class 9 ganit manjari part 1 ncert solution End of Chapter Exercises

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    <title>Class 9 Maths · Ganita Manjari · Chapter 2 End-of-Chapter Exercises</title>
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<body>

    <div class="container">

        <!-- ===== HEADER ===== -->
        <div class="header">
            <h1>
                End-of-Chapter Exercises
                <span class="chapter-tag">Class 9 · Ganita Manjari</span>
            </h1>
            <div class="subhead">
                <span>📘 Chapter 2 — Polynomials</span>
                <span class="badge">📌 Complete Solutions</span>
            </div>
            <div class="info-note">
                <span class="icon">📐</span>
                <span>
                    This set covers <strong>polynomials</strong>, <strong>linear equations</strong>,
                    <strong>graphs</strong>, <strong>slopes and intercepts</strong>, and
                    <strong>word problems</strong> based on linear patterns.
                </span>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 1 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">1</span>
            </div>
            <div class="question-text">
                Write a polynomial of degree 3 in the variable x, in which the coefficient of the x² term is −7.
            </div>
            <div class="solution">
                <p>A polynomial of degree 3 has the general form: <span class="math">ax³ + bx² + cx + d</span></p>
                <p>Given that coefficient of x² is −7, so <span class="math">b = −7</span></p>
                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>One such polynomial is <span class="math">x³ − 7x² + 2x + 1</span>.</span>
                </div>
                <p style="margin-top:0.5rem; font-size:0.95rem; color:#4a6f92;">
                    (Any polynomial of degree 3 with −7 as the coefficient of x² is correct)
                </p>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 2 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">2</span>
            </div>
            <div class="question-text">
                Find the values of the following polynomials at the indicated values of the variables.
                <br><strong>(i)</strong> 5x² − 3x + 7 if x = 1
                <br><strong>(ii)</strong> 4t³ − t² + 6 if t = a
            </div>
            <div class="solution">
                <!-- (i) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(i)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        5x² − 3x + 7 if x = 1
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Substitute x = 1:</p>
                        <div class="math-block">
                            = 5(1)² − 3(1) + 7<br>
                            = 5 − 3 + 7<br>
                            = <strong>9</strong>
                        </div>
                    </div>
                </div>

                <!-- (ii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(ii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        4t³ − t² + 6 if t = a
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Substitute t = a:</p>
                        <div class="math-block">
                            = <strong>4a³ − a² + 6</strong>
                        </div>
                    </div>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 3 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">3</span>
            </div>
            <div class="question-text">
                If we multiply a number by 5/2 and add 2/3 to the product, we get −7/12. Find the number.
            </div>
            <div class="solution">
                <p>Let the number be <span class="math">x</span>.</p>
                <p>According to the question:</p>
                <div class="math-block">
                    (5/2)x + 2/3 = −7/12
                </div>

                <p>Subtract 2/3 from both sides, we get:</p>
                <div class="math-block">
                    (5/2)x = −7/12 – 2/3<br>
                    ⇒ (5/2)x = −7/12 – 8/12<br>
                    ⇒ (5/2)x = −15/12<br>
                    ⇒ (5/2)x = −5/4
                </div>

                <p>Now multiply both sides by 2/5, we get:</p>
                <div class="math-block">
                    x = (−5/4) × (2/5)<br>
                    ⇒ x = <strong>−1/2</strong>
                </div>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Therefore, the number is <span class="math">−1/2</span>.</span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 4 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">4</span>
            </div>
            <div class="question-text">
                A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?
            </div>
            <div class="solution">
                <p>Let the smaller number be <span class="math">x</span>.</p>
                <p>Then the larger number = <span class="math">5x</span></p>

                <p>After adding 21 to both numbers: New numbers are <span class="math">x + 21</span> and <span class="math">5x + 21</span></p>

