MCQOPTIONS
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MCQOPTIONS
This section includes 82 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
A chord subtends an angle 120° at the centre of a unit circle. What is the length of the chord? |
| A. | √2 - 1 units |
| B. | √3 - 1 units |
| C. | √2 units |
| D. | √3 units |
| Answer» E. | |
| 2. |
If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is |
| A. | x2 + y2)2 = 4R2 x2y2 |
| B. | (x2 + y2)3 = 4R2 x2y2 |
| C. | (x2 + y2)(x + y) = R2xy |
| D. | (x2 + y2)2 = 4Rx2y2 |
| Answer» C. (x2 + y2)(x + y) = R2xy | |
| 3. |
Equation of the common tangent, with positive slope, to the circle x2 + y2 - 8x = 0 as well as to the hyperbola \(\rm \frac{x^2}{9}-\frac{y^2}{4}=1\), is: |
| A. | \(\rm2x-\sqrt{5}y-20=0\) |
| B. | \(\rm 2x-\sqrt{5}y + 4 = 0\) |
| C. | 3x - 4y + 8 = 0 |
| D. | 4x - 3y + 4 = 0 |
| Answer» C. 3x - 4y + 8 = 0 | |
| 4. |
If m be the slope of a tangent to the curve ey = 1 + x2, then |
| A. | |m| > 1 |
| B. | m < 1 |
| C. | |m| < 1 |
| D. | |m| ≤ 1 |
| Answer» E. | |
| 5. |
If a variable line, 3x + 4y – λ = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x - 2y + 78 = 0 are on its opposites sides, then the set of all values of λ is the interval |
| A. | [13, 23] |
| B. | (2, 17) |
| C. | [12, 21] |
| D. | (23, 31) |
| Answer» D. (23, 31) | |
| 6. |
Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2 – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq units) of the quadrilateral PC1 QC2 is |
| A. | 8 |
| B. | 4 |
| C. | 6 |
| D. | 9 |
| Answer» C. 6 | |
| 7. |
Radius of the circle x2 + y2 – 4x + 2y – 31 = 0 is |
| A. | 4 units |
| B. | 2 units |
| C. | 6 units |
| D. | 31 units |
| Answer» D. 31 units | |
| 8. |
If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y – 1 = 0, (K ∈ R), intersect at the points P and Q, then the line 4x + 5y – K = 0 passes through P and Q, for: |
| A. | Infinitely many values of K |
| B. | No value of K |
| C. | Exactly two values of K |
| D. | Exactly one value of K |
| Answer» C. Exactly two values of K | |
| 9. |
If AB is diameter of the circle x2 + y2 + 2x + 4y - 3 = 0. If co-ordinates of A are (1, 0), then co-ordinates of B are |
| A. | (-3, 1) |
| B. | (-3, 2) |
| C. | (-3, 3) |
| D. | (-3, -4) |
| Answer» E. | |
| 10. |
Equation of the circle with centre on the y-axis and passing through the origin and (2, 3) is |
| A. | x2 + y2 + 13y = 0 |
| B. | 3x2 + 3y2 - 13y = 0 |
| C. | x2 + y2 + 13y + 3 = 0 |
| D. | 6x2 + 6y2 - 13y = 0 |
| Answer» C. x2 + y2 + 13y + 3 = 0 | |
| 11. |
If the equation of a circle is ax2 + (2a - 3)y2 - 4x - 1 = 0, then its centre is: |
| A. | (1, 1) |
| B. | (2, 0) |
| C. | \(\left( {\frac{2}{3},0} \right)\) |
| D. | \(\left( { - \frac{2}{3},0} \right)\) |
| Answer» D. \(\left( { - \frac{2}{3},0} \right)\) | |
| 12. |
In the conic \(\frac{l}{r} = 1 + ecos\theta \), the sum of the reciprocals of the segments of any focal chord is |
| A. | 1/l |
| B. | 3/l |
| C. | 4/l |
| D. | 2/l |
| Answer» E. | |
| 13. |
Find the length of tangent, drawn from a point which is at a distance of 5 cm from the centre of the circle of radius 3 cm. |
| A. | 3 cm |
| B. | 4 cm |
| C. | 2 cm |
| D. | None of these |
