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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 11451. |
If cos alpha + cos beta + cos gamma = 0 and alos sin alpha + sin beta + sin gamma= 0, then provethat.(a)cos 2 alpha + cos 2 beta + cos 2gamma = sin 2alpha +sin2beta+sin2gamma=0(b)sin 3 alpha+ sin 3 beta + sin3 gamma = 3 sin (alpha + beta + gamma)(c)cos 3 alpha + cos 3beta + cos 3gamma = 3 cos (alpha + beta + gamma) |
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Answer» Solution :Let `z_(1) = cos ALPHA + isin alpha,z_(2) = cos beta + isin beta`, `z_(3) = cos gamma +isin gamma` `thereforez_(1) +z_(2)+z_(3) = (cos alpha + cos beta + cos gamma)+i(sinalpha +sin beta + singamma)` ` = 0+ ixx 0 =0` (a) Now, `(1)/(z_(1)) = (cos alpha + isin alpha)^(-1) = cos alpha- isin alpha` `(1)/(z_(1)) =cos beta- isin beta` `(1)/(z_(2)) =cos gamma-isingamma ` `therefore(1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3))` `=(cos alpha + cos beta + cos gamma) -i(sin alpha + sin beta + sin gamma) (2) ` `=0-ixx 0 =0` `z_(1)^(2) + z_(2)^(2) + z_(3)^(2) =(z_(1) + z_(2) +z_(3))^(2) -2(z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1))` `=0-2z_(1)z_(2)z_(3)((1)/(z_(3))+(1)/(z_(1)) +(1)/(z_(2)))` `RARR (cos alpha + isin alpha)^(2) + (cos beta + isin beta)^(2) + (cosgamma+isin gamma)^(2) =0` `rArr (cos 2alpha + isin 2alpha)+(cos 2beta+isin2)+(cos 2gamma +isin 2gamma)=0+ixx0` Equating realand imaginary parts on both sides, `cos 2alpha + cos 2beta + cos2gamma =0` `and sin 2alpha + sin 2beta + sin2gamma = 0` (b) `z_(1)^(3) +z_(2)^(3) +z_(3)^(3) =(z_(1) +z_(2))^(3) -3z_(1)z_(2)(z_(1) +z_(2))+z_(3)^(3)` ` = (-z_(3))^(3) -3z_(1)z_(2)(-z_(3))+z_(3)^(3)""["Using (1)"]` `=3z_(1)z_(2)z_(3)` `rArr (cos alpha+sin alpha)^(3)+(cos beta+ isinbeta)^(3) +(cos gamma + isin gamma)^(3)` `= 3(cos alpha + isin alpha) (cos beta+isin beta)(cos gamma + isin gamma)^(3)` `cos 3alpha +isin 3alpha +cos 3beta +isin 3beta + cos3gamma+ isin 3gamma` `= 3{cos (alpha +beta+ gamma)+ isin (alpha + beta+gamma)}` Equaiting imaginary parts on bothsides, `sin 3alpha +sin 3 beta +sin 3gamma =3SIN(alpha + beta + gamma)` (c) Equating real parts on both sides, `cos 3alpha + cos3beta + cos 3gamma = 3cos (alpha+beta+gamma)` |
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| 11452. |
From a point on the horizontal plane, the elevation of the top of a hill is 45^(@). After walking 500 m towards its summit up a slope inclined at an angle of 15^(@) to the horizon the elevation is 75^(@), the height of the hill is |
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Answer» `500 SQRT(6)` m |
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| 11453. |
The top of a pole, placed against a wall at an angle alpha with the horizon, just touches the coping, and when its foot is moved a m, away from the wall and its angle of inclination is beta, it rests on the sill of a window, the vertical distance of the sill from the coping is |
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Answer» `a sin (( alpha + beta)//2)` |
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| 11454. |
If the standard deviation of numbers 2,4 5 and 6 is a constant alphathen the standard deviation of the number 4,6, 7 and 8 is |
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Answer» ` ALPHA +2` |
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| 11456. |
Find the distance of plane 3x + 4z + 15 = 0 from (5, 0, 0) |
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Answer» 6 |
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| 11457. |
Prove that(cos9^@+sin9^@)/(cos9^@-sin9^@)=tan54^@ |
