objective Ques (153 results)
SSC CGL Mains 20241)
11 : 15
SSC CGL Mains 20242)In a triangle HJK, HJ = HK. G is a point on HJ such that HG = GK = JK. What is the degree measure of two-third of (∠HGK + ∠GKJ)?
96°
SSC CGL Mains 20243)∆ABC is inscribed in a circle with Centre O. If AB = 21 cm, BC = 20 cm and AC = 29 cm, then what is the length of the circumradius of the triangle?
14.5 cm
SSC CGL Mains 20244)Let PQR be a right angled triangle, right-angled at R. Let PQ = 29 cm, QR = 21 cm and ∠Q = θ. Find the value of cos 2 θ – sin 2 θ.
A)
B)
C)
D)
.
SSC CGL 20225)If the areas of two isosceles triangles are in the ratio of x2 ∶ y2, then the ratio of their corresponding heights is:
x ∶ y
SSC CGL 20226)What is the ASA congruence rule of triangles, where A and S represents angle and side of triangle respectively?
Two triangles are said to be congruent if 2 angles and the included side of one triangle are equal to 2 angles and the included side of the other triangle.
SSC CGL 20227)
△ABC is a right-angle triangle at B and tan A = \(\frac{3}{4} \), then sin A + sin B + sin C will be equal to:
\(2\frac{2}{5}\)
SSC CGL 20228)If ∆ABC ~ ∆EDF such that AB = 6 cm, DF = 16 cm and DE = 8 cm, then the length of BC is:
12 cm
SSC CGL 20229)In ∆ ABC, D and E are points on sides AB and AC, such that DE ΙΙ BC. If AD = x + 3, DB = 2x − 3, AE = x + 1 and EC = 2x − 2, then the value of x is:
\({{3} \over 5}\)
SSC CGL 202210)In ΔPQR, ∠Q = 90°, PQ = 8 cm and ∠PRQ = 45°. Find the length of QR.
8 cm
SSC CGL 202211)ΔABC ~ ΔDEF and the perimeters of these triangles are 32 cm and 12 cm, respectively. If DE = 6 cm, then what will be the length of AB?
16 cm
SSC CGL 202212)ΔABC and ΔDEF are similar triangles and their areas are 49 cm2 and 144 cm2 respectively. If EF = 16.80 cm, then find BC.
9.8 cm
SSC CGL 202213)If ΔABC ≅ ΔPQR and ∠ABC = (x + 60)°, ∠PQR = (85 - 4x)°, and ∠RPQ = (3x + 65)°, then the value of ∠ABC in degree is:
65
SSC CGL 202214)An airplane goes 14 km due east and then 48 km due north. How far is it from its initial position?
50 km
SSC CGL 202215)The side of an equilateral triangle is 9 cm. What is the radius of the circle circumscribing this equilateral triangle?
\(3 \sqrt{3}\) cm
SSC CGL 202216)'O' is a point in the interior of an equilateral triangle. The perpendicular distance from 'O' to the sides are\( \sqrt3 cm\), 2 cm, 5 cm. The perimeter of the triangle is :
48 cm
SSC CGL 202217)ΔPQR is right-angled at Q. The length of PQ is 5 cm and ∠PRQ = 30°. Determine the length of side QR.
\(5\sqrt3 cm\)
SSC CGL 202218)Find the area of the shaded portion of an equilateral triangle with sides 6 units shown in the following figure. A circle of radius 1 unit is centred at midpoint of a side of the triangle.
\(\frac{1}{2}\left( {9\sqrt 3 - \frac{{11}}{7}} \right)\) \(unit^{2}\)
SSC CGL 202219)In a Δ ABC, ∠B + ∠C = 110°, then find the measure of ∠A.
70°
SSC CGL 202220)The area of a triangle is 480 cm2 and the ratio of its sides is 10 ∶ 24 26. What is the perimeter of the triangle?
120 cm
SSC CGL 202221)In the given triangle, CD is the bisector of ∠BCA. CD = DA. If ∠BDC = 76°, What is the degree measure of ∠CBD?
66°
SSC CGL 202222)In triangle ABC, the bisector of angle BAC cuts the line BC at D. If BD = 6 and BC = 14 then what is the value of AB : AC?
3 : 4
SSC CGL 202223)From the following figure find x+ y + z.
120°
SSC CGL 202224)If the figure, AB = AD = 7 cm and AC = AE and BC = 11 cm, then find the length of ED.
11
SSC CGL 202225)If the angles of a triangle are (x - 46) degrees, (x + 96) degrees and 8x degrees, then what is the value of 2x?
