Competency Focused Questions for CBSE and ICSE Class X

Class X Mathematics Competency-Based Questions

Focusing on real-life application, analysis, and conceptual clarity for CBSE & ICSE.

Real Numbers

1. HCF & LCM Application: Three alarm clocks ring at intervals of 12 minutes, 18 minutes, and 24 minutes respectively. If they ring together at 8:00 AM, at what time will they next ring together? Also, find the total number of times they ring together between 8:00 AM and 8:00 PM.
2. Irrationality Proof: Prove that \( \sqrt{5} \) is irrational. Hence, show that \( 3 + 2\sqrt{5} \) is also irrational.
3. Rational Numbers as Decimals: Without actual division, determine if the rational number \( \frac{129}{2^3 \times 5^4 \times 7} \) has a terminating or non-terminating decimal expansion. Justify.
4. Word Problem on LCM: A rectangular courtyard measures 18 m 72 cm × 13 m 20 cm. Exactly square tiles of the same size are required to pave it. Find the least possible number of such tiles.
5. Competency: Reasoning: Can two numbers have 18 as their HCF and 450 as their LCM? Give reasons.

Polynomials

6. Graphical Interpretation: The graph of a quadratic polynomial \( p(x) \) passes through \( (0, -6) \), \( (1, 0) \), and \( (2, 0) \). Find the polynomial and the sum of its zeroes.
7. Relation between zeroes & coefficients: If \( \alpha \) and \( \beta \) are zeroes of \( x^2 - 6x + k \) such that \( 3\alpha + 2\beta = 20 \), find the value of \( k \).
8. Division Algorithm: When \( x^3 - 2x^2 + ax - b \) is divided by \( x^2 - 2x - 3 \), the remainder is \( x - 6 \). Find \( a \) and \( b \).
9. Forming a polynomial: Find a quadratic polynomial whose zeroes are \( \frac{3\alpha}{\beta} \) and \( \frac{3\beta}{\alpha} \), where \( \alpha, \beta \) are zeroes of \( x^2 - 5x + 6 \).
10. Complex zeroes (ICSE): If one zero of the polynomial \( (a^2+4)x^2 + 13x + 4a \) is reciprocal of the other, find \( a \).

Pair of Linear Equations in Two Variables

11. Condition for infinite solutions: Find the value of \( k \) for which the system \( kx + 3y = k-3 \), \( 12x + ky = k \) has infinitely many solutions.
12. Cross-multiplication method: Solve:
    \[ \frac{x}{a} + \frac{y}{b} = a+b, \quad \frac{x}{a^2} + \frac{y}{b^2} = 2 \]
13. Real-life: Age problem: Five years ago, a mother was three times as old as her daughter. Ten years from now, the mother will be twice as old as her daughter. Find their present ages.
14. Geometry application: The line \( 3x + 4y = 12 \) meets the x-axis at A and y-axis at B. Find the equation of the line through the midpoint of AB and perpendicular to AB.
15. Word problem (competitive exam): A man rowing downstream covers 24 km in 4 hours. Against the current, he covers 12 km in 6 hours. Find the speed of the boat in still water and the speed of the current.

Quadratic Equations

16. Nature of roots: For what value of \( k \) does the equation \( (k+1)x^2 + 2(k+3)x + (k+8) = 0 \) have equal roots? Also, find the roots.
17. Word problem: Time and work: Two pipes together fill a tank in \( 12\frac{8}{9} \) minutes. The larger pipe fills it in 8 minutes less than the smaller. Find the time taken by each alone.
18. Equation from roots: If one root of \( x^2 - 4x + a = 0 \) is twice the other, find \( a \) and the roots.
19. Fraction problem: The denominator of a fraction is 2 more than the numerator. If 3 is added to both numerator and denominator, the fraction becomes \( \frac{5}{6} \). Find the original fraction.
20. Profit-loss application: A shopkeeper buys a number of books for ₹ 1800. If he had bought 15 more books for the same amount, each book would have cost ₹ 20 less. How many books did he buy?

Arithmetic Progressions

21. Sum of terms: The sum of first \( n \) terms of an AP is \( 4n^2 - 3n \). Find the 20th term and the common difference.
22. Divisibility: How many three-digit numbers are divisible by 7? Find their sum.
23. Word problem: Installments: A man buys a TV on installment basis. He pays ₹ 1000 at the end of the first month, ₹ 1100 at the end of second, ₹ 1200 at the end of third, and so on. The total amount paid in 20 months is ₹ 39,000. Check if this is correct. If not, find the correct total.
24. Interleaved APs: If \( a, b, c \) are in AP, prove that \( \frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab} \) are also in AP.
25. Competency: Critical thinking: The sum of three numbers in AP is 36 and their product is 1620. Find the numbers.

