Mathematics Problem Solving
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Questions and Answers

Determine the zeroes of the quadratic polynomial after analyzing its intersection points: (-6, 0), (0, -30), (4, -20), and (6, 0).

-6 and 6

For the system of equations given by $3x - ky - 7 = 0$ and $6x + 10y - 3 = 0$, what value of $k$ results in inconsistency?

-10

Explain why the statement 'From a point inside a circle, only two tangents can be drawn' is true or false.

False, only one tangent can be drawn from a point inside a circle.

If the nth term of an arithmetic progression (AP) is given as $7n - 4$, calculate the common difference.

<p>7</p> Signup and view all the answers

If a right circular cone's radius is 5 cm and its volume equals that of a sphere with the same radius, calculate the height of the cone.

<p>20 cm</p> Signup and view all the answers

In a circle, if PT is a tangent and ∠TPO = 35°, what is the value of ∠x?

<p>55°</p> Signup and view all the answers

What shape is formed if the diagonals of a quadrilateral divide each other proportionally?

<p>A parallelogram.</p> Signup and view all the answers

In triangle ABC, with DE parallel to BC and given AD = 2 cm, BD = 3 cm, and BC = 7.5 cm, find the length of DE.

<p>5 cm</p> Signup and view all the answers

If the HCF of 2520 and 6600 is 40, and the LCM is 252k, what is the value of k?

<p>66</p> Signup and view all the answers

What is the probability of randomly choosing an odd prime number from the digits 1 to 9?

<p>1/3</p> Signup and view all the answers

How can you determine the length of the segment connecting two intersection points of chords in a circle?

<p>You can use the power of a point theorem, which states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.</p> Signup and view all the answers

What is the quadratic formula used to find the roots of a quadratic equation?

<p>The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are coefficients of the quadratic equation $ax^2 + bx + c = 0$.</p> Signup and view all the answers

How do you find the median of a data set that is organized in a frequency distribution?

<p>You need to identify the cumulative frequency to locate the median class and then use interpolation if the median falls within that class.</p> Signup and view all the answers

In the context of probability, how would you determine the chances of drawing an Ace from a standard deck of playing cards?

<p>The probability is calculated by dividing the number of favorable outcomes (4 Aces) by the total number of outcomes (52 cards), resulting in $P(Ace) = \frac{4}{52} = \frac{1}{13}$.</p> Signup and view all the answers

What relationship exists between the angles formed by a tangent and a chord at their intersection point on a circle?

<p>The angle formed by the tangent and the chord is equal to the angle in the alternate segment of the circle.</p> Signup and view all the answers

What is the HCF of 480 and 720 using the prime factorization method?

<p>The HCF of 480 and 720 is 240.</p> Signup and view all the answers

What value of m satisfies the equation HCF(85, 238) = 85m - 238?

<p>The value of m is 3.</p> Signup and view all the answers

How would you calculate the perimeter of a sector if the radius is halved and the angle is doubled?

<p>The perimeter remains the same due to the proportional changes in radius and angle.</p> Signup and view all the answers

Find the point(s) on the X-axis at a distance of $ rac{ ext{approximately}}{ ext{sqrt{41}}}$ units from the point (8, -5).

<p>The points are (8 + 6.4, 0) and (8 - 6.4, 0), which are approximately (14.4, 0) and (1.6, 0).</p> Signup and view all the answers

Demonstrate that points A(-5, 6), B(3, 0), and C(9, 8) form an isosceles triangle.

<p>AB = AC = 10 units.</p> Signup and view all the answers

What formula expresses the area of the triangle with vertices at A(-5, 6), B(3, 0), and C(9, 8)?

<p>Area = $0.5 * |(-5)(0-8) + (3)(8-6) + (9)(6-0)|$.</p> Signup and view all the answers

What is the result of evaluating $2 imes ext{sin}(60^ ext{o}) imes ext{tan}^2(30^ ext{o}) / ext{sec}(45^ ext{o})$?

<p>The result is 2.</p> Signup and view all the answers

How can you form a quadratic polynomial whose zeroes are $1/α$ and $1/β$ from the polynomial $6x^2 - 5x + 1$?

<p>The new polynomial is $x^2 - rac{5}{6}x + rac{1}{6}$.</p> Signup and view all the answers

If cosθ + sinθ = 1, what does this imply about the values of cosθ and sinθ?

