HSC Science इयत्ता १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

Assume that water issuing from the end of a horizontal pipe, 7 5. m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2 5. m below the line of the pipe, the flow of water has curved outward 3 m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?

On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4 m when it is 6m away from the point of projection. Finally it reaches the ground 12 m away from the starting point. Find the angle of projection

Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

Choose the correct alternative:

An ellipse has OB as semi-minor axes, F and F’ its foci and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is

Choose the correct alternative:

The eccentricity of the ellipse (x – 3)² + (y – 4)² = `y^2/9` is

Choose the correct alternative:

The locus of a point whose distance from (– 2, 0) is `2/3` times its distance from the line x = `(-9)/2` is

Choose the correct alternative:

The values of m for which the line y = `"m"x + 2sqrt(5)` touches the hyperbola 16x² – 9y² = 144 are the roots of x² – (a + b)x – 4 = 0, then the value of (a + b) is

Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector `4hat"i" + 3hat"j" - 7hat"k"` and parallel to the vector `2hat"i" - 6hat"j" + 7hat"k"`

Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (– 2, 3, 4) and parallel to the straight line `(x - 1)/(-4) = (y + 3)/5 = (8 - z)/6`

Find the points where the straight line passes through (6, 7, 4) and (8, 4, 9) cuts the xz and yz planes

Find the direction cosines of the straight line passing through the points (5, 6, 7) and (7, 9, 13). Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points

Find the acute angle between the following lines.

`vec"r" = (4hat"i" - hat"j") + "t"(hat"i" + 2hat"j" - 2hat"k")`

Find the acute angle between the following lines.

`(x + 4)/3 = (y - 7)/4 = (z + 5)/5, vec"r" = 4hat"k" + "t"(2hat"i" + hat"j" + hat"k")`

Find the acute angle between the following lines.

2x = 3y = – z and 6x = – y = – 4z

The vertices of ΔABC are A(7, 2, 1), 5(6, 0, 3), and C(4, 2, 4). Find ∠ABC

f the straight line joining the points (2, 1, 4) and (a – 1, 4, – 1) is parallel to the line joining the points (0, 2, b – 1) and (5, 3, – 2) find the values of a and b

If the straight lines `(x - 5)/(5"m" + 2) = (2 - y)/5 = (1 - z)/(-1)` and x = `(2y + 1)/(4"m") = (1 - z)/(-3)` are perpendicular to ech other find the  value of m

Show that the points (2, 3, 4), (– 1, 4, 5) and (8, 1, 2) are collinear

Find the parametric form of vector equation and Cartesian equations of straight line passing through (5, 2, 8) and is perpendicular to the straight lines `vec"r" = (hat"i" + hat"j" - hat"k") + "s"(2hat"i" - 2hat"j" + hat"k")` and `vec"r" = (2hat"i" - hat"j" - 3hat"k") + "t"(hat"i" + 2hat"j" + 2hat"k")`

Show that the lines `vec"r" = (6hat"i" + hat"j" + 2hat"k") + "s"(hat"i" + 2hat"j" - 3hat"k")` and `vec"r" = (3hat"i" + 2hat"j" - 2hat"k") + "t"(2hat"i" + 4hat"j" - 5hat"k")` are skew lines and hence find the shortest distance between them