Step 1: Understanding the Concept:

The question requires us to find the Least Common Multiple (LCM) of two numbers, given their Highest Common Factor (HCF). There is a fundamental relationship between the HCF and LCM of any two positive integers.


Step 2: Key Formula or Approach:

For any two positive integers 'a' and 'b', the product of the numbers is equal to the product of their HCF and LCM.

The formula is: \[ HCF(a, b) \times LCM(a, b) = a \times b \]

Step 3: Detailed Explanation:

Here, the two numbers are \(a = 26\) and \(b = 91\).

We are given that HCF(26, 91) = 13.

Using the formula from Step 2: \[ 13 \times LCM(26, 91) = 26 \times 91 \]
To find the LCM, we can rearrange the equation: \[ LCM(26, 91) = \frac{26 \times 91}{13} \]
We can simplify the expression. We know that \(26 = 2 \times 13\). \[ LCM(26, 91) = \frac{(2 \times 13) \times 91}{13} \]
Cancel out the common factor 13 from the numerator and the denominator: \[ LCM(26, 91) = 2 \times 91 \]
Now, we perform the multiplication: \[ 2 \times 91 = 182 \]

Step 4: Final Answer:

Therefore, the LCM of (26, 91) is 182.
Quick Tip: This formula, HCF \(\times\) LCM = Product of numbers, is extremely useful and frequently tested. Always remember it to quickly solve such problems without having to find the prime factorization for the LCM from scratch.

Step 1: Understanding the Concept:

We need to understand two key terms:

1. Trinomial: A polynomial that consists of exactly three terms, separated by plus or minus signs.

2. Degree of a polynomial: The highest exponent (or power) of the variable in the polynomial.


Step 2: Detailed Explanation:

To construct the required polynomial, we need to satisfy both conditions:


It must have three terms. For example, \(T_1 + T_2 + T_3\).
The highest power of the variable in any of these three terms must be 6.

Let's create an example. We can choose the first term to have the variable raised to the power of 6. Let's say \(x^6\).

Now, we need two more terms with powers less than 6. We can pick any powers and any coefficients. For instance, we can choose a term with \(x^3\) and a constant term.

An example could be: \(x^6 + 4x^3 - 7\).

Let's verify this example:

Number of terms: It has three terms: \(x^6\), \(4x^3\), and \(-7\). So, it is a trinomial.
Degree: The powers of the variable \(x\) are 6 and 3. The highest power is 6. So, the degree is 6.

Both conditions are met. Other examples could be \(3y^6 - 10y^2 + 1\), \(z^6 + z^5 + z^4\), etc.


Step 3: Final Answer:

A valid example of a trinomial with degree 6 is \(x^6 + 4x^3 - 7\).
Quick Tip: Remember the definitions: a \textbf{monomial} has one term, a \textbf{binomial} has two terms, and a \textbf{trinomial} has three terms. The degree is simply the highest power of the variable. You can create infinite examples by changing the coefficients and the lower-degree terms.

Step 1: Understanding the Concept:

The question asks for the possible range of values for the probability of any event E, denoted as P(E).

Probability is a measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1.


Step 2: Detailed Explanation:

The probability of an event is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes. \[ P(E) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]
Since the number of favorable outcomes can never be negative, the minimum value is 0 (for an impossible event). This means \(P(E) \ge 0\).

The number of favorable outcomes can never be more than the total number of possible outcomes. Therefore, the maximum value of the ratio is 1 (for a certain event). This means \(P(E) \le 1\).

Combining these two conditions, the probability of any event E must lie in the range from 0 to 1, inclusive.

This is written as \(0 \le P(E) \le 1\).

Let's analyze the given options:

(A) \(P(E) \ge 1\): This is incorrect. Probability cannot be greater than 1.

(B) \(P(E) \le 0\): This is incorrect. Probability cannot be negative.

(C) \(0 \le P(E) \le 1\): This is the correct range for any probability value.

(D) This is identical to option (A) and is incorrect.


Step 3: Final Answer:

The correct statement is that the probability of an event E is always greater than or equal to 0 and less than or equal to 1. Thus, option (C) is the correct answer.
Quick Tip: A simple way to remember this is to think about percentages. Probability can be expressed as a percentage from 0% (impossible) to 100% (certain). In decimal form, this corresponds to the range [0, 1]. Any answer outside this range in a probability question is incorrect.

Step 1: Understanding the Concept:

The question asks for the definition of a tangent in the context of a circle. A tangent is a specific type of line related to a curve.


Step 2: Detailed Explanation:

In geometry, a tangent to a circle is a line that lies in the same plane as the circle and intersects the circle at precisely one point.

Key properties of a tangent are:

It touches the circle at only a single point. This point is called the point of tangency or point of contact.
The tangent line is always perpendicular to the radius of the circle at the point of tangency. This means the angle between the tangent and the radius at the point of contact is \(90^\circ\).
A tangent line does not pass through the interior of the circle.


Step 3: Final Answer:

A tangent to a circle is a line that intersects the circle at exactly one point.
Quick Tip: Visualize a bicycle wheel rolling on a flat road. The road acts as a tangent to the wheel at any given moment, touching it at just one point. This real-world analogy can help you remember the definition.

Step 1: Understanding the Concept:

This is a problem based on trigonometry, specifically involving the angle of elevation. The angle of elevation is the angle formed between the horizontal line of sight and the upward line of sight to an object above the horizontal.


Step 2: Key Formula or Approach:

We can model this situation using a right-angled triangle. We will use the tangent trigonometric ratio: \[ \tan(\theta) = \frac{Opposite Side}{Adjacent Side} \]

Step 3: Detailed Explanation:

Let's visualize the scenario.

Let AB be the tower of height 30 m.
Let CD be the observer of height 1.5 m.
The distance between the observer and the tower is BD = 28.5 m.
The angle of elevation is measured from the observer's eyes (point C). We need to find the angle \(\theta\).

Draw a horizontal line CE from the observer's eyes to the tower. This forms a right-angled triangle ACE.


The adjacent side of the triangle is CE. Since CE is parallel to BD, its length is the same: CE = BD = 28.5 m.
The opposite side of the triangle is AE. This is the height of the tower above the observer's eye level.
Total height of the tower AB = 30 m.
Height of the observer CD = 1.5 m. So, EB = 1.5 m.
The height AE = AB - EB = 30 m - 1.5 m = 28.5 m.

Now, we can use the tangent ratio in the right-angled triangle ACE: \[ \tan(\theta) = \frac{Opposite (AE)}{Adjacent (CE)} \]
Substitute the values we found: \[ \tan(\theta) = \frac{28.5}{28.5} = 1 \]
To find the angle \(\theta\), we need to determine which angle has a tangent of 1. \[ \theta = \arctan(1) \quad or \quad \theta = \tan^{-1}(1) \]
We know from standard trigonometric values that \(\tan(45^\circ) = 1\).

So, \(\theta = 45^\circ\).


Step 4: Final Answer:

The angle of elevation of the top of the tower from her eyes is \(45^\circ\).
Quick Tip: Always draw a diagram for height and distance problems. It helps in visualizing the right-angled triangle and correctly identifying the opposite, adjacent, and hypotenuse sides with respect to the required angle. Remember to subtract the observer's height from the total height of the object.

Step 1: Understanding the Concept:

This question tests the knowledge of standard trigonometric values and identities. We need to evaluate each expression on the left and match it with the correct value on the right.


Step 2: Key Formula or Approach:

We will use the following:

Standard values: \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\).
Reciprocal identity: \(\sec\theta = \frac{1}{\cos\theta}\).
Pythagorean identity: \(\csc^2\theta - \cot^2\theta = 1\), which can be factored as \((\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1\).


Step 3: Detailed Explanation:

Let's evaluate each item one by one.

i. \( \sin 90^\circ \times \cos 90^\circ \)

We know that \(\sin 90^\circ = 1\) and \(\cos 90^\circ = 0\). \[ \sin 90^\circ \times \cos 90^\circ = 1 \times 0 = 0 \]
So, i matches with q.


ii. \( \cos\theta \times \sec\theta \)

Using the reciprocal identity, \(\sec\theta = \frac{1}{\cos\theta}\). \[ \cos\theta \times \sec\theta = \cos\theta \times \frac{1}{\cos\theta} = 1 \]
So, ii matches with r.


iii. If \( \csc\theta + \cot\theta = \frac{1}{2} \), then find \( \csc\theta - \cot\theta \)

We use the Pythagorean identity: \[ \csc^2\theta - \cot^2\theta = 1 \]
This is a difference of squares, \(a^2 - b^2 = (a-b)(a+b)\). \[ (\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1 \]
We are given that \( \csc\theta + \cot\theta = \frac{1}{2} \). Substituting this into the equation: \[ (\csc\theta - \cot\theta) \left( \frac{1}{2} \right) = 1 \]
To find \( \csc\theta - \cot\theta \), we multiply both sides by 2: \[ \csc\theta - \cot\theta = 1 \times 2 = 2 \]
So, iii matches with p.