| Answer» SOLUTION :R.H.S.`=tan54^@=TAN(45^@+9^@)=(TAN45^@+tan9^@)/(1-tan45^@tan9^@)=(1+sin9^@/cos9^@)/(1+sin9^@/cos9^@)=(cos9^@+sin9^@)/(cos9^@-sin9^@)`=L.H.S | |
| 11458. |
Let A and B be two non empty subsets of a set X such that A is not a subset of B then |
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Answer» A is ALWAYS a SUBSET of the complement of B |
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| 11459. |
A die is thrown three times,E : 4 appears on the third tossF : 6 and 5 appears, respectively on first two tosses.Find P(E//F) |
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| 11460. |
Verify Mean Value Theorem if f(x)= x^(3) - 5x^(2) - 3x in the interval [1,3]. |
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| 11461. |
The revenue R from sale of x units of a commodity is given by R = 20 x - 0.5 x^2 . Percentage rate of change of R when x = 10 is(i) 15%(ii) 6(2)/3%(iii) 1/15%(iv) 20% |
| Answer» ANSWER :B | |
| 11462. |
There are 4 letters and 5 boxes in a row. Number of ways of postingthese letters if all the letters are not posted in the same box is |
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Answer» 600 |
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| 11463. |
Through a point P(f,g,h) a plane is drawn at right angles to bar(OP), to meet the axes in A, B and C. If OP = r, the centroid of the triangle ABC is........... |
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Answer» `((f)/(3R),(g)/(3r),(h)/(3r))` |
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| 11464. |
Which of the following is a solution of cos3x=(1)/(2)? |
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Answer» `60^(@)` |
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| 11465. |
-(2)/(3)-(1)/(2)((4)/(9))-(1)/(3)((8)/(27))-(1)/(4)((16)/(81))-…..oo= |
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Answer» `-log_(E )3` |
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| 11467. |
Without expandingthe determinant prove the following. |[2,7,65],[3,8,75],[5,9,86]|=0 |
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Answer» SOLUTION :`|[2,7,65],[3,8,75],[5,9,86]|=|[2+63,7,65],[3+72,8,75],[5+81,9,86]|=|[65,7,65],[75,8,75],[86,9,86]|`(by`C_1rarrC_1+9C_2`) `=0 (becauseC_1=c_3)` |
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| 11468. |
If A and B are two independent events with P(A)= (3)/(5) and P(B)= (4)/(9), then P(A' cap B')equals to ……. |
| Answer» Answer :D | |
| 11469. |
In 2000, the total number of dollars of gift shop revenue was how many times as great as the aveage daily number of full - price tickets sold? |
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Answer» 400 |
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| 11470. |
Write Minors and Cofactors of the elments of following determinants : |{:(1,0,1),(0,1,0),(0,0,1):}| |
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Answer» `A_(11)=1,A_(12)=0,A_(13)=0,A_(21)=0,A_(22)=1,A_(23)=0,A_(31)=0,A_(32)=0,A_(33)=1` |
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| 11472. |
Equation of theplaneperpendicular to the line (x )/(1) = (y )/(2) = ( z )/(3) andpassing throughthe point (2,3,4)is : |
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Answer» 1.`2x + 3y +Z = 17` |
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| 11473. |
If(x + a_(1)) (x + a_(2)) (x + a_(3)) …(x + a_(n)) = x^(n) + S_(1) x^(n-1) + S_(2) x^(n-2) + …+ S_(n) where ,S_(1) = sum_(i=0)^(n) a_(i), S_(2) = (sumsum)_(1lei lt j le n) a_(i) a_(j) , S_(3) (sumsumsum)_(1le i ltk le n) a_(i) a_(j) a_(k) and so on . If (1 + x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n) x^(n) the cefficient ofx^(n) in the expansion of (x + C_(0))(x + C_(1)) (x + C_(2))...(x + C_(n))is |
