UP Board Class 10 Mathematics Question Paper 2025 (Code 822 BY) with Answer Key and Solutions PDF is Available to Download

For a polynomial \(f(x)\), the graph of \(y=f(x)\) is given. The number of zeroes of \(f(x)\) in the graph will be:

• (A) 1
• (B) 3
• (C) 2
• (D) 4

Prime factorisation of 156 will be:

• (A) \(2^2 \times 3 \times 13\)
• (B) \(2 \times 3^2 \times 13\)
• (C) \(2 \times 3 \times 13\)
• (D) \(2 \times 3 \times 13^2\)

The HCF of the numbers 175 and 91 will be:

• (A) 5
• (B) 7
• (C) 9
• (D) 25

Which of the following numbers will be a rational number?

• (A) \(\dfrac{\sqrt{3}}{\sqrt{5}}\)
• (B) \(\sqrt{2} \times \sqrt{7}\)
• (C) \((\sqrt{5} + \sqrt{7})(\sqrt{5} - \sqrt{7})\)
• (D) \(\sqrt{12}\)

The solution of a pair of linear equations \(2x + y = 5\) and \(3x + 2y = 8\) will be:

• (A) Unique
• (B) Two
• (C) Infinitely many
• (D) None of these

25th term of an A.P. 6, 10, 14, ... will be:

• (A) 100
• (B) 102
• (C) 101
• (D) 151

If one root of the quadratic equation \(x^2 + 2x - p = 0\) is \(-2\), then the value of \(p\) will be:

• (A) 0
• (B) 1
• (C) 2
• (D) 3

Distance between two points \((2,3)\) and \((4,1)\) will be:

• (A) \(2\)
• (B) \(2\sqrt{2}\)
• (C) \(2\sqrt{3}\)
• (D) \(12\)

Area of sector of angle \(\theta\) of a circle with radius \(r\) will be:

• (A) \(\dfrac{\theta}{180} \times 2\pi r\)
• (B) \(\dfrac{\theta}{180} \times \pi r^2\)
• (C) \(\dfrac{\theta}{360} \times 2\pi r\)
• (D) \(\dfrac{\theta}{720} \times 2\pi r^2\)

The length of a tangent of a circle of radius \(3 \,cm\) drawn from a point at a distance of \(5 \,cm\) from the centre will be:

• (A) \(3 \,cm\)
• (B) \(4 \,cm\)
• (C) \(5 \,cm\)
• (D) \(6 \,cm\)

In the figure, the value of \(\angle P\) will be:

• (A) \(20^\circ\)
• (B) \(40^\circ\)
• (C) \(60^\circ\)
• (D) \(80^\circ\)

Median class of the following frequency distribution will be:

Class Interval & Frequency
0-10 & 7
10-20 & 12
20-30 & 18
30-40 & 15
40-50 & 10
50-60 & 3

• (A) 20-30
• (B) 30-40
• (C) 40-50
• (D) 10-20

If \(\sin \theta = \cos \theta\), \(0 \leq \theta \leq 90^\circ\), then the value of \(\theta\) will be:

• (A) \(60^\circ\)
• (B) \(45^\circ\)
• (C) \(30^\circ\)
• (D) \(0^\circ\)

The value of \(\cos 0^\circ\) will be:

• (A) 1
• (B) 0
• (C) 2
• (D) -1

If \(\tan 2A = 1\), where \(2A\) is an acute angle, the value of \(A\) will be:

• (A) \(36^\circ\)
• (B) \(22\dfrac{1}{2}^\circ\)
• (C) \(38^\circ\)
• (D) \(45^\circ\)

The value of \((\sec A + \tan A)(1 - \sin A)\) will be:

• (A) \(\sec A\)
• (B) \(\sin A\)
• (C) \(\csc A\)
• (D) \(\cos A\)

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

• (A) \(24.2\)
• (B) \(25.1\)
• (C) \(25.2\)
• (D) \(26.4\)

The modal class of the following table will be:

Class Interval & Frequency
0-5 & 5
5-10 & 8
10-15 & 12
15-20 & 10
20-25 & 7

• (A) 5-10
• (B) 15-20
• (C) 20-25
• (D) 10-15

The area of a rectangular field is \(30 \, m^2\). If its length is \(1 \, m\) greater than its breadth \(x\), then the quadratic equation to find them will be:

• (A) \(x^2 + x + 30 = 0\)
• (B) \(x^2 - x + 30 = 0\)
• (C) \(x^2 + x - 30 = 0\)
• (D) \(x^2 - x - 30 = 0\)

When a die is thrown once, the probability of getting an even number will be:

• (A) \(\tfrac{1}{4}\)
• (B) \(\tfrac{1}{3}\)
• (C) \(\tfrac{1}{2}\)
• (D) \(\tfrac{1}{6}\)

Prove that \(7\sqrt{5}\) is an irrational number.

Find the length of an arc of a circle of radius \(14\) cm which subtends an angle of \(30^\circ\) at the centre.

Find the co-ordinates of the point which divides the line segment joining the points \(A(3,4)\) and \(B(-2,-1)\) in the ratio \(3:2\).

Find the value of \(x\) for which the distance between the points \((x,2)\) and \((6,5)\) is 5 units.

Prove that \(\dfrac{1+\sec\theta}{\sec\theta}=\dfrac{\sin^2\theta}{1-\cos\theta}\).

Find mean of the following frequency table:

Two concentric circles are of radii \(8\ cm\) and \(5\ cm\). Find the length of the chord of the larger circle which touches (is tangent to) the smaller circle.

Prove that if a straight line is drawn parallel to one side of a triangle to intersect the other two sides in two distinct points, then the other two sides are divided by those points in the same ratio.

Find two consecutive positive integers, sum of whose squares is \(365\).

If the roots of the equation \(x^2+2(m-1)x+(m+5)=0\) are real and equal, find the value of \(m\).

Find the unknown frequency if 24 is the median of the following frequency distribution:

Class-interval & 0--10 & 10--20 & 20--30 & 30--40 & 40--50

Frequency & 5 & 25 & 25 & \(p\) & 7

 

A bag contains 5 black, 7 red and 3 white balls. One ball is drawn at random. Find the probability of drawing the ball is (i) red (ii) black (iii) not black.

If the sum of first 7 terms of an A.P. is 49 and the sum of first 17 terms is 289, find the sum of first \(n\) terms.

The altitude of a right-angled triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

The shadow of a tower, when the angle of elevation of the sun is \(30^\circ\), is \(50\) m longer than when the angle of elevation was \(60^\circ\) on the plane ground. Find the height of the tower.

Prove that \(\displaystyle \frac{\sin A-\cos A+1}{\sin A+\cos A-1}=\frac{1}{\sec A-\tan A}\).

A toy is in the form of a hemisphere surmounting a cone whose radius is \(3.5\) cm. If the total height of the toy is \(15.5\) cm, find its total surface area and volume.

A spherical glass vessel has a cylindrical neck \(8\) cm long, \(2\) cm in diameter, and the diameter of the spherical part is \(8.5\) cm. Find the volume of water that can be filled in it.