26 degrees
SSC CGL 202226)The side of an equilateral triangle is 12 cm. What is the radius of the circle circumscribing this equilateral triangle?
4√3 cm
SSC CGL 202227)If areas of similar triangles ΔABC and ΔDEF are x2 cm2 and y2 cm2 respectively, and EF = a cm, then BC (in cm) is:
\(\frac{a x}{y}\)
SSC CGL 202228)In the triangle ABC, AB = 12 cm and AC = 10 cm, and ∠BAC = 60°. What is the value of the length of the side BC?
11.13 cm
SSC CGL 202229)In a right-angled triangle PQR, right-angled at Q, the length of the side PR is 17 units, length of the base QR is 8 units, and length of the side PQ is 15 units. If ∠RPQ = α, then sin α + cos α is:
\(\frac{23}{17}\)
SSC CGL 202230)In ∆ABC, the perpendiculars drawn from A, B and C meet the opposite sides at points D, E and F, respectively. AD, BE and CF intersect at point P. If ∠EPD = 110° and the bisectors of ∠A and ∠B meet at point Q, then ∠AQB = ?
125°
SSC CGL 202231)In the following figure, AD bisects angle BAC. Find the length (in cm) of BD.
4
SSC CGL 202232)In a triangle ABC, points P and Q are AB and AC, respectively, such that AP = 4 cm, PB = 6 cm, AQ = 5 cm and QC = 7.5 cm. If PQ = 6 cm, then find BC (in cm).
15
SSC CGL 202233)In triangle ABC, the bisector of angle BAC meets BC at point D in such a way that AB = 10 cm, AC = 15 cm and BD = 6 cm. Find the length of BC (in cm).
15
SSC CGL 202234)In a ΔABC, D, E and F are the mid-points of side BC, CA and AB respectively. If BC = 25.6 cm, CA = 18.8 cm and AB = 20.4 cm, what is the perimeter (in cm) of the ΔDEF?
32.4
SSC CGL 202235)In a right-angled triangle PQR, ∠Q = 90°. A and B are the mid-points of PQ and PR, respectively. If PQ = 16 cm, QR = 30 cm and PR = 34 cm, what is perimeter (in cm) of the trapezium ABRQ?
70
SSC CGL 202236)In △ABC, ∠A = 68°. If I is the incentre of the triangle, then the measure of ∠BIC is:
124°
SSC CGL 202237)In triangle ABC, X and Y are the points on sides AB and AC, respectively, such that XY is parallel to BC. If XY : BC = 2.5 : 7, what is the ratio of the area of the trapezium BCYX to that of the ΔAXY?
\(171\over25\)
SSC CGL 202238)In ΔPQR, ∠Q = 66° and ∠R = 34. T is a point on QR , and S is a point between Q and T such that PS ⊥ QR and PT is the bisector of ∠QPR. What is the measure of ∠SPT?
16
SSC CGL 202239)A triangle with the lengths of its sides proportional to the numbers 7, 24 and 30 is:
obtuse angled
SSC CGL 202240)In a right triangle ABC, right angled at B, altitude BD is drawn to the hypotenuse AC of the triangle. If AD = 6 cm, CD = 5 cm, then find the value of AB2 + BD2 (in cm).
96
SSC CGL 202241)In a right-angled triangle, the lengths of the medians from the vertices of acute angles are 7 cm and \( 4\sqrt 6\) cm. What is the length of the hypotenuse of the triangle (in cm)?
\(2\sqrt{29}\)
SSC CGL 202242)In a triangle ABC, the bisector of angle BAC meets BC at point D such that DC = 2BD. If AC - AB = 5 cm, then find the length of AB (in cm).
5
SSC CGL 202243)A circle is inscribed in ΔABC, touching AB, BC and AC at the points P, Q and R, respectively: If AB - BC = 4 cm, AB - AC = 2 cm and the perimeter of ΔABC = 32 cm, then \(BC \over 2\) (in cm) = ?
\(13 \over 3\)
SSC CGL 202244)In ΔABC, ∠A = 66°, BD ⊥ AC and CE ⊥ AB. BD and EC intersect at P. The bisectors ∠PBC and ∠PCB meet at Q. What is the measure of ∠BQC?
147°
SSC CGL 202245)The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
9.6
SSC CGL 202246)Let ΔABC ~ ΔQPR and (Area of ΔABC) : (Area of ΔPQR) = 121 : 64. If QP = 14.4 cm, PR = 12 cm and AC = 18 cm, then what is the length of AB?
19.8 cm
SSC CGL 202247)In a ΔABC, D, E and F are the mid-points of side BC, CA and AB respectively. If BC = 14.4 cm, CA = 15.2 cm and AB = 12.4 cm, what is the perimeter (in cm) of the ΔDEF?