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Answer» `2^(2n-1) - (1)/(2) ""^(2n)C_(n-1)` ` = x^(n+1) + (sum_(r=0)^(n) C_(r))^(n) + ( UNDERSET(0 le i j le n )(sumsum)C_(i) C_(j)) x^(n-1) + ... ` ` therefore ` COEFFICIENTOF ` x^(n-1)" in" underset(0 le i j le n )(sumsum)C_(i) C_(j)` `= (1)/(2){ (sum_(r=0)^(n) C_(r))^(2) - ( sum_(r=0)^(n) C_(r)^(2) )} = (1)/(2) = { 2^(2n) - ""^(2n)C_(n)} ` ` = 2^(2n-1)- (1)/(2) . ""^(2n)C_(n) ` . |
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| 11474. |
Coefficient of x^(99) in the expansion of (x-1) (x-3)(x-5)..(x-1999) is |
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Answer» `-100` |
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| 11475. |
Kelvin takes 3 minutes to inspect a car , and John takes 4 minutes to inspect a car. If they both start inspecting different cars at 8 : 30 AM, what would be the ratio of the number of cars inspected by Kelvin and John by 8 : 54 AM of the same day? |
| Answer» ANSWER :D | |
| 11476. |
If(a+bx ) ^ ( -3 )=(1 ) /(27) +(1 ) /(3)x+…,thentheorderedpair(a, b )equalsto |
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Answer» ` ( 3 ,- 27) ` `a ^( -3)( 1+(bx ) /(a)) ^(-3)= (1 )/(27)+(x) /(3)+ … ` `(1 ) /(a ^(3))[ 1 - ""^ 3c_1((bx )/(a )) +….] =(1 ) /(27)+(x) /(3) ` `(1)/(a ^3 )-""^ 3c _ 1(bx ) /(a ^ 4)+ ... = (1 )/(27)+(x ) /(3) ` `(1 )/(a ^ 3 ) = (1 )/(27 ) "" [ BECAUSEA =3 ] ` `-""^3c _ 1(bx ) /(a ^ 4 )=(x ) /(3) ` `therefore-3 ((B))/( 3 ^ 4 ) =(1 )/(3) ` ` b= -9 ` |
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| 11477. |
If ( p ^^ ~ q ) ^^ ( p ^^ r) rarr ~ P ^^ ris false, then truth values of p, q, and rare respectively: |
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Answer» T, T, T |
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| 11478. |
If the real numbers x, y, z are such that x^2 + 4y^2 + 16z^2 = 48 and xy + 4yz + 2zx = 24. what is the value of x^(2) +y^(2) z^(2)=? |
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| 11479. |
Discuss the relative position of the fol- lowing pair of circles. x^(2) + y^(2) -2x + 4y - 4 = 0 x^(2) y^(2) + 4 x - 6y -3 = 0 |
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| 11480. |
One of the complex roots of the equation x^(11)-x^(6)-x^(5)+1 =0 is |
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Answer» `"CIS"(3pi)/(5)` |
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| 11481. |
Using the Lagrange theorem estimate has valuein (1+e). |
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| 11482. |
The value C of the Lagrange's mean value theorem for the function f (x) =x (x - 2) in the interval [0,(1)/(2)] is |
| Answer» Answer :A | |
| 11483. |
(i) IfC isa givennon-zeroscalarand overset(to)(A)" and" overset(to)(B) be givennon-zerovectorssuch thatoverset(to)(A) bot overset(to)(B) then findthevectorsoverset(to)(X) whichsatisfies theequationsoverset(to)(A) "."overset(to)(X) =c" and" overset(to)(A) xxoverset(to)(X)= overset(to)(B) (ii) overset(to)(A) vectors A hascomponents A_(1), A_(2) , A_(3) in a right -handedrectangular cartesiancoordinate system OXYZ. Thecoordinate systemis rotated about theX-axis through an anlge(pi)/(2) . Findthecomponentsof Ain thenewcoordinatesystemin termsof A_(1),A_(2),A_(3) |