Step 4: Final Answer:

The correct matching is:

i \(\rightarrow\) q
ii \(\rightarrow\) r
iii \(\rightarrow\) p

This corresponds to option (D).
Quick Tip: The identity \(\csc^2\theta - \cot^2\theta = 1\) is very powerful. If you are given the value of \(\csc\theta + \cot\theta\), you can immediately find \(\csc\theta - \cot\theta\) because they are reciprocals of each other. The same applies to the identity \(\sec^2\theta - \tan^2\theta = 1\).

Step 1: Understanding the Concept:

The question asks to draw a 3D composite shape made of two basic geometric solids: a cone and a hemisphere.


Step 2: Detailed Explanation:

The description specifies the arrangement of these shapes:

Hemisphere: This is half of a sphere. It has a flat circular base and a curved surface. We should draw this as the bottom part of the toy.
Cone: This has a circular base and a vertex.
Mounted on: This means the cone is placed on top of the hemisphere.
Same radius: This is a crucial detail. The radius of the cone's circular base must be the same as the radius of the hemisphere's circular base. This ensures they fit together perfectly.

Steps to draw the figure:

Draw a semi-circle for the curved part of the hemisphere.
Draw an ellipse (representing the circular base in perspective) to close the semi-circle at the top. This completes the hemisphere.
The base of the cone is this same ellipse.
From the center of the ellipse, draw a vertical line upwards to mark the height of the cone. The endpoint is the vertex of the cone.
Draw two straight lines from the vertex to the outer edges of the ellipse. This forms the slant surface of the cone.

The resulting shape looks like a spinning top or an ice cream cone.


Step 3: Final Answer:

The figure would be a composite solid with a hemispherical base and a conical top, joined at their circular bases of the same radius. (A textual description is provided as drawing is not possible). Quick Tip: When sketching 3D shapes, use dotted lines to represent hidden edges. For this figure, if you were to draw the radius, the line segment from the center of the circular base to its edge would be the common radius for both the cone and the hemisphere.

Step 1: Understanding the Concept:

The question tests the concept of similarity in geometric figures. Two figures are similar if they have the same shape, but not necessarily the same size. This means one can be obtained from the other by uniform scaling (enlarging or shrinking), possibly with additional translation, rotation and reflection.


Step 2: Detailed Explanation:

Analysis of Statement I: Any two circles are similar.

All circles have the same shape. The only thing that distinguishes one circle from another is its radius. If you have a circle of radius \(r_1\) and another of radius \(r_2\), you can always make them congruent by scaling the first circle by a factor of \(k = r_2 / r_1\). Since any circle can be transformed into any other circle by scaling, all circles are similar to each other.

Therefore, Statement I is true.


Analysis of Statement II: Any two equilateral triangles are similar.

An equilateral triangle is a triangle in which all three sides have the same length, and all three internal angles are each \(60^\circ\).

Consider any two equilateral triangles, \(T_1\) and \(T_2\).

The angles of \(T_1\) are \(60^\circ, 60^\circ, 60^\circ\).
The angles of \(T_2\) are \(60^\circ, 60^\circ, 60^\circ\).

According to the Angle-Angle (AA) similarity criterion (or AAA), if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Since all corresponding angles of any two equilateral triangles are equal (all are \(60^\circ\)), any two equilateral triangles are similar.

Therefore, Statement II is true.


Step 3: Final Answer:

Since both Statement I and Statement II are true, the correct option is (C).
Quick Tip: For any type of regular polygon (like equilateral triangles, squares, regular pentagons, etc.), all figures of that type are similar to each other. This is because all their corresponding angles are equal by definition, and the ratio of corresponding sides is constant.

Step 1: Understanding the Concept:

This question involves analyzing a pair of linear equations in two variables. We need to determine the nature of their solution (unique, no solution, or infinitely many) based on the ratios of their coefficients.


Step 2: Key Formula or Approach:

For a pair of linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), the conditions for the nature of their solutions are:

Unique solution (intersecting lines): \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \)
No solution (parallel lines): \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \)
Infinitely many solutions (coincident lines): \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)


Step 3: Detailed Explanation:

Verify the Assertion:

The given equations are:

\(2x + 3y + 6 = 0\)
\(4x + 6y + 7 = 0\)

Here, \(a_1 = 2, b_1 = 3, c_1 = 6\) and \(a_2 = 4, b_2 = 6, c_2 = 7\).

Let's compute the ratios of the coefficients: \[ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \] \[ \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \] \[ \frac{c_1}{c_2} = \frac{6}{7} \]
Comparing the ratios, we find that: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \quad \left(since \frac{1}{2} \ne \frac{6}{7}\right) \]
This is the condition for parallel lines, which never intersect. Therefore, the pair of equations has no solution.

So, the Assertion is true.


Verify the Reason:

The Reason states that if \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), the lines are parallel and have no solution. This is the correct mathematical condition for a system of linear equations to have no solution.

So, the Reason is true.


Check if the Reason supports the Assertion:

Our verification of the Assertion used the exact condition mentioned in the Reason. We calculated the ratios for the given equations and found they satisfy the condition \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), which led to the conclusion that there is no solution. Thus, the Reason is the correct explanation for the Assertion.


Step 4: Final Answer:

Both Assertion and Reason are true, and the Reason is the correct explanation for the Assertion. Therefore, option (A) is the correct choice.
Quick Tip: Remember the geometric interpretation: \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \) \(\Rightarrow\) Lines intersect at one point. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \) \(\Rightarrow\) Lines are parallel and never meet. \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) \(\Rightarrow\) Lines are coincident (the same line).

Step 1: Understanding the Concept:

The question asks to form a quadratic polynomial given the sum and product of its zeroes (roots).


Step 2: Key Formula or Approach:

A quadratic polynomial with zeroes \(\alpha\) and \(\beta\) can be written in the form: \[ P(x) = k \left( x^2 - (sum of zeroes)x + (product of zeroes) \right) \] \[ P(x) = k \left( x^2 - (\alpha + \beta)x + (\alpha\beta) \right) \]
where \(k\) is any non-zero constant. Usually, we take \(k=1\) unless specified otherwise.


Step 3: Detailed Explanation:

We are given the following information:

Sum of zeroes (\(\alpha + \beta\)) = -3
Product of zeroes (\(\alpha\beta\)) = 2

Using the formula from Step 2 with \(k=1\): \[ P(x) = x^2 - (sum of zeroes)x + (product of zeroes) \]
Substitute the given values into the formula: \[ P(x) = x^2 - (-3)x + (2) \]
Simplify the expression: \[ P(x) = x^2 + 3x + 2 \]
This matches option (A).


Step 4: Final Answer:

The required quadratic polynomial is \(x^2 + 3x + 2\). Therefore, option (A) is correct.
Quick Tip: Pay close attention to the signs in the formula \(x^2 - (sum)x + (product)\). A common mistake is to forget the minus sign before the sum of the zeroes.

Step 1: Understanding the Concept:

For any quadratic equation, there is a relationship between its coefficients and its roots (zeroes). We can use this relationship to find the second root if one root is known.


Step 2: Key Formula or Approach:

For a quadratic equation of the form \(ax^2 + bx + c = 0\), if the roots are \(\alpha\) and \(\beta\), then:

Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
Product of roots: \(\alpha\beta = \frac{c}{a}\)

We can use either of these relationships.


Step 3: Detailed Explanation:

The given quadratic equation is \(x^2 - 7x + 12 = 0\).

Comparing this to the standard form \(ax^2 + bx + c = 0\), we have: \(a = 1\), \(b = -7\), \(c = 12\).

Let the roots be \(\alpha\) and \(\beta\).

We are given that one root is 4. Let's say \(\alpha = 4\). We need to find the other root, \(\beta\).


Method 1: Using the sum of roots
\[ \alpha + \beta = -\frac{b}{a} \] \[ 4 + \beta = -\frac{-7}{1} \] \[ 4 + \beta = 7 \] \[ \beta = 7 - 4 \] \[ \beta = 3 \]

Method 2: Using the product of roots
\[ \alpha\beta = \frac{c}{a} \] \[ 4 \times \beta = \frac{12}{1} \] \[ 4\beta = 12 \] \[ \beta = \frac{12}{4} \] \[ \beta = 3 \]
Both methods give the same result.


