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MATHS True False[1 Marks ]

Circles · Rajasthan Board (RBSE) STD 9 (Rajasthan - English Medium). 18 questions.

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Question 11 Mark
If $\text{AOB}$ is a diameter of a circle and $C$ is a point on the circle, then $AC^2 + BC^2 = AB^2.$
Answer
Since, any diameter of the circle subtends a right angle to any point on the circle. If $\text{AOB}$ is a diameter of a circle and $C$ is a point on the circle, then $\triangle\text{ACB}$ is right angled at $C$. In right angled $\triangle\text{ACB}, [$use Pythagoras theorem$] AC^2 + BC^2= AB^2$
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Question 21 Mark
ABCD is a cyclic quadrilateral such that $\angle\text{A}=90^\circ,\angle\text{B}=70^\circ,\angle\text{C}=95^\circ$ and $\angle\text{D}=105^\circ.$
Answer
False.
Solution: In a cyclic quadrilateral, the sum of opposite angles is 180°. Now, $\angle\text{A}+\angle\text{C}=90^\circ+95^\circ=185^\circ\neq180^\circ$ and $\angle\text{B}+\angle\text{D}=70^\circ+105^\circ=175^\circ\neq180^\circ$ Here, we see that, the sum of opposite angles is not equal to 180°. So, it is not a cyclic quadrilateral.
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Question 31 Mark
The degree measure of a semi-circle is 180°
Answer
True. Explanation: Given that the degree measure of a semi-circle is 180°. As we know that the diameter ofa circle divides into two equal parts and each of these two arcs are known as semi-circle. $\widehat{\text{PQ}}$ and $\widehat{\text{QP}}$ are semi circle.
Hence, $\text{m}\big(\widehat{\text{PQ}}\big)=\text{m}\big(\widehat{\text{QP}}\big)=180^\circ$
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Question 41 Mark
A circle is a plane figure.
Answer
True.
Explanation: Given that a circle is a plane figure. As we know that a circle is a collection of those points in a plane that are at a given constant distance from a fixed point in the plane.
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Question 51 Mark
If A, B, C, D are four points such that $\angle\text{BAC}=30^\circ$ and $\angle\text{BDC}=60^\circ$ then D is the centre of the circle through A, B and C.
Answer
False.
Solution: Because, there can be many points D, such that $\angle\text{BDC}=60^\circ$ and each such point cannot be the centre of the circle through A, B and C.
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Question 61 Mark
Two congruent circles with centres O and O′ intersect at two points A and B. Then $\angle\text{AOB}=\angle\text{AO'B}.$
Answer
True.
Solution:

Join AB and OB, O'A and BO'. In $\triangle\text{AOB}$ and $\triangle\text{AO'B,}$ OA = AO' [both circles have same radius] OB = BO' [both circles have same radius] and AB = AB [common chord] $\triangle\text{AOB}=\triangle\text{AO'B}$ [by SSC congruence rule] $\Rightarrow\angle\text{AOB}=\angle\text{AO'B}$ [by CPCT]
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Question 81 Mark
If a circle is divided into three equal arcs each is a major arc.
Answer
True.
Explanation:
Given that if a circle is divided into three equal arcs each is a major arc.  As we know that if points P, Q and R lies on the given circle C(O, r) in such a way that: $1\big(\widehat{\text{PQ}}\big)=1\big(\widehat{\text{QR}}\big)=1\big(\widehat{\text{RP}}\big)$ Then each arc is called major arc.
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Question 91 Mark
Through three collinear points a circle can be drawn.
Answer
False.
Solution:
 Because, circle can pass through only two collinear points but not through three collinear points.
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Question 101 Mark
A circle has only finite number of equal chords.
Answer
False.
Explanation:
It is given that a circle has only finite number of equal chords. As we know that a circle having infinite number of unequal chords.
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Question 111 Mark
Line segment joining the center to any point on the circle is a radius of the circle,
Answer
True.
Explanation:
Given that line segment joining the centre to any point on the circle is a radius of the circle.
As we know that line segment joining the centre to any point on the circle is a radius of the circle.
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Question 121 Mark
Sector is the region between the chord and its corresponding arc.
Answer
True.
Explanation:
It is given that sector is the region between the chord and its corresponding arc.
As we know that the region between the chord and its corresponding arc is called sector.
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Question 131 Mark
A circle of radius 3cm can be drawn through two points A, B such that AB = 6cm.
Answer
True.
Solution:
Suppose, we consider diameter of a circle is AB = 66m.
Then, radius of a circle $=\frac{\text{AB}}{2}=\frac{6}{2}=3\text{cm},$ which is true.
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Question 141 Mark
A chord of a circle, which is twice as long is its radius is a diameter of the circle.
Answer
True.
Explanation:
 Given that a chord of the circle, which is twice as long as its radius is diameter of the circle. As we know that a chord of a circle which is largest to others and passing through the centre of the circle and twice as long as its radius is called diameter of the circle.
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Question 151 Mark
Two chords AB and AC of a circle with centre O are on the opposite sides of OA. Then $\angle\text{OAB}=\angle\text{OAC}.$
Answer
False. Solution: In figure, AB and AC are two chords of a circle. Join OB and OC.

In $\triangle\text{OAB}$ and $\triangle\text{OAC},$ OA = OA [common side] OB = OC [both are the radius of circle] Here, we are not able to show that either the any angle or third side is equal and $\triangle\text{OAB}$ is, $\therefore\angle\text{OAB}\neq\angle\text{OAC}.$
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Question 161 Mark
In Fig. if AOB is a diameter and $\angle\text{ADC}=120^\circ,$ then $\angle\text{CAB}=30^\circ.$
Answer
True.Solution: Join CA and CB.

Since, ADCB is a cyclic quadrilateral. $\angle\text{ADC}+\angle\text{CBA}=180^\circ$ [sum of opposite angles of cyclic quadrilateral is 180°] $\Rightarrow\angle\text{CBA}=180^\circ-120^\circ=60^\circ\ \ [\therefore\angle\text{ADC}=120^\circ]$ In $\triangle\text{ACB,}\ \angle\text{CAB} + \angle\text{CBA} + \angle\text{ACB} = 180^\circ$ [by angle sum property of a triangle] $\angle\text{CAB}+60^\circ+90^\circ=180^\circ$ $\big[$triangle formed from diameter to the circle is 90° i.e., $\angle\text{ACB}=90^\circ\big]$ $\Rightarrow\angle\text{CAB}=180^\circ-150^\circ=30^\circ.$
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Question 181 Mark
The degree measure of an arc is the complement of the central angle containing the arc.
Answer
False.
Explanation:
Given that the degree measure of an arc is the complement of the central angle containing the arc. As we know that the degree measure of a minor arc is the measure of the central angle containing the arc and that of a major arc is 360° minus the degree measure of the corresponding minor arc. Let degree measure of an arc $\widehat{\text{PQ}}$ is $\theta$ of a given circle C(O, r) is denoted by $\text{m}\big(\widehat{\text{PQ}}\big)=\theta.$
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True False[1 Marks ] - MATHS STD 9 Questions - Vidyadip