21
SSC CGL 202248)In Δ PQR, S is a point on the side QR such that PS is the bisector of ∠QPR. If PQ = 12 cm, QS = 3 cm and QR = 7 cm, then what is the length of side PR?
18 cm
SSC CGL 202249)The area of similar triangles PQR and MNT are 196 cm2 and 169 cm2 respectively. If the longest side of the larger Δ PQR be 28 cm then what is the length (in cm) of the longest side of the smaller Δ MNT?
26
SSC CGL 202250)In ΔABC, AB = 7 cm, BC = 10 cm, and AC = 8 cm. If AD is the angle bisector of ∠BAC, where D is a point on BC, then \( \frac{DC}{4}\) (in cm) is equal to:
\(4\over 3\)
SSC CGL 202251)The sides AB and AC of ΔABC are produced to points D and E, respectively. The bisectors of ∠CBD and ∠BCE meet at P. If ∠A = 88°, the measure of ∠P is :
46°
SSC CGL 202252)In a ΔABC, the bisector of ∠A meets BC at D. If AB = 9.6 cm, AC = 11.2 cm and BD = 4.8 cm, the perimeter (in cm) of ΔABC is:
31.2
SSC CGL 202253)A circle is inscribed in ΔABC, touching AB, BC and AC at the points P, Q and R, respectively. If AB - BC = 4 cm, AB - AC = 2 cm, and the perimeter of ΔABC = 32 cm, then AC (in cm) = ?
\(\frac{32}{3}\)
SSC CGL 202254)Let ΔABC ~ ΔPQR and \(\frac{\operatorname{ar}(\triangle \mathrm{ABC})}{\operatorname{ar}(\triangle \mathrm{QPR})}=\frac{64}{169} \text { }\). If AB = 10 cm, BC = 7 cm and AC = 16 cm, then PR (in cm) is equal to:
26
SSC CGL 202255)The sides PQ and PR of ΔPQR are produced to points S and T, respectively. The bisectors of ∠SQR and ∠TRQ meet at point U. If ∠QUR = 69°, then the measure of ∠P is:
42°
SSC CGL 202256)The base of a triangle is increased by 40%. By what percentage (correct to two decimal places) should its height be increased so that the area increases by 60%?
14.29%
SSC CGL 202257)In ΔABC, D is a point on side BC such that ∠ADC = ∠BAC. If CA = 15 cm and CD = 9 cm, then CB (in cm) = ?
25
SSC CGL 202258)In ΔABC, AB = 7 cm, BC = 10 cm, and AC = 8 cm. If AD is the angle bisector of ∠BAC, where D is a point on BC, then DC (in cm) = ?
\(\frac{16}{3}\)
SSC CGL 202259)In a triangle ABC, D and E are points on BC such that AD = AE and ∠BAD = ∠CAE. If AB = (2p + 3), BD = 2p, AC = (3q - 1) and CE = q, then find the value of (p + q).
3
SSC CGL 202260)In Δ ABC, D is a point on side BC such that ∠ADC = ∠BAC. If CA = 12 cm, CD = 8 cm, then CB (in cm) = ?
18
SSC CGL 202261)In Δ ABC, AD is perpendicular to BC and AE is the bisector of ∠ BAC. If ∠ABC = 58° and ∠ACB = 34°, then find the measure of ∠DAE.
12°
SSC CGL 202262)The angles of a triangle are (8x - 15)°,(6x - 11)° and ( 4x – 10)°. What is the value of x ?
12
SSC CGL 202263)Sides AB and AC of ∆ ABC are produced to points D and E, respectively. The bisectors of ∠CBD and ∠BCE meet at P. If ∠A = 78°, then the measure of ∠P is:
51°
SSC CGL 202264)In ∆ ABC, ∠A = 88°. If I is the incentre of the triangle, then the measure of ∠BIC is:
134°
SSC CGL 202265)The bisector of ∠B in ΔABC meets AC at D. If AB = 12 cm, BC = 18 cm and AC = 15 cm, then the length of AD (in cm) is:
6
SSC CGL 202266)The difference between the two perpendicular sides of a right-angled triangle is 17 cm and its area is 84 cm2. What is the perimeter (in cm) of the triangle?
56
SSC CGL 202267)In Δ LMN, the bisector of ∠L and ∠N intersect at and angle of 112°. What is the measure (in degree) of ∠M?