Step 4: Final Answer:

The other root of the quadratic equation is 3.
Quick Tip: Alternatively, you can solve the quadratic equation completely by factoring. For \(x^2 - 7x + 12 = 0\), we need two numbers that multiply to 12 and add up to -7. These are -3 and -4. So, \((x-3)(x-4)=0\). The roots are \(x=3\) and \(x=4\). Since one root is 4, the other must be 3.

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (\(d\)).


Step 2: Key Formula or Approach:

The common difference can be found by subtracting any term from its succeeding term: \(d = a_{n} - a_{n-1}\). We can find the first few terms of the sequence using the given formula for the nth term and then calculate the difference.


Step 3: Detailed Explanation:

We are given the formula for the nth term of the A.P.: \[ a_n = 6n \]
We are also given that the first term is 6. Let's check if the formula holds for n=1:
For \(n=1\), \(a_1 = 6(1) = 6\). This matches the given first term, so the formula is correct for the entire sequence.


To find the common difference, let's find the first two terms of the sequence.

First term (\(a_1\)): \[ a_1 = 6(1) = 6 \]
Second term (\(a_2\)): \[ a_2 = 6(2) = 12 \]
The common difference \(d\) is the difference between the second term and the first term: \[ d = a_2 - a_1 \] \[ d = 12 - 6 = 6 \]
We can verify this by finding the third term: \[ a_3 = 6(3) = 18 \] \[ d = a_3 - a_2 = 18 - 12 = 6 \]
The difference is constant.


Step 4: Final Answer:

The common difference in the A.P. is 6. Therefore, option (C) is correct.
Quick Tip: For any A.P. whose nth term is given by a linear expression like \(a_n = An + B\), the common difference is always the coefficient of \(n\), which is \(A\). In this case, \(a_n = 6n\), so the common difference is 6. This is a very fast shortcut.

Step 1: Understanding the Concept:

The question requires us to simplify a trigonometric expression involving cosecant and cotangent, and express the final result using only the cosine function. This involves using fundamental trigonometric identities.


Step 2: Key Formula or Approach:

We will use the following reciprocal and quotient identities: \[ \csc\theta = \frac{1}{\sin\theta} \] \[ \cot\theta = \frac{\cos\theta}{\sin\theta} \]
And the Pythagorean identity: \[ \sin^2\theta + \cos^2\theta = 1 \implies \sin^2\theta = 1 - \cos^2\theta \]

Step 3: Detailed Explanation:

Start with the given expression: \[ (\csc\theta - \cot\theta)^2 \]
Substitute the identities for \(\csc\theta\) and \(\cot\theta\): \[ \left(\frac{1}{\sin\theta} - \frac{\cos\theta}{\sin\theta}\right)^2 \]
Combine the terms inside the parenthesis since they have a common denominator: \[ \left(\frac{1 - \cos\theta}{\sin\theta}\right)^2 \]
Apply the square to both the numerator and the denominator: \[ \frac{(1 - \cos\theta)^2}{\sin^2\theta} \]
Now, use the Pythagorean identity \(\sin^2\theta = 1 - \cos^2\theta\) to replace the denominator: \[ \frac{(1 - \cos\theta)^2}{1 - \cos^2\theta} \]
The denominator is a difference of squares, \(1 - \cos^2\theta = (1 - \cos\theta)(1 + \cos\theta)\). Substitute this into the expression: \[ \frac{(1 - \cos\theta)(1 - \cos\theta)}{(1 - \cos\theta)(1 + \cos\theta)} \]
Cancel the common factor \((1 - \cos\theta)\) from the numerator and denominator: \[ \frac{1 - \cos\theta}{1 + \cos\theta} \]

Step 4: Final Answer:

The expression \((\csc\theta - \cot\theta)^2\) in terms of \(\cos\theta\) is \(\frac{1-\cos\theta}{1+\cos\theta}\).
Quick Tip: When asked to express a trigonometric expression in terms of a specific function (like sin or cos), the first step is almost always to convert all other functions into sin and cos using their fundamental identities.

Step 1: Understanding the Concept:

This problem requires finding the distance between two points in a Cartesian coordinate system. The coordinates of the points are given in terms of trigonometric functions.


Step 2: Key Formula or Approach:

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We will also use the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\).


Step 3: Detailed Explanation:

Let the two points be \(P_1 = (a \cos\theta, 0)\) and \(P_2 = (0, a \sin\theta)\).
Here, \(x_1 = a \cos\theta\), \(y_1 = 0\), \(x_2 = 0\), and \(y_2 = a \sin\theta\).
Substitute these values into the distance formula: \[ d = \sqrt{(0 - a \cos\theta)^2 + (a \sin\theta - 0)^2} \]
Simplify the terms inside the parentheses: \[ d = \sqrt{(-a \cos\theta)^2 + (a \sin\theta)^2} \]
Square the terms: \[ d = \sqrt{a^2 \cos^2\theta + a^2 \sin^2\theta} \]
Factor out the common term \(a^2\): \[ d = \sqrt{a^2 (\cos^2\theta + \sin^2\theta)} \]
Apply the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\): \[ d = \sqrt{a^2(1)} = \sqrt{a^2} \]
Assuming \(a\) represents a distance or positive value, the distance is: \[ d = a \]

Step 4: Final Answer:

The distance between the two given points is \(a\).
Quick Tip: Whenever you see \(\sin^2\) and \(\cos^2\) terms added together in a distance or magnitude calculation, immediately think of the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\). This is a very common simplification step.

Step 1: Understanding the Concept:

The "zeroes" of a polynomial are the values of the variable for which the polynomial evaluates to zero. For a quadratic polynomial, these are also known as the roots of the corresponding quadratic equation.


Step 2: Key Formula or Approach:

To find the zeroes, we set the polynomial equal to zero and solve the resulting equation: \(x^2 + 7x + 10 = 0\). We can solve this by factoring the quadratic expression. This involves finding two numbers that multiply to the constant term (10) and add up to the coefficient of the x-term (7).


Step 3: Detailed Explanation:

The quadratic equation is: \[ x^2 + 7x + 10 = 0 \]
We need to find two numbers that have a product of 10 and a sum of 7. Let's list the factors of 10: (1, 10) and (2, 5). The pair (2, 5) adds up to 7.

We use these numbers to split the middle term, \(7x\), into \(2x + 5x\): \[ x^2 + 2x + 5x + 10 = 0 \]
Now, we factor by grouping. Factor out the greatest common factor from the first two terms and the last two terms: \[ x(x + 2) + 5(x + 2) = 0 \]
The term \((x + 2)\) is a common factor, so we can factor it out: \[ (x + 2)(x + 5) = 0 \]
For the product of two factors to be zero, at least one of the factors must be zero. So we set each factor to zero: \[ x + 2 = 0 \implies x = -2 \] \[ x + 5 = 0 \implies x = -5 \]

Step 4: Final Answer:

The zeroes of the quadratic polynomial \(x^2 + 7x + 10\) are -2 and -5.
Quick Tip: Factoring by splitting the middle term is a quick method for simple quadratics. Always check the signs. If the constant term is positive and the middle term is positive, both numbers you are looking for will be positive.

Step 1: Understanding the Concept:

This problem involves visualizing the 3D shape formed by joining two identical cubes and then calculating its total surface area. First, we need to find the dimensions of the cube from its volume.


Step 2: Key Formula or Approach:

1. Volume of a cube with side 's' is \(V = s^3\).

2. Total Surface Area (TSA) of a cuboid with length 'l', breadth 'b', and height 'h' is \(TSA = 2(lb + bh + hl)\).


Step 3: Detailed Explanation:

Find the side of the cube:

Given the volume of one cube is 64 cm\(^3\). \[ s^3 = 64 \] \[ s = \sqrt[3]{64} = 4 cm \]
So, each cube has a side length of 4 cm.


Determine the dimensions of the resulting cuboid:

When two such cubes are joined end to end, they form a cuboid.

The length of the cuboid will be the sum of the side lengths of the two cubes: \(l = 4 + 4 = 8\) cm.
The breadth of the cuboid will remain the same as the side of the cube: \(b = 4\) cm.
The height of the cuboid will also remain the same as the side of the cube: \(h = 4\) cm.

So, the dimensions of the resulting cuboid are 8 cm \(\times\) 4 cm \(\times\) 4 cm.


Calculate the total surface area of the cuboid:

Using the TSA formula: \[ TSA = 2(lb + bh + hl) \]
Substitute the dimensions: \[ TSA = 2((8)(4) + (4)(4) + (4)(8)) \] \[ TSA = 2(32 + 16 + 32) \] \[ TSA = 2(80) \] \[ TSA = 160 cm^2 \]

Step 4: Final Answer:

The total surface area of the resulting cuboid is 160 cm\(^2\).
Quick Tip: When cubes are joined, only the length changes (if joined along the length). The breadth and height remain the same. Visualize the joining process to correctly identify the new dimensions.