<p>cosθ = 0 and sinθ = 1.</p> Signup and view all the answers

Calculate the area of the face of a clock described by a minute hand of length 18 cm in 35 minutes.

<p>The area is 660 cm².</p> Signup and view all the answers

Show that √3 is an irrational number.

<p>Assume √3 = p/q leads to a contradiction since p and q must be integers.</p> Signup and view all the answers

Find the speeds of two cars if one is 180 km from the other and they meet in 1 hour when traveling towards each other.

<p>Speeds are 20 km/h and 10 km/h.</p> Signup and view all the answers

What do the lengths of tangents drawn from an external point to a circle being equal imply geometrically?

<p>They show that the tangents are congruent.</p> Signup and view all the answers

If a boy sees a balloon from a height of 1.35 m, how can one find the height of the balloon given the angle of elevation?

<p>By applying trigonometry to the angle of elevation.</p> Signup and view all the answers

For the given frequency distribution of monthly expenditures, how do you compute the mean?

<p>Use the formula: Mean = Σ(fx) / Σf, where f is frequency and x is the mid-point.</p> Signup and view all the answers

Determine if it is feasible to arrange jars in an arithmetic progression if the number pattern is consistent.

<p>Yes, if the common difference is a positive integer.</p> Signup and view all the answers

What can be concluded about the solution set of the linear equations $x + 2y + 5 = 0$ and $-3x + 6y - 1 = 0$?

<p>These equations have infinitely many solutions because they are dependent.</p> Signup and view all the answers

Calculate the common difference of the arithmetic progression given by $ rac{1}{2x}, rac{1 - 4x}{2x}, rac{1 - 8x}{2x},...$.

<p>The common difference is $-2$.</p> Signup and view all the answers

What is the probability that two dice thrown together show different numbers?

<p>The probability is $ rac{5}{6}$.</p> Signup and view all the answers

If the probability of guessing the correct answer to a question is $ rac{1}{3}$, what is the probability of guessing incorrectly?

<p>The probability of guessing incorrectly is $ rac{2}{3}$.</p> Signup and view all the answers

Determine the value of $x$ if the probability of not guessing the correct answer to a question, given as $ rac{1}{3}$, implies certain conditions.

<p>The value of $x$ is 2.</p> Signup and view all the answers

What is the arithmetic progression (AP) representing the number of jars in each layer, and what is the common difference?

<p>The AP is 3, 6, 9, ..., with a common difference of 3.</p> Signup and view all the answers

Is it possible to arrange 34 jars in a layer? Justify your answer.

<p>No, 34 is not achievable as it does not fit the arithmetic progression of layers with a common difference of 3.</p> Signup and view all the answers

Find the expression for the total number of jars in a layer if there are n rows.

<p>The expression for total jars is $3n$, hence $S_n = 3 + 6 + 9 + ... + 3n = \frac{3n(n+1)}{2}$.</p> Signup and view all the answers

How many jars are present in the 5th layer if 3 more jars are added to each layer?

<p>In the 5th layer, there will be 18 jars (15 original + 3 additional).</p> Signup and view all the answers

Given the data on expenditures, what is the range of expenditure for the unfilled bracket?

<p>The expenditure range for the unfilled bracket is $2500 - 3000$.</p> Signup and view all the answers

How can you prove that $ riangle ADPQ ext{ is similar to } riangle ADEF$?

<p>By showing that corresponding angles of the triangles are equal, due to the parallel lines.</p> Signup and view all the answers

Calculate the slant height of the conical part if the radius is 1.5 m and height is 7 m.

<p>The slant height is approximately 7.5 m, calculated using $l = \sqrt{r^2 + h^2} = \sqrt{1.5^2 + 7^2}$.</p> Signup and view all the answers

What is the total capacity of the silo to store grains, combining both parts?

<p>The total capacity is approximately $81.3 m^3$.</p> Signup and view all the answers

What are the coordinates of the point that divides the line segment joining A(2, 3) and B(5, 7) internally in the ratio 1:2?

<p>(3, 5)</p> Signup and view all the answers

Calculate the probability of drawing the queen of hearts from a standard deck of cards.

