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10th Class Mathematics Areas Related to Circles Question Bank MCQs - Areas Related to Circles

  • question_answer
      Following question has four statements (A, B, C and D) given in Column I and five statements (p, q, r, s, t) in Column - II. Any given statement in Column - I. can have correct matching with one or more statement(s) given in Column - II.
    Column - I Column - II
    A. OA = 26 cm, OC = 23 m Area of shaded region = ? (p) \[30\,c{{m}^{2}}\]
    B. AD = 6.5 cm, CA = 5 cm Area of \[\Delta ABC=?\] (q) \[115\,{{m}^{2}}\]
    C. BC = 6m, AB = 8 cm Area of the shaded region = ? (r) \[35\,c{{m}^{2}}\]
    D. OB = 14.8 m Area of the shaded region = ? (s) \[0.003\,{{m}^{2}}\]
    (t) \[30\,\,{{m}^{2}}\]

    A) Codes \[\left( A \right)\to p,\,s;\,\left( B \right)\to q;\,\left( C \right)\to q;\,\left( D \right)\to r,s\]

    B) \[\left( A \right)\to q;\,\left( B \right)\to p,s;\,\left( C \right)\to p,s;\left( D \right)\to q\]

    C) \[\left( A \right)\to p,s;\,\left( B \right)\to q;\,\left( C \right)\to p,s;\,\left( D \right)\to r,s\]

    D) None of the above

    Correct Answer: D

    Solution :

    [A] Area of the shaded region
    \[=\frac{90}{360{}^\circ }\pi \left[ {{\left( 26 \right)}^{2}}-{{\left( 23 \right)}^{2}} \right]\]
    \[=\frac{1}{4}\times \frac{22}{7}\left[ {{\left( 26 \right)}^{2}}-{{\left( 23 \right)}^{2}} \right]\]
    \[=\frac{1}{4}\times \frac{22}{7}\left[ \left( 26+23 \right)\left( 26-33 \right) \right]\]
    \[=115.5\simeq 115\,{{m}^{2}}\]
    [B] \[AB=2\times AD\]
    \[=2\times 6.5=13\,cm\]
    \[BC=\sqrt{{{\left( AB \right)}^{2}}-{{\left( AC \right)}^{2}}}\]
    \[=\sqrt{{{\left( 13 \right)}^{2}}-{{\left( 5 \right)}^{2}}}=12\,cm\]
    Area of \[\Delta ABC=\frac{1}{2}\times AC\times BC\]
    \[=\frac{1}{2}\times 5\times 12=30\,c{{m}^{2}}\]
    \[=0.003\,\,{{m}^{2}}\]
    [C] \[A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}=64+36=100\]
    \[AC=10\,cm\]
    Area of the shaded region
    = (area of the circle)
    - (area of the rectangle ABCD)
    \[=\left[ \frac{22}{7}\times {{\left( \frac{10}{2} \right)}^{2}} \right]-\left( 8\times 6 \right)\]
    \[=\left( 78.57-48 \right)\text{ }=30.57\text{ }c{{m}^{2}}\]
    [D] Area of the shaded region
    \[=\frac{60}{360{}^\circ }\times \frac{22}{7}\times {{\left( 14.8 \right)}^{2}}\]
    \[\left[ \because \,\,area\,of\,\sec tor\,=\frac{\theta }{360{}^\circ }\times \pi {{r}^{2}} \right]\]
    \[=114.7\simeq 115{{m}^{2}}\]


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