Step 1: Understanding the Concept:

A quadratic equation is a polynomial equation of the second degree. The standard form is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and, crucially, \(a \ne 0\). To check if the given equation is quadratic, we must simplify it and see if it can be written in this standard form.


Step 2: Key Formula or Approach:

The approach is to expand both sides of the equation, move all terms to one side, and simplify to check if the highest power of \(x\) is 2.


Step 3: Detailed Explanation:

The given equation is: \[ (2x - 1)(x - 3) = (x + 5)(x - 1) \]
Expand the Left Hand Side (LHS): \[ (2x - 1)(x - 3) = 2x(x - 3) - 1(x - 3) = 2x^2 - 6x - x + 3 = 2x^2 - 7x + 3 \]
Expand the Right Hand Side (RHS): \[ (x + 5)(x - 1) = x(x - 1) + 5(x - 1) = x^2 - x + 5x - 5 = x^2 + 4x - 5 \]
Now, set the expanded sides equal to each other: \[ 2x^2 - 7x + 3 = x^2 + 4x - 5 \]
Move all terms from the RHS to the LHS to set the equation to zero: \[ (2x^2 - x^2) + (-7x - 4x) + (3 + 5) = 0 \]
Combine like terms: \[ x^2 - 11x + 8 = 0 \]
This equation is in the standard form \(ax^2 + bx + c = 0\), with \(a = 1\), \(b = -11\), and \(c = 8\).
Since the coefficient of the \(x^2\) term (\(a\)) is 1, which is not zero, the equation is a quadratic equation.


Step 4: Final Answer:

Yes, the given equation simplifies to \(x^2 - 11x + 8 = 0\), which is a quadratic equation.
Quick Tip: Don't assume an equation is quadratic just because it contains \(x^2\) terms on both sides. You must simplify fully. Sometimes, the \(x^2\) terms might cancel out, leaving a linear equation.

Step 1: Understanding the Concept:

This problem uses a key property of tangents to a circle: the tangent at any point on a circle is perpendicular to the radius through the point of contact. This property allows us to form a right-angled triangle and use the Pythagoras theorem.


Step 2: Key Formula or Approach:

1. The radius (OP) is perpendicular to the tangent (PQ) at the point of contact (P). Thus, \(\angle OPQ = 90^\circ\).
2. For the right-angled triangle \(\triangle OPQ\), the Pythagoras theorem states: \(OP^2 + PQ^2 = OQ^2\).


Step 3: Detailed Explanation:

Let's identify the parts of the right-angled triangle \(\triangle OPQ\):

OP is the radius of the circle, given as 5 cm. This is one of the legs of the triangle.
PQ is the length of the tangent, which we need to find. This is the other leg.
OQ is the distance from the center to the external point Q, given as 12 cm. Since it is opposite the right angle, it is the hypotenuse.

Apply the Pythagoras theorem: \[ OP^2 + PQ^2 = OQ^2 \]
Substitute the known values: \[ 5^2 + PQ^2 = 12^2 \] \[ 25 + PQ^2 = 144 \]
Isolate \(PQ^2\) by subtracting 25 from both sides: \[ PQ^2 = 144 - 25 \] \[ PQ^2 = 119 \]
Take the square root of both sides to find the length of PQ: \[ PQ = \sqrt{119} cm \]

Step 4: Final Answer:

The length of the tangent PQ is \(\sqrt{119}\) cm.
Quick Tip: Always draw a diagram for circle geometry problems. It helps you visualize the right-angled triangle formed by the radius, the tangent, and the line from the center to the external point.

Step 1: Understanding the Concept:

The question asks for the definition of "similar triangles". Similarity in geometry refers to figures having the same shape but not necessarily the same size.


Step 2: Detailed Explanation:

Two triangles, say \(\triangle ABC\) and \(\triangle DEF\), are defined as being similar if they satisfy two fundamental conditions:

Corresponding angles are equal. This means:

\(\angle A = \angle D\), \(\angle B = \angle E\), and \(\angle C = \angle F\).

Corresponding sides are in the same ratio (proportional). This means:

\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \]
where \(k\) is the constant of proportionality or scale factor.

It's important to note that for triangles, if one of these conditions is met, the other is automatically satisfied. For example, if all corresponding angles are equal (AAA similarity), then the sides must be proportional.


Step 3: Final Answer:

Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio.
Quick Tip: Remember the main similarity criteria for triangles: AAA (Angle-Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side). If you can prove any one of these, you have proved the triangles are similar.

Step 1: Understanding the Concept:

The question requires translating a word problem involving an angle of depression into a geometric diagram. The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object.


Step 2: Detailed Explanation:

Here is a step-by-step description of how to draw the diagram:

Draw a vertical line segment and label it AB. This represents the tower. Let A be the top of the tower and B be the base on the ground. Label the height as 20 m.
From point B, draw a horizontal line segment BC. This represents the ground. Let C be the position of the ball.
At the top of the tower (point A), draw a horizontal dashed line, say AD, parallel to the ground (BC). This represents the player's horizontal line of sight.
Draw the line of sight from the player at A to the ball at C. This is the line segment AC.
The angle of depression is the angle formed between the horizontal line AD and the line of sight AC. Label \(\angle DAC\) as \(60^\circ\).
The triangle formed by the tower, the ground, and the line of sight is \(\triangle ABC\). This is a right-angled triangle with \(\angle B = 90^\circ\).
Because the horizontal line AD is parallel to the ground BC, the angle of elevation from the ball to the top of the tower (\(\angle ACB\)) is equal to the angle of depression (\(\angle DAC\)). This is due to the alternate interior angles property. So, you can also label \(\angle ACB = 60^\circ\).


Step 3: Final Answer:

The final diagram is a right-angled triangle ABC, where AB is the vertical tower (20m), BC is the horizontal ground, and AC is the hypotenuse. A horizontal line from A is used to show the angle of depression of 60° to the point C.
Quick Tip: A common mistake is to place the angle of depression inside the triangle at the top vertex (i.e., labeling \(\angle BAC\)). Remember, the angle of depression is always measured from the horizontal line downwards.

Step 1: Understanding the Concept:

Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, we are rolling two six-sided dice.


Step 2: Key Formula or Approach:
\[ P(Event) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

Total Possible Outcomes:

Each die has 6 faces (1, 2, 3, 4, 5, 6). When two dice are thrown, the total number of possible outcomes is \(6 \times 6 = 36\).


Favorable Outcomes for a sum of 6:

We need to find all pairs of numbers (one from each die) that add up to 6. Let the outcome be (blue die, grey die).
The pairs are:

(1, 5)
(2, 4)
(3, 3)
(4, 2)
(5, 1)

There are 5 favorable outcomes.


Calculate the Probability:
\[ P(sum is 6) = \frac{5}{36} \]

Step 4: Final Answer:

The probability that the sum of the two numbers is 6 is \(\frac{5}{36}\).
Quick Tip: For two-dice problems, it can be helpful to visualize or quickly sketch a 6x6 grid to see all 36 possible outcomes. This makes it easier to count the favorable outcomes for any given sum.

Step 1: Understanding the Concept:

We need to find the probability of getting a specific sum (12) when rolling two six-sided dice.


Step 2: Key Formula or Approach:
\[ P(Event) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

Total Possible Outcomes:

As before, the total number of outcomes when rolling two dice is \(6 \times 6 = 36\).


Favorable Outcomes for a sum of 12:

We need to find pairs of numbers that add up to 12. The maximum number on a die is 6.
The only possible pair is:

(6, 6)

There is only 1 favorable outcome.


Calculate the Probability:
\[ P(sum is 12) = \frac{1}{36} \]

Step 4: Final Answer:

The probability that the sum of the two numbers is 12 is \(\frac{1}{36}\).
Quick Tip: The sums of 2 and 12 are the least likely outcomes when rolling two dice, as each can only be formed in one way ((1,1) and (6,6) respectively). The sum of 7 is the most likely outcome.

Step 1: Understanding the Concept:

This part asks for the probability of obtaining a sum of 9 from the roll of two dice.


Step 2: Key Formula or Approach:
\[ P(Event) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

Total Possible Outcomes:

The total number of outcomes remains \(6 \times 6 = 36\).


Favorable Outcomes for a sum of 9:

We list the pairs of numbers (blue die, grey die) that sum to 9:

(3, 6)
(4, 5)
(5, 4)
(6, 3)

There are 4 favorable outcomes.


Calculate the Probability:
\[ P(sum is 9) = \frac{4}{36} \]
Simplifying the fraction gives: \[ \frac{4}{36} = \frac{1}{9} \]

Step 4: Final Answer:

The probability that the sum of the two numbers is 9 is \(\frac{1}{9}\).
Quick Tip: Always simplify your probability fractions to their lowest terms unless the question specifies otherwise.

