Madhyamik WBSSE Board Mathematics Question Paper

 Are you preparing for the Madhyamik Mathematics Board Exam? We've got you covered! 🎯 In this video, we provide a complete step-by-step solution to the official board question papers. This will help you understand the best methods to solve tricky problems, manage time effectively, and score higher in your exams.

🔹 What you'll learn in this video: ✅ Full solutions to all questions ✅ Easy and shortcut methods for quick solving ✅ Important formulas and concepts ✅ Common mistakes to avoid ✅ Time management strategies for the exam 🚀 Boost your confidence and ace the exam! Don't forget to like, share, and subscribe for more educational content. 🎓📚 🔔 Stay tuned for more board exam tips & solution

CLICK ON THE LINK BELOW

Madyamik Mathematics Board Paper 2025

Madyamik Mathematics Board Paper 2024

Madyamik Mathematics Board Paper 2023

Madyamik Mathematics Board Paper 2022

#Madhyamik2025 #MathematicsBoardExam #QuestionPaperSolved #ExamPreparation #MATH #Solution #WestBengal #WBBSE2025 #MadhyamikMaths #MathsTricks #BoardExamSuccess #TopperStrategy #TimeManagementTip #FullPaperSolution #MathForStudents #Education2025 #StudyWithMe

ICSE MATHEMATICS BOARD QUESTION PAPER 2025

ICSE Mathematics 2025 Board Exam with our detailed analysis of the question paper! This blog provides step-by-step solutions, expert tips, and strategic insights to help you tackle every section of the exam with confidence. Whether you're looking to master quadratic equations, understand probability, or solve complex geometry problems, we’ve got you covered. Stay updated with the latest trends, important topics, and frequently asked questions to ensure you’re fully prepared for the big day. Ace your ICSE Mathematics 2026 exam with our expert guidance and practice resources!

Analysis of the ICSE Mathematics 2025 Board Question Paper:

The ICSE Mathematics 2025 question paper is divided into two sections: Section A (40 marks) and Section B (40 marks). Students are required to attempt all questions from Section A and any four questions from Section B. The paper is designed to test a wide range of mathematical concepts, including algebra, geometry, trigonometry, probability, and statistics.

Key Features of the Paper:

1. Section A (Compulsory):

   • Question 1 consists of 15 multiple-choice questions (MCQs) covering topics like quadratic equations, probability, matrices, geometry, and arithmetic progressions. These questions test fundamental concepts and require quick, accurate responses.

   • Question 2 and Question 3 involve problem-solving and application-based questions, including solving quadratic equations, calculating interest in recurring deposits, and proving geometric theorems. These questions require clear working steps and logical reasoning.

2. Section B (Optional - Attempt any 4 questions):

   • This section includes higher-order thinking questions that test deeper understanding and application of concepts. Topics include:

     • Algebra: Solving inequalities, arithmetic progressions, and polynomial factorization.

     • Geometry: Constructions, similarity of triangles, and properties of circles.

     • Trigonometry: Proving trigonometric identities and solving problems using trigonometric ratios.

     • Mensuration: Calculating surface areas and volumes of composite shapes like capsules.

     • Statistics and Probability: Drawing histograms, finding mean and median, and calculating probabilities.

     • Graphical Problems: Plotting points, reflecting shapes, and analyzing graphs.

3. Difficulty Level:

   • The paper is moderately challenging, with a mix of straightforward MCQs and complex problem-solving questions.

   • Section A is relatively easier, focusing on basic concepts, while Section B requires more analytical thinking and application of concepts.

4. Marking Scheme:

   • Marks are allocated based on the complexity of the questions, with detailed working required for full credit.

   • Rough work must be shown clearly, and omission of essential steps can lead to loss of marks.

5. Key Topics to Focus On:

   • Algebra: Quadratic equations, polynomials, and arithmetic progressions.

   • Geometry: Circle theorems, similarity of triangles, and constructions.

   • Trigonometry: Identities and angle calculations.

   • Mensuration: Surface areas and volumes of 3D shapes.

   • Statistics and Probability: Mean, median, histograms, and probability calculations.

