Class 9 Mathematics - Ch 10: Practical Geometry Chapter Mock Test

<p><strong>Step-by-step Solution:</strong></p> <ol> <li>The rational number is $\frac{3}{4} = 0.75$, which lies between $0$ and $1$ on the positive real axis.</li> <li>Divide the unit interval between $0$ and $1$ into $4$ equal segments (each representing $\frac{1}{4} = 0.25$).</li> <li>Count $3$ parts from $0$ towards the right to locate and mark the point $\frac{3}{4}$.</li> </ol> <div style="margin: 16px 0; text-align: center;"> <svg viewBox="0 0 600 120" style="max-width: 100%; height: auto; font-family: system-ui, sans-serif;"> <rect width="600" height="120" rx="8" fill="#f8fafc" stroke="#e2e8f0" stroke-width="1.5"/> <!-- Axis --> <line x1="40" y1="70" x2="560" y2="70" stroke="#334155" stroke-width="2.5" stroke-linecap="round"/> <polygon points="560,65 572,70 560,75" fill="#334155"/> <polygon points="40,65 28,70 40,75" fill="#334155"/> <!-- Major Ticks --> <line x1="120" y1="60" x2="120" y2="80" stroke="#334155" stroke-width="2"/> <text x="120" y="100" font-size="14" font-weight="bold" fill="#1e293b" text-anchor="middle">-1</text> <line x1="240" y1="55" x2="240" y2="85" stroke="#0284c7" stroke-width="2.5"/> <text x="240" y="100" font-size="14" font-weight="bold" fill="#0284c7" text-anchor="middle">0 (Origin)</text> <line x1="360" y1="60" x2="360" y2="80" stroke="#334155" stroke-width="2"/> <text x="360" y="100" font-size="14" font-weight="bold" fill="#1e293b" text-anchor="middle">1</text> <line x1="480" y1="60" x2="480" y2="80" stroke="#334155" stroke-width="2"/> <text x="480" y="100" font-size="14" font-weight="bold" fill="#1e293b" text-anchor="middle">2</text> <!-- Sub-divisions between 0 and 1: 1/4 (270), 2/4 (300), 3/4 (330) --> <line x1="270" y1="64" x2="270" y2="76" stroke="#94a3b8" stroke-width="1.5"/> <text x="270" y="58" font-size="10" fill="#64748b" text-anchor="middle">1/4</text> <line x1="300" y1="64" x2="300" y2="76" stroke="#94a3b8" stroke-width="1.5"/> <text x="300" y="58" font-size="10" fill="#64748b" text-anchor="middle">2/4</text> <line x1="330" y1="64" x2="330" y2="76" stroke="#94a3b8" stroke-width="1.5"/> <!-- Target Point: 3/4 --> <circle cx="330" cy="70" r="6" fill="#dc2626" stroke="#fff" stroke-width="2"/> <rect x="295" y="15" width="70" height="26" rx="4" fill="#dc2626"/> <text x="330" y="33" font-size="12" font-weight="bold" fill="#ffffff" text-anchor="middle">3/4 = 0.75</text> <line x1="330" y1="41" x2="330" y2="62" stroke="#dc2626" stroke-width="2" stroke-dasharray="2,2"/> </svg> </div>

<p><strong>Step-by-step Solution:</strong></p> <ol> <li>Find the decimal approximation: $\sqrt{8} \approx 2.8284$, so $-\sqrt{8} \approx -2.828$.</li> <li>Since $-\sqrt{8}$ is negative, it lies on the left of $0$, specifically between $-2$ and $-3$ (closer to $-3$).</li> <li>Divide the segment between $-2$ and $-3$ into tenths and plot the point at approximately $-2.83$.</li> </ol> <div style="margin: 16px 0; text-align: center;"> <svg viewBox="0 0 650 130" style="max-width: 100%; height: auto; font-family: system-ui, sans-serif;"> <rect width="650" height="130" rx="8" fill="#f8fafc" stroke="#e2e8f0" stroke-width="1.5"/> <!-- Axis --> <line x1="40" y1="75" x2="610" y2="75" stroke="#334155" stroke-width="2.5" stroke-linecap="round"/> <polygon points="610,70 622,75 610,80" fill="#334155"/> <polygon points="40,70 28,75 40,80" fill="#334155"/> <!-- Ticks -4 to 4 --> <line x1="90" y1="65" x2="90" y2="85" stroke="#334155" stroke-width="2"/> <text x="90" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">-4</text> <line x1="150" y1="65" x2="150" y2="85" stroke="#334155" stroke-width="2"/> <text x="150" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">-3</text> <line x1="210" y1="65" x2="210" y2="85" stroke="#334155" stroke-width="2"/> <text x="210" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">-2</text> <line x1="270" y1="65" x2="270" y2="85" stroke="#334155" stroke-width="2"/> <text x="270" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">-1</text> <line x1="330" y1="60" x2="330" y2="90" stroke="#0284c7" stroke-width="2.5"/> <text x="330" y="107" font-size="14" font-weight="bold" fill="#0284c7" text-anchor="middle">0</text> <line x1="390" y1="65" x2="390" y2="85" stroke="#334155" stroke-width="2"/> <text x="390" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">1</text> <line x1="450" y1="65" x2="450" y2="85" stroke="#334155" stroke-width="2"/> <text x="450" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">2</text> <line x1="510" y1="65" x2="510" y2="85" stroke="#334155" stroke-width="2"/> <text x="510" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">3</text> <line x1="570" y1="65" x2="570" y2="85" stroke="#334155" stroke-width="2"/> <text x="570" y="105" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">4</text> <!-- Target Point: -sqrt(8) = -2.828 --> <circle cx="160.3" cy="75" r="6" fill="#dc2626" stroke="#fff" stroke-width="2"/> <rect x="110" y="15" width="100" height="28" rx="4" fill="#dc2626"/> <text x="160.3" y="34" font-size="12" font-weight="bold" fill="#ffffff" text-anchor="middle">-√8 ≈ -2.828</text> <line x1="160.3" y1="43" x2="160.3" y2="67" stroke="#dc2626" stroke-width="2" stroke-dasharray="2,2"/> </svg> </div>

