lines x+2y-3=0, 3x-y-1=0 and lambdax +y-2 =0 cannot form a triangle, is equal to
Answer» ANSWER :`2.35`

Discussion

27.	Match the entries of Column I with those of Column-II
Answer» SOLUTION :A(p,R,s),B(s,t),C(p),D(p,q,r,s,t)

Discussion

28.	On the Argand plane point' A 'denotes a complex number z_1 . A triangle OBQ is made directly similar to the triangle OAM ,where OM = 1as shown in the figure . If the point B denotes the complex number z_2,then find the complex number corresponding to the point ' Q ' in terms ofz_1 & z_2
Answer» ANSWER :`z_2/z_1`

Discussion

29.	Evaluate the following integrals. int(2x-3)/(sqrt(2x^(2)+5x-6))dx
Answer» Answer :`sqrt(2x^(2)+5x-6)-(11)/(2sqrt(2))COSH^(-1)((4x+5)/(sqrt(73)))+c`

Discussion

30.	Match the entries of Column I with those of Column-II
Answer» SOLUTION :A(p,q,R,s,t),B(p,q,r,t),C(r,s,t),D(r)

Discussion

31.	if A =[(2,-2),(-2,2)]," then " A^(n)=2^(k)A, where k=
Answer» `2^(n-1)` n+1 n-1 `2(n-1)` ANSWER :D

Discussion

32.	If{:[( 2,-3,5),( 3,2,-4),( 1,1,-2) ]:} ,find A^(-1)Using A^(-1)solve the system of equations 2x -3y +5z =11 3x+2y -4z=-5 x+ y -2z =-3
Answer» ANSWER :` {:(0,1,-2),(-2,9,-23),(-1,5,-13):},x=1,y=2,z=3`

Discussion

33.	Ifthe equation ay^(2)+bxy+ex+dy=0 represents a pair of line, then
Answer» `e=0, a=B` `e=0, bd=ac` `e=0, d=e` `e=0, be=ad` ANSWER :B

Discussion

34.	If p in (-1, 1) , then roots of the quadratic equation (p - 1)x^(2) + px + sqrt(1 - p^(2)) = 0 are
Answer» PURELY imaginary non-real COMPLEX combers real and equal real and distinct Answer :D

Discussion

35.	Let nge3. A list of numbers x_(1),x_(2),....,x_(n) has mean mu and standard deviation sigma . A new list of numbers y_(1),(y_(2),....,y_(n) is made as follows : y_(1)=(x_(1)+ x_(2))/(2),y_(2)=(x_(1)+x_(2))/(2) and y_(j)"for"j=3,4,....,n. The mean and the standard deviation of the new list are hatmuand hatsigma . Then whcih of the following is necessarily true?
Answer» `mu=hatmu and sigmalehat SIGMA` `mu=hatmu and sigma ge hat sigma` `sigma-hat sigma` `munehat mu` Solution :`mu=(x _(1)+x_(2)+.....x_(n))/(n)` `hatmu=(y_(1)+y_(2)+....+y_(n))/(n)=((x_(1)+x_(2))/(2)+(x_(1)+x_(2))/(2)+x_(3)+....+x_(n))/(n)` `hatmu=(x_(1)+x_(2)+....+x_(n))/(n)=murArrhatmu=mu` `sigma^(2)=(sumx_(i)^(2))/(n)-mu^(2)` `sigma^(2)=(x_(1)^(2)+x_(2)^(2)+....+x_(n)^(2))/(n)-mu^(2)""....(1)` `hatsigma^(2)=(x_(1)^(2)+y_(2)^(2)+....+y_(n)^(2))/(n)-mu^(2) (((x_(1)+x_(2))/(2))^(2)+((x_(1)+x_(2))/(2))^(2)+x_(3)^(2)+....x+_(n)^(2))/(n)-mu^(2)` `hatsigma^(2)=((x_(1)^(2)+x_(2)^(2))/(2)+x_(1)x_(2)+x_(3)^(2)+....+x_(n)^(2))/(n)-mu^(2)"" ....(2)` `sigma^(2)-hatsigma^(2)=(x_(1)^(2)+x_(2)^(2))/(n)-((x_(1)^(2)+x_(2)^(2)+2x_(1)x_(2))/(2N))=(x_(1)^(2)+x_(2)^(3)-2x_(1)x_(2))/(2n)` `(x _(1)-x_(2))^(2)/(2n)ge0rArrgehat sigma&mu=hatmu`