6. Time Management:

   • Students should allocate 15 minutes to read the paper and plan their approach.

   • Section A should be completed within 60-75 minutes, leaving the remaining time for Section B.

Conclusion:

The ICSE Mathematics 2025 paper is well-balanced, testing both conceptual understanding and problem-solving skills. Students should focus on practicing a variety of problems, especially in algebra, geometry, and trigonometry, to perform well. Time management and clear presentation of working steps are crucial for scoring high marks.

ICSE Mathematics Competency Based Questions Detailed Solution

Struggling with ICSE Mathematics Competency-Based Questions? Don't worry, we've got you covered! In this video, we break down some of the most challenging competency-based questions from the ICSE Mathematics syllabus, providing step-by-step solutions to help you understand the concepts and ace your exams. 📚✅

What You'll Learn:

• Key strategies to approach competency-based questions.
• Detailed explanations for solving complex problems.
• Tips and tricks to save time and avoid common mistakes.
• Clear understanding of important mathematical concepts.
• How to approach competency-based questions that test understanding, application, analysis, creation, and evaluation.
• Step-by-step solutions to the latest questions released by the ICSE board.
• Tips to develop critical thinking and problem-solving skills for higher-order questions.
• Strategies to tackle real-life application-based problems effectively.

Topics Covered:

• Algebra
• Geometry
• Trigonometry
• Mensuration
• Statistics and Probability

Whether you're preparing for your board exams or just want to strengthen your math skills, this video is perfect for you! Hit the like button if you find this video helpful, and don't forget to subscribe for more educational content. Share this video with your friends who might need some extra help with their math studies. Let's conquer ICSE Mathematics together! 🚀📖

Why Watch This Video?

• These questions are directly from the board and reflect the latest exam patterns.
• Perfect for students aiming to score high by mastering competency-based learning.
• Clear explanations to help you build confidence and improve your problem-solving skills.

Or download the pdf

50 Important Questions for CBSE and ICSE

 "50 Important Questions for ICSE and CBSE - All Chapters" contains a collection of important mathematics problems for Class 10 students. It includes a variety of questions covering different topics from the ICSE and CBSE syllabus.

Key Topics Covered:

1. Algebra:

   • Arithmetic Progression (AP) and Geometric Progression (GP)
   • Quadratic equations
   • Polynomials and factorization
   • Matrices and determinants

2. Geometry:

   • Triangles (Similarity and Congruence)
   • Circles (Tangents, Chords, and Theorems)
   • Coordinate Geometry (Finding slopes, equations of lines, midpoints, etc.)

3. Trigonometry:

   • Trigonometric identities and proofs
   • Heights and distances problems
   • Simplification of trigonometric expressions

4. Mensuration:

   • Volume and surface area of solids (cones, spheres, and cylinders)
   • Problems involving melting and reshaping of solids

5. Statistics & Probability:

   • Mean, mode, and median
   • Probability of drawing cards, marbles, or coin tosses

6. Commercial Mathematics:

   • Shares and dividends
   • Simple and compound interest
   • Recurring deposits

Types of Questions Included:

• Proof-based questions
• Numerical problem-solving
• Conceptual and application-based problems
• Word problems involving real-life scenarios
• Questions requiring graphical representation

The document is a good resource for exam preparation, covering both theoretical and practical aspects of mathematics for ICSE and CBSE Class 10 students. Let me know if you need detailed solutions or explanations for any specific questions! 

National Means-cum-Merit Scholarship Examination (NMMSE)

The National Means-cum-Merit Scholarship Examination (NMMSE) in India is a prestigious scholarship program conducted by the Ministry of Education, Government of India, aimed at identifying and supporting economically disadvantaged but academically talented students. The examination is typically held at the state/UT level for students studying in Class 8, and successful candidates receive financial assistance to continue their education until Class 12. The NMMSE assesses students' mental ability, scholastic aptitude, and proficiency in subjects like Mathematics and Science. By providing scholarships, the program aims to reduce dropout rates at the secondary level and encourage meritorious students to pursue higher education, thereby contributing to the nation's educational and socio-economic development.

