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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.
| 351. |
Find the angles of a triangle whose verties are `A(3,2,1), B(35,2)` and `C(5,-2,3)`. |
| Answer» Correct Answer - `90^(@), cos^(-1) (1/(sqrt(3))), cos^(-1)(sqrt((2)/(3)))` | |
| 352. |
Find the point on `y-a xi s`which is equidistant from the points `(3,1,2)a n d(5,5,2)dot` |
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Answer» Correct Answer - (0,3,0) Any poin of the y-axis is of the from P(0,y,0) |
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| 353. |
Find the point which equisdistant from points `O(0,0,0),A(a,0,0)B (0,b,0) and (0,0,c)` |
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Answer» Correct Answer - `((a)/(2),(b)/(2),(c)/(2))` Let the required point be `P(x,y,z)` . Then, `PA^(2)=PB^(2)=PC^(2)hArrPA^(2)-PO^(2)=0,PB^(2)-PO^(2)=0 and PC^(2)-PO^(2)=0` |
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| 354. |
Determine the points in i. xy-plan e ii. yz-plane and iii zx-planewhich re equidistant from the points `A(1,-1,0), B(2,1,2), a n d C(3,2,-1`) |
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Answer» Correct Answer - `(0,(31)/(16),(-3)/(16))` Any point in the yz-plane is of the from P(0,y,z). `|AP|=|BP|=|CP|hArrAP^(2)=BP^(2)=and BP^(2)=CP^(2)` `(0-3)^(2)+(y-2)^(2)+(z+1)^(2)=(01)^(2)+(y+1)^(2)+(z-0)^(2)` and `(0-1^(2))+(y-1)^(2)+(z-0)^(2)=(0-2)^(2)+(y-1)^(2)+(z-2)^(2)` `rArr 3y-z-6=0 and 4y+4x-7=0` On solving, we get `y=(31)/(16) and z=(-3)/(16)` Hence, the requied points is `(0,(32)/(16),(-3)/(16))` |
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| 355. |
A line makes `30^(@), 120^(@)` and `90^(@)` angles from the positive direction of x-axis, y-axis and `pi`-axis respectively. Find its direction cosines. |
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Answer» Let direction cosines be `l, m, n` ` :. l = cos 30^(@) = (sqrt(3))/(2)` `m = cos 120^(@) = - cos 60^(@) = -1/2` `n = cos 90^(@) = 0` `:.` Direction cosines are `(sqrt(3))/(2), - 1/2 , 0`. |
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| 356. |
Show that `a x+b y+r=0,b y+c z+p=0a n dc z+a x+q=0`are perpendicular to `x-y ,y-za n dz-x`planes, respectively. |
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Answer» The planes `a_(1)x+b_(1)y+c_(1)z+d_(1)=0 and a_(2)x+b_(2)y +c_(2)z +d_(2)=0` are perpendicular to each other if and only if `a_(1)a+b_(1)b+c_(1)c=0`. The equation of `x-y, y-z, and z-x` planes are `z=0, x=0 and y=0`, respectively. Now we have to show that `z=0` is perpendicular to `ax+by+r=0`. It follows immediately, since `a(0)+b(0)+(0)(1)=0`, other parts can be done similarlhy. |
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| 357. |
Find the points where line `(x-1)/2=(y+2)/(-1)=z/1`intersects `x y ,y za n dz x`planes. |
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Answer» Line meets `xy`-plane where z=0 Hence, from the given equation of line, `(x-1)/(2)=(y+2)/(-1)=(0)/(1)` `rArr" "x=1 and y=-2`. `rArr" "` Line meets `xy`-plane at (1, -2, 0). Line meets `yz`-plane where `x`=0 Hence, from the given equation of line, `(0-1)/(2)=(y+2)/(-1)=(z)/(1)` `rArr" "z=(-1)/(2) and y=-(3)/(2)` `rArr" "` Line meets `yz`-plane at `(0, -(3)/(2), (-1)/(2))` Line meets `zx`-plane where y=0 Hence, from the given equation of line `(x-1)/(2)=(0+2)/(-1)=(z)/(1)` `rArr" "z=-2, x=-3` `rArr" "` Line meets `zx`-plane at (-3, 0, -2) |
