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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.
| 51. |
`(x+1)/(2) =(y-1)/(1)=(z-9)/(-3) "and " (x-3)/(2) =(y+15)/(-7) =(z-9)/(5)` |
| Answer» Correct Answer - `4sqrt(3)" units " x=y =z` | |
| 52. |
Find the image of the point (5,9,3) in the line `(x-1)/(2)=(y-2)/(3)=(z-3)/(4)` |
| Answer» Correct Answer - (1,1,11) | |
| 53. |
Find the value of `lambda ` or which the points A(2,5,1) ,B(1,2,-1) and C(3,`lambda`,3) are collinear . |
| Answer» Correct Answer - `lambda=8` | |
| 54. |
Using the vector method , find the values of `lambda " and " mu` for which the points `A(3, lambda ,mu) B(2,0,-2) " and " C(1,-2,-5)` are collinear |
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Answer» Let `vec( a) , vec(b) " and " vec( c) ` be the position vectors of the given points A,B and C respectively . Then `vec( a) =3hat(i) +lambdahat(j) + mu hat(k) , vec( b) =2hat(i) - 3hat(k) " and " vec (c ) =hat(i) -2hat(j) -5hat(k)` Now the vector equations of te line BC is given by , `vec (r )= vec (b) +alpha(vec( c) -vec( b)) ` for some scalar `alpha` `rArr vec (r ) =(1-alpha) vec( b) + alpha vec( c)` `rArr vec(r ) =(1-alpha) (2hat(i) -3hat(k)) +alpha(hat(i) -2hat(j)-5hat(k))` `rArr vec(r)=(2-2alpha+alpha)hat(i) -2alpha hat(j)+ (-3+3alpha-5alpha)hat(k)` `rArrvec( r)=(2-alpha)hat(i) -2alpha hat(j) +(-3-2alpha)hat(k)` If this line passes through the point A then we must have `3hat(i)+lambdahat(j)+mu hat(k) =(2-alpha)hat(i)-2alpha hat(j)+(-3 -2alpha) hat(k)` `hArr 2-alpha =3, -2alpha =lambda " and " -3-2alpha=mu` `hArr alpha =-1 , lambda =2 " and " mu =(-3+2) =-1` Hence `lambda=2 " and " mu=-1` |
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| 55. |
If the points `A(-1,3,2),B(-4,2,-2)a n dC(5,5,lambda)`are collinear, find the value of `lambda`.A. 5B. 7C. 8D. 10 |
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Answer» Correct Answer - D Equations of the line AB are `(x+1)/(-4+1) =(y-3)/(2-3) =(z-2)/(-2-2) rArr (x+1)/(-3) =(y-3)/(-1) =(z-2)/(-4)` `rArr (x+1)/(3) =(y-3)/(1)=(z-2)/(4)` If the point C (5,5, `lambda`) line on AB, we have `(5+1)/(3)=(5-3)/(1)=(lambda -2)/(4) rArr (lambda-2)/(4)=2 rArr lambda= (8+2) =10` |
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| 56. |
If the points `A(-1,3,2),B(-4,2,-2)a n dC(5,5,lambda)`are collinear, find the value of `lambda`. |
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Answer» The equations of the line AB are `(x+1)/(-4+1) =(y-3)/(2-3) =(z-2)/(-2-2)` `rArr (x+1)/(-3) =(y-3)/(-1) =(z-2)/(-4)` Since the points A ,B and C are collinear so the point `C(5,5,lambda)` lies on (i) `:. (5+1)/(-3) =(5-3)/(-1) =(lambda-2)/(-4) rArr (lambda-2)/(-4) =-2 rArr lambda-2 =8 rArr lambda =10` Hence the required value of `lambda ` is 10. |
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| 57. |
Find the values of `lambda` ad `mu` if the points `A(-1,4,-2),B(lambda,mu,1)` and `C(0,2,-1)` are collinear. |
| Answer» Correct Answer - `lambda=2 , mu=-2` | |
| 58. |
Find the values of `lambda` and `mu` so that the points A(3,2-4) ,B(9,8,-10) and C(`lambda,mu,-6`) are colliinear . |
| Answer» Correct Answer - `lambda=5, mu=4` | |
| 59. |
The direction rations of two lines are a,b,c and (b-c) , (c-a),(a-b) respectively . The angle between these lines isA. `(pi)/(3)`B. `(pi)/(2)`C. `(pi)/(4)`D. `(3pi)/(4)` |
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Answer» Correct Answer - B `" cos " theta = (a(b-c)+b(c-a)+c(a-b))/({sqrt(a^(2)+b^(2)+c^(2)}}{sqrt((b-c)^(2)+(c-a)^(2)+(a-b)^(2))}}=0rArr theta =(pi)/(2)` |