The National Means-cum-Merit Scholarship Examination (NMMSE) in India is a significant initiative by the Ministry of Education to support economically underprivileged students who demonstrate exceptional academic potential. Conducted annually at the state/UT level for Class 8 students, the exam evaluates candidates through two tests: the Mental Ability Test (MAT), which assesses logical and analytical thinking, and the Scholastic Aptitude Test (SAT), which focuses on subjects like Mathematics, Science, and Social Science. Students who qualify are awarded a scholarship of Rs. 12,000 per annum to continue their education from Class 9 to Class 12, provided they maintain a minimum academic performance. The NMMSE not only alleviates financial burdens for deserving students but also motivates them to excel academically, thereby fostering a culture of merit and reducing dropout rates at the secondary school level. This initiative plays a crucial role in empowering young minds and contributing to the nation's educational and socio-economic progress.

A blog post featuring the last five years' solved NMMSE (National Means-cum-Merit Scholarship Examination) Maths papers is an invaluable resource for students preparing for this competitive scholarship exam. Such a post would provide detailed, step-by-step solutions to the Mathematics sections of previous years' question papers, helping students understand the problem-solving techniques, concepts, and patterns frequently tested in the exam. By analyzing these solved papers, students can identify important topics, improve their time management skills, and build confidence in tackling challenging problems. Additionally, the blog post could include tips and strategies for approaching the Maths section, common mistakes to avoid, and insights into the exam's difficulty level over the years. This comprehensive resource would serve as an essential study tool, enabling students to practice effectively and enhance their chances of qualifying for the prestigious NMMSE scholarship.

Chapter Test : Miscellaneous Mathematics Problems for Class X (ICSE & CBSE)

 Mathematics plays a crucial role in strengthening problem-solving skills and logical reasoning. This set of miscellaneous mathematics problems is designed for Class X students studying under ICSE and CBSE curricula, covering a wide range of important topics. Let’s take a closer look at the types of questions included and how they help in mastering key concepts.

---

1. Trigonometry – Identities and Applications

Trigonometry is a core topic in Class X, and this problem set includes challenges like:

• Evaluating expressions involving sin, cos, sec, and cosec functions.
• Proving trigonometric identities and solving equations.
• Finding values of complex trigonometric expressions, such as sin² θ + cos² θ relationships.

Example:

• If sin θ + cos θ = 2, find the value of sin⁷⁶⁸⁵ θ + cos⁷⁶⁸⁵ θ.

These problems test conceptual clarity and the ability to simplify trigonometric functions efficiently.

---

2. Algebra – Quadratic Equations, Factorization & Progressions

Algebraic problem-solving is a fundamental skill. This set includes:

• Quadratic Equations: Proving relationships between coefficients and roots.
• Factorization: Completely factorizing polynomials using algebraic identities and theorems.
• Arithmetic and Geometric Progressions (AP & GP): Finding specific terms and their sums.

Example:

• If the pᵗʰ term of an AP is q and the qᵗʰ term is p, find tₚ₊q and Sₚ₊q.

These problems build a strong foundation in algebra and sequence calculations.

---

3. Geometry – Triangles, Circles & Coordinate Geometry

Geometry questions test understanding of:

• Triangle Properties: Proving area and ratio relationships in right-angled triangles.
• Circle Theorems: Using tangents, chords, and angle properties to prove identities.
• Coordinate Geometry: Finding slopes and relationships between different line segments.

Example:

• Prove that AF + CE + BD = AE + CD + BF = ½ Perimeter (∆ABC) when a circle touches the sides of a triangle.

Mastering these problems enhances logical reasoning and visualization skills.

---

4. Statistics – Mean Transformations

Understanding mean and its transformations is crucial for data interpretation. The given question involves:

• Computing the mean when data points are multiplied by a factor.

Example:

• If x₁, x₂, x₃ … xₙ have a mean x̄, find the mean of (2a/3) * x₁, (2a/3) * x₂, …

Such problems improve conceptual understanding of averages and statistical operations.