<p>$\frac{1}{52}$</p> Signup and view all the answers

For the linear equations 2x + y = 13 and 4x - y = 17, what are the values of x and y?

<p>x = 5, y = 3</p> Signup and view all the answers

What relation exists for point P(x, y) to be equidistant from points A(7, 1) and B(3, 5)?

<p>(x - 7)² + (y - 1)² = (x - 3)² + (y - 5)²</p> Signup and view all the answers

Given that the sum of the first 7 terms of an AP is 49, can you find the sum of its first 20 terms?

<p>Sum = 140</p> Signup and view all the answers

What are the zeroes of the polynomial x² - 15?

<p>$\pm \sqrt{15}$</p> Signup and view all the answers

How would you graphically solve the system of equations x - y + 1 = 0 and x + y = 5?

<p>Plot the lines and find their intersection point (2, 3).</p> Signup and view all the answers

If the ratio of the 10th term to the 30th term of an AP is 1:3, what does that tell you about the common difference?

<p>Common difference d = (3a / 20)</p> Signup and view all the answers

What is the ratio in which the line segment joining the points (5, 3) and (-1, 6) is divided by the y-axis?

<p>The ratio is 3:2.</p> Signup and view all the answers

Find the coordinates of point R on line segment PQ where PR = 2QR, given points P(-2, 5) and Q(3, 2).

<p>The coordinates of R are (1, 3).</p> Signup and view all the answers

Prove the identity $ an^{2} heta = an^{2} heta$ using the expression $ an^{2} heta = rac{ ext{sin}^{2} heta}{ ext{cos}^{2} heta}$.

<p>By using the identity $ ext{sin}^{2} heta + ext{cos}^{2} heta = 1$, the proof confirms the equation.</p> Signup and view all the answers

Why do tangents drawn at the endpoints of a chord of a circle make equal angles with that chord?

<p>This occurs due to the properties of tangents and the inscribed angle theorem.</p> Signup and view all the answers

What is the original duration of a flight that experienced a 100 km/h speed reduction resulting in a 30 min increase in flight time?

<p>The original duration of the flight was 6 hours.</p> Signup and view all the answers

Find the fraction if the denominator is one more than twice the numerator and the sum of the fraction and its reciprocal is $ rac{16}{3}$.

<p>The fraction is $ rac{4}{5}$.</p> Signup and view all the answers

Calculate the height of the tower if the angle of elevation to its top is 60° and to its base is 45°, with the building height being 20 m.

<p>The height of the tower is 20 m.</p> Signup and view all the answers

What is the surface area of a medicine capsule which is a cylinder with hemispherical ends, given the total length is 14 mm and diameter is 4 mm?

<p>The surface area is $98 ext{ mm}^{2}$.</p> Signup and view all the answers

What is the value of $a$ in the expression $H = 2 imes 3 imes 5^a imes 7^b$ if $LCM(H, C) = 3780$?

<p>1</p> Signup and view all the answers

List the zeroes of the quadratic polynomial $2x^2 - 3x - 9$.

<p>$ rac{3 ext{ } ext{±} ext{ } ext{sqrt}(93)}{4}$</p> Signup and view all the answers

Calculate the height of a tower if the angle of elevation from a point 30m away is 60°.

<p>$30 ext{ } ext{sqrt}(3)$</p> Signup and view all the answers

If $ an( heta + rac{ ext{PI}}{3})$ gives a certain value based on the angles $ heta$ where $ rac{1}{2}$ and $ rac{ ext{sqrt}(3)}{2}$, compute this value.

<p>$ ext{not defined}$</p> Signup and view all the answers

What is the length of segment DE in triangle ABC if $ rac{AD}{AB} = rac{DE}{BC}$?

<p>2.5</p> Signup and view all the answers

Determine the value of $k$ if $HCF(2520, 6600) = 40$ and $LCM(2520, 6600) = 252k$.

<p>165</p> Signup and view all the answers

Find a pair of irrational numbers that when multiplied yield a rational number.

<p>($ rac{ ext{sqrt}(3)}{3}, rac{ ext{sqrt}(3)}{ ext{sqrt}(27)})$</p> Signup and view all the answers

Calculate the probability of selecting an odd prime number from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.

<p>$ rac{2}{9}$</p> Signup and view all the answers

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