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| 358. |
A line AB in three-dimensional space makes angles `45^o ` and `120^o`with the positive x-axis and the positive y-axisrespectively. If AB makes an acute angle q with the positivez-axis, then q equals |
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Answer» `AB` makes angle `( alpha , beta, gamma) `with (x,y,z) axis then, `cos^2 alpha + cos^2 beta + cos^2 gamma = 1` `alpha = 45^@` `beta= 120^@` `(1/sqrt2)^2 + (-1/2)^2 + cos^2 gamma = 1` `= cos^2 gamma = 1/4` `cos gamma = 1/2or -1/2` `cos gamma = 1/2` `gamma = pi/3` Answer |
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| 359. |
`(i+j+3k)x+(3i-3j+k)y+(-4i+5j)z=lambda(x i+yj+zk)` then lambda equal to |
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Answer» `(hati+hatj+3hatk)x+(3hati-3hatj+hatk)y +(-4hati+5hatj)z = lambda(xhati+yhatj+zhatk)` Comparing coefficient of `hati, hatj and hatk` `x+3y-4z = lambda x =>(1-lambda)x+3y-4z = 0->(1)` `x-3y+5z = lambda y => x+(-3-lambda)y+5z = 0->(2)` `3x+y = lambda z => 3x+y-lambdaz = 0->(3)` Now, solving (1),(2),(3) using determinant, `|[1-lambda,3,-4],[1,-3-lambda,5],[3,1,-lambda]| = 0` `=>[(1-lambda)(3lambda+lambda^2-5)-3(-lambda-15)-4(1+9+3lambda)] = 0` `=>lambda^2+3lambda-5-lambda^3-3lambda^2+5lambda+3lambda+45-40-12lambda = 0` `=>-lambda^3-2lambda^2-lambda = 0` `=>lambda(lambda^2+2lambda+1) = 0` `=>lambda(lambda+1)^2 = 0` `=>:. lambda = 0 and lambda = -1` |
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| 360. |
If the volume oftetrahedron `A B C D`is 1 cubic units, where `A(0,1,2),B(-1,2,1)a n dC(1,2,1),`then the locus of point `D`isa. `x+y-z=3`b. `y+z=6`c. `y+z=0`d. `y+z=-3`A. `x+y-z=0`B. `y+x=6`C. `y+z=0`D. `y+z=-3` |
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Answer» Correct Answer - b, c Volume of tetrehedron `ABCD= (1)/(6) |[vec(AB)vec(AC)vec(AD)]|= 1` cubic units. `rArr" " |{:(-1,,1,,-1),(1,,1,,-1),(x-0,,y-1,,z-2):}|= pm 6` or `" "-2(y-1)-2(z-2)= pm 6` or `" "y-1+z-2= pm 3` `rArr" "y+z=6 or y+z=0` |
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| 361. |
Find the distance between(i) A(5, 1, 2) and B(4, 6, -1) (iii) R(1, -3, 4) and S (4,-2, -3) |
| Answer» Correct Answer - (ii) `sqrt(35)` units (ii) 9 units (iii) `sqrt(59)` units (iv) 17units | |
| 362. |
In three dimensional space , the equation xy = 0 representsA. a pair of linesB. a planeC. a pair of perpendicular planesD. a parie of parallel planes |
| Answer» Correct Answer - c | |
| 363. |
The volume of tetrahedron included between the plane `2x-3y+ 4z-12=0` and three co-ordinate planes is |
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Answer» when y=0,z=0 2x-12=0,x=6 when x=0,y=0 4z-12=0,z=3 when x=0,z=0 -3-12=0,y=-4 `vec(AB)=6hati,vec(AC)=3hatk,vec(AD)=-4hatj` `V=1/6|[vec(AB)*vec(AC)*vec(AD)]|` `=1/6[6hati*(3hatk*-4hatj)|` `=1/6|6hati*12hati|=72/6=12`. |
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| 364. |
If `|bar r|=9` and `bar r` is equally inclined to the co-ordinate axes, then `bar r=`A. `pm 3(hati+hatj+hatk)`B. `pm9(hati+hatj+hatk)`C. `pm sqrt(3)(hati+hatj+hatk)`D. `pm3 sqrt(3)(hati+hatj+hatk)` |