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| 60. |
The directions rations of two lines are 3,2,-6 and , 1,2,2 respectively . The acute angle between these lines isA. `cos ^(-1) .((5)/(18))`B. `cos^(-1) .((3)/(20))`C. `cos^(-1).((5)/(21))`D. `cos^(-1)((8)/(21))` |
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Answer» Correct Answer - C cos `theta =(|(3xx1) +(2xx2)+(-6)xx2|)/({sqrt(3^(2) +2^(2) +(-6)^(2)}}{sqrt(1^(2)+2^(2)+2^(2))}}=(|-5|)/((sqrt(49)xxsqrt(49)))=(5)/((7xx3)) =(5)/(21)` `=(1-51)/((sqrt(49)xxsqrt(49)))=(5)/((7xx3)) =(5)/(21)` `rArr theta =cos^(-1) ((5)/(21))` |
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| 61. |
`(x-1)/(-1) =(y+2)/(1)=(z-3)/(-2) " and " (x-1)/(1)=(y+1)/(2)=(z+1)/(-2)` |
| Answer» Correct Answer - `(8sqrt(29))/(29)` units | |
| 62. |
Find the value of `lambda`so that the following lines are perpendicular to each other.`(x-5)/(5lambda+2)=(2-y)/5=(1-z)/(-1), x/1=(2y+1)/(4lambda)=(1-z)/(-3)` |
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Answer» `(x-5)/(5lambda+2)=(y-2)/-5=(z-1)/1` & `x/1=(y+1/2)/(2lambda)=(z-1)/3` are perpendicular to each other,So, `(5lambda+2)*1+(2lambda)(-5)+3=0`Hence, `lambda=1` |
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| 63. |
A line passes through the point A(5,-2.4) and it is parallel to the vector `(2hat(i) -hat(j) +3hat(k))` .The vector equations of the line isA. `vec(r) =(2hat(i) -hat(j) +3hat(k)) +lambda (5hat(i)-2hat(j) +4hat(k))`B. `vec(r ) =(5hat(i) -2hat(j) +4hat(k)) + lambda (2hat(i) -hat(j) +3hat(k))`C. `vec(r ) .(5hat(i) -2hat(j) +4hat(k)) =sqrt(14)`D. none of these |
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Answer» Correct Answer - B Clearly the required vector equation of the line is `vec(r ) =(5hat(i) -2hat(j)+4hat(k)) +lambda (2hat(i)-hat(j) +3hat(k))` |
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| 64. |
A line passes through the point (2,1,-3) and is parallel to the vector `(hat(i) -2hat(j) +2hat(k))` . Find the equations of the line in vector and Cartesian forms . |
| Answer» Correct Answer - `vec(r )=(2hat(i)-hat(j)-3hat(k)) +lambda (hat(i) -2hat(j)+3hat(k)) ,(x-2)/(1)=(y-1)/(-2)=(z+3)/(3)` | |
| 65. |
A line is drawn in the direction of `(hat(i) +hat(j) -2hat(k))` and it passes through a point with position vector `(2hat(i) -hat(j) - 4hat(k))` .Find the equations of the line in the vector as well as Cartesian forms. |
| Answer» Correct Answer - `vec(r )=(2hat(i) -hat(j) +4hat(k)) +lambda(hat(i)-hat(j)-2hat(k)) ,(x-2)/(1)=(y+1)/(1)=(z-4)/(-2)` | |
| 66. |
Write the vector equations of each of the following lines and hence determine the distance between them: `(x-1)/(2)=(y-2)/(3)=(z+4)/(6) " and " (x-3)/(4) =(y-3)/(6)=(z+5)/(12)` |
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Answer» Correct Answer - `underset(L_(2) :vec(r) " " (3hat(i) +3hat(j) -5hat(k))" " +" " 2mu (2hat(i) +3hat(j) +6hat(k)))(L_(1) : vec( r) =(hat(i) +2hat(j) -4hat(k)) + lambda(2hat(i) +3hat(j) +6hat(k)))} ,SD =(sqrt(293))/(7) ` units The given lines are `L_(1) : vec( r) =(hat(i) +2hat(j)-4hat(k)) + lambda (2hat(i) +3hat(j) +6hat(k))` `L_(2) : vec(r ) =(3hat(i) + 3hat(j) -5hat(k)) + 2mu (2hat(i) +3hat(j) +6hat(k))` Now find the distance between the parallel lines `L_(1) " and " L_(2)` |
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| 67. |