Discussion

36.	Show that 5x le 8 sin x-sin 2x le 6x for x le x le (pi)/(3)
Answer» Solution :Let f(x) =8 sinx - sin 2x f(x)=8 cos x -2 cos 2x f(x) =-8sin x +4 sin 2x =-8 sinx (1-cosx) `therefore f(x)lt0in [(0,(pi)/(3))]` f(x) is decreasing function in `[(0,(pi)/(3))]` `f(pi)/(3)lef(x)lef(0)` `5lef(x)LE6 in [(0,(ppi)/(3))]` `5xlef(x)le6x in [(0,(pi)/(3))`

Discussion

37.	Let a = I + j + k, b = I - j + 2k and c = xi + (x - 2) j - k. If the vector c lies in the plane of a and b. then x equals
Answer» 0 1 `-4` `-2` ANSWER :D

Discussion

38.	The sine of the angle between the vectors a=3hati+hatj+2hatk and b=2hati-2hatj+4hatk is
Answer» `SQRT((2)/(7))` `(2)/(SQRT7)` `(sqrt2)/(7)` None of these Solution :Given that, `a=3 hat(i)+hat(j)+2HAT(k)` and `b=2 hat(i)-2hat(j)+4hat(k)` Let `theta` be the angle between them. `:. COS theta=(a.b)/(\|a\|\|b\|)=((3hat(i)+hat(j)+2hat(k)))/sqrt(9+1+4)((2hat(i)-2hat(j)+4hat(k)))/sqrt(4+4+16)` `implies cos theta=(6+(-2)+8)/(sqrt(14)sqrt(24))=12/(4sqrt(21))` `:. SIN theta=sqrt(1-cos^(2) theta)` `=sqrt(1-144/336)=sqrt(192/336)` `implies sin theta =2/sqrt(7)`

Discussion

39.	If hatu and hatv are unit vectors and theta is the acute angle between them, then 2hatuxx3hatv is a unit vector for
Answer» exactly TWO value of `THETA` more than two values of `theta` no value of `theta` exactly one value of `theta` Solution :Since, `( 2hatu XX 3 hatv) ` is a UNIT vector. `rArr \|2 HATU xx 3 hatv\|= 1 rArr 6\|hat u\| \|hatv\|\|sin theta\|=1 ` `rArr sin theta = (1)/(6) "" [ because \|hatu \| = \| hatv\|=1] ` Since `theta` ios an acute angle, so there is exactly one value of `theta` for which `(2 hatu xx 3hatv)` is unit vector .

Discussion

40.	The subnormal at any pointion the curve y' = a is constant. Then the value of x is
Answer» 1 2 3 4 Answer :A

Discussion

41.	int (3x+4)/(x^(2)+2x+3)dx=
Answer» `(3)/(2) LOG \|x^(2) + 2x + 3\| + (1)/(sqrt(2)) tan^(-1) ((x + 1)/(sqrt(2))) + c` `(3)/(2) log \| x^(2) + 2x + 3\| - (1)/(sqrt(2)) tan^(-1) ((x + 1)/(sqrt(2))) + c` `(3)/(2) log \| x^(2) + 2x + 3 \| - (1)/(sqrt(2)) sin^(-1) ((x + 1)/(sqrt(2))) + c` `(3)/(2) log \|x^(2) + 2x + 3\| + (1)/(sqrt(2)) sin^(-1) ((x + 1)/(sqrt(2))) + c` ANSWER :A

Discussion

42.	At two high schools, those planning to attend college after graduation were polled. The sizes of the colleges they planned to attend based on student body sizes were tabulated in the table above. The 255 polled students from Manhattan high east Had an average SAT score above 1100, and the 249 polled students from Manhattan High west had an average SAT score below 1100. IF a poll respondent were chosen at random from those planning to attend a college with atleast 5,000 students , what is the probability that the respondent would be enrolled at Manhattan High west?
Answer» `210/249` `210/440` `230/440` `440/504` ANSWER :B