                <p>According to the question:</p>
                <div class="math-block">
                    5x + 21 = 2(x + 21)<br>
                    ⇒ 5x + 21 = 2x + 42<br>
                    ⇒ 5x – 2x = 42 – 21<br>
                    ⇒ 3x = 21<br>
                    ⇒ <strong>x = 7</strong>
                </div>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>So, the smaller number = <strong>7</strong> and the larger number = 5 × 7 = <strong>35</strong>.<br>
                    Therefore, the two numbers are <strong>7</strong> and <strong>35</strong>.</span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 5 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">5</span>
            </div>
            <div class="question-text">
                If you have ₹800 and you save ₹250 every month, find the amount you have after
                (i) 6 months (ii) 2 years. Express this as a linear pattern.
            </div>
            <div class="solution">
                <p>Initial amount = <span class="math">₹800</span></p>
                <p>Saving every month = <span class="math">₹250</span></p>
                <p>Linear pattern: Amount after n months = <span class="math">800 + 250n</span></p>

                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(i)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Amount after 6 months
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <div class="math-block">
                            = 800 + 250(6)<br>
                            = 800 + 1500<br>
                            = <strong>₹2300</strong>
                        </div>
                    </div>
                </div>

                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(ii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Amount after 2 years
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>2 years = 24 months</p>
                        <div class="math-block">
                            = 800 + 250(24)<br>
                            = 800 + 6000<br>
                            = <strong>₹6800</strong>
                        </div>
                    </div>
                </div>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Amount after 6 months = <strong>₹2300</strong><br>
                    Amount after 2 years = <strong>₹6800</strong><br>
                    Linear pattern: <span class="math">A = 800 + 250n</span>, where n is the number of months.</span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 6 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">6</span>
            </div>
            <div class="question-text">
                The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers.
            </div>
            <div class="solution">
                <p>Let the tens digit be <span class="math">x</span> and the units digit be <span class="math">y</span>.</p>
                <p>Then the original number = <span class="math">10x + y</span></p>
                <p>The interchanged number = <span class="math">10y + x</span></p>

                <p>According to the question:</p>
                <div class="math-block">
                    (10x + y) + (10y + x) = 143<br>
                    ⇒ 11x + 11y = 143<br>
                    ⇒ 11(x + y) = 143<br>
                    ⇒ x + y = 13 …(1)
                </div>

                <p>The digits differ by 3.</p>
                <div class="math-block">
                    So, x – y = 3 …(2)
                </div>

                <p>Now adding (1) and (2), we get:</p>
                <div class="math-block">
                    2x = 16<br>
                    ⇒ <strong>x = 8</strong>
                </div>

                <p>Substitute x = 8 in (1), we have:</p>
                <div class="math-block">
                    8 + y = 13<br>
                    ⇒ <strong>y = 5</strong>
                </div>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Therefore, the original number = <strong>85</strong> and the interchanged number = <strong>58</strong>.<br>
                    Hence, the two numbers are <strong>85</strong> and <strong>58</strong>.</span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 7 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">7</span>
            </div>
            <div class="question-text">
                Draw the graph of the following equations, and identify their slopes and y-intercepts.
                Also, find the coordinates of the points where these lines cut the y-axis.
                <br><strong>(i)</strong> y = −3x + 4
                <br><strong>(ii)</strong> 2y = 4x + 7
                <br><strong>(iii)</strong> 5y = 6x – 10
                <br><strong>(iv)</strong> 3y = 6x – 11
                <br>Are any of the lines parallel?
            </div>
            <div class="solution">
                <!-- (i) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(i)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        y = −3x + 4
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Comparing with <span class="math">y = ax + b</span>:</p>
                        <div class="coord-grid">
                            <div class="coord-item"><span class="name">Slope a =</span> −3</div>
                            <div class="coord-item"><span class="name">y-intercept b =</span> 4</div>
                        </div>
                        <p>So, the point, where the line cuts the y-axis, is <span class="math">(0, 4)</span>.</p>
                    </div>
                </div>

                <!-- (ii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(ii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        2y = 4x + 7
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Dividing both sides by 2: <span class="math">y = 2x + 7/2</span></p>
                        <div class="coord-grid">
                            <div class="coord-item"><span class="name">Slope a =</span> 2</div>
                            <div class="coord-item"><span class="name">y-intercept b =</span> 7/2</div>
                        </div>
                        <p>So, the point, where the line cuts the y-axis, is <span class="math">(0, 7/2)</span>.</p>
                    </div>
                </div>

