HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.

If 2 tan^–1(cos x) = tan^–1(2 cosec x). then find the value of x.

Find the general solution of sin θ + sin 3θ + sin 5θ = 0

If –1 ≤ x ≤ 1, the prove that sin^–1 x + cos^–1 x = `π/2`

If a line drawn from the point A( 1, 2, 1) is perpendicular to the line joining P(1, 4, 6) and Q(5, 4, 4) then find the co-ordinates of the foot of the perpendicular.

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

The Cartestation equation of  line is `(x-6)/2=(y+4)/7=(z-5)/3` find its vector equation.

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Find the value of p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l₁.

Find p and q if the equation px² – 8xy + 3y² + 14x + 2y + q = 0 represents a pair of prependicular lines.

Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r=(mbarb+nbara)/(m+n)`. Hence find the position vector of R which divides the line segment joining the points A(1, –2, 1) and B(1, 4, –2) internally in the ratio 2 : 1.

Find the vector equation of the lines which passes through the point with position vector `4hati - hatj +2hatk` and is in the direction of `-2hati + hatj + hatk`

 The equation of a line is 2x -2 = 3y +1 = 6z -2 find the direction ratios and also find the vector equation of the line. 

Find the combined equation of the following pair of lines:

2x + y = 0 and 3x − y = 0

Find the combined equation of the following pair of lines passing through point (2, 3) and parallel to the coordinate axes.

Find the combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0

Find the separate equation of the line represented by the following equation:

3y² + 7xy = 0 

Find k, the slope of one of the lines given by kx² + 4xy – y² = 0 exceeds the slope of the other by 8.

If one of the lines given by ax² + 2hxy + by² = 0 bisects an angle between the coordinate axes, then show that (a + b)² = 4h².