P is a point on the line joining the points (0,5, -2) and B 3, -1,2) . If the x-coordinate of P is 6, then its z-coordinate is

If (a, b), (c, d) and (e, f) are the vertices of ∆ABC and ∆ denotes the area of ∆ABC, then 2 1 1 1 a c e b d f is equal to:

Assertion(A): 2 1 2 2 3 1 x y z+ - -= =- and 3 1 3 2 2 x y z- + = =- - are coplanar. Reason (R) : Let line passes through the point ( ) and parallel to the vector whose direction ratios are Let line passes through the point ( ) and parallel to the vector whose direction ratios are . Then both lines are coplanar if and only if 2 1 2 1 2 1 1 1 1 2 2 2 0 x x y y z z a b c a b c - - - =

Find the coordinates of points on line 1 1 1 2 2 x y z - += = which are at a distance of √11 units from origin.

A particle moves along the curve 3y = ax 3 + 1 such that at a point with x -coordinate 1, y - coordinate is changing twice as fast at x-coordinate. Find the value of a.

Solve the following Linear Programming Problem graphically: Maximize: Z = Subject to : 5

Equation of a line passing through point (1, 1, 1) and parallel to z-axis is:

Position vector of the mid-point of line segment AB is 3î + 2ĵ - 3k̂ . If position vector of the point A is 2î + 3ĵ - 4k̂ , then position vector of the point B is:

The corner points of the feasible region determined by the following system of linear inequalities: 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0,0), (5,0), (3,4), (0,5). Let Z= px + qy, where p,q > 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is

Assertion (A): If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin2α + sin2β + sin2γ = 2. Reason (R): The sum of squares of the direction cosines of a line is 1.

Solve the following linear programming problem graphically : Minimize : Z = 5x + 10y subject to constraints : x + 2y ≤ 120 x +y ≥ 60 x - 2y ≥ 0 x ≥ 0,y ≥ 0.

Find the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines ^ ^ ^ ^r (2 2 )i j k i j k λ= + - + - +⃗ ^ ^ and ^ ^ ^ ^r 2 3 ( 2 2 )i j k i j k μ= - - + + + ^⃗ ^

The direction ratios of the line 1 2 3 3 1 2 x y z- -= = are :

The Cartesian equation of the line passing through the point (1, - 3, 2) and parallel to the line ^ ^ ^2 ( 2 )r i k i j k λ= - + + + ⃗ ^ ^ is

Assertion: If the cartesian equation of a line is 5 4 6 3 7 2 x y z- + -= = , then its vector form is ^ ^ ^ ^5 4 6 (3 7 2 )r i j k i j k λ= - + + + + ⃗ ^ ^ Reason: The cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by 3 4 8 3 5 6 x y z- - += = is 3 4 8 2 4 5 x y z+ - += =- - .

Find the angle between the lines 5 2 7 5 1 x y z- + = =- - and 1 2 3 x y z= = .

(a) Find the shortest distance between the lines 8 9 10 3 16 7 x y z- + -= =- and 15 29 5 3 8 5 x y z- - -= = -

The area (in sq. units) of the region bounded by the curve y = x, x-axis, x = 0 and x = 2 is :

If the equation of a line AB is 3 2 5 1 2 4 x y z- + -= =- , find the direction ratios of a line parallel to A

If the direction cosines of a line are k, k, k then

Find the area of the region in the first quadrant enclosed by the x -axis, the line y = x and the circle x2 + y2 = 32.

An insect is crawling along the line 3 5 7 1 2 1 x y z- - -= =- and another insect is crawling along the line 1 1 1 7 6 1 x y z+ + += =- . At what points on the lines should they reach so that the distance between them is the shortest? Find the shortest possible distance between them.

Solve the following linear programming problem graphically: Minimise Z = 6x + 7y, subject to constraints x + 2y ≥ 240 3x + 4y ≤ 620 2x + y ≥ 180 x, y ≥ 0.