| Answer» Correct Answer - D | |
| 365. |
The points (2, -1, -1), (4, -3, 0) and (0, 1, -2) areA. collinearB. non-coplanarC. non-collinearD. non-collinear and non-coplanar |
| Answer» Correct Answer - A | |
| 366. |
In which octant does each of the given point lie ? (i) (-4,-1,-6) `" "` (ii) (2,3,-4) `" "` (iii) (-6,5,-1) (iv) (4,-3,-2) `" "` (v) (-1,-6,5) `" "` (vi) (4,6,8) |
| Answer» Correct Answer - (i) VIII (ii) V (iii) VI (iv) VIII (v) III (vi) I | |
| 367. |
STATEMENT-1 : The direction ratios of a line joining the points (0, 0, 0) and (x, y, z) must be x, y, z. and STATEMENT-2 : If P(x, y, z) is a point is space and `OP=r` then direction cosines of OP are `x/r, y/r, z/r`,.A. `r/x,r,y,r,z`B. `rx,ry,rz`C. `x/r,y/r,z/r`D. `r cos alpha,r cos beta,r cos gamma ` |
| Answer» Correct Answer - c | |
| 368. |
Prove that the lines `(x+1)/3=(y+3)/5=(z+5)/7a n d(x-2)/1=(y-4)/4=(z-6)/7`are coplanar . Aslo, find the plane containing these two lines. |
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Answer» Here, `(x_(1),y_(1),z_(1)) = (-1,-3,-5)` `(x_(2),y_(2),z_(2)) = (2,4,6)` `a_(1),b_(1),c_(1) = 3,5,7` and `a_(2),b_(2),c_(2)=1,4,7` Now, `|{:(x_(2)-x_(1),y_(2)-y_(1),z_(2)-z_(1)),(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)):}|` `= |{:(3,7,11),(3,5,7),(1,4,7):}|` `= 3(35-28)-7(21-7)+11(12-5)` `= 21-98+77 = 0` Therefore the given lines are coplanar. Equation of plane containing the given lines `|{:(x-x_(1),y-y_(1),z-z_(1)),(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)):}|` `rArr |{:(x+1,y+3,z+5),(3,5,7),(1,4,7):}| = 0` `rArr (x+1)(35-28)-(y+3)(21-7)+(z+5)(12-5)=0` `rArr 7(z+1)-14(y+3)+7(x+5)=0` `rArr (x+1)-2(y+3)+(z+5)=0` `rArr x-2y+z=0` |
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| 369. |
Which of the following is false ?A. `30^(@), 45^(@), 60^(@)` can be the direction angles of a line in space.B. `90^(@), 135^(@), 45^(@)` can be the direction angles of a line in space.C. `120^(@), 60^(@), 45^(@)` can be the direction angles of a line in space.D. `60^(@), 45^(@), 60^(@)` can be the direction angles of a line in space. |
| Answer» Correct Answer - A | |
| 370. |
For what value (s) of awill the two points `(1,a ,1)a n d(-3,0,a)`lie on opposite sides of the plane `3x+4y-12 z+13=0?`a. `a >-1ora >1//3`b. `a=0`onlyc. `0A. `alt-1oragt1//3`B. a=0 onlyC. `0ltalt1`D. `-1ltalt1` |
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Answer» Correct Answer - a We must have `(3+ 4a -12+13)(-9-12a+13) lt 0`. or `" "(a+1)(12a-4) gt 0` or `" " a lt -1 or a gt 1//3` |
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| 371. |
If a linemakes angles `90o," "135o," "45o`with the x, y and z-axes respectively, findits direction cosines. |
| Answer» Correct Answer - `(1)/(sqrt(2)), - (1)/(sqrt(2)),0` | |
| 372. |
l = m = n = 1 are the direction Cosines ofA. X-axisB. Y-axisC. Z-axisD. no line |
| Answer» Correct Answer - d | |
| 373. |
Which of the following is true ?A. A line can make angle `30^(@), 45^(@)` with the X-axis, Y-axis respectively.B. A line can not make angle `30^(@), 60^(@)` with the X-axis , Y-axis respectively.C. A line can not make angle `30^(@), 45^(@)` with the X-axis , Y-axis respectively.D. A line can not make angle `45^(@), 60^(@)` with the X-axis , Y-axis respectively. |
