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Maths Assertion (A) & Reason (B) MCQ

Circles · CBSE STD 10 (CBSE - English Medium). 10 questions.

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Assertion (A) & Reason (B) MCQ

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10 questions · timed · auto-graded

MCQ 11 Mark
Statement-1 (A): The tangents drawn at the end points of a diameter of a circle are parallel.
Statement-2 (R): Diameter of a circle is the longest chord.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.
Answer
Correct option: B.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B
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MCQ 21 Mark
Statement-1 (A): A tangent to a circle is perpendicular to the radius through the point of contact.
Statement-2 (R): The lengths of tangents drawn from an external point to a circle are equal.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.
Answer
Correct option: B.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B 
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MCQ 31 Mark
Statement-1 (A): In Fig. O is the centre of a circle and $P Q$ is a chord. If the tangent $P R$ at $P$ makes an angle of $50^{\circ}$ with $P Q$, then $\angle P O Q=100^{\circ}$.
Statement-2 (R): A tangent to a circle is perpendicular to the radius through the point of contact.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • D
    Statement- 1 is False, Statement- 2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
Image
Statement-2 is true.
Using statement -2 , we find that $\angle O P R=90^{\circ}$. Therefore, $\angle O P Q=40^{\circ}$.
In $\triangle O P Q$, we have
$
O P=O Q \Rightarrow \angle O P Q=\angle O Q P \Rightarrow \angle O Q P=40^{\circ}
$
Thus, in $\triangle O P Q$, we have $\angle O P Q=\angle O Q P=40^{\circ}$
$
\angle P O Q=180^{\circ}-\left(40^{\circ}+40^{\circ}\right)=100^{\circ}
$
Thus, statement- 1 is true. Also, statement- 2 is a correct explanation of statement- 1 . Hence option (a) is correct.
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MCQ 41 Mark
Statement-1 (A): In Fig. if AT is tangent to the circle, with centre $O$, at point $A$ such that $O T=4 cm$, and $\angle O T A=30^{\circ}$, then $A T=4 \sqrt{3} cm$.
Statement-2 (R) : A tangent to a circle is perpendicular to the radius through the point of contact.
  • A
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • ✓
    Statement- 1 is False, Statement- 2 is True.
Answer
Correct option: D.
Statement- 1 is False, Statement- 2 is True.
(D)Statement- 1 is False, Statement- 2 is True.
Image
Statement-2 is true (See Theorem 1 on page 8.2 of main book). Using statement-2, we find that $O A \perp A T$. Consequently, $\triangle O A T$ is a right triangle. In this triangle, we obtain
$
\cos 30^{\circ}=\frac{A T}{O T} \Rightarrow \frac{\sqrt{3}}{2}=\frac{A T}{4} \Rightarrow A T=2 \sqrt{3} cm
$
So, statement-1 is not true. Hence, option (d) is correct.
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MCQ 51 Mark
Statement-1 (A): The base of an isosceles triangle is bisected at the point of contact of its incircle.
Statement-2 (R): The lengths of two tangents drawn from an external point to a circle are equal.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • D
    Statement- 1 is False, Statement- 2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
Statement-2 is true (See Theorem 3 on page 8.9 of main book). Let $A B C$ be an isosceles triangle with base $B C$ such that $A B=A C$. Let $O$ be the centre of the incircle touching sides $B C, C A$ and $A B$ at $D, E$ and $F$ respectively. Since tangents drawn from an external point to a circle are equal. Therefore, $A F=A E, B F=B D$ and $C D=C E$.
Image
$\begin{array}{ll}\text { Now, } & A B=A C \text { and } A F=A E \\ \Rightarrow & A B-A F=A C-A E \\ \Rightarrow & B F=C E \\ \Rightarrow & B D=C D \qquad [\because B F=B D &and& C E=C D]\\ \Rightarrow & B C \text { is bisected at } D .\end{array}$
Thus, statment- 1 is true and statement- 2 is a correct explanation for statement-1. Hence, option (a) is correct.
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MCQ 61 Mark
Statement-1 (A) : The Length of the tangent PT drawn from an external point $P$ to a circle is 24 cm . If the distance between the point $P$ and the centre $O$ of the circle is 25 cm , then the leng th of diameter of the circle is 14 cm .
Statement-2 (R): A tangent to a circle is perpendicular to the radius through the point of contact.
  • ✓
    Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement- 1 is True, Statement- 2 is False.
  • D
    Statement- 1 is False, Statement- 2 is True.
Answer
Correct option: A.
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
Image
Statement-2, being a standard result, is true. Using statement-2, we obtain a right triangle right angled at $T$. Applying Pythagoras theorem in $\triangle P T O$, we obtain
$
\begin{array}{ll}
& O P^2=O T^2+P T^2 \\
\Rightarrow \quad & O T=\sqrt{O P^2-P T^2}=\sqrt{25^2-24^2}=\sqrt{49}=7 cm \\
\therefore & \text { Length of the diameter }=14 cm .
\end{array}
$
Thus, statement-1 is also true and statement-2 is a correct explanation for statement-1. Hence, option (a) is correct.
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MCQ 71 Mark
Statement- 1 (A) : In Fig. $ P A$ and $P B$ are tangents drawn from an external point $P$ to a circle with centre $O$ such that $\angle A P B=40^{\circ}$. If $C$ is a point on the circle, then $\angle A C B=70^{\circ}$.
Statement-2 (R): The lengths of tangents drawn from $P$ to the circle are equal.
Image
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.
Answer
Correct option: B.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
B 
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MCQ 81 Mark
Statement-1 (A): If $P A$ and $P B$ are tangents drawn from an external point $P$ to a circle with centre $O$, then the quadrilateral $A O B P$ is cyclic.
Statement-2 (R): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.
  • D
    Statement- 1 is False, Statement- 2 is True.
Answer
Correct option: A.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
A 
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MCQ 91 Mark
Statement-1 (A): In Fig. (P A$ and $P B$ are tangents drawn from an external point $P$ to a circle with centre $O$. If $\angle A P B=80^{\circ}$, then $\angle A O B=100^{\circ}$.
Image
Statement-2 (R): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.
  • ✓
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.
Answer
Correct option: A.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
A 
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MCQ 101 Mark
Statement-1 (A): The length of the tangent drawn from a point at a distance of 13 cm from the centre of a circle of radius 5 cm is 10 cm .
Statement-2 (R): A tangent to a circle is perpendicular to the radius through the point of contact.
  • A
    Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
  • B
    Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
  • C
    Statement-1 is true, Statement- 2 is false.
  • ✓
    Statement- 1 is false, Statement- 2 is true.
Answer
Correct option: D.
Statement- 1 is false, Statement- 2 is true.
D 
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Assertion (A) & Reason (B) MCQ - Maths STD 10 Questions - Vidyadip