The area bounded by the shaded region as shown in the figure below is:

The area of a parallelogram whose one diagonal is 2î + 2ĵ - 1k̂ and one side is 3î + 1ĵ - 1k̂ is

The area bounded by the curve y = f(x), the y-axis, y = c and y = d is:

Find the area bounded by the curve y = cos x, x ∈ [0, π]

Find the points on the line 2 1 3 3 2 2 x y z+ + -= = at a distance of 5 units from the point P(1, 3, 3).

Find the shortest distance between the following lines : ^ ^ ^ ^ ^ ^ ^ ^ 1 2 : ( 2 4 ) (2 3 6 ) : (3 3 5 ) (4 6 12 ) l r i j k i j k l r i j k i j k λ μ = + - + + + = + - + + + ⃗ ^ ^ ⃗ ^ ^

A set of values of decision variables that satisfies the linear constraints and non-negativity conditions of an L.P.P. is called its:

Direction ratios of a line are 2, 3, -6. Then direction cosines of a line making obtuse angle with the y-axis are

Assertion (A): The angle between the straight lines 1 2 3 2 5 4 x y z+ - += = and 1 2 3 1 2 3 x y z- + -= = - is 90° Reason (R): Skew lines are lines in different planes which are parallel and intersecting.

Find the angle between the straight lines 1 2 3 2 5 4 x y z+ - += = and 1 2 3 1 2 3 x y z- + -= =- .

Find the shortest distance between the lines ^ ^ ^(4 ) ( 2 3 )r i j i j k λ= - + + - ⃗ ^ ^ and ^ ^ ^ ^( 2 ) (2 4 5 )r i j k i j k μ= - + + + - ⃗ ^ ^

The area of the region bounded by the ellipse 2 2 125 16 x y+ = is

The vector equation of the line joining the points (3, -2, -5) and (3, -2, 6) is:

The angle between two lines having direction ratios 1, 1, 2 and (√3 - 1), (-√3 - 1), 4 is

The area (in sq. units) enclosed by the curve shown in the given figure is:

The cartesian equation of a line passing through the point with position vector ^a i j= - ⃗ ^ and parallel to the line ^ ^(2 )r i k i j μ= + + - ⃗ ^ ^ is:

If 1î + 1ĵ + 1k̂ , 2î + 5ĵ , 3î + 2ĵ - 3k̂ and 6î - 1ĵ - 1k̂ are the position vectors of points A, B, C and D respectively, then find the angle between AB and CD . Deduce that AB and CD are collinear.

Find the vector equation of the line through the point (1, 2, -4) and perpendicular to the two lines ^ ^ ^ ^(8 19 10 ) (3 16 7 )r i j k i j k λ= - + + - + ⃗ ^ ^ and ^ ^ ^ ^(15 29 5 ) (3 8 5 )r i j k i j k μ= + + + + - ⃗ ^ ^

Case-Study 1: Mohini purchased a rectangular parallelopiped shaped box and a spherical ball inside it as a showpiece. The sides of the box are x, 2x and x/3, and the radius of the sphere is y. The sum of the surface area of the parallelopiped and sphere is given to be constant. Based on the above information,

The area of a triangle with vertices A, B, C is given by

The direction ratios of the line 6x - 2 = 3y + 1 = 2z - 2 are:

The area enclosed by the circle x2 + y2 = 8 is

Find the area of a parallelogram ABCD whose side AB and the diagonal AC are given by the vectors 3î + 4ĵ + 1k̂ and 4î + 5k̂ respectively.

Find the shortest distance between the lines ^ ^ ^(4 ) ( 2 3 )r i j i j k λ= - + + - ⃗ ^ ^ and ^ ^ ^ ^( 2 ) (2 4 5 )r i j k i j k μ= - + + + - ⃗ ^ ^

Find the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines: ˆ ˆˆ ˆ ˆ ˆ( ) (2 2 )r i j k i j k λ= + - + - +⃗ and (2 3 ) ˆ ( 2 2 ˆˆ ˆ )ˆ ˆr i j k i j k μ= - - + + +⃗

Find the area of the smaller region bounded by the ellipse 2 2 116 9 x y+ = and the straight line 3x + 4y = 12.