                <!-- (iii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(iii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        5y = 6x – 10
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Dividing both sides by 5: <span class="math">y = (6/5)x – 2</span></p>
                        <div class="coord-grid">
                            <div class="coord-item"><span class="name">Slope a =</span> 6/5</div>
                            <div class="coord-item"><span class="name">y-intercept b =</span> −2</div>
                        </div>
                        <p>So, the point, where the line cuts the y-axis, is <span class="math">(0, −2)</span>.</p>
                    </div>
                </div>

                <!-- (iv) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(iv)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        3y = 6x – 11
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Dividing both sides by 3: <span class="math">y = 2x – 11/3</span></p>
                        <div class="coord-grid">
                            <div class="coord-item"><span class="name">Slope a =</span> 2</div>
                            <div class="coord-item"><span class="name">y-intercept b =</span> −11/3</div>
                        </div>
                        <p>So, the point, where the line cuts the y-axis, is <span class="math">(0, −11/3)</span>.</p>
                    </div>
                </div>

                <div class="result-box info">
                    <span class="icon">📐</span>
                    <span><strong>Parallel lines:</strong> Two lines are parallel if they have the same slope.<br>
                    Equation (ii): slope = 2 &nbsp;|&nbsp; Equation (iv): slope = 2<br>
                    Therefore, lines <strong>(ii)</strong> and <strong>(iv)</strong> are parallel.</span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 8 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">8</span>
            </div>
            <div class="question-text">
                If the temperature of a liquid can be measured in Kelvin units as x K and in Fahrenheit units as y °F,
                the relation between the two systems of measurement of temperature is given by the linear equation
                <strong>y = (9/5)(x – 273) + 32</strong>.
                <br><strong>(i)</strong> Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313 K.
                <br><strong>(ii)</strong> If the temperature is 158 °F, then find the temperature in Kelvin.
            </div>
            <div class="solution">
                <!-- (i) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(i)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        x = 313 K, find y
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Using equation <span class="math">y = (9/5)(x – 273) + 32</span>:</p>
                        <div class="math-block">
                            y = (9/5)(313 – 273) + 32<br>
                            ⇒ y = (9/5)(40) + 32<br>
                            ⇒ y = 72 + 32<br>
                            ⇒ y = <strong>104</strong>
                        </div>
                        <div class="result-box">
                            <span class="icon">🌡️</span>
                            <span>Therefore, the temperature is <strong>104 °F</strong>.</span>
                        </div>
                    </div>
                </div>

                <!-- (ii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(ii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        y = 158 °F, find x
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Using equation <span class="math">y = (9/5)(x – 273) + 32</span>:</p>
                        <div class="math-block">
                            158 = (9/5)(x – 273) + 32
                        </div>
                        <p>Subtracting 32 from both sides, we get:</p>
                        <div class="math-block">
                            158 – 32 = (9/5)(x – 273)<br>
                            ⇒ 126 = (9/5)(x – 273)
                        </div>
                        <p>Multiply both sides by 5/9, we get:</p>
                        <div class="math-block">
                            126 × (5/9) = x – 273<br>
                            ⇒ 70 = x – 273<br>
                            ⇒ x = <strong>343</strong>
                        </div>
                        <div class="result-box">
                            <span class="icon">🌡️</span>
                            <span>Therefore, the temperature is <strong>343 K</strong>.</span>
                        </div>
                    </div>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 9 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">9</span>
            </div>
            <div class="question-text">
                The work done by a body on the application of a constant force is the product of the constant force
                and the distance travelled by the body in the direction of the force.
                Express this in the form of a linear equation in two variables (work w and distance d),
                and draw its graph by taking the constant force as 3 units.
                What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph.
            </div>
            <div class="solution">
                <p>We know that:</p>
                <p><span class="math">Work done = Force × Distance</span></p>