44
SSC CGL 202268)In ΔACD, B and E are two points on side AC and AD respectively, such that BE is parallel to CD. CD = 9 cm, BE = 6 cm, AB = 5 cm and ED = 2 cm. What are the measures of the lengths (in cm) of AE and BC?
4, 2.5
SSC CGL 202269)What is the height (in cm) of an equilateral triangle whose each side is 8 cm?
4√3
SSC CGL 202270)The lengths of the three sides of a right-angled triangle are (x - 1) cm, (x + 1) cm and (x + 3) cm, respectively. The hypotenuse of the right-angled triangle (in cm) is:
10
SSC CGL 202271)An equilateral triangle ABC is inscribed in a circle with centre O. D is a point on the minor arc BC and ∠CBD = 40º. Find the measure of ∠BCD.
20º
SSC CGL 202272)In a ΔABC, points P, Q and R are taken on AB, BC and CA, respectively, such that BQ = PQ and QC = QR. If ∠BAC = 75º, what is the measure of ∠PQR (in degrees)?
30
SSC CPO 202073)The sides PQ and PR of ΔPQR are produced to points S and T, respectively. The bisectors of ∠SQR and ∠TRQ meet at U. If ∠QUR = 59°, then the measure of ∠P is:
62°
SSC CPO 202074)In ΔABC, ∠A = 54°. If I is the incentre of the triangle, then the measure of ∠BIC is:
117°
SSC CPO 202075)A circle is inscribed in a triangle ABC. It touches side AB, BC, and AC at points R, P, and Q respectively. If AQ = 2.6 cm, PC = 2.7 cm and BR = 3 cm, then the perimeter (in cm) of the triangle ABC is:
16.6
SSC CPO 202076)In ΔABC, BD ⊥ AC at D, E is a point on BC such that ∠BEA = x°. If ∠EAC = 62° and ∠ EBD = 60°, then the value of x is:
92°
SSC CPO 202077)In ΔABC, ∠A = 68°. If I is the incentre of the triangle, then the measure of ∠BIC is:
124°
SSC CPO 202078)In ΔABC, D is the median from A to BC. AB = 6 cm, AC = 8 cm, and BC = 10 cm. The length of median AD (in cm) is:
5
SSC CPO 202079)The perimeter of a right triangle is 60 cm and its hypotenuse is 26 cm. What is the area (in cm2) of the triangle?
120
SSC CPO 202080)Let Δ ABC ∼ Δ RPQ and \(\frac{{ar(\Delta ABC)}}{{ar(\Delta RPQ)}} = \frac{4}{9}\) . If AB = 3 cm, BC = 4 cm and AC = 5 cm, then RP (in cm) is equal to:
4.5 cm
SSC CPO 202081)In ΔABC, AB and AC are produced to points D and E, respectively. If the bisectors of ∠CBD and ∠BCE meet at the point O, and ∠BOC = 57°, then ∠A is equal to:
66°
SSC CPO 202082)In ΔABC,∠A = 66°. AB and AC are produced to points D and E, respectively. If the bisectors of angle CBD and angle BCE meet at the point O, then∠BOC is equal to:
57°
SSC CPO 202083)LetΔABC ~ΔRPQ and \(\frac {ar (\Delta ABC)}{ar(\Delta RPQ)} = \frac 4 9 \). If AB = 3 cm, BC = 4 cm and AC = 5 cm, then PQ (in cm) is equal to:
6
SSC CPO 202084)In ΔABC, BD ⊥ AC at D, E is a point on BC such that ∠BEA = x°. If ∠EAC = 46° and ∠EBD = 60°, then the value of x is:
76°
SSC CHSL 202185)In Δ ABC, AC = BC, and the length of the base AB is 10 cm. If CG = 8 cm, where G is the centroid, then what is the length of AC?
