HSC Arts Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions

A tunnel through a mountain for a four-lane highway is to have a elliptical opening. The total width of the highway (not the opening) is to be 16 m, and the height at the edge of the road must be sufficient for a truck 4 m high to clear if the highest point of the opening is to be 5 m approximately. How wide must the opening be?

At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.

An engineer designs a satellite dish with a parabolic cross-section. The dish is  5m wide at the opening, and the focus is placed 1 2. m from the vertex. Position a coordinate system with the origin at the vertex and the x-axis on the parabola’s axis of symmetry and find an equation of the parabola

An engineer designs a satellite dish with a parabolic cross-section. The dish is 5 m wide at the opening and the focus is placed 1.2 m from the vertex. Find the depth of the satellite dish at the vertex

Parabolic cable of a 60 m portion of the roadbed of a suspension bridge are positioned as shown below. Vertical Cables are to be spaced every 6m along this portion of the roadbed. Calculate the lengths of first two of these vertical cables from the vertex.

Cross-section of a Nuclear cooling tower is in the shape of a hyperbola with equation `x^2/30^2 - y^2/44^2` = 1. The tower is 150 m tall and the distance from the top of the tower to the centre of the hyperbola is half the distance from the base of the tower to the centre of the hyperbola. Find the diameter of the top and base of the tower

A rod of length 1 2. m moves with its ends always touching the coordinate axes. The locus of a point P on the rod, which is 0 3. m from the end in contact with x-axis is an ellipse. Find the eccentricity

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

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

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

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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")`