                <p>According to question:</p>
                <ul>
                    <li>work done = <span class="math">w</span></li>
                    <li>distance travelled = <span class="math">d</span></li>
                    <li>constant force = <span class="math">3 units</span></li>
                </ul>

                <div class="math-block">
                    Therefore, w = 3d
                </div>

                <p>This is the required linear equation in two variables.</p>

                <p>If <span class="math">d = 2 units</span>, force <span class="math">w = 3 × 2 = 6 units</span>.</p>

                <p>Taking w on y-axis and d on x-axis, we can plot the graph.</p>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>The point <span class="math">(2, 6)</span> lies on a straight line, so it is verified by graph.<br>
                    So, the work done when distance travelled is 2 units: <span class="math">w = 3 × 2 = 6 units</span>.</span>
                </div>

                <p><strong>Verification from graph:</strong></p>
                <p>The point corresponding to <span class="math">d = 2</span> is <span class="math">(2, 6)</span>, so the work done is <strong>6 units</strong>.</p>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 10 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">10</span>
            </div>
            <div class="question-text">
                The graph of a linear polynomial p(x) passes through the points (1, 5) and (3, 11).
                <br><strong>(i)</strong> Find the polynomial p(x).
                <br><strong>(ii)</strong> Find the coordinates where the graph of p(x) cuts the axes.
                <br><strong>(iii)</strong> Draw the graph of p(x) and verify your answers.
            </div>
            <div class="solution">
                <!-- (i) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(i)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Find the polynomial p(x)
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>Let <span class="math">p(x) = ax + b</span></p>

                        <p>Since the graph passes through (1, 5):</p>
                        <div class="math-block">
                            a(1) + b = 5<br>
                            ⇒ a + b = 5 …(1)
                        </div>

                        <p>Since the graph passes through (3, 11):</p>
                        <div class="math-block">
                            a(3) + b = 11<br>
                            ⇒ 3a + b = 11 …(2)
                        </div>

                        <p>Subtracting (1) from (2), we get:</p>
                        <div class="math-block">
                            3a + b – (a + b) = 11 – 5<br>
                            ⇒ 2a = 6<br>
                            ⇒ <strong>a = 3</strong>
                        </div>

                        <p>Substituting a = 3 in (1), we have:</p>
                        <div class="math-block">
                            3 + b = 5<br>
                            ⇒ <strong>b = 2</strong>
                        </div>

                        <div class="result-box">
                            <span class="icon">✅</span>
                            <span>Therefore, <span class="math">p(x) = 3x + 2</span>.</span>
                        </div>
                    </div>
                </div>

                <!-- (ii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(ii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Coordinates where the graph cuts the axes
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p><strong>To find the y-axis intercept:</strong> Put x = 0</p>
                        <div class="math-block">
                            y = p(0) = 3(0) + 2 = 2
                        </div>
                        <p>So, the graph cuts the y-axis at <span class="math">(0, 2)</span>.</p>

                        <p><strong>To find the x-axis intercept:</strong> Put y = 0</p>
                        <div class="math-block">
                            3x + 2 = 0<br>
                            ⇒ 3x = −2<br>
                            ⇒ x = −2/3
                        </div>
                        <p>So, the graph cuts the x-axis at <span class="math">(−2/3, 0)</span>.</p>
                    </div>
                </div>

                <!-- (iii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(iii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Draw the graph and verify
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>From the graph, it verifies that the line passes through the given points and cuts:</p>
                        <ul>
                            <li>y-axis at <span class="math">(0, 2)</span></li>
                            <li>x-axis at <span class="math">(−2/3, 0)</span></li>
                        </ul>
                    </div>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 11 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">11</span>
            </div>
            <div class="question-text">
                Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that:
                <br><strong>(i)</strong> p(0) = 5
                <br><strong>(ii)</strong> The polynomial p(x) − q(x) cuts the x-axis at (3, 0).
                <br><strong>(iii)</strong> The sum p(x) + q(x) is equal to 6x + 4 for all real x.
                <br>Find the polynomials p(x) and q(x).
            </div>
            <div class="solution">
                <p>Let <span class="math">p(x) = ax + b</span> and <span class="math">q(x) = cx + d</span></p>

