HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions

Find k, if one of the lines given by 6x² + kxy + y² = 0 is 2x + y = 0.

Find the joint equation of the pair of lines through the origin and making an equilateral triangle with the line x = 3.

Find the combined equation of bisectors of angles between the lines represented by 5x² + 6xy - y² = 0.

Find a if the sum of the slopes of lines represented by ax² + 8xy + 5y² = 0 is twice their product.

If the line 4x - 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0, then show that 25a + 40h + 16b = 0

Show that the following equation represents a pair of line. Find the acute angle between them:

2x² + xy - y² + x + 4y - 3 = 0

Show that the following equation represents a pair of line. Find the acute angle between them:

(x - 3)² + (x - 3)(y - 4) - 2(y - 4)² = 0

Show that the combined equation of the pair of lines passing through the origin and each making an angle α with the line x + y = 0 is x² + 2(sec 2α)xy + y² = 0

Find the condition that the equation ay² + bxy + ex + dy = 0 may represent a pair of lines. 

If the lines given by ax² + 2hxy + by² = 0 form an equilateral triangle with the line lx + my = 1, show that (3a + b)(a + 3b) = 4h².

If the line x + 2 = 0 coincides with one of the lines represented by the equation x² + 2xy + 4y + k = 0, then prove that k = - 4. 

Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax² + 2hxy + by² = 0 is bx² - 2hxy + ay² = 0.

If equation ax² - y² + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.

Find k if the slope of one of the lines given by 3x² + 4xy + ky² = 0 is three times the other.

Find the joint equation of the line passing through the origin and having slopes 1 + `sqrt3` and 1 - `sqrt3`

Find the vector equation of the line passing through the point having position vector `-2hat"i" + hat"j" + hat"k"  "and parallel to vector"  4hat"i" - hat"j" + 2hat"k"`.

Find the vector equation of the line passing through points having position vector `3hati + 4hatj - 7hatk and 6hati - hatj + hatk`.

Find the vector equation of line passing through the point having position vector `5hat"i" + 4hat"j" + 3hat"k"` and having direction ratios  –3, 4, 2.

Find the vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k"  "and perpendicular to vectors"  hat"i" + hat"j" + hat"k" and 2hat"i" - hat"j" + hat"k"`.

Find the vector equation of the line passing through the point having position vector `-hat"i" - hat"j" + 2hat"k"  "and parallel to the line" bar"r" = (hat"i" + 2hat"j" + 3hat"k") + λ(3hat"i" + 2hat"j" + hat"k").`