The direction ratios of the line 6x - 2 = 3y + 1 = 2z - 2 are:

Two vector ^ ^ 1 2 3a a i a j a k= + + ⃗ ^ and ^ ^ 1 2 3b b i b j b k= + + ⃗ ^ are collinear if:

Two-line 3 1 6 1 3 1 x y z- + -= = - and 5 2 3 7 6 4 x y z+ - -= =- intersect at the point R. The reflection of R in the xy plane has coordinates

Assertion (A): Lines 2 1 2 3 1 x y z z+ - -= =- and 3 1 3 2 2 x y z- + = =- - are coplanar. Reason (R): Let line l1 passes through the point (x1, y1, z1) and parallel to the vector whose direction ratios are a1, b1 and c1; and let line l2 passes through the point (x2, y2, z2) and parallel to the vector whose direction ratios are a2, b2 and c2. Then both lines l1 and l2 are coplanar if and only if 2 1 2 1 2 1 1 1 1 2 2 2 x x y y z z a b c a b c - - - = 0

Prove that the points A, B and C with position vectors a ⃗ , b ⃗ and c ⃗ respectively are collinear if and only if 0a b b c c b× + × + × = ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗

Find the area of a parallelogram whose adjacent side are determined by the vectors a⃗ = 3î - 1ĵ + k̂ and b⃗ = 2î - 7ĵ + k̂ .

Solve the following Linear Programming Problem graphically : Maximize: P = 70x + 40y Subject to: 3x + 2y ≤ 9, 3x + y ≤ 9 x ≥ 0, y ≥ 0

Find the vector equation of the line passing through (1, 2, - 4) and perpendicular to the two lines: 8 19 10 3 16 7 x y z- + -= =- and 15 29 5 3 8 5 x y z- - -= = -

The area of the region bounded by the ellipse 2 2 125 16 x y+ = is

The area enclosed by the circle x2 + y2 = 2 is equal to

A point that lies on the line 1 3 1 2 4 7 x y z- + -= =- is:

Find the coordinates of the foot of the perpendicular drawn from the point P(0 , 2, 3) to the line 3 1 4 5 2 3 x y z+ - += =

Solve the following Linear Programming Problem graphically: Maximise z = 8x + 9y subject to the constraints: 2x + 3y ≤ 6, 3x - 2y ≤ 6, y ≤ 1; x, y ≥ 0

Find the shortest distance between the lines whose vector equations are ^ ^(1 ) ( 2) (3 2 )r t i t j t k= - + - + - ⃗ ^ and ^ ^( 1) (2 1) (2 1)r s i s j s k= + + - + + ⃗ ^

The straight line 3 2 1 3 1 0 x y z- - -= = is:

Assertion (A) : The angle between the straight lines 1 2 3 2 5 4 x y z+ - += = and 1 2 3 1 2 3 x y z- + -= = - is 90°. Reason (R) : Skew lines are lines in different planes which are parallel and intersecting.

Solve the following Linear Programming Problem graphically: Maximise Z = x + 2y subject to the constraints: x + 2y ≥ 100; 2x - y < 0; 2x + y ≤ 200; x, y ≥ 0

Find the vector equation of the line through the point (1, 2, -4) and perpendicular to the two lines ^ ^ ^ ^(8 19 10 ) (3 16 7 )r i j k i j k λ= - + + - + ⃗ ^ ^ and ^ ^ ^ ^(15 29 5 ) (3 8 5 )r i j k i j k μ= + + + + - ⃗ ^ ^

If a line makes angles α, β, γ with the positive direction of co -ordinates axes, then find the value of sin2α + sin2β + sin2γ.

Assertion(A) : The pair of lines given by ^ ^(2 )r i j i k λ= - + + ⃗ ^ ^ and ^ ^ ^2 ( )r i k i j k μ= - + + - ⃗ ^ ^ intersect . Reason(R) : Two lines intersect each other, if they are not parallel and shortest distance = 0.

Find the angle between the vectors a⃗ = î - ĵ + k̂ and b⃗ = î - ĵ + k̂ . OR Find the coordinates of the point where the line 3 1 5 3 1 5 x y z+ - -= =- - cuts the XY plane.