Discussion

43.	If a=hat(i)+hat(j)-2hat(k), b=2hat(i)-hat(j)+hat(k) and c=3hat(i)-hat(k) and c=ma+nb, then m+n is equal to
Answer» 0 1 2 `-1` Solution :Given , ` a =HATI +hatj -2hatk, b =2hati -hatj+HATK, c=3hati and c=m +a+nb` `THEREFORE 3hati-hatk=m(hati+hatj-2hatk)+n(2hati-hatj+hatk)` `implies3hati-hatk=(m+2n)hati+(m-n)hatj+(-2m+n)hatk` On equationg the coefficients of `hati,hatj and hatk,` RESPECTIVELY, we get `3=m+2n,0=m-n` `and -1=-2m+n` `implies3=n+2n` `impliesn=1` `impliesm=1and n=1` ` impliesm+n=1+1=2`

Discussion

44.	A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?
Answer» ANSWER :`(8)/(3)` CM /s

Discussion

45.	((cosA+cosB)/(sinA-sinB))^(2012)+((sinA+sinB)/(cosA-cosB))^(2012)=
Answer» `2cos^(2012)((A+B)/2)` `2COT^(2012)((A-B)/2)` `2tan2012((A+B)/2)` `2tan^(2012)((A-B)/2)` Answer :B

Discussion

46.	Find the probability of getting both even if two dice are rolled
Answer» ANSWER :`(1)/(4)`

Discussion

47.	A man on the ground observes that the angle of elevation of the top of a tower is 68^(@) 11', and a flagstaff 24 m high on the summit of the tower subtends an angle of 2^(@) 10' at the observer's eye. If tan 70^(@) 21' = 2.8 and cot 68^(@) 11' = 0.4, the height of the tower is
Answer» 120 m 168 m 200 m 300 m Answer :C

Discussion

48.	Find area of the triangle with vertices at the point given in each of the following (2,7),(1,1),10,8)
Answer» ANSWER :23.5

Discussion

49.	IfA= {:[( 2,-1,1),(-1,2,-1),(1,-1,2) ]:} Verify thatA^(3) -6A^(2) +9A -4I=Oand hence find A^(-1)
Answer» ANSWER :`(1)/(4){:[( 3,1,-1),(1,3,1),(-1,1,3)]:}`

Discussion

50.	How many 4 letter words can be formed using the letters of the word ARTICLEsuch that each word begin with vowel
Answer» ANSWER :360

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

If (7x-1)/(1-5x+6x^(2)) is expanded in ascending powers of x when abs(x) lt 1/3 then the coefficient of x^(n) is

Show that the relation R in the set A of all the books in a library of a college , given by R = {(x,y) : x and y have same number of pages} is an equivalence relation.

Find the sum of all odd numbers between 1 and 100 which are divisible by 3

If vec(a).vec(a)=0 and vec(a).vec(b)=0 then what can be concluded about the vector vec(b) ?

If |z-1| = 2 then the locus of z is

If alpha, beta, gammaare roots of the equation3x^(3) + 6x^(2) - 9x + 2 = 0then sum (alpha//beta) =

If a curve y=asqrtx+bx passes through the point (1,2) and the area bounded by the curve, line x=4 and x-axis is 8 square units, then

If |z - (2)/(z)| = 1 then find the maximum value of |z|.

Find the value of k if the line 2y=5x+k is a tangent to the parabola y^(2)=6x

Differentiate the following with respect to x: x log x

Let n(A)=m" and "n(B)=n. Then the total number of non-empty relations that can be defined from A to B is :

The slope of one of the common tangent to circle x^(2)+y^(2)=1 and ellipse x^(2)/4+2y^(2)=1 is sqrt(a/b) where gcd(a, b)=1 then (a+b)/2 is equal to

matrix A=({:(-i,0),(0,-i):}) then A^2 =

By Simpson's rule, value of int_(1)^(2)(dx)/(x) dividing the interval (1,2) into four equal parts, is

(i) Let z be a non-real complex number lying on the circle |z|=1. Then prove that z= (1+ I tan (("arg" z)/(2)))/(1-I tan(("arg"z )/(2))) (ii) Find the modulus and the argument of the complex number z_1, where z_1 = z^2 - z and z = cos theta+I sin theta.

