HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Important Questions

The combined equation of the lines through origin and perpendicular to the pair of lines 3x² + 4xy − 5y² = 0 is ______

Find the value of h, if the measure of the angle between the lines 3x² + 2hxy + 2y² = 0 is 45°. 

Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0

If θ is the acute angle between the lines given by ax² + 2hxy + by² = 0 then prove that tan θ = `|(2sqrt("h"^2) - "ab")/("a" + "b")|`. Hence find acute angle between the lines 2x² + 7xy + 3y² = 0 

If the angle between the lines represented by ax² + 2hxy + by² = 0 is equal to the angle between the lines 2x² − 5xy + 3y² = 0, then show that 100(h² − ab) = (a + b)²

Equation of line passing through the points (0, 0, 0) and (2, 1, –3) is ______.

Write the separate equations of lines represented by the equation 5x² – 9y² = 0

Find the value of k. if 2x + y = 0 is one of the lines represented by 3x² + kxy + 2y² = 0

Write the joint equation of co-ordinate axes.

If ax² + 2hxy + by² = 0 represents a pair of lines and h² = ab ≠ 0 then find the ratio of their slopes.

If θ is the acute angle between the lines represented by ax² + 2hxy + by² = 0 then prove that tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`

Prove that the acute angle θ between the lines represented by the equation ax² + 2hxy+ by² = 0 is tanθ = `|(2sqrt(h^2 - ab))/(a + b)|` Hence find the condition that the lines are coincident.

If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

If the vectors `-3hati+4hatj-2hatk, hati+2hatk, hati-phatj` are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment  AB.

Find the volume of the parallelopiped whose coterminus edges are given by vectors

`2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`

(A) 100

(B) 101

(C) 110

(D) 109

If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4