---

5. Mensuration – Cones & Volume-Surface Area Relations

Mensuration problems focus on:

• The relationship between the volume and curved surface area of a cone.
• Finding values of expressions involving height (h) and radius (r).

Example:

• If the numerical values of volume and curved surface area of a right circular cone are equal, find (1/h² + 1/r²).

These questions refine students’ ability to work with formulas and spatial reasoning.

---

6. Financial Mathematics – Shares & Investments

Real-life application of financial mathematics includes:

• Calculating dividends on shares.
• Determining returns on investments after selling shares at different prices.
• Finding the percentage increase in returns.

Example:

• Mr. Ram Gopal invested ₹8000 in 7% shares at ₹80. He later sold them at ₹75 and reinvested in 18% ₹25 shares at ₹41. Find:
  • First-year dividend.
  • Second-year annual income.
  • Percentage increase in return on the original investment.

Such problems build financial literacy and real-world application skills.

---

7. Miscellaneous Problem-Solving

Some additional challenges include:

• Solving rational equations.
• Working with sec, tan, and sine transformations.
• Finding values of trigonometric summations like sin²10° + sin²20° + … + sin²90°.
• Proving complex algebraic expressions.

These problems push students to think creatively and develop strong problem-solving abilities.

---

Conclusion

This set of miscellaneous problems provides a comprehensive revision of key Class X topics across Algebra, Geometry, Trigonometry, Statistics, and Mensuration. Whether preparing for board exams or sharpening mathematical skills, these questions serve as excellent practice material.

---

Circles, Similarity & Trigonometry

Here are some challenging circle-related problems for a Grade 10 level. These exercises involve concepts like circle theorems, tangents, chords, angles, and more. 

Exercise 1: Circle Theorems

In the diagram below, $O$ is the center of the circle. Points $A$, $B$, and $C$ lie on the circumference. Angle $ABC={50}^{\circ }$, and angle $OAB={30}^{\circ }$. Find:

1. Angle $AOC$.

2. Angle $OCB$.

Exercise 2: Tangents and Chords

A circle has a chord $AB$ of length 12 cm. The tangent at point $A$ makes an angle of ${60}^{\circ }$ with the chord $AB$. Find:

1. The radius of the circle.

2. The length of the arc $AB$.

Exercise 3: Cyclic Quadrilaterals

In a cyclic quadrilateral $ABCD$, angle $A={70}^{\circ }$, angle $B={110}^{\circ }$, and angle $C={80}^{\circ }$. Find:

1. Angle $D$.

2. The measure of the arc $ADC$.

Exercise 4: Intersecting Chords

Two chords $AB$ and $CD$ intersect at point $P$ inside the circle. If $AP=6\text{cm}$, $PB=4\text{cm}$, and $CP=3\text{cm}$, find the length of $PD$.

Exercise 5: Tangent-Secant Theorem

A tangent $PA$ and a secant $PBC$ are drawn to a circle from an external point $P$. If $PA=8\text{cm}$ and $PB=4\text{cm}$, find the length of $BC$.

Exercise 6: Angle in a Semicircle

In a circle with diameter $AB$, point $C$ lies on the circumference such that angle $ACB={90}^{\circ }$. If $AC=6\text{cm}$ and $BC=8\text{cm}$, find:

1. The radius of the circle.

2. The area of the circle.

Exercise 7: Concentric Circles

Two concentric circles have radii $5\text{cm}$ and $10\text{cm}$. A chord of the larger circle is tangent to the smaller circle. Find the length of the chord.

Exercise 8: Sector Area and Arc Length

A circle has a radius of $7\text{cm}$. A sector of the circle has an angle of ${120}^{\circ }$. Find:

1. The area of the sector.

2. The length of the arc of the sector.

Exercise 9: Inscribed Angles

In a circle, two chords $AB$ and $CD$ intersect at point $E$. If angle $AEC={40}^{\circ }$ and arc $AC={100}^{\circ }$, find:

1. Angle $BED$.

2. The measure of arc $BD$.

Exercise 10: Complex Circle Geometry

In the diagram below, $O$ is the center of the circle. $AB$ is a chord, and $OC$ is perpendicular to $AB$, intersecting it at point $D$. If $OD=3\text{cm}$ and $CD=4\text{cm}$, find:

1. The radius of the circle.

2. The length of chord $AB$.

Here are some challenging  problems for Grade 10 students. These exercises involve concepts like similar triangles, proportionality, and applications of similarity theorems. Let me know if you need hints or solutions!