                <p>Using condition (i), we have <span class="math">p(0) = 5</span></p>
                <div class="math-block">
                    ⇒ a(0) + b = 5<br>
                    ⇒ <strong>b = 5</strong><br>
                    So, p(x) = ax + 5
                </div>

                <p>Now, using condition (iii), we have</p>
                <div class="math-block">
                    p(x) + q(x) = 6x + 4<br>
                    ⇒ (ax + 5) + (cx + d) = 6x + 4<br>
                    ⇒ (a + c)x + (5 + d) = 6x + 4
                </div>

                <p>Comparing coefficients, we get:</p>
                <div class="math-block">
                    a + c = 6 …(1)<br>
                    5 + d = 4<br>
                    ⇒ <strong>d = –1</strong><br>
                    So, q(x) = cx – 1
                </div>

                <p>Using condition (ii), <span class="math">p(x) − q(x)</span> cuts x-axis at (3, 0)</p>
                <div class="math-block">
                    ⇒ p(3) − q(3) = 0
                </div>
                <p>Here:</p>
                <div class="math-block">
                    p(3) = 3a + 5<br>
                    q(3) = 3c − 1
                </div>

                <div class="math-block">
                    So, (3a + 5) − (3c − 1) = 0<br>
                    ⇒ 3a + 5 − 3c + 1 = 0<br>
                    ⇒ 3a − 3c + 6 = 0<br>
                    ⇒ a − c = –2 …(2)
                </div>

                <p><strong>Solving equations:</strong></p>
                <div class="math-block">
                    From (1): a + c = 6<br>
                    From (2): a − c = –2<br>
                    Adding both: 2a = 4 ⇒ <strong>a = 2</strong><br>
                    Then: 2 + c = 6 ⇒ <strong>c = 4</strong>
                </div>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Hence:<br>
                    <span class="math">p(x) = 2x + 5</span><br>
                    <span class="math">q(x) = 4x − 1</span></span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 12 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">12</span>
            </div>
            <div class="question-text">
                Look at the first three stages of a growing pattern of hexagons made using matchsticks.
                A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage.
                <br><strong>(i)</strong> Draw the next two stages of the pattern. How many matchsticks will be required at these stages?
                <br><strong>(ii)</strong> Complete the following table.
                <br><strong>(iii)</strong> Find a rule to determine the number of matchsticks required for the nᵗʰ stage.
                <br><strong>(iv)</strong> How many matchsticks will be required for the 15th stage of the pattern?
                <br><strong>(v)</strong> Can 200 matchsticks form a stage in this pattern? Justify your answer.
            </div>
            <div class="solution">
                <p>A single hexagon needs 6 matchsticks.</p>
                <p>Since each new hexagon shares one side with the previous hexagon, only 5 new matchsticks are added at every new stage.</p>

                <p>So the pattern is:</p>
                <div class="math-block">
                    Stage 1 = 6<br>
                    Stage 2 = 6 + 5 = 11<br>
                    Stage 3 = 11 + 5 = 16
                </div>

                <!-- (i) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(i)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Next two stages
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <div class="math-block">
                            Stage 4: Number of matchsticks = 16 + 5 = <strong>21</strong><br>
                            Stage 5: Number of matchsticks = 21 + 5 = <strong>26</strong>
                        </div>
                        <div class="result-box">
                            <span class="icon">✅</span>
                            <span>Stage 4 requires <strong>21 matchsticks</strong><br>
                            Stage 5 requires <strong>26 matchsticks</strong></span>
                        </div>
                    </div>
                </div>

                <!-- (ii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(ii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Complete the table
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <div class="table-wrap">
                            <table>
                                <thead>
                                    <tr>
                                        <th>Stage (n)</th>
                                        <th>Matchsticks</th>
                                    </tr>
                                </thead>
                                <tbody>
                                    <tr><td>1</td><td>6</td></tr>
                                    <tr><td>2</td><td>11</td></tr>
                                    <tr><td>3</td><td>16</td></tr>
                                    <tr><td>4</td><td>21</td></tr>
                                    <tr><td>5</td><td>26</td></tr>
                                </tbody>
                            </table>
                        </div>
                    </div>
                </div>