13 cm
SSC CHSL 202186)The sum of all the three sides of an equilateral triangle is 15√3 cm. The height of the triangle is:
7.5 cm
SSC CHSL 202187)The area of an equilateral triangle is 10.24√3 m2. Its perimeter (in m) is:
19.2
SSC CHSL 202188)The side QR of a triangle PQR is extended to a point S. If ∠PRS = 104° and ∠RQP =\(\frac{3}{5}\)∠QPR, then the value of ∠QPR is:
65°
SSC CHSL 202189)In Δ ABC, AD is the bisector of ∠A meeting BC at D. If AB = 15 cm, BC = 10 cm and the length of BD is 2 cm less than that of DC , then the length of AC is:
22.5 cm
SSC CHSL 202190)In ΔABC, P and Q are the mid-points of the sides AB and AC, respectively. R is a point on the segment PQ such that PR ∶ RQ = 1 ∶ 3. If PR = 4 cm, then BC is equal to:
32 cm
SSC CHSL 202191)The sum of three sides of an isosceles triangle is 20 cm, and the ratio of an equal side to the base is 3 ∶ 4. The altitude of the triangle is:
2√5 cm
SSC CHSL 202192)In a triangle ABC, the length of side AC is 4 cm less than five times the length of side AB. The length of side BC exceeds four times the length of side AB by 4 cm. If the perimeter of Δ ABC is 90 cm, then its area is:
180 cm2
SSC CHSL 202193)If Δ ABC∼Δ QPR,\( \rm \frac{ar(\Delta ABC)}{ar(\Delta PQR)}=\frac{4}{9}\), AC = 12 cm, AB = 18 cm and BC = 10 cm, then PR (in cm) is equal to:
15
SSC CHSL 202194)If O is the centroid and RP is the median with length 24 cm of Δ RST , where P is a point on ST, then the value of RO is:
16 cm
SSC CHSL 202195)The sides AB, BC and AC of ΔABC are 12 cm, 8 cm and 10 cm, respectively. A circle is inscribed in the triangle touching AB, BC and AC at D, E and F, respectively. The ratio of the lengths of AD to CE is:
7 ∶ 3
SSC CHSL 202196)In ΔPQR, QT ⊥ PR and S is a point on QR such that ∠PSQ = p°. If ∠TQR = 44° and ∠SPR = 32°, then the value of p is:
78°
SSC CHSL 202197)The sides of a triangle are in the ratio \(\frac{1}{3},\frac{1}{5},\frac{1}{4}\) and its perimeter is 141 cm. The difference between the greatest side and the smallest side is:
24 cm
SSC CHSL 202198)In any triangle, if the angles are in the ratio 1 : 2 : 3, then what will be the ratio of the sides opposite to them?
1 : √3 : 2
SSC CHSL 202199)If S is a point on side QR of a triangle PQR such that QS = 10 cm, QR = 18 cm and ∠PSR = ∠QPR, then the length of PR will be:
12 cm
SSC CHSL 2021100)In Δ ABC, points D and E are on AB and AC, respectively, such that DE is parallel to BC. IE AD = 3 cm, BD = 6 cm and AE = 2 cm, then find the length of CE.
4 cm
SSC CHSL 2021101)ABC is a triangle inscribed in a circle and ∠ACB is equal to 35°. P is a point on the circle on the side AB, opposite to C. What is the value of ∠APB in degrees?
145
SSC CHSL 2021102)In triangle ABC, AD is the internal bisector of \(\angle A\) meeting BC at D. If BD = 3.6 cm and BC = 8 cm, then the ratio of AB to AC will be:
9 ∶ 11
SSC CHSL 2021103)In a circle, two chords UV and WX intersect each other at a point Z within the circle. If UV = 18 cm, ZV = 6 cm and WZ = 9 cm, then the length of ZX is:
8 cm
SSC CHSL 2021104)Chords AB and CD of a circle intersect externally at P. If AB = 8.8 cm, PB = 7.2 cm, PD = 6.4 cm, then CD is equal to:
11.6 cm
SSC CHSL 2021105)AB is the diameter of a circle of radius 9 cm. PQ is a chord (not a diameter) that intersects AB at M perpendicularly. If AM : BM = 5 : 4, then the length of chord PQ will be:
8√5 cm
SSC CHSL 2021106)AB is 12 cm long chord of a circle with centre O and radius 10 cm. The tangents at A and B intersect at P. What is the length of OP?
12.5 cm
SSC CHSL 2021107)A circle touches the side BC of ΔABC at P and also touches AB and AC produced at Q and R, respectively. If the perimeter of ΔABC = 14.1 cm, then the length (in cm) of AQ will be:
7.05
SSC CHSL 2021108)Two circle touch each other externally. The distance between their centres is 14 cm. If the radius of one circle is 8 cm, then the radius of the other circle is:
6 cm
SSC CHSL 2021109)The circumference of a circle is 'aπ' units and the area of the circle is 'bπ' square units. If a ∶ b is equal to 4 ∶ 5, then the radius of the circle is:
2.5 cm
SSC CHSL 2021110)Chord AB of a circle with radius 5 cm is at a distance of 4 cm from the centre O. If tangents drawn at A and B intersect at P, then find the length of the tangent AP.
3.75 cm
SSC CHSL 2021111)In a circle with center O, APB is a tangent at P. If MN is a diameter such that ∠ BPN = 52°, then what is the measure of ∠ PNM?
38°
SSC CHSL 2021112)Points A, B, C and D are concyclic points of a circle with centre O, such that ∠DOC = 73°. The measure of ∠AOC is 215°. What is the measure of ∠ AOD?
