28278 + Interview Questions in Maths in Class 12 Page 9 InterviewSolution

401.	x+2y+4z=12 y+2z=-1 3x+2y+4z=4
Answer»

Discussion

402.	xa^3+ya^2+za=0 In the equation above,x,y and z are constants. IF the equations has roots -6,0 and 4, which of the following is a factor of xa^3+ya^2+za?
Answer» a-2 a+4 a-6 a+6 Answer :D

Discussion

403.	A letter is known to have come from either 'MAHARASTRA' or 'MADRAS' on the post mark only conseutive letter 'RA' can be read clearly. What is the chance that the letter came from 'MAHARASTRA'.
Answer» ANSWER :`(10)/(19)`

Discussion

404.	If f(x)=sqrt("cosec"^2x-2sinx cosx - 1/(tan^2x)) x in ((7pi)/4,2pi), then f.((11pi)/6)=
Answer» `(1-sqrt3)/2` `(sqrt3-1)/2` `(1+sqrt3)/2` NONE of these SOLUTION :`f(x)=sqrt("cosec"^2x-cot^2x-2sin x cos x)` `=sqrt((cosx-sinx)^2)` f(x)=cos x -sin x `RARR` f.(x)=-sin x - cos x `f.((11pi)/6)=-[sin (11pi)/6 + cos(11pi)/6]` `rArr f.((11pi)/6)=-[(-1)/2+sqrt3/2]=(1-sqrt3)/2`

Discussion

405.	The longest distance of the point (a, 0) from the curve 2x^2+y^2=2x is
Answer» `1+a` `\|1-a\|` `ROOT()(1-2a+2a^2)` `root()(1-2a+3a^2)` ANSWER :C

Discussion

406.	The set of points where the function f given by f(x) = \|2x – 1\| sin x is differentiable is
Answer» R `R-{(1)/(2)}` `(0,OO)` None of these Answer :B

Discussion

407.	There are 5 bags, each containing 5 white balls and 3 black balls. Also, there are 6 bags, each containing 2 white balls and 4 black balls. A white balls is drawn at random. Find the probability that this white ball is from a bag of the first group.
Answer» <P> SOLUTION :Let `E_1` = event of selecting a bag from the first group, `E_2` = event of selecting a bag from the second group, and E = event of drawing a WHITE ball. Then, `P(E_1)=5/11and P(E_2)=6/11`. `P(E//E_1)`= probability of GETTING a white ball, given that it is from a bag of the first group `=5/8`. `P(E//E_2)` = probability of getting a white ball, given that it is from a bag of the second group `=2/6=1/3` Probability of getting the ball from a bag of the first group, given that it is white `P(E_1//E)` `=(P(E//E_1).P(E_1))/(P(E//E_1).P(E_1)+P(E//E_2).P(E_2))`[by Bayes's theorem] `=((5/8xx5/11))/((5/8xx5/11)+(1/3xx6/11))=75/123`.

Discussion

408.	If veca, vecb and vecc are non-zero vectors, then vecaxxvecb = vecaxxvecc
Answer» `VECB` = `VECC` `VECA\|\|(vecb-vecc) `vecb\|\|vecc` `vecbbotvecc` ANSWER :B

Discussion

409.	Find the vector cartesian equation of the plane passing through the points (1, -2, 3) and (-1, 2, -1) and is parallel to the line vecr=(2hati-hatj+hatk)+t(2hati+3hatj+4hatk)
Answer» SOLUTION :N/A

Discussion

410.	IF a,b,care thenumberof4digitednumberthat canbeformedusingthe digits2,4,5,7,8that aredivisibleby3,4,5 respectively thenascendingorderof a,b,c, is
Answer» a,B,C b,c,a c,b,a c,a,b Answer :C

Discussion

411.	If veca and vecb are two nonzero vectors such that \|vecaxxvecb\|=veca.vecb then find the angle between veca and vecb.
Answer» ANSWER :`45^(@)`