Prove that the circle on a focal radius of a parabola, as diameter touches the tangent at the vertex.

Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area ?

The scalartriple product[veca+vecb-vec c" "vecb+vec c -veca" "vec c+veca-vecb] is equal to :

1+ cos 2x + cos 4 x+ cos 6x- 4 cos x cos 2x cos 3x =

Given that the 4th term in the expansion of (2+(3)/(8)x)^(10) has the maximum numerical value, find the range of x for which the statement will be true.

x^(3)/((2x-1)(x+2)(x-3))=A+B/(2x-1)+C/(x+2)+D/(x-3) rArr A =

f:R rarrR , f(x) =x/(sqrt(1+x^2)),AAx inR.Then find (fofof) (x).

The position vectors of the points A, B, C and D are 4hati+3hatj-hatk, 5hati+2hatj+2hatk, 2hati-2hatj-3hatk and 4hati-4hatj+3hatk respectively. Show that AB and CD are parallel.

The product of all the values of |lambda|, such that the lines x+2y-3=0, 3x-y-1=0 and lambdax +y-2 =0 cannot form a triangle, is equal to

Match the entries of Column I with those of Column-II

On the Argand plane point' A 'denotes a complex number z_1 . A triangle OBQ is made directly similar to the triangle OAM ,where OM = 1as shown in the figure . If the point B denotes the complex number z_2,then find the complex number corresponding to the point ' Q ' in terms ofz_1 & z_2

Evaluate the following integrals. int(2x-3)/(sqrt(2x^(2)+5x-6))dx

Match the entries of Column I with those of Column-II

if A =[(2,-2),(-2,2)]," then " A^(n)=2^(k)A, where k=

If{:[( 2,-3,5),( 3,2,-4),( 1,1,-2) ]:} ,find A^(-1)Using A^(-1)solve the system of equations 2x -3y +5z =11 3x+2y -4z=-5 x+ y -2z =-3

Ifthe equation ay^(2)+bxy+ex+dy=0 represents a pair of line, then

If p in (-1, 1) , then roots of the quadratic equation (p - 1)x^(2) + px + sqrt(1 - p^(2)) = 0 are

Show that 5x le 8 sin x-sin 2x le 6x for x le x le (pi)/(3)

Let a = I + j + k, b = I - j + 2k and c = xi + (x - 2) j - k. If the vector c lies in the plane of a and b. then x equals

The sine of the angle between the vectors a=3hati+hatj+2hatk and b=2hati-2hatj+4hatk is

If hatu and hatv are unit vectors and theta is the acute angle between them, then 2hatuxx3hatv is a unit vector for

The subnormal at any pointion the curve y' = a is constant. Then the value of x is

int (3x+4)/(x^(2)+2x+3)dx=

If a=hat(i)+hat(j)-2hat(k), b=2hat(i)-hat(j)+hat(k) and c=3hat(i)-hat(k) and c=ma+nb, then m+n is equal to

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

((cosA+cosB)/(sinA-sinB))^(2012)+((sinA+sinB)/(cosA-cosB))^(2012)=

Find the probability of getting both even if two dice are rolled

A man on the ground observes that the angle of elevation of the top of a tower is 68^(@) 11', and a flagstaff 24 m high on the summit of the tower subtends an angle of 2^(@) 10' at the observer's eye. If tan 70^(@) 21' = 2.8 and cot 68^(@) 11' = 0.4, the height of the tower is

Find area of the triangle with vertices at the point given in each of the following (2,7),(1,1),10,8)

IfA= {:[( 2,-1,1),(-1,2,-1),(1,-1,2) ]:} Verify thatA^(3) -6A^(2) +9A -4I=Oand hence find A^(-1)

How many 4 letter words can be formed using the letters of the word ARTICLEsuch that each word begin with vowel