72°
SSC CHSL 2021113)PA and PB are tangents drawn to a circle with centre O from an external point P. If A and B are points on the circle and ∠OBA = 42°, then ∠APB is:
84°
SSC CHSL 2021114)Two chords PQ and RS of a circle meet at A when produced. AT is a tangent to the circle meeting it at T. The ratio PA : SA is equal to which of the following?
RA : AQ
SSC CHSL 2021115)The distance between two equal parallel chords of a circle is 10 cm. If the chords are 24 cm long, then what is the length of the radius?
13 cm
SSC CHSL 2021116)Two circles of radii 4 cm and 3 cm, respectively, touch each other externally. What is the distance (in cm) between their centres?
7
SSC CHSL 2021117)A circle is inscribed in a triangle ABC. It touches sides AB, BC and AC at points R, P and Q, respectively. If AQ = 6.5 cm, PC = 7.5 cm and BR = 9 cm, then the perimeter (in cm) of the triangle ABC will be:
46
SSC CHSL 2021118)If PA and PB are tangents drawn to a circle with centre O at A and B from external point P such that ∠APB = 78°, then ∠OAB is equal to:
39°
SSC CHSL 2021119)In a circle, AB and DC are two chords. When AB and DC are produced, they meet at P. If PC = 2.8 cm, PB = 3.15 cm and AB = 3.85 cm, then CD = ?
5.075 cm
SSC CHSL 2021120)A chord PQ of a circle C1 of radius 9.25 cm touches another circle C2 that is concentric to C1, and the radius of C2 is 3 cm. What is the length (in cm) of PQ?
17.5
SSC CHSL 2021121)Chords AB and CD of a circle are produced to meet at the point P, outside the circle, and AD is the diameter of the circle. If ∠DAP = 36° and ∠APC = 30°, then what will be the measure of ∠CBD?
24°
SSC CHSL 2021122)Two concentric circles are of radii 15 cm and 6 cm. What is the length (in cm) of the chord of the larger circle that is tangent to the smaller circle?
6√21
SSC CHSL 2021123)AB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 112°, then what is the measure of ∠OAB ?
22°
SSC CHSL 2021124)A and B are two points on a circle with centre O. C is a point on the minor arc of the circle between points A and B. The tangents to the circle at A and B meet each other at a point D. If ∠ ADB = 25°, then ∠ ACB (in degrees) is equal to :
102.5
SSC CHSL 2021125)AB is a diameter of the circle with centre 0. The tangent at the point C on the circle meets AB produced at Q. If ∠BAC = 34°, then the measures of ∠CQA (in degrees) will be:
22
SSC CHSL 2021126)AB is a diameter of a circle with centre O. If C is any point on the circle such that ∠BAC = 42°, then find the measure of ∠BOC.
84°
SSC CHSL 2021127)P, Q and R are three points on the circumference of a circle such that QR is a diameter and PQ = PR. If the radius of the circle is 7 cm. then the length of PQ is:
7√2 cm
SSC CHSL 2021128)AB and CD are two chords of a circle that intersects at E inside the circle. If ∠BEC = 125° and ∠EBD = 28°, then what is the measure of ∠BAC?
97°
SSC CHSL 2021129)Let O be the centre of a circle. PA and PB are tangents to the circle from a point P outside the circle and A and B are points on the circle. If angle APB = 50°, then angle OAB is equal to:
25°
SSC CHSL 2021130)AB and AC are tangents to a circle with centre O from an external point A. The tangents AB and AC touch the circle at B and C, respectively, such that ∠BAC = 118°. Find the measures of ∠OCB.
59°
SSC CHSL 2021131)In a circle with centre O, a 6 cm long chord is at a distance 4 cm from the centre. Find the length of the diameter.
