The Math Factory

1. Quadratic equation in one variable
2. Simple interest calculation
3. Theorems related to circles
4. Cuboid
5. Ratio and proportion
6. Compound interest and growth rate
7. Theorem related to angle in a circle
8. Right circular cylinder
9. Quadratic factorization
10. Theorem related to cyclic quadrilateral
11. Construction: Drawing the circumcircle and incircle of a triangle
12. Sphere
13. Variation
14. Partnership business
15. Theorem related to tangent to a circle
16. Right circular cone
17. Construction: Drawing a tangent to a circle
18. Similarity
19. Real-life problems related to different solid shapes
20. Trigonometry: Concept of angle measurement
21. Construction: Finding the mean proportional
22. Pythagoras theorem
23. Trigonometric ratios and trigonometric identities
24. Trigonometric ratios of complementary angles
25. Application of trigonometric ratios: Height and distance
26. Statistics: Mean, Median, Ogive, Mode

🧠 Challenge | Solve Random 10 | Refresh To Regenerate

Q1. Find the value of: \( \cot^2 30^\circ - 2 \cos^2 60^\circ - \cfrac{3}{4} \sec^2 45^\circ - 4 \sin^2 30^\circ \)
Q2. State and prove the Pythagorean Theorem.
Q3. If the sum of the roots of the quadratic equation \[ x^2 - x = k(2x - 1) \] is zero, then what is the value of \(k\)?
Q4. Two circles intersect each other at points P and Q. If PA and PB are the diameters of the respective circles, then prove that the points A, Q, and B lie on a straight line.
Q5. If \(a : b = c : d\), prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).
Q6. At the beginning of the year, Srikanta and Saifuddin purchased a minibus and started operating it with investments of ₹2,40,000 and ₹3,00,000 respectively. After four months, their friend Peter joined them by contributing ₹81,000. Srikanta and Saifuddin withdrew amounts from the business in proportion to their original capital. At the end of the year, if there is a profit of ₹39,150, calculate how much profit each person will receive.
Q7. If the annual interest rate and time period are equal, in how many years will the interest on 144 rupees become 25 rupees?
Q8. Translate: The condition for the roots of the quadratic equation \[ ax^2 + bx - c = 0 \quad (a \ne 0) \] to be equal is:
Q9. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?
Q10. Two chords of a circle have their lengths in the ratio 4:3. If their distances from the center are 9 cm and 12 cm respectively, what is the length of the chord closer to the center?