Step 1: Understanding the Concept:

This part asks for the probability of an impossible event. We need to determine if a sum of 13 is possible when rolling two standard six-sided dice.


Step 2: Key Formula or Approach:
\[ P(Event) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

Total Possible Outcomes:

The total number of outcomes is 36.


Favorable Outcomes for a sum of 13:

The maximum score on a single die is 6. The maximum possible sum when rolling two dice is \(6 + 6 = 12\).
It is impossible to get a sum of 13.
Therefore, the number of favorable outcomes is 0.


Calculate the Probability:
\[ P(sum is 13) = \frac{0}{36} = 0 \]

Step 4: Final Answer:

The probability that the sum of the two numbers is 13 is 0.
Quick Tip: The probability of an impossible event is always 0. The probability of a certain event is always 1. All other probabilities lie between 0 and 1.

Step 1: Understanding the Concept:

The median is the middle value in a dataset. For grouped data (data presented in class intervals), we cannot find the exact middle value, so we use a formula to estimate the median.


Step 2: Key Formula or Approach:

The formula for the median of grouped data is: \[ Median = l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h \]

Step 3: Detailed Explanation of Terms:

To use this formula, we first need to identify the median class, which is the class interval where the \((n/2)\)-th observation falls.

The terms in the formula are defined as follows:

l: This is the lower limit of the median class.
n: This is the total number of observations, which is the sum of all frequencies (\(n = \sum f_i\)).
cf: This is the cumulative frequency of the class preceding the median class.
f: This is the frequency of the median class.
h: This is the class size or class width (assuming all class intervals have the same size). It is calculated as (Upper limit - Lower limit) of any class.


Step 4: Final Answer:

The formula to find the median of grouped data is Median \(= l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h\), where \(l\) is the lower limit of the median class, \(n\) is the total frequency, \(cf\) is the cumulative frequency of the preceding class, \(f\) is the frequency of the median class, and \(h\) is the class size.
Quick Tip: The most common error in median calculation is using the cumulative frequency of the median class itself instead of the class preceding it. Always be careful to use the correct 'cf' value.

Step 1: Understanding the Concept:

The volume of the remaining wood in the pen stand is the initial volume of the wooden cuboid minus the volume of the wood removed to create the four conical depressions.


Step 2: Key Formula or Approach:

1. Volume of a cuboid = length \(\times\) breadth \(\times\) height (\(V_{cuboid} = l \times b \times h\)).
2. Volume of a cone = \(\frac{1}{3}\pi r^2 h_{cone}\), where \(r\) is the radius and \(h_{cone}\) is the height/depth.
3. Volume of wood = \(V_{cuboid} - 4 \times V_{cone}\).


Step 3: Detailed Explanation:

Calculate the volume of the cuboid:

Dimensions are \(l = 15\) cm, \(b = 10\) cm, \(h = 3.5\) cm. \[ V_{cuboid} = 15 \times 10 \times 3.5 = 150 \times 3.5 = 525 cm^3 \]

Calculate the volume of one conical depression:

Dimensions are radius \(r = 0.5\) cm and depth \(h_{cone} = 1.4\) cm.
Using \(\pi = \frac{22}{7}\): \[ V_{cone} = \frac{1}{3} \pi r^2 h_{cone} = \frac{1}{3} \times \frac{22}{7} \times (0.5)^2 \times 1.4 \] \[ V_{cone} = \frac{1}{3} \times \frac{22}{7} \times 0.25 \times 1.4 \]
Notice that \(7 \times 0.2 = 1.4\), so we can simplify: \[ V_{cone} = \frac{1}{3} \times 22 \times 0.25 \times 0.2 = \frac{1}{3} \times 22 \times 0.05 = \frac{1.1}{3} cm^3 \]

Calculate the total volume of wood removed:

There are four conical depressions, so the total volume removed is: \[ V_{removed} = 4 \times V_{cone} = 4 \times \frac{1.1}{3} = \frac{4.4}{3} cm^3 \]

Calculate the volume of wood in the stand:
\[ V_{wood} = V_{cuboid} - V_{removed} = 525 - \frac{4.4}{3} \] \[ \frac{4.4}{3} \approx 1.467 cm^3 \] \[ V_{wood} \approx 525 - 1.467 = 523.533 cm^3 \]

Step 4: Final Answer:

The volume of wood in the entire stand is approximately 523.53 cm\(^3\).
Quick Tip: In problems involving combined solids where material is removed, the logic is always (Volume of original shape) - (Volume of removed shape). Read the question carefully to identify the shapes and whether they are added or removed.

Step 1: Understanding the Concept:

The nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is determined by its discriminant, \(D = b^2 - 4ac\). For the equation to have two equal real roots, the discriminant must be equal to zero (\(D = 0\)).


Step 2: Key Formula or Approach:

1. Convert the given equation into the standard quadratic form \(ax^2 + bx + c = 0\).
2. Identify the coefficients \(a\), \(b\), and \(c\).
3. Set the discriminant \(D = b^2 - 4ac\) to zero and solve for \(k\).


Step 3: Detailed Explanation:

Convert to standard form:

The given equation is \(kx(x - 2) + 6 = 0\).
Distribute \(kx\) into the parenthesis: \[ kx^2 - 2kx + 6 = 0 \]
This is now in the standard form.


Identify coefficients:

Comparing with \(ax^2 + bx + c = 0\), we have:

\(a = k\)
\(b = -2k\)
\(c = 6\)


Set the discriminant to zero:

The condition for two equal real roots is \(D = 0\). \[ b^2 - 4ac = 0 \]
Substitute the coefficients: \[ (-2k)^2 - 4(k)(6) = 0 \] \[ 4k^2 - 24k = 0 \]
Now, solve this equation for \(k\). Factor out the common term \(4k\): \[ 4k(k - 6) = 0 \]
This gives two possible solutions: \[ 4k = 0 \implies k = 0 \] \[ k - 6 = 0 \implies k = 6 \]

Check for validity:

A quadratic equation requires the coefficient of the \(x^2\) term (\(a\)) to be non-zero. In our case, \(a=k\). If we take \(k=0\), the equation becomes \(0 \cdot x^2 - 0 \cdot x + 6 = 0\), which simplifies to \(6=0\). This is not a quadratic equation. Therefore, \(k=0\) is not a valid solution.

The only valid solution is \(k=6\).


Step 4: Final Answer:

The value of k for which the equation has two equal real roots is 6.
Quick Tip: Always remember to check if your solution for \(k\) makes the equation non-quadratic. If a value of \(k\) makes the \(x^2\) term disappear (\(a=0\)), it must be rejected.

Step 1: Understanding the Concept:

The problem requires proving a trigonometric identity. The strategy is to simplify the Left Hand Side (LHS) until it becomes equal to the Right Hand Side (RHS). The presence of a square root suggests rationalizing the expression inside the root.


Step 2: Key Formula or Approach:

We will use the following identities:

Rationalization by multiplying the numerator and denominator by the conjugate of the denominator.
Pythagorean identity: \( \cos^2 A = 1 - \sin^2 A \).
Reciprocal and quotient identities: \( \sec A = \frac{1}{\cos A} \) and \( \tan A = \frac{\sin A}{\cos A} \).


Step 3: Detailed Explanation:

Starting with the Left Hand Side (LHS): \[ LHS = \sqrt{\frac{1-\sin A}{1+\sin A}} \]
To remove the square root, we rationalize the denominator by multiplying the numerator and denominator by \( (1 - \sin A) \): \[ LHS = \sqrt{\frac{(1-\sin A)}{(1+\sin A)} \times \frac{(1-\sin A)}{(1-\sin A)}} \] \[ LHS = \sqrt{\frac{(1-\sin A)^2}{(1+\sin A)(1-\sin A)}} \]
Using the identity \( (a+b)(a-b) = a^2 - b^2 \) for the denominator: \[ LHS = \sqrt{\frac{(1-\sin A)^2}{1 - \sin^2 A}} \]
Using the Pythagorean identity \( 1 - \sin^2 A = \cos^2 A \): \[ LHS = \sqrt{\frac{(1-\sin A)^2}{\cos^2 A}} \]
Now, we can take the square root of the numerator and the denominator: \[ LHS = \frac{1 - \sin A}{\cos A} \]
Split the fraction into two parts: \[ LHS = \frac{1}{\cos A} - \frac{\sin A}{\cos A} \]
Using the reciprocal and quotient identities: \[ LHS = \sec A - \tan A \]
This is equal to the Right Hand Side (RHS).


Step 4: Final Answer:

Thus, we have shown that \( LHS = RHS \), and the identity is proved.
Quick Tip: When you see expressions like \( \frac{1 \pm \sin A}{1 \mp \sin A} \) or \( \frac{1 \pm \cos A}{1 \mp \cos A} \) inside a square root, rationalization is almost always the correct first step.

