10th Grade Math Problems

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Questions and Answers

Explain how Euclid's division lemma is used to find the HCF of two numbers. Provide a brief example.

Euclid's division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. To find the HCF of two numbers, apply the division lemma repeatedly until the remainder is zero. The divisor at this stage is the HCF. Example: To find the HCF of 45 and 15, 45 = 15 * 3 + 0, so the HCF is 15.

If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = x^2 - 5x + 6$, find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$.

$\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}$. From the polynomial, $\alpha + \beta = 5$ and $\alpha \beta = 6$. Thus, $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{5}{6}$.

For the pair of linear equations $2x + 3y = 7$ and $4x + ky = 14$, find the value of $k$ for which the lines are coincident.

For coincident lines, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$. So, $\frac{2}{4} = \frac{3}{k} = \frac{7}{14}$. From $\frac{2}{4} = \frac{3}{k}$, we get $k = 6$.

Determine the nature of the roots of the quadratic equation $2x^2 - 4x + 3 = 0$ without explicitly solving for the roots.

To determine the nature of roots without explicitly solving, calculate the discriminant, $D = b^2 - 4ac$. Here, $a = 2, b = -4, c = 3$, so $D = (-4)^2 - 4(2)(3) = 16 - 24 = -8$. Since $D < 0$, the roots are imaginary. Signup and view all the answers

The sum of the first 7 terms of an AP is 49, and the sum of the first 17 terms is 289. Find the common difference of the AP.

Given $S_7 = 49$ and $S_{17} = 289$. Using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$, we have $49 = \frac{7}{2}[2a + 6d]$ or $2a + 6d = 14$, and $289 = \frac{17}{2}[2a + 16d]$ or $2a + 16d = 34$. Solving these two equations gives $d = 2$. Signup and view all the answers

In triangle ABC, D and E are points on sides AB and AC respectively such that DE || BC. If AD = 2 cm, DB = 3 cm, and AE = 3 cm, find the length of EC.

By Basic Proportionality Theorem (Thales' Theorem), $\frac{AD}{DB} = \frac{AE}{EC}$. So, $\frac{2}{3} = \frac{3}{EC}$. Therefore, $EC = \frac{9}{2} = 4.5$ cm. Signup and view all the answers

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

Using the section formula, the coordinates of the point are given by $(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n})$. Here, $m = 3, n = 1, (x_1, y_1) = (4, -3), (x_2, y_2) = (8, 5)$. So, the point is $(\frac{3(8) + 1(4)}{3 + 1}, \frac{3(5) + 1(-3)}{3 + 1}) = (7, 3)$. Signup and view all the answers

Given that $\sin A = \frac{3}{5}$, find the value of $\tan A + \cos A$.

Since $\sin A = \frac{3}{5}$, we can consider a right-angled triangle where the opposite side is 3 and the hypotenuse is 5. Using Pythagoras theorem, the adjacent side is 4. Therefore, $\cos A = \frac{4}{5}$ and $\tan A = \frac{3}{4}$. Hence, $\tan A + \cos A = \frac{3}{4} + \frac{4}{5} = \frac{15 + 16}{20} = \frac{31}{20}$. Signup and view all the answers

A tower stands vertically on the ground. From a point on the ground, which is 30 m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. Find the height of the tower.

Let the height of the tower be h. Using the tangent of the angle of elevation, $\tan 60° = \frac{h}{30}$. Since $\tan 60° = \sqrt{3}$, we have $h = 30\sqrt{3}$ m. Signup and view all the answers

A sector of a circle with radius 6 cm has an angle of 60°. Find the area of the sector.

The area of a sector is given by the formula $A = \frac{\theta}{360°} \pi r^2$. Here, $r = 6$ cm and $\theta = 60°$. So, $A = \frac{60°}{360°} \pi (6)^2 = \frac{1}{6} \pi (36) = 6\pi$ cm$^2$. Signup and view all the answers

Flashcards

Euclid's Division Lemma

A statement that proves if 'a' divides 'b' and 'a' divides 'c', then 'a' divides (b+c). It's used to find the HCF of two numbers.

Prime Factorization Method

A method of finding the HCF and LCM by expressing numbers as products of prime factors.

Zeroes and Coefficients Relationship

A relationship stating that for a quadratic polynomial ax² + bx + c, the sum of zeroes is -b/a, and the product of zeroes is c/a.

Substitution Method

A method to solve linear equations where one variable is isolated in one equation and then substituted into the other equation.