Madhyamik Practice Test Paper

Mathematics Test Examination Paper

Class X (Madhyamik)

3 Hours 15 Minutes

Full Marks: 90

West Bengal Board Format | Test Paper I - Vinod Singh

About This Examination Paper

This is a Mathematics Test Examination paper from Prime Maths, designed for Class X (Madhyamik) students. It follows the standard West Bengal Board format with a total of 90 marks and a duration of 3 hours and 15 minutes.

Arithmetic

Problems on simple interest, compound interest, and partnership business

Algebra

Quadratic equations, ratio and proportion, variation, and surds

Geometry

Theorems related to circles, tangents, circumcenters, and construction

Mensuration

Problems involving cubes, cylinders, cones, spheres, and cuboids

Trigonometry

Solving identities, trigonometric ratios, heights and distances problems

Statistics

Calculation of Mean, Median, and Mode from frequency distribution tables

This paper serves as excellent practice material for students preparing for their final Madhyamik examination, testing their understanding of both objective (MCQ, True/False, Fill in the blanks) and subjective (long answer) type questions.

MADHYAMIK PRACTICE TEST PAPER

Time: 3 Hours 15 Minutes  |  Full Marks: 90

1. Select the correct answer in each of the following cases: [1 × 6 = 6]

1. If the difference between compound interest and simple interest on a certain principal for 3 years at 20% compound interest rate per annum is ₹48, the principal will be:

   • a) ₹350
   • b) ₹375
   • c) ₹400
   • d) ₹425

2. For what value(s) of \(k\) will the quadratic equation \(2x^2 - kx + k = 0\) have equal roots?

   • a) Only 0
   • b) Only 8
   • c) 4
   • d) 0 and 8

3. If \(A:B=2:3\), \(B:C=5:8\), and \(C:D=6:7\), what is the value of \(A:D\)?

   • a) 2:7
   • b) 5:8
   • c) 7:2
   • d) 5:14

4. \(O\) is the circumcenter of \(\triangle ABC\). Given \(\angle BAC=85^\circ\) and \(\angle BCA=55^\circ\), the value of \(\angle AOC\) will be:

   • a) \(65^\circ\)
   • b) \(45^\circ\)
   • c) \(50^\circ\)
   • d) \(100^\circ\)

5. When two circles touch each other internally, what is the maximum number of common tangents that can be drawn?

   • a) 1
   • b) 3
   • c) 4
   • d) 2

6. The diameters of two solid cylinders are 8 cm and 12 cm respectively, and their heights are \(3(n + 1)\) cm and \((n + 3)\) cm respectively. If the volumes of the two cylinders are equal, what is the value of \(n\)?

   • a) 13
   • b) 7
   • c) 8
   • d) 5

2. Fill in the blanks (Any five): [1 × 5 = 5]

1. If \(P_1: P_2: P_3\) is the ratio of profits and \(t_1:t_2:t_3\) is the ratio of time, then the ratio of invested capital is ______________.
2. \(7\sqrt{\frac{101}{404}}\) is a ______________ number.
3. If \(x \propto z\) and \(y \propto z\), then \(x+y \propto\) ______________.
4. The value of \(\cos 28^\circ \csc 62^\circ + \tan 1^\circ \cot 89^\circ\) = ______________.
5. If the ratio of three consecutive angles of a cyclic quadrilateral is \(1 : 3 : 4\), the measure of the fourth angle = ______________.
6. If the volume of a cube is \(V\), the total surface area of the cube = ______________ square units.

3. Write True or False (Any five): [1 × 5 = 5]

1. A partnership business requires at least 4 people.
2. If the annual simple interest rate is 12%, the interest for \(8\frac{1}{3}\) years will be equal to the principal.
3. The two roots of the equation \(x^2 = -2024\) are real numbers.
4. In two triangles ABC and DEF, if \(\angle A= \angle D\), \(\angle B= \angle F\), and \(\angle C = \angle E\), then \(\frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF}\).
5. Two concentric circles of different radius will have one common tangent.
6. If the radius of the base of a right circular cone is halved and the height is doubled, the volume of the cone remains the same.

4. Answer the following questions (Any ten): [2 × 10 = 20]

1. If the simple interest on a principal for \(n\) years at \(r\%\) per annum is \(\frac{pnr}{25}\), show that the Principal = \(4p\).
2. If ₹8,000 is invested for 3 years at 10% compound interest per annum, how much interest will be earned at the end of 2 years?
3. In a business, A invested ₹1800 and B invested ₹1000 for 9 months. If both their profit shares are equal, for how long was A's money invested?
4. What is the ratio of the sum and product of the roots of the equation \(-x^2 - 6x + 2 = 0\)?
5. Rationalize the denominator: \(\frac{2\sqrt{2}-3}{\sqrt{2}}\).
6. AB and AC are two chords of a circle that are perpendicular to each other. If AB = 5 cm and AC = 12 cm, find the length of the radius of the circle.
7. AD is perpendicular to the hypotenuse BC of a right-angled triangle ABC. If BC = 16 cm and BD = 9 cm, find the length of AB.
8. The curved surface area of a right circular cone is \(\sqrt{5}\) times its base area. What is the ratio of the height to the radius of the cone?
9. If \(x = a \cos \theta + b \sin \theta\) and \(y = a \sin \theta - b \cos \theta\), what is the value of \(x^2 + y^2\)?
10. Find the median and arithmetic mean of the numbers: 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11.
11. The length of the diagonal of a cube is \(4\sqrt{3}\) cm. Find the total surface area of the cube.