Discussion

412.	Match the statements/expressions given in List I with the values given in List II.
Answer» <P>`{:(a,b,c,d),(s,r,Q,p):}` `{:(a,b,c,d),(q,s,r,p):}` `{:(a,b,c,d),(s,r,p,q):}` `{:(a,b,c,d),(q,s,p,r):}` Solution :`y=COS (3 cos^(-1)x)` `rArr""(3 sin (3 cos^(-1)x))/(sqrt(1-x^(2)))` `rArr""sqrt(1-x^(2))y'=3 sin (3 cos ^(-1)x)` `rArr""(-x)/(sqrt(1-x^(2)))y'+sqrt(1+x^(2))y''=3COS (3cos ^(-1) x)CDOT(-3)/(sqrt(1-x^(2)))` `rArr""-xy'+(1-x^(2))y''=-9y` `rArr""(1)/(y)[(x^(2)-1)y''+xy']=9`

Discussion

413.	Let triangle ABC be right angled at C.If tanA+tanB2, then-
Answer» `/_A = 30^(@)` TRIANGLE ABC must be ISOSCLES sides of the triangle ABC are 3,4,5 triangle must be scalene Answer :B

Discussion

414.	If int (3 cos x - 2 sin x )/(4 sin x + 5 cos x )dx = Ax + B log \| 5cos x + 4 sin x \| then (A, B ) =
Answer» `((7)/(41), (22)/(41)) ` `((22)/(41), (7)/(41))` `((15)/(41), (7)/(21))` `((7)/(21), (15)/(41))` ANSWER :A

Discussion

415.	If S is defined on R by (x,y) inR hArr xy ge 0 . Then S is ...........
Answer» an EQUIVALENCE RELATION REFLEXIVE only SYMMETRIC only TRANSITIVE only Solution :N/A

Discussion

416.	If for a continuous function f(x), int_(-pi)^(t) (f(x)+x)dx=pi^(2)-t^(2), forall t ge -pi, thenf(-(pi)/(3)) is equal to :
Answer» `PI` `(pi)/(2)` `(pi)/(3)` `(pi)/(6)` ANSWER :1

Discussion

417.	If A and B are coefficients of x^(n)in the expansions of (1+x)^(2n) and (1+x)^(2n-1) respectively, then A/B is equal to
Answer» 4 2 9 6 Answer :B

Discussion

418.	Find the direction cosines of the line whose direction ratios are 12, 4, -8.
Answer» ANSWER :`(6)/(11),(2)/(11),(-9)/(11)`

Discussion

419.	Show that if f :R {(7)/(5)} to R - {(3)/(5)} is defined by f (x) = (3x +4)/( 5x -7) and g :R - {(3/(5)} to R - {(7)/(5)} is defined by g (x) = (7x +4)/(5x -3), then fog = I _(A) andgof =I _(B), where, A = R - {(3)/(5)}, B+R -{(7)/(5)},I_(A)(x) =x, AA x in A, I_(B)\|(x) =x , AA x in B are called identity functions on sets A and B, respectively.
Answer» ANSWER :`I _(A)`

Discussion

420.	Given that x is a real number satisfying(5x^(2)-26x+5)/(3x^(2)-10x+3) lt 0, then
Answer» `x LT (1)/(5)` `(1)/(5) lt x lt 3` `x gt 5` `(1)/(5) lt x lt (1)/(3) or 3 lt x lt 5` ANSWER :d

Discussion

421.	Find the eccentricity of the ellipse (a) whose latus rectum is equal to
Answer» ANSWER :`(SQRT(3))/(2)`

Discussion

422.	int_(0)^(1) (x^(5)dx)/(1+x^(4))=
Answer» `(2-pi)/(4)` `(2+pi)/(4)` `(4+pi)/(8)` `(4-pi)/(8)` ANSWER :D

Discussion

423.	Let A and B be independent events with P(A) = 0 . 3 and P(B) = 0 . 4, . FindP(Acap B)
Answer» ANSWER :(i) 0.12, (II) 0.58, (III) 0.3, (IV) 0.4

