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This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Find the distance from the origin to (6,6,7). |
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Answer» Distance from origin to the point (6,6,7) `sqrt((0-6)^(2)+(0-6)^(2)+(0-7)^(2)) " " d= sqrt((x_(1)-x_(2))^(2)+(y_(1)-y_(2))^(2)+(z_(1)-z_(2))^(2))` `sqrt(36+36+49)` `sqrt121=11` |
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| 2. |
Find the ratio in which the line segment joining the points `(4, 8, 10)`and `(6, 10 , -8)`is divided by the YZplane. |
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Answer» Let `P(4,8,10) ` and `Q(6,10,-8)` are the given points and YZ plane divides the line segment joining these points in ratio `k:1`. Coordinates of a point which divides the line internally in ratio `m:n` are given by `((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n),(mz_2+nz_1)/(m+n))` So, `R(x,y,z) = ((k(6)+1(4))/(k+1),(k(10)+1(8))/(k+1),(k(-8)+1(10))/(k+1))` `=((6k+4)/(k+1),(10k+18)/(k+1),(-8k+10)/(k+1))` As this is divided by YZ plane, x-coordinate will be `0`. `:. (6k+4)/(k+1) = 0=> k = -2/3` So, required ratio is `2:3` and line segment is divided externally. |
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| 3. |
Show that the points `A (1, 2, 3)`, `B (1, 2, 1)`, `C (2, 3, 2)`and `D (4, 7, 6)`are the vertices of a parallelogram ABCD, but it is not a rectangle. |
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Answer» With the given vertices, `AB = sqrt((-2)^2+(-4)^2+(-4)^2 ) = sqrt(4+16+16) = sqrt 36 = 6` `BC = sqrt((3)^2+(5)^2+(3)^2 ) = sqrt(9+25+9) = sqrt 43 ` `CD = sqrt((2)^2+(4)^2+(4)^2 ) = sqrt(4+16+16) = sqrt 36 = 6` `AD = sqrt((3)^2+(5)^2+(3)^2 ) = sqrt(9+25+9) = sqrt 43 ` Here, `AB = CD` and `BC = AD`. So, `ABCD` can be a parallelogram or a rectangle. For `ABCD` to be a rectangle, all angles should be `90^@`. So, it should satisfy pythagoras theorem. Now, `BD = sqrt((5)^2+(9)^2+(7)^2 ) = sqrt(25+81+49) = sqrt 155` Now, `BC^2 + CD^2 = 43+36 = 79` `BD^2 = 155` As, `BD^2 != BC^2 + CD^2` So, `Delta BCD` is not a right angle triangle.Thus, `ABCD` is a parallelogram but not a rectangle. |
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| 4. |
Show that , if `x^(2)+y^(2)=1`, then the point ` (x,y,sqrt(1-x^(2)-y^(2)))` is at is distance 1 unit form the origin. |
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Answer» Given that `x^(2) + y^(2) =1` Distance of the point `(x,ysqrt(1-x^(2)-y^(2)))` from collinear, then AB+ BC=AC. `d=|sqrt(x^(2)+y^(2)+(sqrt(1-x^(2)-y^(2)))^(2))|` `=|sqrt(x^(2)+y^(2)+1-x^(2)-y^(2))|=1` |
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| 5. |
Find the coordinates of the point which divides the line segment joining the points `( 2, 3, 5)`and `(1, 4, 6)`in the ratio (i) `2 : 3`internally, (ii) `2 : 3`externally. |
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Answer» Let `P(-2,3,5) ` and `Q(1,-4,6)` are the given points and we have to find coordinates of point `R(x,y,z)`. (i) Coordinates of a point which divides the line internally in ratio `m:n` are given by `((mx_2+nx_1)/(m+n),(my_2+ny_1)/(m+n),(mz_2+nz_1)/(m+n))` Here, `m:n` = `2:3`. So, `R(x,y,z) = ((2(1)+3(-2))/(2+3),(2(-4)+3(3))/(2+3),(2(6)+3(5))/(2+3))` `= (-4/5,1/5,27/5)` (ii)Coordinates of a point which divides the line externally in ratio `m:n` are given by `((mx_2-nx_1)/(m-n),(my_2-ny_1)/(m-n),(mz_2-nz_1)/(m-n))` Here, `m:n` = `2:3`. So, `R(x,y,z) = ((2(1)-3(-2))/(2-3),(2(-4)-3(3))/(2-3),(2(6)-3(5))/(2-3))` `= (-8,17,3)`. |
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| 6. |
Name the octant in which each of the following points lie. (i)(1,2,3), (ii) (4,-2,3) (4,-2,-5), (iv)(4,2,-5), (v)(-4,2,5), (iv)(-3,-1,6), (vii)(2,-4,-7), (viii),(-4,2,-5) |
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Answer» (i) point (1,2,3) lies in first quadrant. (ii) (4,-2,3) in forth octant. (iii)(4,-2,-5) in eight octant. (iv) (4,2,-5) in fifth octant. (v) (-4, 2,5) in second octant. (vi) (-3,-1,6) in third octant. (vii) (2,-4,-7) in eigh octant. (viii) (-4,2,-5) in sixth octant. |
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| 7. |
A point R with xcoordinate 4 lies on the line segment joining the points `P (2, 3, 4)`and `Q (8, 0, 10)`. Find the coordinates of the point R. |