                <!-- (iii) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(iii)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Rule for the nᵗʰ stage
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>The number of matchsticks forms an arithmetic pattern: 6, 11, 16, 21, 26, …</p>
                        <ul>
                            <li>First term = 6</li>
                            <li>Common difference = 5</li>
                        </ul>
                        <div class="math-block">
                            Number of matchsticks in the nᵗʰ stage<br>
                            = 6 + (n – 1) × 5<br>
                            = 6 + 5n – 5<br>
                            = <strong>5n + 1</strong>
                        </div>
                        <div class="result-box">
                            <span class="icon">✅</span>
                            <span>Hence, the rule is <span class="math">Mₙ = 5n + 1</span>.</span>
                        </div>
                    </div>
                </div>

                <!-- (iv) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(iv)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Matchsticks for the 15th stage
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <div class="math-block">
                            M₁₅ = 5(15) + 1<br>
                            = 75 + 1<br>
                            = <strong>76</strong>
                        </div>
                        <div class="result-box">
                            <span class="icon">✅</span>
                            <span>Therefore, <strong>76 matchsticks</strong> will be required for the 15th stage.</span>
                        </div>
                    </div>
                </div>

                <!-- (v) -->
                <div class="sub-question">
                    <div class="q-header">
                        <span class="q-label sub">(v)</span>
                    </div>
                    <div class="question-text" style="border-left-color:#a6c4e4; font-weight:500;">
                        Can 200 matchsticks form a stage?
                    </div>
                    <div class="solution" style="border-left-color:#6b9fd4;">
                        <p>For some stage n,</p>
                        <div class="math-block">
                            5n + 1 = 200<br>
                            ⇒ 5n = 199<br>
                            ⇒ n = 199/5<br>
                            ⇒ n = 39.8
                        </div>
                        <p>Since n is not a whole number, 200 matchsticks cannot form any stage in this pattern.</p>
                        <div class="result-box warning">
                            <span class="icon">❌</span>
                            <span>Therefore, <strong>200 matchsticks cannot form a stage</strong> in this pattern.</span>
                        </div>
                    </div>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 13 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">13</span>
            </div>
            <div class="question-text">
                Let p(x) = ax + b and q(x) = cx + d be two linear polynomials such that:
                <br><strong>(i)</strong> The graph of p(x) passes through the points (2, 3) and (6, 11).
                <br><strong>(ii)</strong> The graph of q(x) passes through the point (4, −1).
                <br><strong>(iii)</strong> The graph of q(x) is parallel to the graph of p(x).
                <br>Find the polynomials p(x) and q(x). Also, find the coordinates of the point where these lines meet the x-axis.
            </div>
            <div class="solution">
                <p>Given: <span class="math">p(x) = ax + b</span> and <span class="math">q(x) = cx + d</span></p>

                <p><strong>First, to find p(x):</strong></p>
                <p>Since p(x) passes through (2, 3) and (6, 11), its slope is:</p>
                <div class="math-block">
                    m = (11 – 3) / (6 – 2)<br>
                    = 8/4<br>
                    = <strong>2</strong>
                </div>
                <p>So, <span class="math">p(x) = 2x + b</span></p>

                <p>Using the point (2, 3), we have:</p>
                <div class="math-block">
                    3 = 2(2) + b<br>
                    ⇒ 3 = 4 + b<br>
                    ⇒ <strong>b = −1</strong>
                </div>
                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Therefore, <span class="math">p(x) = 2x – 1</span>.</span>
                </div>

                <p><strong>Now to find q(x):</strong></p>
                <p>Since q(x) is parallel to p(x), it has the same slope.</p>
                <p>So, slope of <span class="math">q(x) = 2</span></p>
                <p>Hence, <span class="math">q(x) = 2x + d</span></p>