| Answer» Correct Answer - C | |
| 374. |
The direction cosines of the line which bisects the angle between positive direction of Y and Z axis areA. `0, (1)/(sqrt(2)), (1)/(sqrt(2))`B. `0, (-1)/(sqrt(2)), (-1)/(sqrt(2))`C. `0, (1)/(sqrt(2)), (-1)/(sqrt(2))`D. `0, (-1)/(sqrt(2)), (1)/(sqrt(2))` |
| Answer» Correct Answer - A | |
| 375. |
Perpendicular distance of point (3, 4, 5) from Y-axis isA. 4B. 5C. `sqrt(34)`D. `sqrt(41)` |
| Answer» Correct Answer - C | |
| 376. |
If `p hati+q hatj+r hatk` is vector along a line, then p, q, r areA. direction ratios of the lineB. direction cosines of the lineC. components of the lineD. co-ordinates of a point on the line |
| Answer» Correct Answer - A | |
| 377. |
A straight line `L`on the xy-plane bisects the angle between `O Xa n dO Ydot`What are the direction cosines of `L ?`a. ``b. ``c. ``d. ``A. `lt(1//sqrt2),(1//sqrt2),0gt`B. `lt(1//2),(sqrt3//2),0gt`C. `lt0,0,1gt`D. `lt(2//3),(2//3),(1//3)gt` |
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Answer» Correct Answer - a The given line makes angles of `pi//4, pi//4 and pi//2` with the `x-, y- and z-`axes, respectively. Hence, direction cosines of the given line are `" "cos(pi//4), cos(pi//4) and cos(pi//2)`, or `" "(1//sqrt2), (1//sqrt2)` and 0. |
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| 378. |
The pairs of rectangular co-ordinate planes have equationsA. xy=yz=zx=0B. x=y=z=0C. xyz=0D. `xy=yz=zx ne 0` |
| Answer» Correct Answer - A | |
| 379. |
If the line OP of length r makes an angle `alpha` with the X - axis , and lies in the ZX - plane , then the co - ordinates of P areA. `(r.cos alpha,0,r.sin alpha)`B. `(r.cos alpha,0,r.sin alpha)`C. `(0,0,r.cos alpha)`D. `(0,r.cos alpha , 0)` |
| Answer» Correct Answer - a | |
| 380. |
If a line makes angles `90^@, 135^@, 45^@` with X,Y and Z axes respectively , then find its direction cosines.A. `0,(1)/(sqrt(2)), (-1)/(sqrt(2))`B. `0, (-1)/(sqrt(2)), (-1)/(sqrt(2))`C. `0, (1)/(sqrt(2)), (1)/(sqrt(2))`D. `0, (-1)/(sqrt(2)), (1)/(sqrt(2))` |
| Answer» Correct Answer - D | |
| 381. |
In space , the equation `3x-4y=0` representsA. the Z - axisB. a plane containing the Z - axisC. the Xy - planeD. the YZ - plane |
| Answer» Correct Answer - b | |
| 382. |
The distance of point (1, 2, 3) from X-axis isA. `sqrt(14)`B. `sqrt(13)`C. `sqrt(10)`D. `sqrt(5)` |
| Answer» Correct Answer - B | |
| 383. |
The ratio in which the plane `vecr.(veci-2 vecj+3veck)=17` divides the line joining the points `-2veci+4vecj+7veckandvec3i-5vecj+8veck` isA. `1:5`B. `1:10`C. `3:5`D. `3:10` |
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Answer» Correct Answer - d Let the plane `vecr.(veci-2vecj+3veck)=17` divide the line joining the points `-2veci+4vecj+7veckand3veci-5vecj+8veck` in the ration t : 1 at point P is Therefore, point P is `(3t-2)/(t+1)veci+(-5t+4)/(t+1)vecj+(8t+7)/(t+1)veck` This lies on the given plane `because(3t-2)/(t+1).(1)+(-5t+4)/(t+1)(-2)+(8t+7)/(t+1)(3)=17` Solving, we get `t=(3)/(10)` |
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| 384. |
The direction cosines of any normal to the xy-plane are (A) 1,0,0 (B) 0,1,0 (C) 1,1,0 (D) 0,01A. 1, 0, 0B. 0, 1, 0C. 0, 0, 1D. 1, 1, 0 |
| Answer» Correct Answer - C | |
| 385. |