10 cm
SSC CHSL 2021132)Chord AB and diameter CD of a circle meet at the point P, outside the circle when the produced, If PB = 8 cm, AB = 12 cm and distance of P from the centre of the circle is 18 cm, the radius (in cm) of the circle is closest to:
12.8
SSC CGL 2020133)Two chords AB and CD of a circle with centre O intersect each other at P. If \(\angle APC = 95^0\) and \(\angle AOD = 110^0\) , then \(\angle BOC\) is:
\(60^0\)
For an equal arc, angle at the centre = 2 \(\times\) angle at the circumference ;
\(\therefore \angle AOD =2\angle ABD\); ⇒ \(\angle ABD = {110^0\over2}= 55^0= \angle PBD\) ;
\(\angle APC = \angle BPD = 95^0\);
In \(\triangle PBD\), \(\angle BPD +\angle PDB+\angle PBD = 180^0\) ; ⇒ \(\angle BDP= 180-95-55= 30^0\) ;
\(\therefore \angle BOC = 2\times \angle BDP = 2\times 30= 60^0\)
SSC CGL 2020134)Diameter AB of a circle with centre O is produced to a point P such that PO = 16.8 cm. PQR is a secant which intersects the circle at Q and R, such that PQ = 12 cm and PR = 19.2 cm.The length of AB (in cm.) is :
14.4
PQ = 12 cm ; PR = 19.2 cm ; PO = 16.8 cm ; Let, OA = OB = x ;
BP x AP = PQ x PR ; ⇒ (16.8 - x)(16.8 + x) = 12 x 19.2 ; x = 7.2 cm.
\(\therefore AB=2\times7.2 = 14.4cm\)
SSC CGL 2020135)In the given figure, \(\angle KLN = 58^0\), then \(\angle KMN=\space?\)
\(58^0\)
\(\angle KLN = 58^0\) ; \(\therefore \angle KMN = 58^0\) ;
Because angles in the same segment are equal.
SSC CGL 2020136)In the figure, two circles with centres P and Q touch externally at R. Tangents AT and BT meet the common tangent TR at T. If AP = 6 cm and PT = 10 cm, then BT = ?
8 cm
In \(\triangle PAT\) \(\angle PAT = 90^0\);
\(\therefore AT= \sqrt{ PT^2-AP^2} = \sqrt{10^2-6^2}= 8cm\);
Tangent drawn from an external point to a circle are equal.
\(\therefore\) AT = BT = 8 cm
SSC CGL 2020137)Two tangents PA and PB are drawn to a circle with centre O from an external point P. If \(\angle OAB = 30^0\), then \(\angle APB\) is:
\(60^0\)
\angle OAB = \(30^0\); OA = OB = radii of circle; \(\angle OAB = \angle OBA = 30^0\); \(\angle AOB = 180^0-(30^0+30^0)=120^0\); In quadrilateral OAPB, \(\therefore\angle OAP = \angle OBP = 90^0\); \(\therefore \angle AOB +\angle APB = 180^0\); ⇒\(\angle APB = 180-120 =60^0\)
SSC CGL 2020138)In the given figure, if \(\angle APO = 35^0\), then which of the following options is correct?
\(\angle BPO = 35^0\)
Join OA and OB. Since OA = OB ; AP = PB and OP is common. Therefore, triangle AOP is similar to triangle BOP. Since angle OAP = 90 & OBP = 90. Therefore, \({OAP\over OBP} ={APO\over BPO}\); ⇒ \({90\over90}={35\over BPO}\); ⇒ BPO = \(35^0\)
SSC CGL 2020139)In the given figure, if AB = 10 cm, CD = 7 cm, SD = 4 cm and AS = 5 cm, then BC = ?
8 cm
SD = 4cm; So DR = 4 cm ; CR = CQ = 7 - 4 = 3 cm; AS = 5 cm ; AS = AP = 5 cm ; AB = 10 cm ; BP = BQ = 10 - 5= 5 cm; So BC = BQ + CQ = 5 + 3 = 8 cm
SSC CGL 2020140)A, B and C are three points on a circle such that the angle subtended by the chords AB and AC at the centre O are \(80^0\) and \(120^0\), respectively. The value of \(\angle BAC\) is:
\(80^0\)
\(\angle AOB = 80^0\); \(\angle AOC = 120^0\); \(\therefore \angle BOC = 360^0-(80^0+120^0)=160^0\) \(\therefore \angle BAC = {\angle BOC\over2}={160^0\over2} = 80^0\)
SSC CGL 2020141)In the given figure, MP is a tangent to a circle with center A and NQ is a tangent to a circle with center B. If MP = 15 cm, NQ = 8 cm, PA = 17 cm and BQ = 10 cm, then AB is:
14 cm.
PA = 17 ; PM = 15 cm ; \(\angle PMA= 90^0\) ; In \(\triangle AMP,\) \(\therefore MA = 8 cm\); Again, BQ = 10 cm. ; NQ = 8 cm; \(\angle BNQ = 90^0\) ; So BN = 6 cm ; Ab = AC + CB = AM + BN = (8 + 6) = 14 cm.
SSC CGL 2020142)AB is a diameter of a circle with centre O. The tangent at a point C on the circle meets AB produced at Q. If \(\angle CAB=42^0\), then what is the measure of \(\angle CQA\)?