Triangles

26. Thales theorem application: In \( \triangle ABC \), \( D \) and \( E \) are points on AB and AC such that \( DE \parallel BC \). If \( AD = 2.4 \) cm, \( DB = 3.6 \) cm, and \( AC = 7.5 \) cm, find AE and EC.
27. Similarity in real life: A 1.5 m tall boy stands at a distance of 2 m from a lamp post and casts a shadow of 2.8 m on the ground. Find the height of the lamp post.
28. Pythagoras theorem: In a right triangle, the hypotenuse is 10 cm more than the smallest side. The third side is 10 cm less than the hypotenuse. Find the sides.
29. Area ratio: Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
30. Competency: Proof: In a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. Prove this using a trapezium-based method (area proof).

Coordinate Geometry

31. Area of triangle: Find the area of the triangle formed by the points \( (2, 3) \), \( (-1, 0) \), and \( (2, -4) \). Also, find the ratio in which the y-axis divides the side joining the first two points.
32. Collinearity: For what value of \( k \) are the points \( (k, 2-2k) \), \( (-k+1, 2k) \), and \( (-4-k, 6-2k) \) collinear?
33. Real-life: Intersection: Two straight roads are represented by the equations \( x + 2y = 5 \) and \( 2x + y = 7 \). A park is located at their intersection. Find the coordinates of the park. Also, find the distance of the park from the point \( (3, 2) \).
34. Competency: Ratio: The line segment joining \( A(2, 3) \) and \( B(6, -5) \) is divided by the point \( P \) in the ratio \( 3:1 \) internally. Find the coordinates of \( P \) and verify that it lies on the line \( 2x + y - 7 = 0 \).
35. Centroid and midpoint: Two vertices of a triangle are \( (1, 4) \) and \( (5, 2) \). If its centroid is \( (3, 2) \), find the third vertex. Also, find the length of the median from the third vertex.

Trigonometry

36. Trigonometric identities: Prove that
    \[ \frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac{1 + \sec\theta}{\tan\theta} \]
37. Heights and distances: From the top of a 120 m high tower, the angle of depression of two cars on opposite sides of the tower are \( 30^\circ \) and \( 60^\circ \). Find the distance between the two cars.
38. Trigonometric ratios: If \( \tan A + \cot A = 2 \), find the value of \( \tan^8 A + \cot^8 A \).
39. Real-life: Navigation: A ship is 40 km from a lighthouse. The angle of elevation from the ship to the top of the lighthouse is \( 30^\circ \). After sailing towards the lighthouse, the angle becomes \( 60^\circ \). How far did the ship sail?
40. Competency: Simplification: Evaluate:
    \[ \frac{\cos^2 20^\circ + \cos^2 70^\circ}{\sin^2 59^\circ + \sin^2 31^\circ} + \frac{\tan 15^\circ \cdot \tan 30^\circ \cdot \tan 45^\circ \cdot \tan 60^\circ \cdot \tan 75^\circ}{3} \]

Circles

41. Tangent properties: From an external point \( P \), two tangents \( PA \) and \( PB \) are drawn to a circle with centre \( O \). If \( \angle APB = 60^\circ \), prove that \( \triangle PAB \) is equilateral. Also, find \( \angle OAB \).
42. Length of tangent: A circle touches all four sides of a quadrilateral \( ABCD \) with sides \( AB = 6 \) cm, \( BC = 7 \) cm, \( CD = 5 \) cm. Find \( DA \).
43. Competency: Real-life: A circular park of radius 20 m has a path of width 2 m around it on the inside. Find the area of the path and the cost of fencing the inner boundary at ₹ 15 per metre.
44. Two circles intersecting: Two circles of radii 5 cm and 3 cm intersect at two points, and the distance between their centers is 4 cm. Find the length of the common chord.
45. Alternate segment theorem: In a circle, a chord \( AB \) subtends \( 90^\circ \) at the centre. \( C \) is a point on the major arc. Find the angle between the tangent at \( A \) and chord \( AC \).

Constructions (Competency – Steps & Justification)

46. Dividing a segment: Construct a triangle \( ABC \) with \( BC = 7 \) cm, \( \angle B = 45^\circ \), \( \angle C = 60^\circ \). Then construct a triangle similar to it whose sides are \( \frac{3}{4} \) of the corresponding sides of \( \triangle ABC \). (Write steps and justification.)
47. Tangent to a circle: Draw a circle of radius 3 cm. From a point 8 cm away from the center, construct a pair of tangents. Measure and verify their lengths.

Areas Related to Circles

48. Composite figure: A square of side 14 cm has four quadrants drawn inside it from each corner (each quadrant radius = 7 cm). Find the area of the region common to all four quadrants (central overlapping region).
49. Real-life: Gardening: A cow is tied with a rope of length 14 m at the corner of a rectangular field of dimensions 30 m × 20 m. Find the area the cow can graze. (Assume \( \pi = \frac{22}{7} \)).

Statistics & Probability

50. Competency: Interpretation: The mean of 20 observations is 25. Later it was found that two observations 23 and 35 were wrongly taken as 32 and 53. Find the correct mean. Also, if each observation is multiplied by 2, what is the new mean?