Discussion

424.	Decide whether each statement Must Be True, Could Be True, or Will Never Be True. pqgt t
Answer» ANSWER :COULD Be TRUE

Discussion

425.	{:("Column A","Kate ate 1/3 of a cake, Fritz ate 1/5 of remaining cake, and what was left was eaten by Emily","Column B"),("Fraction of the cake eaten by Emily",,5//7):}
Answer» If COLUMN A is larger If column B is larger If the columns are EQUAL If there is not ENOUGH information to decide Answer :B

Discussion

426.	\|((a+b)^(2),ca,bc),(ca,(b+c)^(2),ab),(bc,ab,(c+a)^(2))\|=
Answer» `(ab+bc+ca)^(3)` `abc(a+b+C)^(3)` `2ABC(a+b+c)^(3)` `2(ab+bc+ca)^(3)` ANSWER :C

Discussion

427.	The x-coordinate of the incentre of the triangle where the midpoints of the sides are (0, 1) (1, 1) and (1, 0) is
Answer» `2+sqrt2` `1+sqrt2` `2-sqrt2` `1-sqrt2` ANSWER :C

Discussion

428.	Ify = A (x) e^(-intpdx) isa solution of (dy)/(dx) +P(x)y = Q(x) ,then A'(x) =
Answer» `E^(intpdx)` `Q(X)e^(-intpdx)` `INT Q (x)e^(intpdx) dx` `Q(x) e^(intpdx)` Answer :D

Discussion

429.	If the lines (x-1)/-3 = (y-2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z-6)/-5 are perpendicular, then find the value of k.
Answer» Answer :The direction RATIOS of the two lines are -3, 2k, 2 and 3k, 1, -5 respectively SINCE the lines are perpendicular, `-3xx3k+2kxx1+2xx-5=0` `rArr` -9k+2k-10=0 `rArr` 7k=-10 `rArr = (-10)/7`

Discussion

430.	Statement 1 : (A hArr ~B)) a tautology to A hArr B Statement 2: A vv (~A ^^ ~B)) a tautology
Answer» Statement 1 and statement 2 are both false Statement 1 and staement 2 are both TRUE statement 1 is true and statement 2 is false statement 1 is false and statement 2 is true Answer :B

Discussion

431.	If I _(1) = int _(0) ^(1) 2 ^(x ^(2)) dx, I _(2) = int _(0) ^(1) 2 ^(x ^(3)) dx, I _(3) = int _(1) ^(2) 2 ^(x ^(3)) dx and I _(4 ) = int _(1) ^(2) 2 ^(x ^(3)) dx then-
Answer» `I _(2) gt I _(1)` `I _(1) gt I _(2)` `I _(3) =I_(4)` `I _(3) gt I _(4)` ANSWER :B

Discussion

432.	LeT p(X,Y) be the midpoint of the line joining (1,0) to a point on the curve y^(2)=\|(x+1,x+2),(x+3,x+5)\|. The locus of P is symmetrical about
Answer» X-axis Y-axis `y=1` `X+1` ANSWER :A

Discussion

433.	Find number of three digit numbers equal to the sum of the factorials of their digits.
Answer» ANSWER :1

Discussion

434.	When \|x\|lt (1)/(2) , the coefficient of x^(4)in the expansion of(3x^(2) - 5x + 3)/((x - 1)(2x +1) (x +3))is
Answer» `(722)/(27)` `(724)/(27)` `(-722)/(27)` `(-724)/(27)` ANSWER :C

Discussion

435.	Differentiate x^(sin x), x gt 0 w.r.t x
Answer» ANSWER :`X^(SIN x-1).sin x + x^(sin x). COS x.log x`

Discussion

436.	Find the number of 4- letter words that can be formed using the letters of the word. MIRACLE. How many of them begin with an vowel
Answer» ANSWER :360