A line lies in ZX plane and makes `60^(@)` with x - axis ,then direction cosines of the line are :A. ` 1/2,0,(sqrt(3))/2`B. `(sqrt(3))/2,0,1/2`C. `0,1/2,(sqrt(3))/2`D. `1/2 , sqrt(3),0` |
| Answer» Correct Answer - a | |
| 386. |
If `l`, m, n are the direction consines of a line, thenA. `l+m+n=0`B. `l+m+n=1`C. `l^(2)+m^(2)+n^(2)=1`D. `l^(2)+m^(2)+n^(2)=0` |
| Answer» Correct Answer - C | |
| 387. |
The direction cosines of any normal to ZX-plane areA. 1, 0, 0B. 0, 1, 0C. 0, 0, 1D. 1, 1, 0 |
| Answer» Correct Answer - B | |
| 388. |
Direction Cosines of a normal to the XOY - plane areA. `1,0,0`B. `0,1,0`C. `1,1,0`D. `0,0,1` |
| Answer» Correct Answer - d | |
| 389. |
The direction cosines of any normal to YZ-plane areA. 1, 0, 0B. 0, 1, 0C. 0, 0, 1D. 1, 1, 0 |
| Answer» Correct Answer - A | |
| 390. |
The point which divides the line joining the points (2, 4, 5) and (3, 5, -4) in the ratio -`2:3`, lies onA. XOY planeB. YOZ planeC. ZOX planeD. XYZ plane |
| Answer» Correct Answer - B | |
| 391. |
Find the ratio in which the plane `x-2y+3z=5` divides the jion of A(3,-5,4) and B(2,3-7). Find the coordinats of the point intersection of the line and the plane. |
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Answer» Correct Answer - `2:3,((13)/(5),(-9)/(5),(-2)/(5))` D divides BC in the ratio `AB:AC` |
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| 392. |
The acute angle between the lines whose direction ratios are 1, 1, 2 and `sqrt(3)-1, -sqrt(3)-1,4` isA. `45^(@)`B. `30^(@)`C. `90^(@)`D. `60^(@)` |
| Answer» Correct Answer - D | |
| 393. |
The acute angle between the lines whose direction ratios are 1, 2, 2 and -3, 6, -2 isA. `cos^(-1)((13)/(21))`B. `cos^(-1)((5)/(21))`C. `cos^(-1)((11)/(21))`D. `cos^(-1)((5)/(9))` |
| Answer» Correct Answer - B | |
| 394. |
The acute angle between the lines whose direction ratios are 3, 2, 6 and -2, 1, 2 isA. `cos^(-1)((8)/(21))`B. `cos^(-1)((12)/(21))`C. `cos^(-1)((2)/(21))`D. `cos^(-1)((2)/(3))` |
| Answer» Correct Answer - A | |
| 395. |
If direction angles of a line are `alpha, beta, gamma` such that `alpha+beta=90^(@)`, then `(cos alpha+cos beta+cos gamma)^(2)=`A. `1-cos 2 alpha`B. `1+cos 2 alpha`C. `1-sin 2 alpha`D. `1+sin2alpha` |
| Answer» Correct Answer - D | |
| 396. |
Find the measure of a acute angle between the line direction ratios are 5, 12, -13 and 3, -4, 5.A. `cos^(-1)((7)/(65))`B. `cos^(-1)((-7)/(65))`C. `cos^(-1)((49)/(65))`D. `cos^(-1)((-49)/(65))` |
| Answer» Correct Answer - D | |
| 397. |
A line lies in XZ-plane and makes an angle `60^(@)` with Z-axis, find its inclination with X-axis.A. `30^(@)`B. `45^(@)`C. `60^(@)`D. `90^(@)` |
| Answer» Correct Answer - A | |
| 398. |
Find the ratio in which the point C(5,9,-14) divided the join of A(2,-3,4) and B(3,1-2) |
| Answer» Correct Answer - `3:2` (externally) | |
| 399. |
If a line passing through (4, 1, 2) and (5, k, 0) is perpendicular to the line passing through (2, 1, 1) and (3, 3, -1), then k =A. `(1)/(2)`B. `(-1)/(2)`C. `(3)/(2)`D. `(-3)/(2)` |
| Answer» Correct Answer - D | |
| 400. |
Find the ratio in which the line segment having the end points `A(-1, -3, 4)` and `B(4, 2, -1) ` is divided by the `xz-`plane. Also, find the coordinates of the point of division. |
| Answer» Correct Answer - `3:2` (2,0,1) | |