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Unit Wise Class Test 2 | Chapter 7 Trigonometry | Class 10 Mathematics | Punjab Board 2027 Model Papers
Class 10 Chapter 7 Test 2 Trigonometry
Unit Wise Class Test 2 – Chapter 7: Trigonometry for Class 10 Mathematics is designed according to the Punjab Board 2027 Model Papers and new board paper pattern. This Class 10 Mathematics Trigonometry Test provides students with focused practice of important concepts, objective questions, short questions, and mathematical problems from Chapter 7. It is useful for Punjab Board Class 10 Mathematics preparation and chapter-wise test practice.
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Unit 07:
Total Time: 35 min
Total Time: 35 min
CHAPTER WISE CLASS TEST
(Test # 2)
Mathematics
Total Marks: 25
Total Marks: 25
A | B | C | D
1 Ⓐ Ⓑ Ⓒ Ⓓ
2 Ⓐ Ⓑ Ⓒ Ⓓ
3 Ⓐ Ⓑ Ⓒ Ⓓ
A | B | C | D
4 Ⓐ Ⓑ Ⓒ Ⓓ
5 Ⓐ Ⓑ Ⓒ Ⓓ
6 Ⓐ Ⓑ Ⓒ Ⓓ
A | B | C | D
7 Ⓐ Ⓑ Ⓒ Ⓓ
8 Ⓐ Ⓑ Ⓒ Ⓓ
NOTE: Four possible answers A, B, C and D to each question are given. The choice which you think is correct, fill that circle in front of that question with Marker or Pen ink. Cutting or filling two or more circles will result in zero mark in that question.
Q.1:(8)
1. tan (180° − θ) =
Answer: (D) − tan θ
Step 1: 180° − θ lies in the second quadrant, where tan is negative.
Step 2: Using the identity for supplementary angles: tan(180° − θ) = −tan θ.
Check: θ = 45°: tan 135° = −1 = −tan 45° ✓
2. Area of △ABC =
Answer: (D) ½ ac sin β
Step 1: Area = ½ × (product of two sides) × sin(angle between those two sides).
Step 2: The standard forms are ½ ab sin γ, ½ bc sin α and ½ ac sin β.
Step 3: (A) uses sides a, b but angle α (not between them) ✗. (B) uses sides b, c but angle β ✗. (C) has no sides ✗.
Step 4: (D) uses sides a, c and the angle β between them ✓, so the answer is ½ ac sin β.
3. Bearing is written as a:
Answer: (C) 3 figures
Step 1: A bearing is an angle measured clockwise from North.
Step 2: It is always written with three figures, adding zeros in front if needed. For example, 45° is written 045° and 5° is written 005°.
4. What is the value of tan 45°?
Answer: (D) 1
Step 1: In a right triangle with a 45° angle, the other acute angle is also 45°, so the two legs are equal.
Step 2: tan 45° = opposite ÷ adjacent = 1 ÷ 1 = 1.
Check: sin 45° ÷ cos 45° = (1/√2) ÷ (1/√2) = 1 ✓
5. In which quadrant only tan θ and cot θ are positive?
Answer: (C) III
Step 1: Use the ASTC rule: Quadrant I: All positive. II: Sin (and cosec) positive. III: Tan (and cot) positive. IV: Cos (and sec) positive.
Step 2: Only tan θ and cot θ are positive in the third quadrant (III).
Check: In quadrant III both x and y are negative, so tan = y ÷ x = (−) ÷ (−) = positive, while sin and cos are negative ✓
6. Sin 105° = ________?
Answer: (A) sin 75°
Step 1: 105° is in the second quadrant, where sine is positive.
Step 2: sin(180° − θ) = sin θ, so sin 105° = sin(180° − 105°) = sin 75°.
Step 3: (B) cos 50° ≈ 0.643 and (C) sin 95° ≈ 0.996 are different from sin 105° ≈ 0.966, and (D) −2 is impossible since sine lies between −1 and 1.
7. a2 = b2 + c2 − 2bc cos α is called:
Answer: (B) Law of cosines
Step 1: The Law of Cosines is: a² = b² + c² − 2bc cos α (and similarly for b² and c²).
Step 2: The Law of Sines is different: a ÷ sin α = b ÷ sin β = c ÷ sin γ.
Step 3: So the given formula is the Law of cosines.