Step 1: Understanding the Concept:

The two-digit numbers divisible by 3 form an Arithmetic Progression (A.P.). We need to find the total number of terms in this A.P.


Step 2: Key Formula or Approach:

The formula for the \(n\)-th term of an A.P. is \( a_n = a + (n-1)d \), where:

\( a_n \) is the last term.
\( a \) is the first term.
\( n \) is the number of terms.
\( d \) is the common difference.


Step 3: Detailed Explanation:

First, we identify the terms of the A.P.

The first two-digit number divisible by 3 is 12. So, \( a = 12 \).
The last two-digit number is 99, which is divisible by 3. So, \( a_n = 99 \).
The numbers are divisible by 3, so the common difference is 3. So, \( d = 3 \).

Now, we substitute these values into the formula for the \(n\)-th term: \[ 99 = 12 + (n-1)3 \]
Subtract 12 from both sides: \[ 99 - 12 = (n-1)3 \] \[ 87 = (n-1)3 \]
Divide by 3: \[ \frac{87}{3} = n-1 \] \[ 29 = n-1 \]
Add 1 to both sides: \[ n = 29 + 1 = 30 \]

Step 4: Final Answer:

There are 30 two-digit numbers that are divisible by 3.
Quick Tip: A quick way to check this is to find the total numbers up to 99 divisible by 3 (\(99/3 = 33\)) and subtract the count of single-digit numbers divisible by 3 (3, 6, 9 - which is 3 numbers). So, \(33 - 3 = 30\).

Step 1: Understanding the Concept:

This is a fundamental theorem in circle geometry. We need to prove it using the properties of circles, tangents, and the congruence of triangles.


Step 2: Key Formula or Approach:

We will use the following geometric properties:

The tangent at any point of a circle is perpendicular to the radius through the point of contact.
RHS (Right angle-Hypotenuse-Side) congruence rule for triangles.


Step 3: Detailed Explanation:

Given: A circle with center O. An external point P from which two tangents PA and PB are drawn to the circle at points of contact A and B respectively.

To Prove: \( PA = PB \).

Construction: Join OA, OB, and OP.


Proof:
We have two triangles, \(\triangle OAP\) and \(\triangle OBP\).

\( \angle OAP = \angle OBP = 90^\circ \) (Since the radius is perpendicular to the tangent at the point of contact).
\( OA = OB \) (Radii of the same circle).
\( OP = OP \) (Common side to both triangles).

By the RHS congruence criterion, \(\triangle OAP \cong \triangle OBP\).

Since the triangles are congruent, their corresponding parts are equal (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).
Therefore, \( PA = PB \).


Step 4: Final Answer:

Hence, it is proved that the lengths of tangents drawn from an external point to a circle are equal.
Quick Tip: Always start geometry proofs by clearly stating what is given, what needs to be proved, and any constructions made. Drawing a neat diagram is crucial for visualizing the problem and structuring the proof.

Step 1: Understanding the Concept:

The question requires identifying the specific mathematical name for the U-shape shown in the graph.


Step 2: Detailed Explanation:

The graph of any quadratic polynomial (\( y = ax^2 + bx + c \)) is a curve with a characteristic U-shape. This specific shape is called a parabola. If the leading coefficient 'a' is positive, the parabola opens upwards, as seen in this graph. If 'a' were negative, it would open downwards.


Step 3: Final Answer:

The shape of the graph is a parabola.
Quick Tip: Remember that a U-shaped graph is called a parabola and represents a quadratic function. A straight line represents a linear function, and a curve with two turns often represents a cubic function.

Step 1: Understanding the Concept:

The "zeroes" of a polynomial are the x-values where the polynomial's value is zero (\(y=0\)). Graphically, these are the points where the graph intersects the x-axis.


Step 2: Detailed Explanation:

By observing the provided graph, we can see that the curve (the parabola) crosses the horizontal x-axis at two distinct points. Each intersection with the x-axis represents a real zero of the polynomial. Since there are two intersection points, the polynomial has two real zeroes.


Step 3: Final Answer:

There are 2 zeroes of the polynomial shown in the graph.
Quick Tip: The number of real zeroes of a polynomial is equal to the number of times its graph intersects the x-axis. If the graph only touches the x-axis at one point, there is one real zero (with a multiplicity of 2 for a parabola). If it doesn't cross the x-axis at all, there are no real zeroes.

Step 1: Understanding the Concept:

We need to find the specific x-coordinates of the points where the graph intersects the x-axis.


Step 2: Detailed Explanation:

Looking closely at the graph:

The graph crosses the x-axis at a point between -2 and 0. The point is clearly marked as -1.
The graph crosses the x-axis again at a point between 3 and 5. The point is clearly marked as 4.

Therefore, the x-values where \(y=0\) are -1 and 4.


Step 3: Final Answer:

The zeroes of the polynomial are -1 and 4.
Quick Tip: Zeroes are also called roots or x-intercepts. Always read the scale on the axes carefully to determine the correct values of the intercepts.

Step 1: Understanding the Concept:

This question asks for the sum of the zeroes that we identified in the previous part.


Step 2: Key Formula or Approach:

Sum of zeroes = (First zero) + (Second zero).


Step 3: Detailed Explanation:

From part (c), we found that the zeroes of the polynomial are -1 and 4.
To find their sum, we add these two values: \[ Sum = (-1) + 4 \] \[ Sum = 3 \]

Step 4: Final Answer:

The sum of the zeroes is 3.
Quick Tip: For a quadratic polynomial \(ax^2 + bx + c\), the sum of the zeroes is given by the formula \(-b/a\). Based on the zeroes (-1 and 4), the polynomial is of the form \(k(x - (-1))(x - 4) = k(x+1)(x-4) = k(x^2 - 3x - 4)\). The sum of zeroes is \(-(-3)/1 = 3\), which confirms our answer.

Step 1: Understanding the Concept:

We will prove this using the method of contradiction. This involves assuming the opposite of what we want to prove and then showing that this assumption leads to a logical inconsistency.


Step 2: Key Formula or Approach:

The core idea is that if a prime number \(p\) divides \(a^2\), then \(p\) must also divide \(a\).


Step 3: Detailed Explanation:

Let us assume, to the contrary, that \( \sqrt{7} \) is a rational number.
Then, by definition, we can write \( \sqrt{7} = \frac{p}{q} \), where \(p\) and \(q\) are integers, \(q \neq 0\), and \(p\) and \(q\) are coprime (they have no common factors other than 1).

Squaring both sides of the equation, we get: \[ 7 = \frac{p^2}{q^2} \] \[ \implies 7q^2 = p^2 \quad \dots(1) \]
This equation shows that \(p^2\) is a multiple of 7. Therefore, \(p^2\) is divisible by 7.

If \(p^2\) is divisible by 7, then \(p\) must also be divisible by 7.

So, we can write \(p = 7k\) for some integer \(k\).

Now, substitute \(p = 7k\) into equation (1): \[ 7q^2 = (7k)^2 \] \[ 7q^2 = 49k^2 \]
Divide both sides by 7: \[ q^2 = 7k^2 \]
This shows that \(q^2\) is a multiple of 7, which means \(q^2\) is divisible by 7.

Therefore, \(q\) must also be divisible by 7.

From our steps, we have concluded that both \(p\) and \(q\) are divisible by 7. This means that 7 is a common factor of \(p\) and \(q\).
However, this contradicts our initial assumption that \(p\) and \(q\) are coprime.

This contradiction arises because our initial assumption that \( \sqrt{7} \) is rational was wrong.


Step 4: Final Answer:

Therefore, we conclude that \( \sqrt{7} \) is an irrational number.
Quick Tip: The proof by contradiction for the irrationality of \( \sqrt{p} \) (where p is a prime number) always follows the same logical structure. Practice it once or twice, and you can apply it to \( \sqrt{2}, \sqrt{3}, \sqrt{5} \), etc.

Step 1: Understanding the Concept:

A trapezium is a quadrilateral with at least one pair of parallel sides. To prove that ABCD is a trapezium, we need to show that one pair of opposite sides is parallel (e.g., AB || DC). We can do this by proving that a pair of triangles formed by the diagonals are similar, which would imply that alternate interior angles are equal.


Step 2: Key Formula or Approach:

We will use the Side-Angle-Side (SAS) similarity criterion for triangles. If two triangles have a pair of corresponding sides in proportion and the included angle is equal, then the triangles are similar. If triangles are similar, their corresponding angles are equal.


Step 3: Detailed Explanation:

Given: A quadrilateral ABCD where diagonals AC and BD intersect at point O such that \( \frac{AO}{BO} = \frac{CO}{DO} \).