Discussion

437.	Let a=2^(log_35 , b=3^(log_5 2, c=4 the identify the correct option :-
Answer» ab=C ab GT c ac =b ab `LE` c

Discussion

438.	Find the number of 4- letter words that can be formed using the letters of the word. MIRACLE. How many of them begin and end with vowels
Answer» ANSWER :120

Discussion

439.	Find the number of 4-letter words that can be formed using the letters of the word MIRACLE. How many of them end with a consonant ?
Answer» ANSWER :480

Discussion

440.	The xy-plane above shows the two points of intersection of thegraphs of a linear function and a quadratic function. The leftmost point of intersection has coordinates (a, b) and the rightmost point of intersection has coordinates (c, d) . If the vertex of the graph of the quadraticfunction is at (2, -27) , what is the value of b-d ?
Answer» ANSWER :36

Discussion

441.	Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The total number of root(s) of the equation f(x)=g(x) is/ are
Answer» `infty` 1 2 0 Solution :`F(x) le0` and `F^(')(x)=f(x)` Now, `f(x) ge cF(x)` or `e^(-cx)F^(')(x) -ce^(-cx)F(x) ge0` THUS, `e^(-cx)F(x)` is an increasing function `therefore e^(-cx) F(x) ge e^(-c(0))F(0)` or `e^(-cx)F(x) ge` or `F(x) ge0` [as `f(x) ge cF(x)` and c is POSITIVE. Also, `(dg(x))/(dx) lt g (x) AA x gt 0` or `(d/(dx)) e^(-x)g(x) lt 0` Thus, `e^(-x)g(x)` has ONE solution, `x=0`

Discussion

442.	Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The number of solution(s) of the equation \|x^(2)+x-6\|=f(x)+g(x) is/are
Answer» 2 1 0 3 SOLUTION :`\|X^(2)+x-6\|=f(x) + G(x)` or `\|x^(2)+x-6\|=g(x)` Thus, no solution exists.

Discussion

443.	Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The solution set of inequation g(x)(cos^(-1)x-sin^(-1)) le0
Answer» `[-1,1/sqrt(2)]` `[1/sqrt(2),1]` `[0,1/sqrt(2)]` `[0,1/sqrt(2)]` SOLUTION :`G(X) (COS^(-1)x-sin^(-1)x) le0` or `(cos^(-1)x-sin^(-1)x) ge0` or `x in [-1, 1/sqrt(2)]`

Discussion

444.	There are 50 stations on a railway line. The number of different kinds of single 2nd class tickets must be printed so as to enable a passenger to go from one station to another station is
Answer» `""^(50)C_(2)` 2500 2450 `50!` ANSWER :C

Discussion

445.	If the vertices A, B, C of a DeltaABC have position vectors (1, 2, 3), (-1, 0, 0), (0, 1, 2) respectively, then angleABC angleABC is the angle between the vectors BA and BC), is equal to
Answer» `(pi)/(2)` `(pi)/(4)` `cos^(-1)((10)/(sqrt(102)))` `cos^(-1)((1)/(3))` Solution :WE aregiventhe points `A (1,2,3),B(-1,0,0)` and`C (0,1,2)` Also , itgiventhat` angleABC `is theanglebetweenthevectorsBAand BChereBA= PVofofA -PV of B `=( HATI + 2 hatj+3 hatk )- (-hati+ 0 hatj + 0 hatk )` `=[ hati-(-hati )+(2hatj -0 )+ (3 hatk -0 )] = 2 hati+ 2 hatj+ 3hatk` ` IMPLIES\| BA\|= sqrt((2) ^(2)+(2)^(2)+(3)^(2))=sqrt(4+4+9)=sqrt(17)` `implies BC=PVofC-PV of B ` `=(0 hati +1 hatj + 2 hatk)-(-hati +0hatj+0hatk)=hati +hatj+2hatk` `implies \|BC\|=sqrt((1)^(2) +(1)^(2)+(2)^(2))=sqrt(1+1+4)= sqrt(6)` Now , `BA.BC =(2 hati + 2 hatj + 3 hatk ).(hati + hatj + 2 hatk )` `=2xx1+2xx1+3xx2=10` ` thereforecos theta= (BA.BC)/(\|BA\|\|BC\|)implies cos (angle ABC)=(10)/(sqrt(17)sqrt(6))` `impliesangleABC= cos ^(-1) ((10)/(sqrt(102)))`