\(6^0\)
\(\angle OAC = \angle OCA = 42^0 ( \because OA = OC)\); \(\angle OCQ = 90^0 \) (since CQ is tangent). Now in \(\triangle ACQ\), \(\angle A+\angle C+\angle Q =180^0\); ⇒ \(42^0+(42^0+90^0)+x = 180\); x = \(6^0\)
SSC CGL 2020143)PAQ is a tangent to circle with centre O, at a point A on it. AB is a chord such that \(\angle BAQ=x^0 (x<90)\). C is a point on the major arc AB such that \(\angle ACB = y^0\). If \(\angle ABO = 32^0\), then the value of x + y is:
116
OA = OB; \(\angle ABO = \angle BAO =32^0\); \(\angle AOB = 116^0\); so \(y ={ 1\over2}\angle AOB = 58^0\); and \(x= 90^0-32^0= 58^0\); x + y = \(58^0 +58^0 = 116^0\)
SSC CGL 2020144)In a circle, AB is a the diameter and CD is a chord. AB and CD produced meet at a point P, outside the circle. If PD = 15.3 cm, CD = 11.9 cm and AP = 30.6 cm,then the radius of the circle is is:
8.5 cm
From the property,
\(PA \times PB = PC \times PD\);
\(30.6 \times PB = (PD + CD) \times 15.3\);
\(30.6 \times PB = (15.3 + 11.9) \times 15.3\);
\(30.6 \times PB = 27.2 \times 15.3\);
PB = 416.16/30.6 = 13.6 cm;
Diameter (AB) = PA - PB = 30.6 - 13.6 = 17 cm;
Radius = AB/2 = 17/2 = 8.5 cm
SSC CGL 2020145)From an external point P, a tangent PQ is drawn to a circle, with the centre O, touching the circle at Q. If the distance of P from the centre is 13 cm and length of the tangent PQ is 12 cm, then the radius of the circle is:
5 cm
\triangle OPQ is a right angle triangle because \(\angle Q = 90^0\),
By Pythagoras,
\((OQ)^2 + (PQ)^2 = (OP)^2\);
\((OQ)^2 = (13)^2 - (12)^2\);
\((OQ)^2\) = 169 - 144;
\((OQ)^2\) = 25;
OQ = 5 cm.
SSC CGL 2020146)In a circle, chords PQ and TS are produced to meet at R. If RQ = 14.4 cm, PQ = 11.2 cm, and SR = 12.8 cm, then the length of chord TS is:
16 cm
RQ X PR = RS X TR; 14.4 X (14.4+11.2) = 12.8 X TR; TR = \({14.4\times25.6\over12.8}=28.8 cm\) ; TS = TR - SR = (28.8 - 12.8) cm = 16 cm
SSC CGL 2019147)In a circle, AB and DC are two chords. When AB and DC are produced, they meet at P. If PC = 5.6 cm, PB = 6.3 cm and AB = 7.7 cm, then the length of CD is:
10.15 cm.
SSC CGL 2019148)In circle with centre O, AC and BD are two chords. AC and BD meet at E when produced. If AB is the diameter and \(\angle AEB=68^0\), then the measure of \(\angle DOC\) is :
\(44^0\)
SSC CGL 2019149)A circle touches the side BC of \(\triangle ABC\) at D and AB and AC are produced to E and F, respectively. If AB = 10 cm, AC = 8.6 cm and BC =6.4 cm, then BE =?
2.5 cm
SSC CGL 2019150)Two parallel chords on the same side of the centre of a circle are 12 cm and 20 cm long and the radius of the circle is \(5\sqrt{13}cm\). What is the distance (in cm) between the chords?
2
SSC CGL 2019151)Chord AB of a circle is produced to a point P, and is a point on the circle such that PC is a tangent to the circle. If PC = 18 cm, and BP = 15 cm, then AB is equal to:
6.6 cm
By the property,
\(PC^2 = PA \times PB;\)
\((18)^2 = PA \times 15;\)
PA = 324/15 = 21.6 cm;
AB = PA - PB = 21.6 - 15 = 6.6 cm
Class 10 Maths152)If angle of sector is 60°, radius is 3.5 cm then length of the arc is
3.66 cm
SSC CGL 2020153)In the figure, PA is a tangent from an external point P to the circle with centre O. If \(\angle POB = 110 ^{\circ}\), then measure of \(\angle APO\) is:
\(20 ^{\circ}\)
PA is a tangent
\(\angle POA =180^{\circ}- 110 ^{\circ}=70^{\circ}\)
\(\angle PAO =90^{\circ}\)
\(\angle APO =180^{\circ}- 70 ^{\circ}-90^{\circ}=20^{\circ}\)