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Two vectors `veca = a_1 hati + a_2 hatj + a_3 hatk` and `vecb = b_1 hati + b_2 hatj + b_3 hatk` are collinear if ______.
Concept: Components of Vector
The magnitude of the vector `6hati - 2hatj + 3hatk` is ______.
Concept: Magnitude and Direction of a Vector
Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line `(x+3)/3=(4-y)/5=(z+8)/6`
Concept: Equation of a Line in Space
Find the distance between the planes 2x - y + 2z = 5 and 5x - 2.5y + 5z = 20
Concept: Shortest Distance Between Two Lines
The x-coordinate of a point of the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its z-coordinate
Concept: Vector and Cartesian Equation of a Plane
Find the coordinates of the point where the line through the points (3, - 4, - 5) and (2, - 3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2,- 3) and (0, 4, 3)
Concept: Equation of a Plane >> Equation of a Plane in Normal Form
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Concept: Direction Cosines and Direction Ratios of a Line
Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.
Concept: Vector and Cartesian Equation of a Plane
Find the value of λ, so that the lines `(1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5)` are at right angles. Also, find whether the lines are intersecting or not.
Concept: Equation of a Line in Space
Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.
Concept: Vector and Cartesian Equation of a Plane
Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.
Concept: Vector and Cartesian Equation of a Plane
Show that the lines: `(1 - x)/2 = (y - 3)/4 = z/(-1)` and `(x - 4)/3 = (2y - 2)/(-4) = z - 1` are coplanar.
Concept: Coplanarity of Two Lines
Find the distance of the point (1, –2, 0) from the point of the line `vecr = 4hati + 2hatj + 7hatk + λ(3hati + 4hatj + 2hatk)` and the point `vecr.(hati - hatj + hatk)` = 10.
Concept: Distance of a Point from a Plane
Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, – 6), Q(5, – 3, 1), R(12, 4, 5) and S(11, 9, – 2). Use these equations to find the point of intersection of diagonals.
Concept: Equation of a Line in Space
Two tailors, A and B, earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP
Concept: Linear Programming Problem and Its Mathematical Formulation
A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A
require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours and 20 minutes available for cutting and 4 hours available for assembling. The profit is Rs. 50 each for type A and Rs. 60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize profit? Formulate the above LPP and solve it graphically and also find the maximum profit.
Concept: Methods to Find the Solution of L.P.P> Graphical Method
A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.
Concept: Methods to Find the Solution of L.P.P> Graphical Method
The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and `(20/3, 4/3)`. If Z = 30x + 24y is the objective function, then (maximum value of Z – minimum value of Z) is equal to ______.
Concept: Methods to Find the Solution of L.P.P> Graphical Method
How many total matches will be played in a knock-out fixture of 19 teams?
Concept: Procedure for Drawing Knock - Out Fixture
Match the following:
| List I | List II | ||
| I. | Knock Knee/Genu Valgum | 1. | Increase exaggeration of backward curve |
| II. | Kyphosis | 2. | Wide gap between the knees when standing with feet together |
| III. | Lordosis | 3. | Knees touch each other in normal standing position |
| IV. | Bow legs | 4. | Inward curvature of the spine |
Choose the correct option from the following:
Concept: Common Postural Deformities: Knock Knees