| Answer» C. 2 cm | |
| 14. |
If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x – 6y = 12 externally at the point (1, -1), then the radius of C is: |
| A. | 2√5 |
| B. | 4 |
| C. | 5 |
| D. | √57 |
| Answer» D. √57 | |
| 15. |
A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point: |
| A. | (3, 10) |
| B. | (3, 5) |
| C. | (2, 3) |
| D. | (1, 5) |
| Answer» B. (3, 5) | |
| 16. |
Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, -3) and B is (1, 4) |
| A. | (3, -10) |
| B. | (10, 3) |
| C. | (3, -2) |
| D. | (3, 2) |
| Answer» B. (10, 3) | |
| 17. |
Find the radius of the curvature at point (s, ψ) on the curve s = 4 a sin ψ [cycloid] |
| A. | tan ψ |
| B. | 2 a sec3 ψ |
| C. | 4 a cos ψ |
| D. | None of the above |
| Answer» E. | |
| 18. |
Find the equation of the circle which passes through (-1, 1) and (2, 1), and having centre on the line x + 2y + 3 = 0. |
| A. | 2x2 + 2y2 - 2x + 7y - 13 = 0 |
| B. | x2 + y2 - 2x + 7y - 13 = 0 |
| C. | 2x2 + 2y2 + 2x + 7y - 13 = 0 |
| D. | x2 + y2 + 2x + 7y - 13 = 0 |
| Answer» B. x2 + y2 - 2x + 7y - 13 = 0 | |
| 19. |
If two diameters of a circle lie along the lines x - y = 5 and x + y = 7 , and the area of the circle is 50π sq units, find the equation of the circle. |
| A. | \(\rm (x-6)^2+(y-1)^2=50\) |
| B. | \(\rm (x-1)^2+(y-6)^2=16\) |
| C. | \(\rm (x-5)^2+(y-7)^2=12\) |
| D. | \(\rm (x-1)^2+(y-4)^2=25\) |
| Answer» B. \(\rm (x-1)^2+(y-6)^2=16\) | |
| 20. |
From a circular plate of diameter 6 cm, a circle is cut out whose diameter is radius of the plate. The area of the remaining plate is |
| A. | 27π/2 cm2 |
| B. | 27π/4 cm2 |
| C. | 27π2/4 cm2 |
| D. | 27π cm2 |
| Answer» C. 27π2/4 cm2 | |
| 21. |
Let f(x, y) = 0 represents a circle, If f(0, a) has roots a = 2, 3 and f(a, 0) has roots \(a = 12, \frac 1 2,\) then the centre of the circle is: |
| A. | \(\left(\frac {-5}2, \frac 1 6\right)\) |
| B. | \(\left(\frac {4}{25}, \frac 2 5\right)\) |
| C. | \(\left(\frac {2}{7}, 6\right)\) |
| D. | \(\left(\frac {25}{4}, \frac 5 2\right)\) |
| Answer» E. | |
| 22. |
\({\left( {{\rm{x}} - 1} \right)^2} + {\left( {{\rm{y}} - 3} \right)^2} = {{\rm{r}}^2}\) and \({{\rm{x}}^2} + {{\rm{y}}^2} - 8{\rm{x}} + 2{\rm{y}} + 8 = 0\)What is the distance between the centres of the two circles? |
| A. | 5 units |
| B. | 6 units |
| C. | 8 units |
| D. | 10 units |
| Answer» B. 6 units | |
| 23. |
\({\left( {{\rm{x}} - 1} \right)^2} + {\left( {{\rm{y}} - 3} \right)^2} = {{\rm{r}}^2}\) and \({{\rm{x}}^2} + {{\rm{y}}^2} - 8{\rm{x}} + 2{\rm{y}} + 8 = 0\)If the circles intersect at two distinct points, then which one of the following is correct? |
| A. | r = 1 |
| B. | 1 < r < 2 |
| C. | r = 2 |
| D. | 2 < r < 8 |
| Answer» E. | |
| 24. |
If the centre of a circle is (-6,8) and it passes through the origin, then equation to its tangent at the origin is |
| A. | 2y = x |
| B. | 4y = 3x |
| C. | 3y = 4x |
| D. | 3x + 4y = 0 |
| Answer» C. 3y = 4x | |
| 25. |
If y-axis touches the circle x2 + y2 + gx + fy + \(\frac{e}{4}\) = 0, then the normal at this point intersects the circle at the point |
| A. | \(\left( {\frac{g}{2},\frac{f}{2}} \right)\) |
| B. | \(\left( { - g, - \frac{f}{2}} \right)\) |