To Prove: ABCD is a trapezium.


Proof:
First, rearrange the given proportion: \[ \frac{AO}{BO} = \frac{CO}{DO} \implies \frac{AO}{CO} = \frac{BO}{DO} \quad \dots(1) \]
Now, consider the triangles \(\triangle AOB\) and \(\triangle COD\).

From (1), we have the ratio of two pairs of corresponding sides equal: \( \frac{AO}{CO} = \frac{BO}{DO} \).
The angle included between these sides in \(\triangle AOB\) is \(\angle AOB\), and in \(\triangle COD\) is \(\angle COD\).
We know that \( \angle AOB = \angle COD \) because they are vertically opposite angles.

Based on the Side-Angle-Side (SAS) similarity criterion, since two sides are in proportion and the included angle is equal, we can conclude that: \[ \triangle AOB \sim \triangle COD \]
Since the triangles are similar, their corresponding angles must be equal. Therefore: \[ \angle OAB = \angle OCD \quad (or \angle CAB = \angle ACD) \]
These two angles are alternate interior angles formed by the lines AB and DC with the transversal line AC.

When a pair of alternate interior angles are equal, the lines must be parallel.
Therefore, \( AB \parallel DC \).

Since the quadrilateral ABCD has one pair of opposite sides parallel, it is a trapezium.


Step 4: Final Answer:

Hence, it is proved that ABCD is a trapezium.
Quick Tip: To prove a quadrilateral is a trapezium, you need to show one pair of opposite sides are parallel. The converse of the Basic Proportionality Theorem and properties of similar triangles are the key tools for this. Make sure to identify the correct pairs of similar triangles.

Step 1: Understanding the Concept:

"Points of trisection" are two points that divide a line segment into three equal parts. If a line segment is AB, and P and Q are the points of trisection, then AP = PQ = QB. This means point P divides AB in the ratio 1:2, and point Q divides AB in the ratio 2:1.


Step 2: Key Formula or Approach:

We use the section formula. If a point \(P(x, y)\) divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of P are: \[ (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Step 3: Detailed Explanation:

Let the given points be \(A(-3, -5)\) and \(B(-6, -8)\). Let the points of trisection be P and Q.


Finding the coordinates of point P:

Point P divides the line segment AB in the ratio 1:2. Here, \(m=1, n=2\), \(x_1=-3, y_1=-5\), \(x_2=-6, y_2=-8\). \[ x_P = \frac{1(-6) + 2(-3)}{1+2} = \frac{-6 - 6}{3} = \frac{-12}{3} = -4 \] \[ y_P = \frac{1(-8) + 2(-5)}{1+2} = \frac{-8 - 10}{3} = \frac{-18}{3} = -6 \]
So, the coordinates of point P are \((-4, -6)\).


Finding the coordinates of point Q:

Point Q divides the line segment AB in the ratio 2:1. Here, \(m=2, n=1\). \[ x_Q = \frac{2(-6) + 1(-3)}{2+1} = \frac{-12 - 3}{3} = \frac{-15}{3} = -5 \] \[ y_Q = \frac{2(-8) + 1(-5)}{2+1} = \frac{-16 - 5}{3} = \frac{-21}{3} = -7 \]
So, the coordinates of point Q are \((-5, -7)\).


Step 4: Final Answer:

The coordinates of the points of trisection are (-4, -6) and (-5, -7).
Quick Tip: Once you find the first point of trisection (P), you can find the second point (Q) by recognizing that Q is the midpoint of the segment PB. This can sometimes be a faster calculation than using the section formula again with the ratio 2:1.

Step 1: Understanding the Concept:

The cow is tied at a corner of a square field. The area it can graze within the field will be a sector of a circle. Since the corner of a square is a right angle (90°), the shape of the grazing area is a quadrant (one-fourth of a circle). The length of the rope is the radius of this circle.


Step 2: Key Formula or Approach:

The area of a sector of a circle with radius \(r\) and angle \(\theta\) (in degrees) is given by: \[ Area = \frac{\theta}{360^\circ} \times \pi r^2 \]

Step 3: Detailed Explanation:

Given:

Shape of the field: Square
Angle at the corner, \(\theta = 90^\circ\)
Length of the rope (radius), \(r = 6\) m

The grazing area is a quadrant of a circle with radius 6 m.
Using the formula for the area of a sector: \[ Area = \frac{90^\circ}{360^\circ} \times \pi (6)^2 \] \[ Area = \frac{1}{4} \times \pi \times 36 \] \[ Area = 9\pi m^2 \]
To find a numerical value, we can use \(\pi \approx 3.14159\): \[ Area \approx 9 \times 3.14159 \approx 28.27 m^2 \]

Step 4: Final Answer:

The area of the field in which the cow can graze is \(9\pi\) m\(^2\), which is approximately 28.27 m\(^2\).
Quick Tip: For grazing problems at the corner of a square or rectangular field, the area is always a quadrant. The formula simplifies to \( \frac{1}{4}\pi r^2 \).

Step 1: Understanding the Concept:

We need to calculate the new grazing area with the longer rope and then find the difference between the new area and the original area calculated in the previous part.


Step 2: Key Formula or Approach:

1. Calculate the new area using the sector formula with the new radius.
2. Increase in Area = New Area - Old Area.


Step 3: Detailed Explanation:

Calculate the new grazing area:

Given:

New length of the rope (new radius), \(R = 11.5\) m
Angle at the corner, \(\theta = 90^\circ\)
\[ New Area = \frac{90^\circ}{360^\circ} \times \pi R^2 = \frac{1}{4} \times \pi \times (11.5)^2 \] \[ New Area = \frac{1}{4} \times \pi \times 132.25 = 33.0625\pi m^2 \]

Calculate the increase in area:

From part (i), the old area was \(9\pi\) m\(^2\). \[ Increase in Area = New Area - Old Area \] \[ Increase in Area = 33.0625\pi - 9\pi = 24.0625\pi m^2 \]
To find a numerical value, we can use \(\pi \approx 3.14159\): \[ Increase in Area \approx 24.0625 \times 3.14159 \approx 75.59 m^2 \]

Step 4: Final Answer:

The increase in the grazing area is \(24.0625\pi\) m\(^2\), which is approximately 75.59 m\(^2\).
Quick Tip: The increase in area can be calculated more directly using the formula \( \frac{\theta}{360^\circ} \pi (R^2 - r^2) \). This avoids calculating both areas separately and then subtracting, which can be faster.

Step 1: Understanding the Concept:

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. A standard deck has 52 cards, divided into 4 suits (hearts, diamonds, clubs, spades) with 13 cards in each.


Step 2: Key Formula or Approach:
\[ P(Event) = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]
Total number of possible outcomes is 52.


Step 3: Detailed Explanation:

i) Probability of being an ace:

There are 4 aces in a deck (one for each suit). \[ P(ace) = \frac{Number of aces}{Total cards} = \frac{4}{52} = \frac{1}{13} \]

ii) Probability of not being an ace:

This is a complementary event to drawing an ace. \[ P(not an ace) = 1 - P(ace) = 1 - \frac{1}{13} = \frac{12}{13} \]
Alternatively, there are \(52 - 4 = 48\) non-ace cards. \[ P(not an ace) = \frac{48}{52} = \frac{12}{13} \]

iii) Probability of being a face card:

Face cards are Kings, Queens, and Jacks. There are 3 face cards in each of the 4 suits.
Total number of face cards = \(3 \times 4 = 12\). \[ P(face card) = \frac{Number of face cards}{Total cards} = \frac{12}{52} = \frac{3}{13} \]

iv) Probability of being a spade:

There are 13 cards in the spade suit. \[ P(spade) = \frac{Number of spades}{Total cards} = \frac{13}{52} = \frac{1}{4} \]

Step 4: Final Answer:

The probabilities are: i) \(\frac{1}{13}\), ii) \(\frac{12}{13}\), iii) \(\frac{3}{13}\), iv) \(\frac{1}{4}\).
Quick Tip: Memorize the structure of a standard 52-card deck: 4 suits, 13 cards per suit, 2 colors (red/black), 12 face cards, 4 aces. This makes solving card-based probability questions much faster.

Step 1: Understanding the Concept:

This problem involves using trigonometry to solve right-angled triangles. The slide, the vertical height, and the ground form a right-angled triangle. The length of the slide is the hypotenuse. We are given the height (opposite side to the angle of inclination) and the angle.