Show that the line through the points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Solve the following Linear Programming Problem graphically: Minimise Z = 13x - 15y subject to the constraints x + y ≤ 7, 2x - 3y + 6 ≥ 0, x ≥ 0 and y ≥ 0.

Find the area of the region bounded by the parabola y2 = 8x and the line x = 2.

Find the shortest distance between the lines ^ ^ ^ ^3 2 4 ( 2 2 )r i j k i j k λ= + - + + + ⃗ ^ ^ and ^ ^ ^5 2 (3 2 6 )r i j i j k μ= - + + + ⃗ ^ ^ . If the lines intersect find their point of intersection.

If the equation of a line AB is 3 2 5 1 2 4 x y z- + -= =- , find the direction ratios of a line parallel to A

Area of parallelogram, whose diagonals are along vectors 2î + 1k̂ and 2ĵ - 3k̂ is

Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stich 6 shirts and 4 pants per day while B can stich 10 shirts and 4 pants per day. Form a linear programming problem to minimise the labour cost to produce at least 60 shirts and 52 pants.

Show that the line through the points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Find the area enclosed between the parabola 4y = 3x2 and the straight line 3x - 2y + 12 = 0.

Find the equation of the line which intersects the lines 2 3 1 1 2 4 x y z+ - += = and 1 2 3 2 3 4 x y z- - -= = passes through the point (1, 1, 1).

Direction ratios of the line 4 1 2 6 3 x y z- - = = are

The area enclosed by the circle x2 + y2 = 2 is equal to

Show that the line through the points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Find the shortest distance between the lines ^ ^ ^ ^( 2 ) ( )r i j k i j k λ= + + + - + ⃗ ^ ^ and ^ ^ ^ ^(2 ) (2 2 )r i j k i j k μ= - - + + + ⃗ ^ ^

Show that the rectangle of maximum area that can be inscribed in a circle of radius 'r' is a square of side √2r.

Find the co-ordinates of the foot of the r and the length of the perpendicular drawn from the point P(5, 4, 2) to the line ^ ^ ^ ^( 3 ) (2 3 )r i j k i j k λ= - + + + + - ⃗ ^ ^ . Also, find the image of P in this line.

If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are:

The angle between the lines 2x = 3y = -z and 6x = -y = - 4z is:

If a ⃗ and b ⃗ are unit vectors inclined at an angle θ, then the value of | a b- ⃗ ⃗ | is

Find the vector and the cartesian equations of a line that passes through the point A(1, 2, - 1) and parallel to the line 5x - 25 = 14 - 7y = 35z.

Find the area of the region {(x, y) : x2 + y2 ≤ 4, x + y ≥ 2}.

Find the distance between the lines: ^ ^ ^ ^( 2 4 ) (2 3 6 );r i j k i j k λ= + - + + + ⃗ ^ ^ ^ ^ ^ ^(3 3 5 ) (4 6 12 )r i j k i j k μ= + - + + + ⃗ ^ ^

The acute angle between two lines whose direction ratios are 2, 3, 6 and 1, 2, 2 is:

The area of a parallelogram whose one diagonal and one side are represented by 2i^ and ^3 j- is:

Assertion (A): The acute angle between the line ^ ^ ^2 ( )r i j k i j λ= + + + - ⃗ ^ ^ and the x-axis is 4 π Reason(R): The acute angle θ between the lines ^ ^ ^ ^ 1 1 1 1 1 1 ( )r x i y j z k a i b j c kλ= + + + + + ⃗ ^ ^ and ^ ^ ^ ^ 2 2 2 2 2 2 ( )r x i y j z k a i b j c kλ= + + + + + ⃗ ^ ^ is given by 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 | |cos a a b b c c a b c a b c θ + += + + + + .

Solve the following Linear Programming Problem graphically: Maximize Z = 400x + 300y subject to x +y ≤ 200, x ≤ 40, x ≥ 20, y ≥ 0

Find the shortest distance between the lines: ^ ^ ^ ^3 5 7 ( 2 )r i j k i j k λ= + + + - + ⃗ ^ ^ and ^ ^ ^ ^(7 6 )r i j k i j k μ= - - - + - + ⃗ ^ ^