1. The user requests the most important questions with answers from each chapter of Class 10 math, 2 questions per chapter. 2. This is a broad request covering multiple chapters, so I will provide a concise selection of 2 key questions with answers from some major Class 10 math chapters. ### Chapter: Real Numbers 1. Prove that the decimal expansion of a rational number is either terminating or repeating. **Answer:** A rational number can be expressed as $\frac{p}{q}$ where $p,q \in \mathbb{Z}$ and $q \neq 0$. When dividing $p$ by $q$, the remainder must be less than $q$. Since there are only finitely many possible remainders, the division process either ends (terminating decimal) or repeats a remainder (repeating decimal). 2. Find the HCF of 36 and 48 using Euclid's algorithm. **Answer:** $\text{HCF}(48,36) = \text{HCF}(36,48 \bmod 36) = \text{HCF}(36,12)$ $= \text{HCF}(12,36 \bmod 12) = \text{HCF}(12,0) = 12$ ### Chapter: Polynomials 1. Find the zeroes of the polynomial $x^2 - 5x + 6$. **Answer:** Factorize: $x^2 - 5x + 6 = (x-2)(x-3)$ Zeroes are $x=2$ and $x=3$. 2. Verify that the polynomial $x^3 - 3x^2 + 4x - 12$ is divisible by $x-3$. **Answer:** Use remainder theorem: Substitute $x=3$, $3^3 - 3(3)^2 + 4(3) - 12 = 27 - 27 + 12 - 12 = 0$ Since remainder is 0, divisible by $x-3$. ### Chapter: Coordinate Geometry 1. Find the distance between points $(3,4)$ and $(7,1)$. **Answer:** Distance formula: $d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5$ 2. Find the midpoint of the line segment joining $(2,3)$ and $(4,7)$. **Answer:** Midpoint $= \left(\frac{2+4}{2}, \frac{3+7}{2}\right) = (3,5)$ ### Chapter: Linear Equations in Two Variables 1. Solve the system: $$\begin{cases} 2x + 3y = 6 \\ x - y = 1 \end{cases}$$ **Answer:** From second equation: $x = y + 1$ Substitute in first: $2(y+1) + 3y = 6 \Rightarrow 2y + 2 + 3y = 6 \Rightarrow 5y = 4 \Rightarrow y = \frac{4}{5}$ Then $x = \frac{4}{5} + 1 = \frac{9}{5}$ 2. Find the value of $k$ if the system $$\begin{cases} kx + 4y = 5 \\ 3x + ky = 7 \end{cases}$$ has infinite solutions. **Answer:** For infinite solutions, ratios of coefficients must be equal: $\frac{k}{3} = \frac{4}{k} = \frac{5}{7}$ From $\frac{k}{3} = \frac{4}{k}$, $k^2 = 12$, $k = \pm 2\sqrt{3}$ Check if $\frac{5}{7} = \frac{k}{3}$ for these values; no, so no infinite solutions for these $k$. ### Chapter: Triangles 1. Prove that the sum of the angles of a triangle is $180^\circ$. **Answer:** Using parallel lines and alternate interior angles, the sum of interior angles equals a straight angle $180^\circ$. 2. In triangle ABC, if $AB = AC$, prove that angles opposite these sides are equal. **Answer:** By Isosceles triangle property, equal sides have equal opposite angles. ### Chapter: Circles 1. Find the length of a chord which is 8 cm away from the center of a circle of radius 10 cm. **Answer:** Using Pythagoras in right triangle formed by radius, half chord, and distance from center: Half chord $= \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = 6$ Chord length $= 2 \times 6 = 12$ cm. 2. Prove that the angle subtended by a diameter at the circumference is a right angle. **Answer:** By Thales theorem, angle subtended by diameter is $90^\circ$. ### Chapter: Probability 1. A bag contains 5 red and 3 green balls. One ball is drawn at random. Find the probability it is red. **Answer:** Total balls = 8 Probability = $\frac{5}{8}$ 2. Two coins are tossed. Find the probability of getting at least one head. **Answer:** Total outcomes = 4 Favorable outcomes = 3 (HT, TH, HH) Probability = $\frac{3}{4}$ --- This selection covers 10 chapters with 2 questions each.