8. Which of the following is considered an “ordinal direction”?
Answer: (D) South − west (SW)
Step 1: The four cardinal directions are North, South, East and West.
Step 2: The ordinal (intercardinal) directions lie halfway between two cardinal directions: NE, SE, SW and NW.
Step 3: N, S and W are cardinal, while South-west (SW) is the ordinal direction, so the answer is (D).
(PART – I)
Q.2 Write short answers to following questions.(6 × 2 = 12)
(i) What are the trigonometric values of sin 30° and cos 60°.
Answer: sin 30° = 12, cos 60° = 12
Step 1: Take a right triangle with angles 30°, 60°, 90° and sides in the ratio 1 : √3 : 2 (the side opposite 30° is 1, opposite 60° is √3, hypotenuse is 2).
Step 2: sin 30° = opposite ÷ hypotenuse = 12.
Step 3: cos 60° = adjacent ÷ hypotenuse = 12 (for 60°, the adjacent side is the one of length 1).
Result: Both values are equal: sin 30° = cos 60° = ½, as expected since 30° + 60° = 90° (complementary angles).
(ii) Subtract the value of cos 30° from the value of sin 180°.
Answer: −√32
Step 1: sin 180° = 0 (the point at 180° is (−1, 0), so y = 0).
Step 2: cos 30° = √32.
Step 3: “Subtract cos 30° from sin 180°” means sin 180° − cos 30° = 0 − √32.
Step 4: The result is −√32 ≈ −0.866.
(iii) Determine the sign of tan (315°).
Answer: Negative (tan 315° = −1)
Step 1: 315° lies between 270° and 360°, so it is in the fourth quadrant.
Step 2: In the fourth quadrant only cos (and sec) are positive, so tan is negative.
Step 3 (value): tan 315° = tan(360° − 45°) = −tan 45° = −1.
(iv) Without calculator, find the value of sec 120°.
Answer: sec 120° = −2
Step 1: sec θ = 1 ÷ cos θ.
Step 2: 120° is in the second quadrant, where cos is negative. Reference angle = 180° − 120° = 60°.
Step 3: cos 120° = −cos 60° = −12.
Step 4: sec 120° = 1 ÷ (−12) = −2.
(v) Define an oblique triangle.
Answer: Definition
An oblique triangle is a triangle that has no right angle (90°).
It is either an acute-angled triangle (all three angles less than 90°) or an obtuse-angled triangle (one angle greater than 90°).
Oblique triangles are solved using the Law of Sines and the Law of Cosines.
(vi) In a triangle if c = 3 cm, β = 60° and γ = 30°, then find b.
Answer: b = 3√3 ≈ 5.20 cm
Step 1: Law of Sines: b ÷ sin β = c ÷ sin γ.
Step 2: Make b the subject: b = c · sin β ÷ sin γ.
Step 3: b = 3 × sin 60° ÷ sin 30° = 3 × √32 ÷ 12.
Step 4: b = 3 × √3 = 3√3 ≈ 5.20 cm.
Check: α = 180° − 60° − 30° = 90°, so it is a right triangle with hypotenuse a. Then a = c ÷ sin 30° = 3 ÷ ½ = 6, and by Pythagoras b = √(6² − 3²) = √27 = 3√3 ✓
(PART – II)
Note: Attempt the following question.(5)
Q.3: In a triangle ABC, β = 45°, γ = 30°, a = 6cm, calculate b and c.
Answer: b = 6√3 − 6 ≈ 4.39 cm, c = 3(√6 − √2) ≈ 3.11 cm
Step 1 (third angle): α = 180° − β − γ = 180° − 45° − 30° = 105°.
Step 2 (Law of Sines): a ÷ sin α = b ÷ sin β = c ÷ sin γ.
Step 3: sin 105° = sin 75° = (√6 + √2) ÷ 4 ≈ 0.9659.
Step 4 (a ÷ sin α): 6 ÷ 0.9659 ≈ 6.212.
Step 5 (find b): b = 6.212 × sin 45° = 6.212 × 0.7071 ≈ 4.39 cm.
Step 6 (find c): c = 6.212 × sin 30° = 6.212 × 0.5 ≈ 3.11 cm.
Exact forms: b = 6√3 − 6 and c = 3(√6 − √2).
Check: The largest angle (105°) faces the largest side a = 6, and the smallest angle (30°) faces the smallest side c = 3.11 ✓