Discussion

446.	In the figure, the area of the quadrilateral ABCD is 75. What is the area of parallelogram EFGH?
Answer» 96 153 157 171 Answer :D

Discussion

447.	Integrate the following functions : int(dx)/(x(x^(5)+1))
Answer» ANSWER :`log\|x\|-(1)/(5)log\|x^(5)+1\|+c`

Discussion

448.	Vertices of a parallelogram are (0, 0), ((1)/(m-n), (m)/(m-n)), ((-1)/(m-n), (m)/(m-n)) and (0, 1) are m,n are the roots of the equation 441x^(2)+42x-8=0. If area of the parallelogram is A then A is equal to
Answer» ANSWER :`3.5`

Discussion

449.	Four rotten oranges are accidentally mixed with 16 good ones. Three oranges are drawn at random from the mixed lot. Let X be the number of rotten oranges drawn. Find the mean and variance of X.
Answer» ANSWER :`"MEAN "=(3)/(5), "VARIANCE "=(68)/(125)`

Discussion

450.	Equation of circle which touch X-axies at (3,0) and making Y-intercept of length 8 units is
Answer» `X^(2)+y^(2)-6x+-10y+9=0` `x^(2)+y^(2)-6x+-10y-9=0` `x^(2)+y^(2)+16x+110y+9=0` None Answer :A

Discussion

Explore topic-wise InterviewSolutions in .

x+2y+4z=12 y+2z=-1 3x+2y+4z=4

xa^3+ya^2+za=0 In the equation above,x,y and z are constants. IF the equations has roots -6,0 and 4, which of the following is a factor of xa^3+ya^2+za?

A letter is known to have come from either 'MAHARASTRA' or 'MADRAS' on the post mark only conseutive letter 'RA' can be read clearly. What is the chance that the letter came from 'MAHARASTRA'.

If f(x)=sqrt("cosec"^2x-2sinx cosx - 1/(tan^2x)) x in ((7pi)/4,2pi), then f.((11pi)/6)=

The longest distance of the point (a, 0) from the curve 2x^2+y^2=2x is

The set of points where the function f given by f(x) = |2x – 1| sin x is differentiable is

There are 5 bags, each containing 5 white balls and 3 black balls. Also, there are 6 bags, each containing 2 white balls and 4 black balls. A white balls is drawn at random. Find the probability that this white ball is from a bag of the first group.

If veca, vecb and vecc are non-zero vectors, then vecaxxvecb = vecaxxvecc

Find the vector cartesian equation of the plane passing through the points (1, -2, 3) and (-1, 2, -1) and is parallel to the line vecr=(2hati-hatj+hatk)+t(2hati+3hatj+4hatk)

IF a,b,care thenumberof4digitednumberthat canbeformedusingthe digits2,4,5,7,8that aredivisibleby3,4,5 respectively thenascendingorderof a,b,c, is

If veca and vecb are two nonzero vectors such that |vecaxxvecb|=veca.vecb then find the angle between veca and vecb.

Match the statements/expressions given in List I with the values given in List II.

Let triangle ABC be right angled at C.If tanA+tanB2, then-

If int (3 cos x - 2 sin x )/(4 sin x + 5 cos x )dx = Ax + B log | 5cos x + 4 sin x | then (A, B ) =

If S is defined on R by (x,y) inR hArr xy ge 0 . Then S is ...........

If for a continuous function f(x), int_(-pi)^(t) (f(x)+x)dx=pi^(2)-t^(2), forall t ge -pi, thenf(-(pi)/(3)) is equal to :

If A and B are coefficients of x^(n)in the expansions of (1+x)^(2n) and (1+x)^(2n-1) respectively, then A/B is equal to

Find the direction cosines of the line whose direction ratios are 12, 4, -8.

