10-ஆம் வகுப்பு கணிதம்:
காலாண்டுத் தேர்வு
வினாத்தாள் மற்றும்
விடைகள் 2026
தமிழகப் பள்ளிகளில் நடைபெறும் 10-ஆம் வகுப்பு மாணவர்களுக்கான 2026-ஆம் ஆண்டு காலாண்டுத் தேர்வு (Common Quarterly Examination) கணித வினாத்தாள் (ஆங்கில வழி) மற்றும் ஒரு மதிப்பெண் வினாக்களுக்கான விடைகள் கீழே கொடுக்கப்பட்டுள்ளது. இதனை மாணவர்கள் எளிதில் படிக்கும் வகையிலும், மொபைல் போனில் பார்ப்பதற்கு வசதியாகவும் வடிவமைத்துள்ளோம்.
| வகுப்பு | 10-ஆம் வகுப்பு |
|---|---|
| பாடம் | கணிதம் (Mathematics - English Medium) |
| தேர்வு | காலாண்டுத் தேர்வு 2026 |
| மொத்த மதிப்பெண்கள் | 100 மதிப்பெண்கள் |
COMMON QUARTERLY EXAMINATION - 2026
MATHEMATICS
Part - I
I. Choose the correct answer: 14 x 1 = 14
1. If $n(A \times B) = 6$ and $A = \{1, 3\}$ then $n(B)$ is
a) 1 b) 2 c) 3 d) 6
விடை: c) 3
2. The range of the relation $R = \{(x, x^2) \mid x \text{ is a prime number less than } 13\}$ is
a) $\{2,3,5,7\}$ b) $\{2,3,5,7,11\}$ c) $\{4,9,25,49,121\}$ d) $\{1,4,9,25,49,121\}$
விடை: c) {4,9,25,49,121}
3. If $f: A \to B$ is a bijective function and if $n(B) = 7$, then $n(A)$ is equal to
a) 7 b) 49 c) 1 d) 14
விடை: a) 7
4. Euclid's division lemma states that for positive integers $a$ and $b$, there exist unique integers $q$ and $r$ such that $a = bq + r$, where $r$ must satisfy.
a) $1 < r < b$ b) $0 < r < b$ c) $0 \le r < b$ d) $0 < r \le b$
விடை: c) $0 \le r < b$
5. If 6 times of 6th term of an A.P. is equal to 7 times the 7th term, then the 13th term of the A.P. is
a) 0 b) 6 c) 7 d) 13
விடை: a) 0
6. If the sequence $t_1, t_2, t_3 \dots$ are in A.P. then the sequence $t_6, t_{12}, t_{18} \dots$ is
a) a Geometric Progression b) an Arithmetic Progression c) neither an Arithmetic Progression nor a Geometric Progression d) a constant sequence
விடை: b) an Arithmetic Progression
7. If $(x - 6)$ is the HCF of $x^2 - 2x - 24$ and $x^2 - kx - 6$ then the value of $k$ is
a) 3 b) 5 c) 6 d) 8
விடை: b) 5
8. The values of $a$ and $b$ if $4x^4 - 24x^3 + 76x^2 + ax + b$ is a perfect square are
a) 100, 120 b) 10, 12 c) -120, 100 d) 12, 10
விடை: c) -120, 100
9. In $\Delta LMN$, $\angle L = 60^\circ$, $\angle M = 50^\circ$. If $\Delta LMN \sim \Delta PQR$ then the value of $\angle R$ is
a) $40^\circ$ b) $70^\circ$ c) $30^\circ$ d) $110^\circ$
விடை: b) 70^\circ
10. In a given figure $ST \parallel QR$, $PS = 2 \text{ cm}$ and $SQ = 3 \text{ cm}$. Then the ratio of the area of $\Delta PQR$ to the area of $\Delta PST$ is
a) 25 : 4 b) 25 : 7 c) 25 : 11 d) 25 : 13
விடை: a) 25 : 4
11. If $(5,7)$, $(3,p)$ and $(6,6)$ are collinear, then the value of $p$ is
a) 3 b) 6 c) 9 d) 12
விடை: c) 9
12. The slope of the line which is perpendicular to a line joining the points $(0,0)$ and $(-8, 8)$ is
a) -1 b) 1 c) 1/3 d) -8
விடை: b) 1
13. $\tan\theta \operatorname{cosec}^2\theta - \tan\theta$ is equal to
a) $\sec\theta$ b) $\cot^2\theta$ c) $\sin\theta$ d) $\cot\theta$
விடை: d) \cot\theta
14. If $f$ is the identity function, then the value of $f(1) - 2f(2) + f(3)$ is
a) 1 b) 0 c) -1 d) -3
விடை: b) 0
Part - II
II. Answer any 10 questions. (Q.No.28 is compulsory) 10 x 2 = 20
15. Find $A \times B$ and $B \times A$ if $A = \{2, -2, 3\}$ and $B = \{1, -4\}$.
16. Let $f = \{(x, y) \mid x, y \in \mathbb{N} \text{ and } y = 2x\}$ be a relation on $\mathbb{N}$. Find the domain, co-domain and range. Is this relation a function?