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Answer» Let `R` divides the line joining `P(2,-3,4)` and `Q(8,0,10)` in ratio `k:1`. The, coordinates of `R(x,y,z) = ((kx_2+x_1)/(k+1),(ky_2+y_1)/(k+1),(kz_2+z_1)/(k+1))` Here, we are given x-coordinate of `R` that is `4`.`:. R(4,y,z) = ((8k+2)/(k+1),(0-3)/(k+1),(10k+4)/(k+1))` `:. (8k+2)/(k+1) = 4=> 8k+2 = 4k+4 => k = 1/2` So, y-coordinate` = -3/(k+1) = -3/(3/2) = -2` z-coordinate ` = (10(1/2)+4)/(3/2) = 9/(3/2) = 6` So, coordinates of point `R` are `(4,-2,6)`. |
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| 8. |
Find the coordinates of the points which trisectthe line segment `A B ,`given that `A(2,1,-3)n d B(5,-8,3)dot` |
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Answer» Let the P `(x_(1)y_(2),z_(1) and Q (x_(2),y_(2),z_(2))` trisect line segment AB. `(2,1,-3) (x_(1),y_(1),z_(1)),(x_(2),y_(2),z_(2)) (5,-8,3)` Since , the point P divded line AB in 1:2 internally then `x_(1)=(2xx2+1xx5)/(1+2)=9/3=3` `y_(1)=(2xx1+1xx(-8))/3= -(-6)/3=-2` `z_(1)=(2xx(-3)+1xx3)/3= (-6+3)/3= (-3)/3=-1` Since the point Q divide the line segment AB in 2:1 then `x_(2)=(1xx2+2xx5)/3=4` `y_(2)=(1xx1+(-8xx2))/3=-5` `z_(2)=(1xx(-3)+2xx3)/3=-1` So, the coordinates of P are ( 3,-2,-1) and the corrdinates of Q are (4,-5,1) |
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| 9. |
The centroid of a triangle ABC is at the point `(1, 1, 1)`. If the coordinates of A and B are `(3, 5, 7)`and `(1, 7, 6)`, respectively, find the coordinates of the point C. |
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Answer» Coordinates of a centroid `G(x,y,z)` of a triangle can be given as, `G(x,y,z) = ((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3,(z_1+z_2+z_3)/3)` Here, coordinates `A(3,-5,7)` and `B(-1,7,-6)` are given and coordinates of centroid `(1,1,1)` are given. We have to find `C(c_1,c_2,c_3)`. `:. (3-1+c_1)/3 = 1=> c_1 = 1` `(-5+7+c_2)/3 = 1=> c_2 = 1` `(7-6+c_3)/3 = 1 => c_3 = 2` So, coordinates of `C` are `(1,1,2)`. |
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| 10. |
Show that the point A (1,-1,3), B ( 2,-4,5) and C ( 5,-13,11) are collinear. |
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Answer» Given points , A (1,-1,3), B ( 2,-4,5) and C ( 5,-13,11) `AB= sqrt((1-2)^(2)+(-1+4)^(2)+(3-5)^(2))` `sqrt(1+9+4)=sqrt14` `BC=sqrt((2-5)^(2)+(-4+13)^2+(5+11)^2)` `sqrt(9+81+36)=sqrt126` `AC= sqrt((1-5)^(2)+(-1+13)^(2)+(3-11)^(2))` ` sqrt(16+144+64)=sqrt224` AB + BC+AC `sqrt14+sqrt126=sqrt224` `sqrt14+3sqrt14=4sqrt14` ltbr. So, the points A,B, and C are collinear. |
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| 11. |
If A,B,C be the feet of perpendiculars from a point p on the X,Y and Z- axes repsectively, then find the coordinates of A,Band C in each of the following where the point P is (i) A (3,4,2) (ii) B (-5,3,7) (iii) C (4,-3,-5) |
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Answer» The coordinates of A,B and C are the following (i) A (3,0,0)B (0,4,0) C(0,0,2) (ii) A (-5,0,0), B (0,3,0), C (0,0,7) (iii) A (4,0,0), B (0,-3,0), C (0,0,-5) |
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| 12. |
L is the foot of the perpendicular drawn from a point (3,4,5) on X-axis. The coordinates of L are.A. 3,0,0B. 0,4,0C. 0,0,5D. none of these |
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Answer» Correct Answer - a On the X-axis , y=0 and z=0 Hence, the required coordinates are (3,0,0). |
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| 13. |
The Three coordiantes planes divide the space into ……. Parts. |
| Answer» Correct Answer - Eight parts. | |
| 14. |
The length of the longest piece of a string that can be stetched straight in a rectangular room whose dimensions are 10,13 and 8 units are ……….. |
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Answer» Given dimesions are a 10,13 and c=8 Required length = `sqrt(a^(2)+b^(2)+c^(2))` `sqrt(100+169+64)=sqrt333` |
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| 15. |
The coordiantes of a point are the perpendicular distance from the ….. On the respectives axes. |
| Answer» Given points. | |
| 16. |
The three planes, determine a rectangular parallelopiped which has ………. Of rectangular faces. |
| Answer» Three points. | |