| C. | \(\left( { - \frac{g}{2},f} \right)\) |
| D. | (-g, -f) |
| Answer» C. \(\left( { - \frac{g}{2},f} \right)\) | |
| 26. |
If two circles \(x^2 + y^2 +2gx + 2fy = 0\) and \(x^2 + y^2 +2g's + 2f'y = 0\) touch each other then which of the following is true? |
| A. | gf = g’f’ |
| B. | g’f = gf’ |
| C. | gg’ = ff’ |
| D. | None of these |
| Answer» C. gg’ = ff’ | |
| 27. |
If the circles x2 + y2 – 16x – 20y + 164 = r2and(x – 4)2 + (y – 7)2 = 36 intersect at two distinct points, then: |
| A. | r > 11 |
| B. | 0 < r < 1 |
| C. | r = 11 |
| D. | 1 < r < 11 |
| Answer» E. | |
| 28. |
A circle cuts a chord of length ‘4a’ on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is: |
| A. | A hyperbola |
| B. | An ellipse |
| C. | A straight line |
| D. | A parabola |
| Answer» E. | |
| 29. |
If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is: |
| A. | 13/5 |
| B. | 120/13 |
| C. | 60/13 |
| D. | 13/2 |
| Answer» C. 60/13 | |
| 30. |
Find the distance between two parallel chords of lengths 24 cm and 32 cm if radius of the circle is 20 cm. |
| A. | 10 cm |
| B. | 28 or 4 cm |
| C. | 5 cm |
| D. | 16 or 12 cm |
| Answer» C. 5 cm | |
| 31. |
If the area of an equilateral triangle inscribed in the circle, x2 + y2 + 10x + 12y + c = 0 is 27√3 sq. units then c is equal to: |
| A. | 13 |
| B. | 20 |
| C. | -25 |
| D. | 25 |
| Answer» E. | |
| 32. |
If a circle of radius b units with center at (0, b) touches the line y = -x +√2, then what is the value of b? |
| A. | 2 + √2 |
| B. | -2 – √2 |
| C. | 2√2 |
| D. | √2 |
| Answer» C. 2√2 | |
| 33. |
For two circles x2 + y2 = 16 and x2 + y2 - 2y = 0, there is / are |
| A. | One pair of common tangent |
| B. | Two pair of common tangents |
| C. | Three pair of common tangents |
| D. | No common tangents |
| Answer» E. | |
| 34. |
A circle is drawn on the chord of a circle x2 + y2 = a2 as diameter. The chord lies on the line x + y = a. What is the equation of the circle? |
| A. | x2 + y2 – ax – ay + a2 = 0 |
| B. | x2 + y2 – ax – ay = 0 |
| C. | x2 + y2 + ax + ay = 0 |
| D. | x2 + y2 + ax + ay – 2a2 = 0 |
| Answer» C. x2 + y2 + ax + ay = 0 | |
| 35. |
Let R be the radius of a circle. What is the angle subtended by an arc of length R at the centre of the circle? |
| A. | 1 degree |
| B. | 1 radian |
| C. | 90 degrees |
| D. | π radians |
| Answer» C. 90 degrees | |
| 36. |
AB is a chord of a circle and AOC is its diameter such that ∠ACB = 50°, if AT is tangent to the circel at the point A, then ∠BAT is equal to |
| A. | 50° |
| B. | 60° |
| C. | 65° |
| D. | None of these |
| Answer» B. 60° | |
| 37. |
Given that two circles x2 + y2 = r2 and x2 + y2 -10x + 16 = 0, the value of r such that they intersect in real and distinct points is given by |
| A. | 2 < r < 8 |
| B. | r = 2 or r = 8 |
| C. | r < 2 or r < 8 |
| D. | None of these |
| Answer» B. r = 2 or r = 8 | |
| 38. |
Area of a circle is 81π and the equations of the normal to the circle are 2y + 3x - 5 = 0 and 2y - 3x + 5 = 0. Find the equation of the circle. |
| A. | (x - \(5\over3\))2 + y2 = 9 |
| B. | (x - \(5\over3\))2 + y2 = 81 |
| C. | (x + \(5\over3\))2 + y2 = 81 |
| D. | (x + \(5\over3\))2 + y2 = 9 |
| Answer» C. (x + \(5\over3\))2 + y2 = 81 | |
