List of top Questions asked in UP Board

When the angle of elevation of the sun becomes \( \theta \) from \( \varphi \), then the shadow of a pillar standing on the horizontal ground is increased by \( \alpha \) metres. Find the length of the pillar.

By observing from the top of a multi-storeyed building, the angle of depression of the top and bottom of a 4 m high building are \( 30^\circ \) and \( 45^\circ \) respectively. Find the height of the multi-storeyed building and the distance between both the buildings.

An arc of a circle of radius 21 cm subtends an angle of 60° at the centre. Find the area of the sector formed by the arc and the area of the segment formed by the corresponding chord.

A cylindrical bucket of 32 cm height and 18 cm base radius is full of sand. This bucket is emptied on the ground and a conical mound of sand is formed. If the height of this mound is 24 cm, then find its radius and slant height.

Which one is a pair of co-prime numbers?

The sum of the powers of prime factors in prime factorization of 144 is:

Find the median of the following frequency distribution:

Given that HCF(255, 867) = 51, find the value of LCM(255, 867).

Prove that \( \sqrt{2} \) is an irrational number.

Find the mode from the following table:

Construct a triangle of sides 4 cm, 5 cm, and 6 cm, and again construct another corresponding triangle whose sides are \( \frac{2}{3} \) of the corresponding sides of the previous triangle.

In a right triangle \( ABC \), angle \( B \) is a right angle and \( BD \) is perpendicular on \( AC \). Prove that: \( AB^2 = AC \cdot AD \).

The product of the age of Punita 2 years ago and her age after 4 years from now is 1 year more than twice her present age. Find her present age.

Solve the following equation: \[ \frac{1}{x} - \frac{1}{x-2} = 3, \quad x \neq 0, 2. \]

Find two positive even consecutive numbers, the sum of whose squares is 340.

The discriminant of the quadratic equation \( x - \frac{1}{x} = 1 \) will be:

The area of a rectangular field is 30 m². If its length is 1 m greater than its breadth, then the quadratic equation to find them will be:

The mean and median of a frequency distribution are 26.1 and 25.8 respectively. The value of mode for the distribution will be :

The measure of central tendency is :

The distance between the points \( (2, y) \) and \( (10, 3) \) is 10 units. Find the value of \( y \).

The mean of \( x \) and \( \frac{1}{x} \) is \( M \). Find the mean of \( x^3 \) and \( \frac{1}{x^3} \).

If \( \tan(2A) = \cot(A - 18^\circ) \), where \( 2A \) is an acute angle, then find the value of \( A \).

A perpendicular is drawn from the vertex to the base of an equilateral triangle of side a. The measure of the perpendicular will be:

The whole surface area of a solid hemisphere of diameter \( \frac{1}{2} \) cm will be :

An arc of a circle of radius 6 cm subtends an angle of \( 60^\circ \) at the centre. The area of the sector will be: