List of practice Questions

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0,4, 0) and (6, 0, 0).
Three vertices of a parallelogram ABCD are A(3, – 1, 2), B (1, 2, – 4) and C (– 1, 1, 2). Find the coordinates of the fourth vertex.
Find the equation of the set of points P, the sum of whose distances from A (4, 0, 0) and B (-4, 0, 0) is equal to 10.
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
If the 4\(^{th}\), 10\(^{th}\) and 16\(^{th}\) terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.
Evaluate \(\displaystyle\sum_{k=1}^{11}\) (2 + 3k).
Find the sum to indicated number of terms in each of the geometric progressions in\(x^3,x^5,x^7\), ... n terms (if x ≠ ± 1).
Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term.

Show that the function given by \(f(x)=e^{2x}\) is strictly increasing on R.

\(Find\) \(\frac {dy}{dx}\): \(2x+3y=sin \ y\)

Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.
\(if\, x-iy=√\frac{a-ib}{c-id}\) prove that \((x^2y^2)^2=\frac{a^2+b^2}{c^2+d^2}.\)

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola  \(9y^2-4x^2=37\)

The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola \(\dfrac{y^2}{9}-\dfrac{x^2}{27}=1\)

Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1) (ii) (–3, 7, 2) and (2, 4, –1)
(iii) (–1, 3, – 4) and (1, –3, 4) (iv) (2, –1, 3) and (–2, 1, 3)
How many terms of G.P. \(3,3^2,3^3\), … are needed to give the sum 120?
Fill in the blanks:
(i) The x-axis and y-axis taken together determine a plane known as_______.
(ii) The coordinates of points in the XY-plane are of the form _______.
(iii) Coordinate planes divide the space into ______ octants.

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola \( \dfrac{x^2}{16}-\dfrac{y^2}{9}=2\)

Name the octants in which the following points lie:
(1, 2, 3), (4, -2, 3), (4, -2, -5), (4, 2, -5), (-4, 2, -5), (-4, 2, 5), (-3, -1, 6), (2, -4, -7).
The sum of first three terms of a G.P. is \(\frac{39}{10}\) and their product is 1. Find the common ratio and the terms.
A point is in the XZ plane. What can you say about its y-coordinate?

Find the equation for the ellipse that satisfies the given conditions:Major axis on the x-axis and passes through the points (4,3) and (6,2).

Prove that.  \(2sin^2 \frac{π}{6}+cosec^2 \frac{7π}{6}–cos^2 \frac{π}{3}=\frac{3}{2}\)

Prove that. \(sin^2 \frac{π}{6}+cos^2 \frac{π}{3}–tan^2 \frac{π}{4}=-\frac{1}{2}\)