Exercise 1: Similar Triangles

In triangle $ABC$, $DE$ is parallel to $BC$. If $AD=4\text{cm}$, $DB=6\text{cm}$, and $DE=5\text{cm}$, find:

1. The length of $BC$.

2. The ratio of the areas of $\mathrm{△}ADE$ to $\mathrm{△}ABC$.

Exercise 2: Proportional Segments

In triangle $PQR$, $S$ and $T$ are points on sides $PQ$ and $PR$, respectively, such that $ST$ is parallel to $QR$. If $PS=3\text{cm}$, $SQ=2\text{cm}$, and $QR=10\text{cm}$, find:

1. The length of $ST$.

2. The ratio of the areas of $\mathrm{△}PST$ to $\mathrm{△}PQR$.

Exercise 3: Midsegment Theorem

In triangle $ABC$, $D$ and $E$ are the midpoints of sides $AB$ and $AC$, respectively. If $BC=12\text{cm}$, find:

1. The length of $DE$.

2. The ratio of the area of $\mathrm{△}ADE$ to the area of quadrilateral $BCED$.

Exercise 4: Right Triangle Similarity

In right triangle $ABC$, $\mathrm{\angle }B={90}^{\circ }$. A perpendicular is drawn from $B$ to the hypotenuse $AC$, meeting it at point $D$. If $AD=4\text{cm}$ and $DC=9\text{cm}$, find:

1. The length of $BD$.

2. The lengths of $AB$ and $BC$.

Exercise 5: Overlapping Triangles

Two triangles $ABC$ and $DEF$ overlap such that $\mathrm{\angle }A=\mathrm{\angle }D$ and $\mathrm{\angle }B=\mathrm{\angle }E$. If $AB=6\text{cm}$, $BC=8\text{cm}$, $DE=9\text{cm}$, and $EF=12\text{cm}$, find:

1. The ratio of the sides of $\mathrm{△}ABC$ to $\mathrm{△}DEF$.

2. The length of $AC$ if $DF=15\text{cm}$.

Exercise 6: Area Ratios

Two similar triangles have areas in the ratio $9:16$. If the side length of the smaller triangle is $12\text{cm}$, find:

1. The corresponding side length of the larger triangle.

2. The ratio of their perimeters.

Exercise 7: Shadow Problem

A vertical pole of height $6\text{m}$ casts a shadow of length $4\text{m}$ on the ground. At the same time, a nearby building casts a shadow of length $20\text{m}$. Find:

1. The height of the building.

2. The distance between the pole and the building if the tip of their shadows coincide.

Exercise 8: Nested Triangles

In triangle $ABC$, $D$ and $E$ are points on sides $AB$ and $AC$, respectively, such that $DE\parallel BC$. If $AD=2\text{cm}$, $DB=3\text{cm}$, and the area of $\mathrm{△}ADE=8{\text{cm}}^2$, find:

1. The area of $\mathrm{△}ABC$.

2. The area of trapezoid $BCED$.

Exercise 9: Proportional Medians

Two triangles are similar, and their corresponding medians are in the ratio $3:5$. If the area of the smaller triangle is $36{\text{cm}}^2$, find:

1. The area of the larger triangle.

2. The ratio of their perimeters.

Exercise 10: Complex Similarity

In quadrilateral $ABCD$, $AB\parallel CD$, and the diagonals $AC$ and $BD$ intersect at point $O$. If $AO=6\text{cm}$, $OC=4\text{cm}$, and $BO=9\text{cm}$, find:

1. The length of $DO$.

2. The ratio of the areas of $\mathrm{△}AOB$ to $\mathrm{△}COD$.

Here are some challenging problems on heights and distances for Grade 10 students. These exercises involve concepts like trigonometry, angles of elevation and depression, and real-life applications. 