Since $0.2 = \frac{1}{5}$, multiplying $\frac{1}{5} \times 5 = 1$ satisfies $a \cdot \frac{1}{a} = 1$.

Multiplying $-3$ across each term: $(-3)(2) - (-3)(y) = -6 + 3y$.

<p><strong>Step-by-step Solution:</strong></p> <ol> <li>Draw a horizontal real number line with markings from $-6$ to $+6$.</li> <li>For the left bound $x > -4$ (strict inequality), plot a <strong>hollow (unfilled) circle</strong> at $-4$ to indicate $-4$ is excluded.</li> <li>For the right bound $x \le 4$ (non-strict inequality), plot a <strong>solid (filled) circle</strong> at $+4$ to indicate $+4$ is included.</li> <li>Shade the continuous line segment between $-4$ and $+4$ to represent all real values in the interval $(-4, 4]$.</li> </ol> <div style="margin: 16px 0; text-align: center;"> <svg viewBox="0 0 650 120" style="max-width: 100%; height: auto; font-family: system-ui, sans-serif;"> <rect width="650" height="120" rx="8" fill="#f8fafc" stroke="#e2e8f0" stroke-width="1.5"/> <!-- Axis --> <line x1="30" y1="65" x2="620" y2="65" stroke="#94a3b8" stroke-width="2" stroke-linecap="round"/> <polygon points="620,60 632,65 620,70" fill="#94a3b8"/> <polygon points="30,60 18,65 30,70" fill="#94a3b8"/> <!-- Shaded Line Segment from -4 to +4 --> <line x1="145" y1="65" x2="505" y2="65" stroke="#0284c7" stroke-width="6"/> <!-- Ticks -6 to 6 --> <line x1="145" y1="55" x2="145" y2="75" stroke="#334155" stroke-width="2"/> <text x="145" y="95" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">-4</text> <line x1="325" y1="52" x2="325" y2="78" stroke="#334155" stroke-width="2.5"/> <text x="325" y="95" font-size="13" font-weight="bold" fill="#64748b" text-anchor="middle">0</text> <line x1="505" y1="55" x2="505" y2="75" stroke="#334155" stroke-width="2"/> <text x="505" y="95" font-size="13" font-weight="bold" fill="#1e293b" text-anchor="middle">+4</text> <!-- Endpoints --> <circle cx="145" cy="65" r="7" fill="#ffffff" stroke="#dc2626" stroke-width="3.5"/> <circle cx="505" cy="65" r="7" fill="#16a34a" stroke="#16a34a" stroke-width="2"/> <!-- Callout labels --> <rect x="70" y="12" width="150" height="24" rx="4" fill="#fee2e2" stroke="#fca5a5"/> <text x="145" y="28" font-size="11" font-weight="bold" fill="#991b1b" text-anchor="middle">Hollow circle (x &gt; -4)</text> <rect x="430" y="12" width="150" height="24" rx="4" fill="#dcfce7" stroke="#86efac"/> <text x="505" y="28" font-size="11" font-weight="bold" fill="#166534" text-anchor="middle">Solid circle (x ≤ 4)</text> </svg> </div>

Multiplying both sides of an inequality by a negative real number ($c = -1 < 0$) reverses the inequality sign: $a < b \implies ac > bc$.

Step 1: $\sqrt[5]{4 \times 8} = \sqrt[5]{32}$. Step 2: Since $32 = 2^5$, $\sqrt[5]{2^5} = 2$.

Step 1: $\sqrt[4]{\frac{x^7}{x^5}} = \sqrt[4]{x^2}$. Step 2: Reduce index $(x^2)^{1/4} = x^{2/4} = x^{1/2} = \sqrt{x}$.

Step 1: $(216)^{-2/3} = \frac{1}{(216)^{2/3}}$. Step 2: Since $216 = 6^3$, $\frac{1}{(6^3)^{2/3}} = \frac{1}{6^2} = \frac{1}{36}$.

Step 1: $16^{\frac{1}{5} + \frac{1}{4} - (-\frac{3}{10})} = 16^{\frac{4+5+6}{20}} = 16^{\frac{15}{20}} = 16^{\frac{3}{4}}$. Step 2: $(2^4)^{\frac{3}{4}} = 2^3 = 8$.