Find the vector equations of the line passing through the point with position vector `(2hat(i) +hat(j) -5hat(k))` and parallel to the vector `(hat(i) +3hat(j) -hat(k))` .Deduce the Cartesian equations of the line . |
| Answer» Correct Answer - `vec(r ) =(2hat(i) +hat(j) -5hat(k)) +lambda(hat(i) +3hat(j)-hat(k)) ,(x-2)/(1)=(y-1)/(3)=(z+5)/(-1)` | |
| 68. |
Find the vector equtions of a line passing through the point (2,3,2) and parallel to the line `vec(r ) =(-2hat(i) +3hat(j) ) + lambda (2hat(i) - 3hat(j) + 6hat(k)) ` Also find the distance between these lines . |
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Answer» Correct Answer - `L_(2): vec(r ) =(2hat(i) +3hat(j) +2hat(k)) + mu (2hat(i)- 3hat(j) +6hat(k))` ,(sqrt(580))/(7)` units The given line is `L_(1) : vec(r ) =(-2hat(i) +3hat(j)) + lambda (2hat(i) -3hat(j) +6hat(k))` The required line is `L_(2) : vec( r) =(2hat(i) +3hat(j) +2hat(k)) + mu (2hat(i) -3hat(j) +6hat(k))` Now , find the distance between the parallel lines `L_(1) " and " L_(2)`. |
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| 69. |
`(x)/(1)=y/1=(z)/(-1) " and " (x)/(3) =(y)/(4)=(z)/(5)` Find the angle between the lines . |
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Answer» Correct Answer - `cos^(-1) .((1)/(5))` Given lines in standard form are : `(x)/(1) =(y-0)/(0)=(z-1)/(-1) "and " (x)/(3)=(y)/(4)=(z)/(5)` |
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| 70. |
Find the angle between two lines whose directionratios are proportional to `1,1,2a n d(sqrt(3)-1),(-sqrt(3)-1),4`.A. `(pi)/(6)`B. `(pi)/(2)`C. `(pi)/(3)`D. `(pi)/(4)` |
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Answer» Correct Answer - C `" cos " theta = (1xx(sqrt(3)-1)+1xx(-sqrt(3)-1) +2xx4)/({sqrt(1^(2)+1^(2)+2^(2)}}{(sqrt(3)-1)^(2)+(-sqrt(3)-1)^(2)+4^(2)}}` `=((sqrt(3)-1)-sqrt(3)-1+8)/((sqrt(6)xxsqrt(24))) =(sqrt(6))/(sqrt(24)) =(1)/(2) rArr theta =(pi)/(3)` |
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| 71. |
Find the vector equation of the line passing through the point A(2,-1,1) and parallel to the line joining the points B (-1,4,1) and C(1,2,2) .Also find the Cartesian equations of the line . |
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Answer» Vector equation of the given line: the p.v of B `=(hat(i) +hat(j) +hat(k)) " and " p.v " of " C = (hat(i) +2hat(j) +2hat(k))` `:. , vec(BC) =(p.v " or " C) =(p .v. " of " B)` `=(hat(i) +2hat(j)+2hat(k)) -(-hat(i) +4hat(j) +hat(k)) =(2hat(i) -2hat(j) +hat(k))` The p.v. of A is `vec( r) =2hat(i) -hat(j) +hat(k)` `:. ` the vector equation of the given line is `vec( r) =vec(r )_(1) +lambda (vec(BC))` `hArr vec(r ) =(2hat(i) -hat(j) +hat(k)) +lambda(2hat(i) -2hat(j) +hat(k))` Cartesian equations of the given line: Taking `vec(r) =x hat(i) +yhat(j) +zhat(k)` equations (i) becomes `(xhat(i) +yhat(j) +zhat(k))=(2hat(i)-hat(j)+hat(k))+lambda(2hat(i) -2hat(j)+hat(k))` `hArr (xhat(i) +yhat(j)+zhat(k))=(2+2lambda)hat(i) +(-1-2lambda) hat(j) +(1+lambda) hat(k)` `hArr x =2 + 2lambda , y= 1- 2lambda " and " z=1+ lambda` `hArr (x-2)/(2)=(y+1)/(-2) =(z-1)/(1)=lambda` Hence `(x-2)/(2)=(y+1)/(-2) =(z-1)/(1)` are the required equation of the given line in the Cartesian form. |
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| 72. |
Find the angle between two lines whose direction ratios are `"(i) 2,1,2 and 4,8,1 "" ""(ii) 5 ,-12 ,13 and -3,4,5"` `"(iii) 1 ,1,2 and" (sqrt(3)-1) ,(-sqrt(3)-1),4 " ""(iv) a,b,c and (b-c) ,(c-a),(a-b)"` |
| Answer» Correct Answer - `cos^(-1) ((2)/(3)) (ii) cos^(-1) ((1)/(65)) (iii) (pi)/(3) (iv) (pi)/(2) ` | |