Step 2: Key Formula or Approach:

We will use the sine trigonometric ratio, which relates the opposite side, the hypotenuse, and the angle: \[ \sin(\theta) = \frac{Opposite}{Hypotenuse} \]

Step 3: Detailed Explanation:

Case 1: Slide for younger children


Height (Opposite side) = 1.5 m
Angle of inclination (\(\theta\)) = 30°
Length of the slide (Hypotenuse) = L1
\[ \sin(30^\circ) = \frac{1.5}{L_1} \]
We know that \(\sin(30^\circ) = \frac{1}{2}\). \[ \frac{1}{2} = \frac{1.5}{L_1} \] \[ L_1 = 1.5 \times 2 = 3 m \]

Case 2: Slide for elder children


Height (Opposite side) = 3 m
Angle of inclination (\(\theta\)) = 60°
Length of the slide (Hypotenuse) = L2
\[ \sin(60^\circ) = \frac{3}{L_2} \]
We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). \[ \frac{\sqrt{3}}{2} = \frac{3}{L_2} \] \[ L_2 = \frac{3 \times 2}{\sqrt{3}} = \frac{6}{\sqrt{3}} \]
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\): \[ L_2 = \frac{6 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} m \]

Step 4: Final Answer:

The length of the slide for younger children should be 3 m, and the length of the slide for elder children should be \(2\sqrt{3}\) m.
Quick Tip: Remember SOH-CAH-TOA to choose the correct trigonometric ratio. SOH (\(\sin = \frac{Opposite}{Hypotenuse}\)) is used here. Always draw a diagram for height and distance problems to correctly identify the sides relative to the angle.

Step 1: Understanding the Concept:

The step-deviation method is a simplified way to calculate the mean for grouped data, especially when the class intervals are of equal size. It reduces the calculation complexity by working with smaller numbers.


Step 2: Key Formula or Approach:

The formula for the mean using the step-deviation method is: \[ Mean (\bar{x}) = A + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h \]
where \(A\) is the assumed mean, \(h\) is the class size, \(f_i\) is the frequency of the \(i\)-th class, and \(u_i = \frac{x_i - A}{h}\) with \(x_i\) being the class mark.


Step 3: Detailed Explanation:

We construct the following table to perform the calculation. Let's choose the assumed mean \(A = 25\) (the class mark of the middle class with high frequency). The class size is \(h = 10\).


\begin{tabular{|c|c|c|c|c|
\hline
Class & Frequency (\(f_i\)) & Class Mark (\(x_i\)) & \(u_i = \frac{x_i - 25}{10}\) & \(f_i u_i\)

\hline
0-10 & 8 & 5 & -2 & -16

10-20 & 16 & 15 & -1 & -16

20-30 & 36 & 25 & 0 & 0

30-40 & 34 & 35 & 1 & 34

40-50 & 6 & 45 & 2 & 12

\hline
Total & \(\sum f_i = 100\) & & & \(\sum f_i u_i = 14\)

\hline
\end{tabular


From the table, we have:

\(\sum f_i = 100\)
\(\sum f_i u_i = (-16) + (-16) + 0 + 34 + 12 = 14\)
Assumed Mean, \(A = 25\)
Class size, \(h = 10\)

Now, substitute these values into the formula: \[ \bar{x} = 25 + \left( \frac{14}{100} \right) \times 10 \] \[ \bar{x} = 25 + \frac{140}{100} \] \[ \bar{x} = 25 + 1.4 = 26.4 \]

Step 4: Final Answer:

The mean for the given data is 26.4.
Quick Tip: When using the step-deviation method, always choose the assumed mean (A) as the class mark of the class with the highest frequency (the modal class). This often simplifies the calculations by centering the 'u' values around zero.

Step 1: Understanding the Concept:

We are given two consecutive terms of an Arithmetic Progression (A.P.) and asked to find the sum of the first 51 terms. First, we need to find the first term (\(a\)) and the common difference (\(d\)) of the A.P.


Step 2: Key Formula or Approach:

1. Common difference, \(d = a_n - a_{n-1}\).
2. The sum of the first \(n\) terms of an A.P. is given by the formula: \[ S_n = \frac{n}{2} [2a + (n-1)d] \]

Step 3: Detailed Explanation:

Given:

Second term, \(a_2 = 14\)
Third term, \(a_3 = 18\)

Find the common difference (d):
\[ d = a_3 - a_2 = 18 - 14 = 4 \]
Find the first term (a):

The second term is given by \(a_2 = a + d\). \[ 14 = a + 4 \] \[ a = 14 - 4 = 10 \]
Now we have \(a = 10\), \(d = 4\), and we need to find the sum of the first 51 terms, so \(n = 51\).

Calculate the sum (S51):

Using the sum formula: \[ S_{51} = \frac{51}{2} [2(10) + (51-1)4] \] \[ S_{51} = \frac{51}{2} [20 + (50)4] \] \[ S_{51} = \frac{51}{2} [20 + 200] \] \[ S_{51} = \frac{51}{2} [220] \] \[ S_{51} = 51 \times 110 \] \[ S_{51} = 5610 \]

Step 4: Final Answer:

The sum of the first 51 terms of the A.P. is 5610.
Quick Tip: The difference between any two consecutive terms in an A.P. is the common difference. This is the quickest way to find 'd' if you are given consecutive terms.

Step 1: Understanding the Concept:

To solve a pair of linear equations graphically, we need to draw the graph for each equation on the same coordinate plane. The point where the two lines intersect is the solution to the system of equations.


Step 2: Key Formula or Approach:

To draw the graph of a linear equation, we find at least two pairs of (x, y) coordinates that satisfy the equation. We then plot these points and draw a straight line through them.


Step 3: Detailed Explanation:

Equation 1: \(2x + y - 6 = 0 \implies y = 6 - 2x\)

Let's find some points:

If \(x=0\), \(y = 6 - 2(0) = 6\). Point: (0, 6)
If \(x=3\), \(y = 6 - 2(3) = 0\). Point: (3, 0)
If \(x=2\), \(y = 6 - 2(2) = 2\). Point: (2, 2)


Equation 2: \(4x - 2y - 4 = 0\)

We can simplify this by dividing by 2: \(2x - y - 2 = 0 \implies y = 2x - 2\)

Let's find some points:

If \(x=0\), \(y = 2(0) - 2 = -2\). Point: (0, -2)
If \(x=1\), \(y = 2(1) - 2 = 0\). Point: (1, 0)
If \(x=2\), \(y = 2(2) - 2 = 2\). Point: (2, 2)


Graphical Representation:

Now, we plot these points on a graph and draw the two lines.

Line 1 passes through (0, 6), (3, 0), and (2, 2).
Line 2 passes through (0, -2), (1, 0), and (2, 2).

When we draw these lines, we observe that they intersect at the point (2, 2).


Step 4: Final Answer:

The point of intersection of the two lines is (2, 2). Therefore, the solution to the pair of linear equations is x = 2 and y = 2.
Quick Tip: Finding a third point for each line is a good way to check your calculations. If all three points do not lie on the same straight line, you have likely made an arithmetic error.

Step 1: Understanding the Concept:

We need to translate the word problem into a system of two linear equations with two variables. Then, we solve this system graphically, similar to the previous question.


Step 2: Key Formula or Approach:

1. Define variables for the unknown prices.
2. Formulate two linear equations based on the given information.
3. Find coordinate points for each equation.
4. Plot the lines and find their point of intersection.


Step 3: Detailed Explanation:

Formulating the Equations:

Let the price of one pencil be \(x\) rupees.
Let the price of one chocolate be \(y\) rupees.

Lata's purchase: \(2x + 3y = 9\) \quad \dots(1)
Suma's purchase: \(x + 2y = 5\) \quad \dots(2)


Finding points for Equation 1: \(2x + 3y = 9 \implies y = \frac{9 - 2x}{3}\)


If \(x=0\), \(y = \frac{9}{3} = 3\). Point: (0, 3)
If \(x=3\), \(y = \frac{9 - 6}{3} = 1\). Point: (3, 1)
If \(x=-3\), \(y = \frac{9 + 6}{3} = 5\). Point: (-3, 5)


Finding points for Equation 2: \(x + 2y = 5 \implies y = \frac{5 - x}{2}\)


If \(x=1\), \(y = \frac{5 - 1}{2} = 2\). Point: (1, 2)
If \(x=3\), \(y = \frac{5 - 3}{2} = 1\). Point: (3, 1)
If \(x=5\), \(y = \frac{5 - 5}{2} = 0\). Point: (5, 0)


Graphical Solution:

Plot the points for each line and draw them on a graph.

Line 1 passes through (0, 3) and (3, 1).
Line 2 passes through (1, 2) and (3, 1).

The two lines intersect at the point (3, 1).

This means the solution is \(x = 3\) and \(y = 1\).


Step 4: Final Answer:

The price of one pencil (\(x\)) is ₹3 and the price of one chocolate (\(y\)) is ₹1.
Quick Tip: When creating points for graphing, try to choose input values (x or y) that result in integer coordinates for the other variable. This makes plotting much easier and more accurate.