HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

The slope of the normal to the curve y = x² + 2e^x + 2 at (0, 4) is ______.

If the line y = 4x – 5 touches the curve y² = ax³ + b at the point (2, 3) then a + b is

If the tangent at (1, 1) on y² = x(2 − x)² meets the curve again at P, then P is

Find the slope of tangent to the curve y = 2x³ – x² + 2 at `(1/2, 2)`

Find the slope of normal to the curve 3x² − y² = 8 at the point (2, 2)

Find the slope of tangent to the curve x = sin θ and y = cos 2θ at θ = `pi/6`

Find the equation of normal to the curve y = 2x³ – x² + 2 at `(1/2, 2)` 

Find points on the curve given by y = x³ − 6x² + x + 3, where the tangents are parallel to the line y = x + 5.

Find the cartesian equation of the plane passing through A(1, 2, 3) and the direction ratios of whose normal are 3, 2, 5.

Find the vector equation of the line `x/1 = (y - 1)/2 = (z - 2)/3`

Verify if the point having position vector `4hat"i" - 11hat"j" + 2hat"k"` lies on the line `bar"r" = (6hat"i" - 4hat"j" + 5hat"k") + lambda (2hat"i" + 7hat"j" + 3hat"k")`

Find the Cartesian equation of the line passing through  A(1, 2, 3) and having direction ratios 2, 3, 7

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

Find the Cartesian equation of the plane passing through the points (3, 2, 1) and (1, 3, 1)

Find the direction ratios of the line perpendicular to the lines

`(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1` and `(x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)`

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

Find Cartesian equation of the line passing through the point A(2, 1, −3) and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `hat"i" + 2hat"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") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter

Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2