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Answer» and `vec(A) xx vec(X) =vec(B) rArr vec(A) ". " vec(B) =0 " and " vec(X)". " vec(B)=0` Now`[vec(X) vec(A) vec(A) xx vec(B)] = vec(X) ". " {vec(A) xx (vec(A) xx vec(B))}` ` =vec(X) .{(vec(A) ". " vec(B))vec(A)-(vec(A) ". " vec(B)) vec(B)}` ` = (vec(A) ". " vec(B))(vec(X) " ." vec(A)) - (vec(A) ". " vec(A)) (vec(X) " ." vec(B))=0` `rArr vec(X) , vec(A) , vec(A)xx vec(B)` are coplanar So `vec(X)` can berepresentedas alinearcombinationof `vec(A)" and"vec(A) xx vec(B)` , Letus consider, `vec(X) = lvec(A) + m (vec(A) xx vec(B))` Since `vec(A)" . " vec(X) = c` `:. vec(A) " ." {(vec(A) +m (vec(A)xx vec(B)) }=c` ` rArr l(vec(A) xx vec(A)) +m {vec(A) xx (vec(A) xx vec(B))}= vec(B)` `rArr 0- m |vec(A)|^(2)vec(B) =vec(B)` `rArrm = -(1)/(|vec(A)|^(2))` `:. vec(X) =((C)/(|vec(A)|^(2)))vec(A) -((1)/(|vec(A)|^(2))) (vec(A) xx vec(B))` (II) Sincevector`vec(A)`hascomponents `A_(1) , A_(2) , A_(3)` in thecoordinatesystemOXYZ `:. vec(A)= A_(1) HAT(i)+A_(2) hat(j)+A_(3) hat(k)` Whenthe givensystemis rotatedaboutan angleof `pi//2` the newX-axisis alongold Y-axisand newY-axisis alongthe oldnegativeX - axis, whereasz remainssame . Hencethe componentsof A in thenew systemare `(A_(2) , -A_(1), A_(3))` `:. vec(A) ` becomes`(A_(2) hat(i)- A_(2)hat(j)+ A_(3) hat(k))` |
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| 11484. |
Write the followingcomplex numberin polarform : (i) -3 sqrt(2) + 3 sqrt(2) i(ii) 1+i(iii) (1+7i)/(2-i)^2 |
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Answer» Solution :(i) Let `z = - 3sqrt(2) + 3sqrt(2)i` Then, `|z| = SQRT(-3sqrt(2)^(2) + (3sqrt(2))^(2)) =6` Let `tan alpha |(Im(z))/(Re(z))| = 1 rArr alpha = (pi)/(4)` Since the pointreqresenting z lies in thesecond quadrant, the agrumentof z is givenby `theta = pi - alpha- ((pi)/(4)) = ((3pi)/(4))` So, the polarform of `z - 3sqrt(2)+3sqrt(2)`i si `z=|z|(cos theta + i SIN theta) = 6 (cos.(3pi)/(4) + i sin .(3pi)/(4))` (II)Let `z = 1+i.` Then `|z| = sqrt(1^(2)+1^(2)) = sqrt(2)`. Let `tan alpha = |(Im(z))/(Re(z))| ` Then, `tan alpha|(1)/(1)| =1 or alpha = (pi)/(4)` Sincethe point(1,1) representing z lies in the first quadrant,the argumentof z is given by `theta = aloha= pi//4`. So, thepolarform of `z =1 + i` is `z =|z| (cos theta + isin theta) = (cos.(pi)/(4) +isin.(pi)/(4))` (iii)Let`z = -1-i`. Then `|z| = sqrt((-1)^(2)+ (-1)^(2)) = sqrt(2)` Let `tan alpha = |(Im(z))/(Re(z))|` Then , `tan alpha = |(-1)/(-1)| = 1 or alpha = (pi)/(4)` Sincethe point(-1,-1) representing z lies in thethird quadrant, theargumentof z is given by `theta = -(pi -alpha) = - (pi-(pi)/(4))= (-3pi)/(4)` So, thepolarform of z = - 1- is `z = |z| (costheta + isin theta) = sqrt(2) {cos ((-3pi)/(4))+isin((-3pi)/(4)) }` (iv) Let `z = 1 - i` . Then `|z|= sqrt(1+(-1)^(2)) = sqrt(2)`. Let `tan alpha =|(Im(z))/(Re(z))|` Then, `tan alpha =|(-1)/(1)| =1 or alpha = (pi)/(4)` sincethe point (1,-1) lies in thefourthquadrant, theargument of z is givenby `theta = alpha = - pi//4`. So the polar formof z = 1 - i si `z =|z|(cos theta + isin theta) = sqrt(2) {cos((-pi)/(4)) +isin((-pi)/(4))} = sqrt(2) (cos.(pi)/(4) -isin.(pi)/(4))` (V) Let`z = (1+7i) //[(2-i)^(2)]`. Then `z = (1+7i)/(4-4i+^(2)) = (1+7i)/(3-4i) = ((1+7i)/(3-4))((3+4i)/(3+4i))=(-25+25i)/(25) = -1+i` `therefore |z| = sqrt((-1)^(2) + (1)^(2) ) = sqrt(2)` Let `alpha` be theacutue ANGLE given by `tan alpha =|(Im(z))/(Re(z))| = |(-1)/(1)| = 1` Then `alpha = pi//4`. Since the point (-1,1) represeting z lies in thesecond quadrant, we have`theta = arg(z) = pi - alpha = pi - pi//4 = 3pi//4` Hence, z in thepolar form is given by `z = sqrt(2) (cos. (3pi)/(4) + isin.(3pi)/(4))` |
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| 11485. |
Let A = {1, 2, 3, ..., n} and B= {a,b}. Then the number of surjections from A into B is |
