List of practice Questions

The sum of the first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the 1st term, the common ratio, and the sum to \( n \) terms of the G.P.

Find the value of 'n' so that \( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \) may be the geometric mean between a and b.

Find the coordinates of the focus, axis of the parabola, the equation of the directrix, and the length of the latus rectum, if \( y^2 = 12x \).

Find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity, and the length of the latus rectum of the ellipse: \[ \frac{x^2}{36} + \frac{y^2}{16} = 1 \]

If \( \frac{2}{11} \) is the probability of an event, what is the probability of the event 'not A'?

Find the multiplicative inverse of \( \sqrt{5} + 3i \)

Express the given complex number in the form of \( a + ib \): \( \left( \frac{1}{3} + 3i \right)^3 \)

Solve the given inequality for real \( x \): \[ \frac{3(x - 2)}{5} \leq \frac{5(2 - x)}{3} \]

Solve the given inequalities and represent the solution graphically on the number line: \[ 2(x - 1)<x + 5, \quad 3(x + 2)>2 - x \]

Using Binomial Theorem, Evaluate: \( (102)^5 \)

Evaluate \( \left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6 \)

Prove that \( \frac{\sin x - \sin y}{\cos x + \cos y} = \tan \frac{x - y}{2} \).

How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

The 4th term of a G.P. is square of its second term, and the 1st term is -3. Determine its 7th term.

Show that the points (-2, 3, 5), (1, 2, 3), and (7, 0, -1) are collinear OR verify that the points (0,7,10), (-1,6,6) and (-4,9,6) are the vertices of a right angled triangle.

Verify that (0,7,10), (-1,6,6) and (-4,9,6) are the vertices of a right angled triangle.

For some constants \( a \) and \( b \), find the derivative of \( \frac{x - a}{x - b} \)

Assertion (A): The radius of a circle in which a central angle of 60 degrees intercepts an arc length of 37.4 cm (using \( \pi = \frac{22}{7} \))
Reason (R): The formula to calculate the length of an arc is \( l = Q \times r \), where \( l \) is the arc length, \( Q \) is the central angle in radians, and \( r \) is the radius of the circle.

Assertion (A): The derivative of the function \( f(x) = x^2 \) with respect to \( x \) is \( 2x \).
Reason (R): The derivative of a power function \( x^n \) is given by the formula \( \frac{d}{dx}(x^n) = nx^{n-1} \).

A collection of most dangerous animals of the world is:

If \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), \( A = \{2, 4, 6, 8\} \), and \( B = \{2, 3, 5, 7\} \), verify that \( (A \cup B)' = A' \cap B' \).

The radian measure of 520° is:

Complex conjugate of \( 3i - 4 \) is:

If \( n = 5 \) and \( r = 3 \), then the value of \( ^nP_r \) is: