List of practice Questions

The sum of the digits of a two-digit number is 9. Nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Find the mode of the following data:
Show that any positive odd integer is of the form \( 4q + 1 \) or \( 4q + 3 \), where \( q \) is some integer.
Find the H.C.F. of 867 and 255 using Euclid's division algorithm.
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at the angle of depression of \( 30^\circ \), which is approaching the foot of the tower with a uniform speed. After 6 seconds the angle of depression of the car is found to be \( 60^\circ \). Find the time taken by the car to reach the foot of the tower from this point.
Find the coordinates of the points which divide the line segment \( AB \) joining points \( A (-2, 2) \) and \( B (2, 8) \) into four equal parts.
The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of new fraction and the original fraction is \( \frac{29}{20} \). Find the original fraction.
In the figure, AB and CD are two perpendicular diameters of a circle with center O. OD is the diameter of the smaller circle. If OA = 7 cm, find the ratio of the areas of the smaller and larger circles.
The roots of the quadratic equation $x^2 - 3x - 10 = 0$ are:
If the length of shadow of a tower on the plane ground is \( \sqrt{3} \) times of its height, the angle of elevation of the sun is:
The number of solid spheres, each of diameter $6$ cm, that can be made by melting a solid metal cylinder of height $45$ cm and diameter $4$ cm is:
If \( \triangle ABC \) is an equilateral triangle such that \( AD \perp BC \), then \( AD^2 \) is equal to:
Find the discriminant of the quadratic equation \( 4x^2 - 6x + 5 = 0 \) and hence find the nature of its roots.
Draw a circle of radius 3.0 cm. From a point 7.0 cm away from its centre, construct the pair of tangents to the circle. Write the steps of the construction in brief.
If \( \cos(A - B) = \frac{\sqrt{3}}{2} \) and \( \sin(A + B) = \frac{\sqrt{3}}{2} \), where \( 0^\circ<(A + B) \leq 90^\circ \) and \( A>B \), then find the values of \( A \) and \( B \).
Find the median of the following frequency distribution:
Prime factors of the number 156 will be:
If mean and mode of a grouped data are 24 and 12 respectively, then its median is:
If the pair of linear equations $2x + 3y = 5$ and $4x + ky = 10$ has infinitely many solutions, then the value of $k$ is:
ABC and BDE are two equilateral triangles such that D is the midpoint of BC. The ratio of the areas of the triangles ABC and BDE is:
If $\cos A = \dfrac{7}{25}$, then the value of $\tan A + \cot A$ is:
If the radii of two circles are 4 cm and 3 cm respectively, then the radius of the circle having area equal to the sum of the areas of these circles is:
The value of $\sin^2 18^\circ - \cos^2 72^\circ$ is:
For any positive even integer \( p \), every positive odd integer will be of the form:
If the mean of the numbers $7, 8, x, 11, 14$ is $x$, then the value of $x$ is: