10th class
Here are the squares of numbers from 1 to 25:1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
Here’s a sample Class 10 Half-Yearly Examination Question Paper on Trigonometry, following the AP State Syllabus.
CLASS: 10
SUBJECT: MATHEMATICS
TOPIC: TRIGONOMETRY
MAXIMUM MARKS: 40
DURATION: 2 HOURS
Instructions:
1. Answer all the questions.
2. Draw neat diagrams wherever necessary.
3. Use π = 22/7, if required.
Section A: Very Short Answer Questions (4 × 1 = 4 Marks)
Answer each question in one or two sentences.
1. Define sin θ and cos θ in a right-angled triangle.
2. Write the values of sin 30° and cos 60°.
3. State the Pythagorean identity involving sin²θ and cos²θ.
4. Find the value of tan 45°.
Section B: Short Answer Questions – I (4 × 2 = 8 Marks)
Answer each question in 2-3 steps.
5. If sin A = 3/5, find cos A. (Assume A is an acute angle.)
6. Prove: sec²θ - tan²θ = 1.
7. Write the trigonometric ratios for 0°, 30°, 45°, 60°, and 90°.
8. If tan A = 5/12, find sin A and cos A.
Section C: Short Answer Questions – II (4 × 3 = 12 Marks)
Answer each question in detailed steps.
9. Evaluate:
(a) sin²30° + cos²60°
(b) tan²45° - cot²45°
10. A ladder is leaning against a wall. The ladder is 10 meters long and makes an angle of 30° with the ground. Find the height of the wall where the ladder touches it.
11. Prove: (1 + tan²A) = sec²A.
12. From the top of a tower 30 m high, the angle of depression of a car on the ground is 60°. Find the distance of the car from the base of the tower.
Section D: Long Answer Questions (4 × 4 = 16 Marks)
Answer each question in a detailed manner with diagrams where necessary.
13. Prove:
14. A kite is flying at a height of 50 m. The string attached to the kite is tied to the ground and makes an angle of 60° with the horizontal. Find the length of the string.
15. Evaluate without using a calculator:
.
16. Find the angles of elevation of the top of a tree, which is 20 m high, at distances of 20 m and 40 m from the base of the tree.
End of Question Paper
This question paper comprehensively covers basic definitions, identities, problems involving heights and distances, and proofs, as required for a half-yearly exam.
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Here are the answers to the given Class 10 Half-Yearly Examination question paper on Trigonometry.
Section A: Very Short Answer Questions (4 × 1 = 4 Marks)
1. Define sin θ and cos θ in a right-angled triangle:
2. Write the values of and :
3. State the Pythagorean identity involving and :
4. Find the value of :
Section B: Short Answer Questions – I (4 × 2 = 8 Marks)
5. If , find :
Using ,
6. Prove :
From the trigonometric identity,
Rearranging:
7. Write the trigonometric ratios for :
\begin{aligned}
&\text{Angle} & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ \\
&\sin \theta & 0 & \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{\sqrt{3}}{2} & 1 \\
&\cos \theta & 1 & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} & 0 \\
&\tan \theta & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & \infty \\
\end{aligned}
8. If , find and :
Opposite side = 5, Adjacent side = 12.
Hypotenuse = .
Section C: Short Answer Questions – II (4 × 3 = 12 Marks)
9. Evaluate:
(a)
(b)
10. Ladder Problm:
Length of ladder = 10 m, Angle = .
Height =
11. Prove :
From the identity,
(Proved)
12. Angle of Depression Problem:
Height = 30 m, Angle = .
Distance =
Section D: Long Answer Questions (4 × 4 = 16 Marks)
13. Prove:
LHS = ,
Taking LCM:
\frac{\sin A (1 - \cos A) + \sin A (1 + \cos A)}{(1 + \cos A)(1 - \cos A)} = \frac{2 \sin A}{\sin^2 A} = 2 \csc A.
14. Kite Problem:
Height = 50 m, Angle = .
Length of string =
15. Evaluate without using a calculator:
16. Angles of Elevation:
Height of tree = 20 m.