17. If $f(x) = 3x - 2$, $g(x) = 2x + k$ and if $f \circ g = g \circ f$, then find the value of $k$.
18. Find the number of terms in the A.P. $3, 6, 9, 12, \dots, 111$.
19. If $3 + k, 18 - k, 5k + 1$ are in A.P, then find $k$.
20. Find the sum of $1 + 3 + 5 + \dots + \text{to } 40 \text{ terms}$.
21. Find the sum : $3 + 1 + \frac{1}{3} + \dots \infty$
22. Find the sum and product of the roots for $x^2 + 3x - 28 = 0$
23. Determine the nature of the roots for $2x^2 - 2x + 9 = 0$
24. If the difference between a number and its reciprocal is $24/5$, find the number.
25. If $\Delta ABC$ is similar to $\Delta DEF$ such that $BC = 3 \text{ cm}$, $EF = 4 \text{ cm}$ and area of $\Delta ABC = 54 \text{ cm}^2$. Find the area of $\Delta DEF$.
26. Find the slope of a line joining the points $(14, 10)$ and $(14, -6)$
27. Prove that $\tan^2\theta - \sin^2\theta = \tan^2\theta \sin^2\theta$
28. Show that the given points are collinear : $(1, 2), (2, 3)$ and $(3, 4)$
Part - III
III. Answer any 10 questions. (Q.No.42 is compulsory) 10 x 5 = 50
29. Let $f: A \to B$ be a function defined by $f(x) = \frac{x}{2} - 1$, where $A = \{2, 4, 6, 10, 12\}$, $B = \{0, 1, 2, 4, 5, 9\}$. Represent $f$ by
i) set of ordered pairs (ii) a table (iii) an arrow diagram (iv) a graph
30. If the function $f : \mathbb{R} \to \mathbb{R}$ is defined by
$$f(x) = \begin{cases} 2x + 7 & ; x < -2 \\ x^2 - 2 & ; -2 \le x < 3 \\ 3x - 2 & ; x \ge 3 \end{cases}$$
then find the values of
i) $f(4)$ ii) $f(-2)$ iii) $f(4) + 2f(1)$ iv) $\frac{f(1) + 3f(4)}{f(-3)}$
31. Find the HCF of $396, 504, 636$.
32. Find the sum of all natural numbers between $300$ and $600$ which are divisible by $7$.
33. Find the sum to $n$ terms of the series $3 + 33 + 333 + \dots \text{to } n \text{ terms}$.
34. Find the square root of $64x^4 - 16x^3 + 17x^2 - 2x + 1$
35. Simplify: $\frac{1}{x^2 - 5x + 6} + \frac{1}{x^2 - 3x + 2} - \frac{1}{x^2 - 8x + 15}$
36. Find the GCD of the following division algorithm.
$2x^4 + 13x^3 + 27x^2 + 23x + 7, \quad x^3 + 3x^2 + 3x + 1, \quad x^2 + 2x + 1$
37. State and prove Thales theorem.
38. Find the area of the quadrilateral whose vertices are at $(-9, 0), (-8, 6), (-1, -2)$ and $(-6, -3)$
39. Without using Pythagoras theorem, show that the points $(1, -4), (2, -3)$ and $(4, -7)$ form a right angled triangle.
40. Find the equation of a straight line passing through $(1, -4)$ and has intercepts which are in the ratio $2:5$
41. Prove the following identity: $$\sqrt{\frac{1 + \sin\theta}{1 - \sin\theta}} + \sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} = 2\sec\theta$$
42. Let $A =$ The set of all natural numbers less than $8$, $B =$ The set of all prime numbers less than $8$, $C =$ The set of even prime number. Verify that $(A \cap B) \times C = (A \times C) \cap (B \times C)$
Part - IV
IV. Answer all the questions. 2 x 8 = 16
43. a) Construct a triangle similar to a given triangle $PQR$ with its sides equal to $7/3$ of the corresponding sides of the triangle $PQR$ (scale factor $7/3 > 1$)
(OR)
b) Draw a triangle $ABC$ of base $BC = 8 \text{ cm}$, $\angle A = 60^\circ$ and the bisector of $\angle A$ meets $BC$ at $D$ such that $BD = 6 \text{ cm}$.
44. a) Varshika drew six circles with different sizes. Draw a graph for the relationship between the diameter and circumference (approximately related) of each circle as shown in the table and use it to find the circumference of a circle when its diameter is $6 \text{ cm}$.
| Diameter (x) cm | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Circumference (y) cm | 3.1 | 6.2 | 9.3 | 12.4 | 15.5 |
(OR)
b) Draw the graph of $xy = 24, x, y > 0$. Using the graph find
i) $y$ when $x = 3$ and ii) $x$ when $y = 6$
