TS SSC 2021 Public Examination

BOARD OF SECONDARY EDUCATION, TELANGANA

TS SSC CLASS 10 MATHEMATICS PUBLIC EXAMINATION 2021

Official Question Paper & Step-by-Step Solutions | Muqeem Tutorial

TIME: 3 Hours 15 Mins PART - A & PART - B MAX MARKS: 80

PART - A (60 MARKS) — SUBJECTIVE QUESTIONS

SECTION - I (6 × 2 = 12 Marks) — Very Short Answer Questions (Answer ALL)

Q1. Use Euclid's division lemma to find the HCF of 90 and 144.

Solution:
By Euclid's Division Lemma: a = bq + r
1. 144 = 90 × 1 + 54
2. 90 = 54 × 1 + 36
3. 54 = 36 × 1 + 18
4. 36 = 18 × 2 + 0
Since remainder is 0, the divisor at this final step is 18.
👉 HCF(90, 144) = 18

Q2. If A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}, find A - B and B - A.

Solution:
• A - B = {1, 3, 5}
• B - A = {6, 8}
👉 A - B = {1, 3, 5}, B - A = {6, 8}

Q3. Find the zeroes of the polynomial p(x) = x² - 7x + 12.

Solution:
• x² - 4x - 3x + 12 = 0 ⇒ (x - 4)(x - 3) = 0
👉 Zeroes are x = 4 and x = 3

Q4. Find the 15th term of the AP: 3, 8, 13, 18...

Solution:
• a = 3, d = 5
• a₁₅ = 3 + 14(5) = 3 + 70 = 73
👉 a₁₅ = 73

Q5. Find the distance between the points A(3, 4) and B(7, 7).

Solution:
• d = √[(7 - 3)² + (7 - 4)²] = √[16 + 9] = √25 = 5
👉 Distance = 5 units

Q6. If tan θ = 3/4, find sin θ and cos θ.

Solution:
• Opposite = 3, Adjacent = 4, Hypotenuse = 5
👉 sin θ = 3/5, cos θ = 4/5

SECTION - II (6 × 4 = 24 Marks) — Short Answer Questions (Answer ALL)

Q7. Prove that 2 + √3 is an irrational number.

Solution:
1. Let 2 + √3 = a/b (a, b coprime integers, b ≠ 0).
2. √3 = (a - 2b) / b.
3. Since a, b are integers, (a - 2b)/b is rational, implying √3 is rational.
4. Contradiction! √3 is irrational.
👉 Hence, 2 + √3 is Irrational. (Proved)

Q8. Find the zeroes of quadratic polynomial x² + 7x + 10 and verify relationship between zeroes and coefficients.

Solution:
• (x + 5)(x + 2) = 0 ⇒ α = -5, β = -2.
• Sum α + β = -7 = -b/a = -7/1. (Verified)
• Product αβ = 10 = c/a = 10/1. (Verified)
👉 Zeroes: -5 and -2. Verified!

Q9. Solve by elimination method: 2x + 3y = 11 and 2x - 4y = -24.

Solution:
• Subtracting equations: 7y = 35 ⇒ y = 5
• Substituting y = 5 in 2x + 3(5) = 11 ⇒ 2x = -4 ⇒ x = -2
👉 Solution: x = -2, y = 5

Q10. Find the coordinates of the point which divides line segment joining (4, -3) and (8, 5) in ratio 3 : 1 internally.

Solution:
• P(x, y) = [(3(8) + 1(4))/4, (3(5) + 1(-3))/4] = [28/4, 12/4] = (7, 3)
👉 Required Point P = (7, 3)

Q11. A card is drawn from a well-shuffled deck of 52 cards. Find probability of getting: (i) a king of red color, (ii) a face card.

Solution:
(i) Red Kings = 2 ⇒ P = 2/52 = 1/26
(ii) Face cards = 12 ⇒ P = 12/52 = 3/13
👉 Probabilities: (i) 1/26, (ii) 3/13

Q12. Prove that (1 + tan² A) / (1 + cot² A) = tan² A.

Solution:
LHS = sec² A / csc² A = (1/cos² A) / (1/sin² A) = sin² A / cos² A = tan² A = RHS
👉 Hence Proved.

SECTION - III (4 × 6 = 24 Marks) — Essay Questions with Internal Choice

Q13 (A). Solve linear equations graphically: 2x + y = 6 and 2x - y = 2.

Solution (13A):
• Line 1: (0, 6) and (3, 0)
• Line 2: (0, -2) and (1, 0)
• Intersection point on graph = (2, 2).
👉 Solution: x = 2, y = 2

--- OR ---

Q13 (B). Draw the graph of polynomial p(x) = x² - 3x - 4 and find its zeroes.

Solution (13B):
• Parabola intersects X-axis at (-1, 0) and (4, 0).
👉 Zeroes are x = -1 and x = 4

Q14 (A). State and prove Pythagoras Theorem.

Solution (14A):
• Statement: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• Proof: In ΔABC right-angled at B, draw BD ⊥ AC.
ΔABD ~ ΔABC ⇒ AB² = AC × AD --- (1)
ΔBCD ~ ΔABC ⇒ BC² = AC × CD --- (2)
Adding (1) and (2): AB² + BC² = AC(AD + CD) = AC².
👉 AB² + BC² = AC². (Proved)

--- OR ---

Q14 (B). A solid metallic sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm. Find the total number of cones formed.