                <p>Since q(x) passes through (4, −1):</p>
                <div class="math-block">
                    −1 = 2(4) + d<br>
                    ⇒ −1 = 8 + d<br>
                    ⇒ <strong>d = −9</strong>
                </div>
                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Therefore, <span class="math">q(x) = 2x – 9</span>.</span>
                </div>

                <p><strong>Now we have to find where these lines meet the x-axis.</strong></p>
                <p>A line meets the x-axis where <span class="math">y = 0</span>.</p>

                <p>For <span class="math">p(x) = 2x – 1</span>:</p>
                <div class="math-block">
                    0 = 2x – 1<br>
                    ⇒ 2x = 1<br>
                    ⇒ x = 1/2
                </div>
                <p>So, p(x) meets the x-axis at <span class="math">(1/2, 0)</span>.</p>

                <p>For <span class="math">q(x) = 2x – 9</span>:</p>
                <div class="math-block">
                    0 = 2x – 9<br>
                    ⇒ 2x = 9<br>
                    ⇒ x = 9/2
                </div>
                <p>So, q(x) meets the x-axis at <span class="math">(9/2, 0)</span>.</p>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Hence, <span class="math">p(x) = 2x – 1</span> and <span class="math">q(x) = 2x – 9</span>.<br>
                    The x-axis intercepts are:<br>
                    For p(x): <span class="math">(1/2, 0)</span><br>
                    For q(x): <span class="math">(9/2, 0)</span></span>
                </div>
            </div>
        </div>

        <!-- ============================================================ -->
        <!-- QUESTION 14 -->
        <!-- ============================================================ -->
        <div class="question-block">
            <div class="q-header">
                <span class="q-label">14</span>
            </div>
            <div class="question-text">
                What do all linear functions of the form f(x) = ax + a, a > 0, have in common?
            </div>
            <div class="solution">
                <p>Given: <span class="math">f(x) = ax + a</span>, where <span class="math">a > 0</span></p>
                <p>We can write it as: <span class="math">f(x) = a(x + 1)</span></p>

                <p><strong>Common properties of all such linear functions are:</strong></p>

                <ul>
                    <li>
                        <strong>Slope:</strong><br>
                        The slope is <span class="math">a</span>, and since <span class="math">a > 0</span>, all the lines have positive slope.<br>
                        So, all these lines rise from left to right.
                    </li>
                    <li>
                        <strong>y-intercept:</strong><br>
                        Putting <span class="math">x = 0</span>,<br>
                        <span class="math">f(0) = a</span><br>
                        So, the y-intercept is <span class="math">(0, a)</span>.<br>
                        Since <span class="math">a > 0</span>, all the lines cut the y-axis above the origin.
                    </li>
                    <li>
                        <strong>x-intercept:</strong><br>
                        To find where the line cuts the x-axis, put <span class="math">f(x) = 0</span><br>
                        <span class="math">ax + a = 0</span><br>
                        <span class="math">a(x + 1) = 0</span><br>
                        Since <span class="math">a > 0</span>, <span class="math">a</span> is not zero.<br>
                        So, <span class="math">x + 1 = 0</span> or <span class="math">x = −1</span><br>
                        Thus, every line cuts the x-axis at the same point <span class="math">(−1, 0)</span>.
                    </li>
                </ul>

                <div class="result-box">
                    <span class="icon">✅</span>
                    <span>Therefore, all linear functions of the form <span class="math">f(x) = ax + a</span>, <span class="math">a > 0</span>, have the following in common:<br>
                    • all have <strong>positive slope</strong><br>
                    • all cut the <strong>y-axis above the origin</strong><br>
                    • all pass through the fixed point <strong>(−1, 0)</strong>.</span>
                </div>
            </div>
        </div>

        <!-- ===== FOOTER ===== -->
        <div class="footer-note">
            <span>⚡ Complete step‑by‑step solutions</span>
            <span class="dot"></span>
            <span>NCERT / Ganita Manjari · Chapter 2</span>
        </div>

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