At 20 m distance:
At 40 m distance:
End of Answer Key
If you are asking for one mark questions related to the topics covered in Class 7 CBSE Mathematics (which you previously requested), here are one mark questions from various chapters:
Integers
1. What is the additive identity of integers?
Answer: 0
Fractions and Decimals
2. What is the decimal form of 3/5?
Answer: 0.6
Data Handling
3. What is the mode in the following data set: 3, 7, 3, 8, 3, 2?
Answer: 3
Simple Equations
4. Solve the equation: .
Answer:
Lines and Angles
5. How many degrees are there in a right angle?
Answer: 90°
The Triangle and Its Properties
6. What is the sum of the interior angles of a triangle?
Answer: 180°
Congruence of Triangles
7. What is the condition for two triangles to be congruent by the RHS rule?
Answer: Right angle, hypotenuse, and one side.
Comparing Quantities
8. What is 50% of 200?
Answer: 100
Rational Numbers
9. What is the reciprocal of 7?
Answer: 1/7
Perimeter and Area
10. What is the formula for the area of a rectangle?
Answer: Length × Breadth
Algebraic Expressions
11. What is the coefficient of in the expression ?
Answer: 3
Exponents and Powers
12. Simplify: .
Answer: 8
Symmetry
13. How many lines of symmetry does an equilateral triangle have?
Answer: 3
Visualising Solid Shapes
14. How many faces does a cube have?
Answer: 6
Practical Geometry
15. What instrument is used to draw a circle?
Answer: Compass
These one-mark questions cover a wide range of topics from the CBSE Class 7 Maths syllabus. They are designed to test basic knowledge and understanding of concepts.
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7TH CLASS
The CBSE (Central Board of Secondary Education) syllabus for Class 7 Mathematics typically covers a range of topics designed to build a strong foundation for higher classes. Below is an overview of the major topics and chapters in Class 7 Maths based on the NCERT (National Council of Educational Research and Training) curriculum.
1. Integers
Properties of Addition and Subtraction
Multiplication and Division of Integers
Word problems involving integers
2. Fractions and Decimals
Addition, Subtraction, Multiplication, and Division of Fractions
Types of Fractions (Proper, Improper, Mixed)
Multiplication and Division of Decimals
3. Data Handling
Collection and Organization of Data
Mean, Median, and Mode
Construction and Interpretation of Bar Graphs
Probability (Simple Introduction)
4. Simple Equations
Formation of Simple Equations
Solving Equations with Variables on One Side
Applications in Word Problems
5. Lines and Angles
Basic Definitions (Point, Line, Line Segment, Ray)
Types of Angles (Acute, Obtuse, Right, Straight, Reflex)
Pair of Angles (Complementary, Supplementary, Adjacent, Linear Pair, Vertically Opposite)
Properties of Parallel Lines and Transversals
6. The Triangle and Its Properties
Types of Triangles (Equilateral, Isosceles, Scalene)
Properties of Triangles (Angle Sum Property, Exterior Angle Property)
Pythagoras Theorem (Introduction)
Medians and Altitudes of a Triangle
7. Congruence of Triangles
Criteria for Congruence of Triangles (SSS, SAS, ASA, RHS)
Applications of Congruence
8. Comparing Quantities
Ratios and Proportions
Unitary Method
Percentages and Their Applications (Profit, Loss, Discount, Simple Interest)
9. Rational Numbers
Representation of Rational Numbers on a Number Line
Operations on Rational Numbers (Addition, Subtraction, Multiplication, Division)
Properties of Rational Numbers
10. Perimeter and Area
Perimeter and Area of Rectangles, Squares, and Triangles
Area of Parallelograms and Circles
Applications in Real Life Problems
11. Algebraic Expressions
Terms, Coefficients, and Constants
Addition and Subtraction of Algebraic Expressions
Multiplication of Algebraic Expressions by a Constant or Variable
12. Exponents and Powers
Laws of Exponents
Standard Form of Numbers
Use of Exponents in Simplifying Expressions
13. Symmetry
Lines of Symmetry in Different Shapes
Rotational Symmetry
Symmetry in Regular Polygons
14. Visualising Solid Shapes
Plane Figures and Solid Shapes
Faces, Edges, and Vertices of 3D Shapes
Drawing and Visualizing 3D Objects (Cubes, Cuboids, Cylinders, etc.)
15. Practical Geometry
Construction of Parallel Lines
Construction of Triangles (Using SSS, SAS, ASA, RHS)
Construction of Special Angles (30°, 45°, 60°, 90°)
These topics provide a well-rounded mathematical understanding that lays the groundwork for more advanced concepts in higher grades. For detailed explanations and exercises, NCERT textbooks are recommended, and they are also aligned with CBSE exam patterns.
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