Solution (14B):
• Volume of Sphere = (4/3)π(10.5)³ = 1543.5π cm³
• Volume of 1 Cone = (1/3)π(3.5)²(3) = 12.25π cm³
• Cones formed = 1543.5π / 12.25π = 126
👉 Total Cones Formed = 126

Q15 (A). A tree breaks due to a storm and the broken part bends so that the top touches the ground making an angle of 30° with it. The distance from foot to top is 8 m. Find tree height.

Solution (15A):
• tan 30° = h / 8 ⇒ h = 8 / √3 m.
• cos 30° = 8 / x ⇒ x = 16 / √3 m.
• Total height = h + x = 24 / √3 = 8√3 m.
👉 Tree Height = 8√3 metres (≈ 13.85 m)

--- OR ---

Q15 (B). Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Solution (15B):
• Radius ⊥ Tangent ⇒ ∠OAL₁ = 90° and ∠OBL₂ = 90°.
• Since alternate interior angles are equal (90° = 90°),
👉 Tangents L₁ and L₂ are parallel. (Proved)

Q16 (A). Calculate the Mean and Mode for the following frequency distribution:

Class Interval	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50
Frequency (fi)	5	8	15	10	2

Solution (16A):
• Mean = Σf_ix_i / Σf_i = 960 / 40 = 24
• Mode = 20 + [(15 - 8)/(30 - 8 - 10)] × 10 = 20 + 5.83 = 25.83
👉 Mean = 24, Mode = 25.83

--- OR ---

Q16 (B). Two dice are thrown at the same time. Find the probability of getting: (i) a doublet, (ii) sum = 8, (iii) sum > 10.

Solution (16B):
(i) Doublet = 6/36 = 1/6
(ii) Sum = 8 = 5/36
(iii) Sum > 10 = 3/36 = 1/12
👉 Probabilities: (i) 1/6, (ii) 5/36, (iii) 1/12

PART - B (20 MARKS) — MULTIPLE CHOICE QUESTIONS (1 Mark Each)

1. The value of log₂ 32 is equal to:

(A) 2      (B) 5      (C) 10      (D) 32

👉 Correct Answer: (B) 5 [2⁵ = 32]

2. If log₁₀ (x - 2) = 1, then x = ?

(A) 8      (B) 10      (C) 12      (D) 20

👉 Correct Answer: (C) 12 [x - 2 = 10¹ ⇒ x = 12]

3. The slope of a line parallel to the X-axis is:

(A) 0      (B) 1      (C) Undefined      (D) -1

👉 Correct Answer: (A) 0

4. Value of tan 30° × tan 60° is:

(A) 0      (B) 1      (C) 2      (D) 1/2

👉 Correct Answer: (B) 1 [(1/√3) × √3 = 1]

5. The mode of 2, 3, 2, 4, 5, 2, 3 is:

(A) 3      (B) 2      (C) 4      (D) 5

👉 Correct Answer: (B) 2

6. Probability of an impossible event is:

(A) 0      (B) 1      (C) 1/2      (D) -1

👉 Correct Answer: (A) 0

7. If P(E) = 0.05, then P(NOT E) is:

(A) 0.95      (B) 0.50      (C) 0.05      (D) 1.05

👉 Correct Answer: (A) 0.95

8. Total surface area of a hemisphere of radius r is:

(A) 2πr²      (B) 3πr²      (C) 4πr²      (D) πr²

👉 Correct Answer: (B) 3πr²

9. Midpoint of class interval 10 - 20 is:

(A) 10      (B) 15      (C) 20      (D) 30

👉 Correct Answer: (B) 15

10. Distance of point (3, 4) from origin is:

(A) 3      (B) 4      (C) 5      (D) 7

👉 Correct Answer: (C) 5

11. HCF of 8, 9, 25 is:

(A) 1      (B) 2      (C) 4      (D) 25

👉 Correct Answer: (A) 1

12. Product of zeroes of 2x² - 8x + 6 is:

(A) 4      (B) -4      (C) 3      (D) -3

👉 Correct Answer: (C) 3

13. Common difference of AP: 5, 2, -1, -4... is:

(A) 3      (B) -3      (C) 2      (D) -2

👉 Correct Answer: (B) -3

14. Value of tan 45° + cot 45° is:

(A) 1      (B) 2      (C) 0      (D) √2

👉 Correct Answer: (B) 2

15. Discriminant of x² - 4x + 4 = 0 is:

(A) 0      (B) 16      (C) 4      (D) -4

👉 Correct Answer: (A) 0

16. Angle of elevation of sun when height equals shadow length:

(A) 30°      (B) 45°      (C) 60°      (D) 90°

👉 Correct Answer: (B) 45°

17. Volume of a cone formula:

(A) πr²h      (B) (1/3)πr²h      (C) (4/3)πr³      (D) 2πrh

👉 Correct Answer: (B) (1/3)πr²h

18. Probability of getting Head in a single coin toss:

(A) 0      (B) 1      (C) 1/2      (D) 1/4

👉 Correct Answer: (C) 1/2

19. Empirical relation: Mode = 3 Median - 2 Mean

(A) Mode = 3 Median - 2 Mean      (B) Mode = 2 Median - 3 Mean      (C) Mode = Median + Mean      (D) None

👉 Correct Answer: (A) Mode = 3 Median - 2 Mean

20. Which of the following is NOT a measure of central tendency?

(A) Mean      (B) Median      (C) Range      (D) Mode

👉 Correct Answer: (C) Range