Exercise 1: Angle of Elevation

A person standing on the ground observes the angle of elevation of the top of a tower to be ${30}^{\circ }$. After walking $20\text{meters}$ closer to the tower, the angle of elevation becomes ${45}^{\circ }$. Find:

1. The height of the tower.

2. The original distance of the person from the tower.

Exercise 2: Angle of Depression

From the top of a cliff $100\text{meters}$ high, the angle of depression of a boat at sea is ${30}^{\circ }$. Find:

1. The distance of the boat from the base of the cliff.

2. The angle of elevation of the top of the cliff from the boat.

Exercise 3: Two Towers

Two towers of heights $20\text{meters}$ and $30\text{meters}$ are standing on the same ground. The angle of elevation of the top of the taller tower from the top of the shorter tower is ${30}^{\circ }$. Find:

1. The distance between the two towers.

2. The angle of elevation of the top of the shorter tower from the base of the taller tower.

Exercise 4: Shadow Problem

A vertical pole of height $10\text{meters}$ casts a shadow of length $10\sqrt{3}\text{meters}$ on the ground. Find:

1. The angle of elevation of the sun.

2. The length of the shadow when the angle of elevation becomes ${45}^{\circ }$.

Exercise 5: Moving Object

A person standing on the ground observes the angle of elevation of a flying airplane to be ${60}^{\circ }$. After $10\text{seconds}$, the angle of elevation becomes ${30}^{\circ }$. If the airplane is flying at a constant height of $3000\text{meters}$, find:

1. The speed of the airplane in $\text{km/h}$.

2. The horizontal distance traveled by the airplane in $10\text{seconds}$.

Exercise 6: Lighthouse and Ship

From the top of a lighthouse $50\text{meters}$ high, the angle of depression of a ship is ${45}^{\circ }$. After some time, the angle of depression becomes ${30}^{\circ }$. Find:

1. The distance traveled by the ship during this time.

2. The time taken by the ship to travel this distance if its speed is $10\text{m/s}$.

Exercise 7: Mountain and Valley

From the top of a mountain $500\text{meters}$ high, the angles of depression of the top and bottom of a valley are ${30}^{\circ }$ and ${60}^{\circ }$, respectively. Find:

1. The depth of the valley.

2. The horizontal distance between the mountain and the valley.

Exercise 8: Kite Flying

A kite is flying at a height of $60\text{meters}$ from the ground. The string attached to the kite makes an angle of ${60}^{\circ }$ with the ground. Find:

1. The length of the string.

2. The horizontal distance of the kite from the person flying it.

Exercise 9: Building and Tree

From the top of a building $20\text{meters}$ high, the angle of elevation of the top of a tree is ${45}^{\circ }$, and the angle of depression of the base of the tree is ${30}^{\circ }$. Find:

1. The height of the tree.

2. The distance between the building and the tree.

Exercise 10: Complex Problem

From a point $P$ on the ground, the angle of elevation of the top of a tower is ${30}^{\circ }$. After walking $20\text{meters}$ towards the tower, the angle of elevation becomes ${60}^{\circ }$. Find:

1. The height of the tower.

2. The distance of point $P$ from the base of the tower.

Exercise 1: Basic Trigonometric Ratios

In a right triangle $ABC$, $\mathrm{\angle }B={90}^{\circ }$, $AB=5\text{cm}$, and $BC=12\text{cm}$. Find:

1. $\sin A$, $\cos A$, and $\tan A$.

2. $\sin C$, $\cos C$, and $\tan C$.

Exercise 2: Complementary Angles

If $\sin \theta =\frac{3}{5}$, find:

1. $\cos \theta$.

2. $\tan \theta$.

3. $\sin ({90}^{\circ }-\theta )$ and $\cos ({90}^{\circ }-\theta )$.

Exercise 3: Pythagorean Identity

If $\tan \theta =\frac{4}{3}$, find:

1. $\sin \theta$ and $\cos \theta$.

2. ${\sin }^2\theta +{\cos }^2\theta$.