Given that x is a real number satisfying(5x^(2)-26x+5)/(3x^(2)-10x+3) lt 0, then

Find the eccentricity of the ellipse (a) whose latus rectum is equal to

int_(0)^(1) (x^(5)dx)/(1+x^(4))=

Let A and B be independent events with P(A) = 0 . 3 and P(B) = 0 . 4, . FindP(Acap B)

Decide whether each statement Must Be True, Could Be True, or Will Never Be True. pqgt t

{:("Column A","Kate ate 1/3 of a cake, Fritz ate 1/5 of remaining cake, and what was left was eaten by Emily","Column B"),("Fraction of the cake eaten by Emily",,5//7):}

|((a+b)^(2),ca,bc),(ca,(b+c)^(2),ab),(bc,ab,(c+a)^(2))|=

The x-coordinate of the incentre of the﻿ triangle where the midpoints of the sides﻿ are (0, 1) (1, 1) and (1, 0) is

Ify = A (x) e^(-intpdx) isa solution of (dy)/(dx) +P(x)y = Q(x) ,then A'(x) =

If the lines (x-1)/-3 = (y-2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z-6)/-5 are perpendicular, then find the value of k.

Statement 1 : (A hArr ~B)) a tautology to A hArr B Statement 2: A vv (~A ^^ ~B)) a tautology

If I _(1) = int _(0) ^(1) 2 ^(x ^(2)) dx, I _(2) = int _(0) ^(1) 2 ^(x ^(3)) dx, I _(3) = int _(1) ^(2) 2 ^(x ^(3)) dx and I _(4 ) = int _(1) ^(2) 2 ^(x ^(3)) dx then-

LeT p(X,Y) be the midpoint of the line joining (1,0) to a point on the curve y^(2)=|(x+1,x+2),(x+3,x+5)|. The locus of P is symmetrical about

Find number of three digit numbers equal to the sum of the factorials of their digits.

When |x|lt (1)/(2) , the coefficient of x^(4)in the expansion of(3x^(2) - 5x + 3)/((x - 1)(2x +1) (x +3))is

Differentiate x^(sin x), x gt 0 w.r.t x

Find the number of 4- letter words that can be formed using the letters of the word. MIRACLE. How many of them begin with an vowel

Let a=2^(log_35 , b=3^(log_5 2, c=4 the identify the correct option :-

Find the number of 4- letter words that can be formed using the letters of the word. MIRACLE. How many of them begin and end with vowels

Find the number of 4-letter words that can be formed using the letters of the word MIRACLE. How many of them end with a consonant ?

Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The total number of root(s) of the equation f(x)=g(x) is/ are

Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The number of solution(s) of the equation |x^(2)+x-6|=f(x)+g(x) is/are

Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The solution set of inequation g(x)(cos^(-1)x-sin^(-1)) le0

There are 50 stations on a railway line. The number of different kinds of single 2nd class tickets must be printed so as to enable a passenger to go from one station to another station is

If the vertices A, B, C of a DeltaABC have position vectors (1, 2, 3), (-1, 0, 0), (0, 1, 2) respectively, then angleABC angleABC is the angle between the vectors BA and BC), is equal to

In the figure, the area of the quadrilateral ABCD is 75. What is the area of parallelogram EFGH?

Integrate the following functions : int(dx)/(x(x^(5)+1))

Vertices of a parallelogram are (0, 0), ((1)/(m-n), (m)/(m-n)), ((-1)/(m-n), (m)/(m-n)) and (0, 1) are m,n are the roots of the equation 441x^(2)+42x-8=0. If area of the parallelogram is A then A is equal to

Four rotten oranges are accidentally mixed with 16 good ones. Three oranges are drawn at random from the mixed lot. Let X be the number of rotten oranges drawn. Find the mean and variance of X.

Equation of circle which touch X-axies at (3,0) and making Y-intercept of length 8 units is

The x-coordinate of the incentre of the triangle where the midpoints of the sides are (0, 1) (1, 1) and (1, 0) is