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Answer» <P>P(n,2) |
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| 11486. |
The partial fractions of (6x^(4)+5x^(3)+x^(2)+5x+2)/(1+5x+6x^(2)) are |
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Answer» `X^(2)-(1)/(1+2x)+(1)/(1+3x)` |
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| 11487. |
The measure of the angle of intersection between y^(2)=x and x^(2)=y other than one at (0, 0) is ………… |
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Answer» `"tan"^(-1)(4)/(3)` |
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| 11488. |
If S_(n) = (.^(n)C_(0))^(2) + (.^(n)C_(1))^(2) + (.^(n)C_(n))^(n), then maximum value of[(S_(n+1))/(S_(n))] is "_____". (where [*] denotes the greatest integer function) |
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Answer» `RARR (S_(n+1))/(S_(n)) = (.^(2n+2)C_(n+1))/(.^(2n)C_(n)) = ((2n+2)(2n+1))/((n+1)(n+1))` `= (2(2n+1))/(n+1)=4-(2)/(n+1)` `:. [(S_(n+1))/(S_(n))]_("max") = 3` |
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| 11489. |
A company has three plants at which it produces a certain item. 30% are produced at plant A, 50% at plant B and remaining at plant C. Suppose that 1% , 4% and 3% of the items produced at plants A, B and C respectively are defective. If an item is selected at random from all of those produced, what is the probability that item was produced at plant B is defective ? |
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Answer» 0.5 |
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| 11490. |
If the volume of the parallelopiped whose coterminus edges are represented by the vectors 5hati -4hatj +hatk , 4hati+3hatj-lambda hatk and hati-2hatj+7hatk is 216 cubic units, find the value of lambda. |
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| 11491. |
Equation 3x^(2)+7xy+2y^(2)+5x+3y+2=0 |
| Answer» Answer :D | |
| 11492. |
The two adjacent sides of a parallelogram are 2hati-4hatj+5hatk and hati-2hatj-3hatk. Find the unit vector parallel to its diagonal. Also, find its area. |
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| 11493. |
For non coplanar vector, bar(a), bar(b) and bar(c ) determine p for which the vector bar(a) + bar(b) +bar(c ), bar(a) + pbar(b) +2bar(c ) and -bar(a)+bar(b)+bar(c ) |
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| 11494. |
(Transportation problem) There are two factories located one place at place P and the other at place Q. From these locations, a certain commodity is to be delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are respectively 8 and 6 units. The cost of transportation per unit is given below : How many units should be transported from each factory to each depot in order that the transportation cost is minimum. What will be the minimum transportation cost ? |
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| 11496. |
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag. |
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| 11497. |
I= int ( dx)/( a^(2) sin^(2) x + b^(2) cos^(2) x) . |
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| 11498. |
1 mole of a molecular species X_2^(-2) has 40 moles neutrons & 36 moles of electrons then atomic mass of specie X is |
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| 11499. |
Find the numerically greatest term (s) in the expansion of (4+3x)^(15)" when "x=(7)/(2) |
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| 11500. |
Evaluate the following integrals intsec^(-1)sqrt(x)dx |
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