| 73. |
`(x-3)/(-3) =(y+1)/(4) =(z-6)/(3) " and " (x)/(3) =(y-1)/(2) =(z+2)/(-1)` Find the angle between the lines |
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Answer» Correct Answer - `cos^(-1).((11)/(14))` Given lines in standard form are : `(x-3)/(2)=(y+5)/(1) =(z-1)/(-3) " and " (x)/(3) =(y-1)/(2)=( z+2)/(-1)` |
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| 74. |
Find the equations of a line passing through the point P(2,-1,3) and perpendicular to the lines `vec(r ) =(hat(i) + hat(j) -hat(k)) +lambda (2hat(i) -2hat(j) +hat(k))` and `vec( r) =(2hat(i) -hat(j) -3hat(k)) +mu (hat(i) +2hat(j) +2hat(k))` |
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Answer» The given lines are `vec(r) =vec(a)_(1) +lambdavec(b)_(1) ......(i)" where " vec(a)_(1)=(hat(i) +hat(j) -hat(k)) " and " vec(b)_(1) =(2hat(i) -2hat(j) +hat(k))" and "` `vec(r ) =vec(a)_(2) +lambda vec(b)_(2) .....(ii) " where "vec(a)_(2) =(2hat(i) -hat(j) -3hat(k)) " and " vec(b)_(2) =(hat(i) +2hat(j) +2hat(k))` The required line is perpendicular to (i) as well as (ii) . Also (i) is parallel to `vec(b)_(1) ` and (ii) is parallel to `vec(b)_(2)` So the required line is perpendicular to both `vec(b)_(1) " and " vec_(b)_(2)` Consequently this line must be parallel to `(vec(b)_(1)xx vec(b)_(2))` Now , `(vec(b)_(1) xx vec(b)_(2)) =|{:(hat(i),,hat(j),,hat(k)),(2,,-2,,1),(1,,2,,2):}| =(-6hat(i) -3hat(j) +6hat(k))` So we have to find the equation of a line passing through the point P(2,-1,3) and parallel to `(vec(b)_(1)xx vec(b)_(2))` Hence the required equation is `vec( r) =(2hat(i) -hat(j) +3hat(k)) + t (-6hat(i) -3hat(j) +6hat(k)) where t is an arbitrary constant. |
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| 75. |
Find the Vector and Cartesianequations of the line passing through the point (1, 2, 4) and perpendicularto the two lines`(x-8)/3=(y+19)/(-16)=(z-10)/7`and `(x-15)/3=(y-29)/8=(z-5)/(-5)` |
| Answer» Correct Answer - `(x-1)/(2)=(y-2)/(3)=(z+4)/(6),vec(r ) =(hat(i) +2hat(j) -4hat(k)) +lambda(2hat(i) +3hat(j) +6hat(k))` | |
| 76. |
If the lines `(x-1)/(-3) =(y-2)/(2lambda) =(z-3)/(2) " and " (x-1)/(3lambda) =(y-1)/(1)=(6-z)/(5)` are perpendicular to each other then find the value of `lambda` |
| Answer» Correct Answer - `lambda =(-10)/(7) ` | |
| 77. |
`(x-4)/(3) =(y+1)/(4) =(z-6)/(5) " and " (x-5)/(1)=(2y+5)/(-2) =(z-3)/(1)` Find angle between the lines |
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Answer» Correct Answer - `cos^(-1).((2sqrt(6))/(15))` given lines is standard form are : `(x-4)/(3)=(y+1)/(4)=(z-6)/(5) " and " (x-5)/(1)=(y+(5)/(2))/(-1) =(z-3)/(1)` |
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| 78. |
Find the length and the foot of the perpendicular drawn from the point `(2, -1,5)` to the line `(x -11)/10=(y + 2)/-4=(x+ 8)/11` |
| Answer» Correct Answer - `sqrt(14) units `(1,2,3)` | |
| 79. |
Find the foot of the perpendicular drawn from thepoint `2 hat i- hat j+5 hat k`to the line ` vec r=(11 hat i-2 hat j-8 hat k)+lambda(10 hat i-4 hat j-11 hat k)dot`Also find the length of the perpendicular. |
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Answer» Correct Answer - (0,5,1) `xhat(i) +yhat(j) +zhat(k) =(11 +10lambda)hat(i) -(2+4lambda)hat(j) -(8+11lambda)hat(k)` `hArr x=11 +10 lambda ,y=-(2 +4lambda)" and " z=-(8 +11lambda)` `hArr (x-11)/(10) =(y+2)/(-4) =(z+8)/(-11)` are the equations of the given line. |
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