| 39. |
On which line does the centre of the circle lie? |
| A. | x + y = 0 |
| B. | x – y = 0 |
| C. | x + y = a + b |
| D. | x – y = a2 – b2 |
| Answer» B. x – y = 0 | |
| 40. |
At the centre of a circle of 10 cm radius, the angle made by an arc of \(12 \dfrac{2}{9}\) cm length is |
| A. | 60° |
| B. | 65° |
| C. | 70° |
| D. | 75° |
| Answer» D. 75° | |
| 41. |
A straight line x = y + 2 touches the circle 4(x2 + y2) = r2. The value of r is |
| A. | \(\sqrt 2\) |
| B. | \(2\sqrt 2\) |
| C. | 2 |
| D. | 1 |
| Answer» C. 2 | |
| 42. |
The centre of a circle is 0 and its diameter is of length 9 cm. Which of the points P, Q, R and S lies on the circle if OP = 9 cm; OQ = 4 cm; OR = 4.5 cm and OS = 6 cm? |
| A. | S |
| B. | R |
| C. | 121 |
| D. | p |
| Answer» C. 121 | |
| 43. |
In the given figure, the value of \[x\]is ____. |
| A. | \[{{60}^{o}}\] |
| B. | \[{{40}^{o}}\] |
| C. | \[{{20}^{o}}\] |
| D. | None of these |
| Answer» C. \[{{20}^{o}}\] | |
| 44. |
An equilateral\[\Delta PQR\]is inscribed in circle with centre O. Find\[\angle QOR.\] |
| A. | \[60{}^\circ \] |
| B. | \[120{}^\circ \] |
| C. | \[30{}^\circ \] |
| D. | \[90{}^\circ \] |
| Answer» C. \[30{}^\circ \] | |
| 45. |
In a circle of radius 5 cm, AB and AC are two chords such that AB = AC = 6 cm. What is the distance of the centre of the circle from BC? |
| A. | 1.4 cm |
| B. | 2.1 cm |
| C. | 2.4 cm |
| D. | 2.7 cm |
| Answer» B. 2.1 cm | |
| 46. |
In \[\Delta \text{ABC},\text{ }\angle \text{B}=90{}^\circ \].If a circle drawn with AB as diameter intersects the hypotenuse AC at P, which of the following is true? |
| A. | The tangent drawn to the circle at P bisects the side BC. |
| B. | The tangent drawn to the circle at A bisects the side AB. |
| C. | The tangent drawn to the circle at B bisects the side AC. |
| D. | The tangent drawn to the circle at C bisects the side BC. |
| Answer» B. The tangent drawn to the circle at A bisects the side AB. | |
| 47. |
In the given figure, ABC is an isosceles triangle in which AB = AC.A circle through B touches AC at its mid-point D and intersects AB at P. Which of the following is correct? |
| A. | \[AP=\frac{3}{4}AB\] |
| B. | \[AP=\frac{2}{3}AB\] |
| C. | \[AP=\frac{4}{5}AB\] |
| D. | \[AP=\frac{1}{4}AB\] |
| Answer» E. | |
| 48. |
A circle with centre 0 and radius 5 cm is inscribed in an equilateral triangle ABC. Find the perimeter of\[\Delta \text{ABC}\]. |
| A. | \[15\sqrt{3}\,cm\] |
| B. | \[25\sqrt{2}\,cm\] |
| C. | \[14\sqrt{2}\,cm\] |
| D. | \[30\sqrt{3}\,cm\] |
| Answer» E. | |
| 49. |
When are two circles said to be concentric? |
| A. | If they have the same radius. |
| B. | If they have different radii. |
| C. | If they have the same centre. |
| D. | If their centres are collinear. |
| Answer» D. If their centres are collinear. | |
| 50. |
Tangents AP and AQ are drawn to a circle with centre 0 from an external point A. Identify the correct statement. |
| A. | \[\angle \text{PAQ}=\text{2}\angle \text{OPQ}\] |
| B. | \[\angle \text{PAQ}=\angle \text{OPQ}\] |
| C. | \[\angle \text{PQA}=\angle \text{QPA}\] |
| D. | \[\angle \text{PQA}=\text{2}\angle \text{OPA}\] |
| Answer» B. \[\angle \text{PAQ}=\angle \text{OPQ}\] | |