5. Answer any one question: [5 × 1 = 5]

1. The simple interest and annual compound interest on a certain capital for 2 years at the same rate are ₹4,000 and ₹4,100 respectively. Find the capital and the rate of interest.
2. The current number of students in all secondary education centers in a district is 3993. If the number of students increases by 10% compared to the previous year every year, determine what the number of students was in all secondary education centers of that district 3 years ago.

6. Solve any one question: [3 × 1 = 3]

1. \(\frac{1}{x} - \frac{1}{x+b} = \frac{1}{a} - \frac{1}{a+b}, \quad x \neq 0, -b.\)
2. If \(x = \frac{\sqrt{a+2} + \sqrt{a-2}}{\sqrt{a+2} - \sqrt{a-2}}\), where \(a > 2\), find the simplest value of \(x + \frac{1}{x}\).

7. Solve any one question: [3 × 1 = 3]

1. Find the simplest value: \(\frac{2(\sqrt{2}+\sqrt{3})}{\sqrt{3}+1} - \frac{2(\sqrt{2}-\sqrt{3})}{\sqrt{3}-1}\).
2. If it takes 9 days for 5 men to cultivate 10 acres of land, use the theory of variation to find how many days it will take for 25 men to cultivate 30 acres of land.

8. Answer any one question: [3 × 1 = 3]

1. If \(a:b = b:c\), prove that \(\frac{(a+b)}{(b+c)^2} = \frac{a^2+b^2}{b^2+c^2}\).
2. If \(\frac{a+b-c}{a+b} = \frac{b+c-a}{b+c} = \frac{c+a-b}{c+a}\) and \(a+b+c \neq 0\), prove that \(a = b = c\).

9. Answer any one question: [5 × 1 = 5]

1. Prove that the angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.
2. Prove that if a perpendicular is drawn from the center of a circle to a chord which is not a diameter, the perpendicular bisects the chord.

10. Answer any one question: [3 × 1 = 3]

1. Prove that a cyclic trapezium is an isosceles trapezium and the lengths of its diagonals are equal.
2. In an isosceles triangle ABC, AB = AC and BE is perpendicular to AC from B. Prove that \(BC^2 = 2AC \times CE\).

11. Answer any one question: [5 × 1 = 5]

1. Draw a square figure with a side length of 14 cm.
2. Draw a right-angled triangle whose sides adjacent to the right angle have lengths 4 cm and 8 cm. Draw the incircle of the triangle. (Only traces of construction are required).

12. Solve any two questions: [3 × 2 = 6]

1. If \(m = \cos \theta - \sin \theta\) and \(n = \cos \theta + \sin \theta\), prove that \(\sqrt{\frac{m}{n}} + \sqrt{\frac{n}{m}} = \frac{2}{\sqrt{1-\tan^2 \theta}}\).
2. Find the value of \(x\): \(x \sin 60^\circ \cos^2 30^\circ = \frac{\tan^2 45^\circ \sec 60^\circ}{\csc 60^\circ}\).
3. The measures of three angles of a quadrilateral are \(\frac{\pi}{3}\), \(\frac{5\pi}{6}\), and \(90^\circ\). Find the circular measure (in radians) of the fourth angle.

13. Answer any one question: [5 × 1 = 5]

1. From an object on the roof of a house \(30\sqrt{3}\) meters high, the angles of depression of the top and bottom of a lamp post are \(30^\circ\) and \(60^\circ\) respectively. Find the height of the lamp post.
2. From the roof of a five-story building 18 meters high, the angle of elevation of the top of a monument is observed to be \(45^\circ\) and the angle of depression of the foot of the monument is observed to be \(60^\circ\). What is the height of the monument?

14. Answer any two questions: [4 × 2 = 8]

1. If the radius of a solid sphere and a solid right circular cylinder are equal and their volumes are also equal, calculate the ratio of the radius and height of the cylinder.
2. The volume of a right circular cone is \(100\pi\) cubic cm and its height is 12 cm. Calculate and write the slant height of the cone.
3. A rectangular pond is 20 m long and 18.5 m wide, and has water 3.2 m deep. How long will it take to irrigate the entire water of the pond using a pump that can irrigate 160 kiloliters of water per hour?

15. Answer any two questions: [4 × 2 = 8]

1. Determine the Arithmetic Mean from the following frequency distribution:

   Value:	Below 10	Below 20	Below 30	Below 40	Below 50	Below 60	Below 70	Below 80
   Frequency:	4	16	40	76	96	112	126	125

2. The frequency distribution of daily wages of 100 workers is as follows. Find the median of this distribution.

   Daily Wage (in ₹):	460-470	470-480	480-490	490-500	500-510	510-520	520-530
   No. of Workers:	6	4	29	23	16	10	2

3. Determine the Mode of the frequency distribution below:

   Class Limit:	0-5	5-10	10-15	15-20	20-25	25-30	30-35
   Frequency:	5	12	18	28	17	12	8