List of practice Questions

If \( 2 \tan^2 \theta - 5 \sec \theta = 1 \) has exactly 7 solutions in the interval \(\left[ 0, \frac{n\pi}{2} \right],\) for the least value of \( n \in \mathbb{N} \) then \(\sum_{k=1}^{n} \frac{k}{2^k}\) is equal to:
A chord of a circle of radius 15 cm subtends an angle $60^\circ$ at the centre. Find the area of minor and major sectors of the circle. \[ (\pi = 3.14, \ \sqrt{3} = 1.73) \]
Find the LCM of the numbers 92 and 510.
Prove that the lengths of tangents drawn from an external point to a circle are equal.
The discriminant of the quadratic equation $x^2 + x - 1 = 0$ will be
The sum of the roots of the quadratic equation $1 - 4x + 4x^2 = 0$ will be
If the distance between the points $(x, 5)$ and $(2, -3)$ is 17 units, then find the value of $x$.
If the points $(1, 4)$, $(a, -2)$, and $(-3, 16)$ are collinear, then find the value of $a$.
The radius of the base of a cylinder is 3.5 cm. If its height is 8.4 cm, then its curved surface area will be
The angle of a sector of a circle of radius 4 cm is $60^\circ$. Its area will be
If \( p \sin \theta = q \cos \theta \), then the value of \(\csc \theta\) will be
The value of \(\tan 1^\circ \tan 2^\circ \tan 3^\circ \dots \tan 88^\circ \tan 89^\circ\) will be
If the roots of the equation \(3x^2 + 5x - q = 0\) are equal, then the value of \(q\) will be
The eleventh term of the A.P. \(-62, -59, \dots, 7, 10\) will be
If \(P(E) = 0.05\), then the value of \(P(\overline{E})\) will be
A bag contains 3 red and 5 black balls. One ball is drawn out at random. The probability of it being a red ball will be
Prove that $(\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 7 + \tan^2 \theta + \cot^2 \theta$.
If a pipe drained the water from a semi-spherical tank filled with water at the rate of $3 \dfrac{4}{7}$ litres per second, how much time will the pipe take to drain half of the water of the tank if the diameter of the tank is 3 m?
The maximum number of tangents drawn from an external point to a circle will be
The distance of point (7, 3) from y-axis will be
If $\csc \theta = 2$, then find the value of $\cot \theta + \dfrac{\sin \theta}{1 + \cos \theta}$.
Prove that the points (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.
Show that the linear equations $3x - y = 2$ and $9x - 3y = 6$ have many solutions.
The angle of elevation of the top of a 10-metre high building from a point P on the earth is 30°. There is a flag on top of the building. The angle of elevation of the top of the flag from point P is 45°. Find the length of the flag.
The sum of the digits of a two-digit number is 9. If the digits of the number be interchanged then the new number is 27 more than the original number. Find the number.