Exercise 4: Solving Triangles

In triangle $ABC$, $\mathrm{\angle }A={30}^{\circ }$, $\mathrm{\angle }B={60}^{\circ }$, and side $AB=10\text{cm}$. Find:

1. The length of side $BC$.

2. The length of side $AC$.

Exercise 5: Angle of Elevation

A ladder leaning against a wall makes an angle of ${60}^{\circ }$ with the ground. If the foot of the ladder is $5\text{meters}$ away from the wall, find:

1. The length of the ladder.

2. The height at which the ladder touches the wall

Exercise 6: Trigonometric Identities

Prove the following identities:

1. ${\sin }^2\theta +{\cos }^2\theta =1$.

2. $\tan \theta =\frac{\sin \theta }{\cos \theta }$.

3. $\sin ({90}^{\circ }-\theta )=\cos \theta$.

Exercise 7: Real-Life Application

A flagpole casts a shadow of $15\text{meters}$ when the angle of elevation of the sun is ${45}^{\circ }$. Find:

1. The height of the flagpole.

2. The length of the shadow when the angle of elevation becomes ${30}^{\circ }$.

Exercise 8: Trigonometric Equations

Solve for $\theta$ in the interval $0^{\circ }\le \theta \le {90}^{\circ }$:

1. $\sin \theta =\frac{1}{2}$.

2. $\tan \theta =\sqrt{3}$.

3. $\cos \theta =\frac{\sqrt{2}}{2}$.

Exercise 9: Heights and Distances

From the top of a building $50\text{meters}$ high, the angle of depression of a car on the ground is ${30}^{\circ }$. Find:

1. The distance of the car from the base of the building.

2. The angle of elevation of the top of the building from the car.

Exercise 10: Complex Problem

In triangle $ABC$, $\mathrm{\angle }A={45}^{\circ }$, $\mathrm{\angle }B={60}^{\circ }$, and side $AC=10\text{cm}$. Find:

1. The length of side $BC$.

2. The length of side $AB$.

Mock test for ICSE Class X students : Mathematics

The 50 marks Mathematics test for ICSE Class X students will assess their understanding of key concepts from the prescribed syllabus. The test will cover a variety of topics such as:

1. Algebra: Solving quadratic equations, polynomials, linear equations, and word problems.
2. Geometry: Theorems involving triangles, circles, and construction problems.
3. Trigonometry: Ratios, identities, and applications in solving problems.
4. Mensuration: Surface area and volume of 3D shapes like cylinders, cones, and spheres.
5. Statistics and Probability: Data interpretation, mean, median, mode, and probability calculations.   

Structure:

1. Total Marks: 50
2. Time Limit: 1 hour 45 minutes
3. Type of Questions: The questions include a mix of short answer, long answer, and proof-based problems, covering a variety of mathematical topics.

Content Overview:

1. Algebra:

   • Questions on factorization, solving quadratic equations, and properties of proportions (e.g., questions 1, 4, 5, and 17).

2. Geometry:

   • Questions involving centroids, triangles, circle tangents, and coordinate geometry (e.g., questions 2, 8, and 12).

3. Probability:

   • Basic probability involving labeled cards (question 7).

4. Trigonometry:

   • Trigonometric proofs and calculations (e.g., questions 13, 14, and 15).

5. Mensuration:

   • Calculations involving conical tents (question 10).

6. Arithmetic/Word Problems:

   • Budget-related word problems and proportional geometry problems (e.g., questions 9 and 11).

7. Sequences and Series:

   • Questions on geometric progression (question 16).

Difficulty Levels:

1. Basic/Recall: Direct formula-based or straightforward calculations (e.g., Q1, Q2).
2. Intermediate: Problems requiring some algebraic manipulation or logical reasoning (e.g., Q6, Q12).
3. Advanced: Proofs, geometry constructions, and real-world applications (e.g., Q8, Q9).

Strengths:

• Comprehensive Coverage: The paper tests a wide range of topics, ensuring diverse evaluation.
• Progressive Difficulty: Questions start simple and gradually increase in complexity, catering to different skill levels.
• Real-Life Applications: Questions like the picnic cost (Q9) and the conical tent (Q10) assess practical problem-solving skills.