MCQ’s and 1 Marks
1. Out of the following which is the Pythagorean triplet?
(A) (1,5,10)
(B) (3,4,5)
(C) (2,2,2)
(D) (5,5,2)
2. Two circles of radii 5.5 cm and 3.3 cm respectively touch each other externally. What is the distance between their centres?
(A) 4.4 cm
(B) 2.2 cm
(C) 8.8 cm
(D) 8.9 cm
3. Distance of point (–3,4) from the origin is
4. Find the volume of a cube of side 3 cm :
(A) 27 cm³
(B) 9 cm³
(C) 81 cm³
(D) 3 cm³
5. If \(\Delta ABC \sim \Delta DEF\) and \(\angle A = 48^\circ\), then \(\angle D =\)
(A) 48°
(B) 83°
(C) 49°
(D) 132°
6. AP is a tangent at A drawn to the circle with centre O from an external point P. \(OP = 12 \text{ cm}\) and \(\angle OPA = 30^\circ\), then the radius of a circle is
(A) 12 cm
(B) \(6\sqrt{3}\) cm
(C) 6 cm
(D) \(12\sqrt{3}\) cm
7. Seg AB is parallel to X-axis and co-ordinates of the point A are \((1,3)\), then the co-ordinates of the point B can be
(A) \((-3,1)\)
(B) \((5,1)\)
(C) \((3,0)\)
(D) \((-5,3)\)
8. The value of \(2\tan 45^\circ - 2\sin 30^\circ\) is
(A) 2
(B) 1
(C) 1/2
(D) 3/4
9. Some question and their alternative answer are given. Select the correct alternative.
If a, b, and c are sides of a triangle and \(a^2 + b^2 = c^2\), name the type of triangle.
(A) Obtuse angled triangle
(B) Acute angled triangle
(C) Right-angled triangle
(D) Equilateral triangle
10. Chords AB and CD of a circle intersect inside the circle at point E. If \(AE = 4, EB = 10, CE = 8\), then find ED.
11. Co-ordinates of origin are ______.
(A) (0, 0)
(B) (0, 1)
(C) (1, 0)
(D) (1, 1)
12. If radius of the base of cone is 7 cm and height is 24 cm, then find its slant height.
(A) 23 cm
(B) 26 cm
(C) 31 cm
(D) 25 cm
13. Out of the dates given below which date constitutes a Pythagorean triplet?
(A) 15/8/17
(B) 16/8/16
(C) 3/5/17
(D) 4/9/15
14. Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet: \(\sin\theta \times \text{cosec}\theta =\)
(A) 1
(B) 0
(C) 1/2
(D) \(\sqrt{2}\)
15. Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:
Slope of X-axis is .
(A) 1
(B) –1
(C) 0
(D) Cannot be determined
16. Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:
A circle having radius 3 cm, then the length of its largest chord is .
(A) 1.5 cm
(B) 3 cm
(C) 6 cm
(D) 9 cm
Solve the Following Questions
1. The ratio of corresponding sides of similar triangles is 3:5, then find the ratio of their areas.
2. Find the diagonal of a square whose side is 10 cm.
3. \(\square ABCD\) is cyclic. If \(\angle B=110^\circ\), then find measure of \(\angle D\).
4. Find the slope of the line passing through the points \(A(2, 3)\) and \(B(4, 7)\).
5. In the figure, 'O' is the centre of the circle, seg PS is a tangent segment and S is the point of contact. Line PR is a secant.
If \(PQ = 3.6, QR = 6.4\), find PS.
6. In \(\Delta PQR\), \(NM \parallel RQ\). If \(PM = 15, MQ = 10, NR = 8\), then find PN.
7. In \(\Delta MNP\), \(\angle MNP = 90^\circ\), seg \(NQ \perp\) seg MP. If \(MQ = 9\), \(QP = 4\), then find NQ.
8. In the figure, M is the centre of the circle and seg KL is a tangent segment. L is a point of contact. If \(MK = 12\), \(KL = 6\sqrt{3}\), then find the radius of the circle.
9. Find the co-ordinates of midpoint of the segment joining the points (22,20) and (0,16).
10. A person is standing at a distance of 80 metres from a Church and looking at its top. The angle of elevation is of \(45^\circ\). Find the height of the Church.
11. In the given figure, X is any point in the interior of the triangle. Point X is joined to the vertices of triangle. seg \(PQ \parallel\) seg DE, seg \(QR \parallel\) seg EF. Complete the activity and prove that seg \(PR \parallel\) seg DF.
12. If \(A(6,1), B(8,2), C(9,4)\) and \(D(7,3)\) are the vertices of \(\square ABCD\), show that \(\square ABCD\) is a parallelogram.
13. In \(\Delta PQR\), point S is the mid-point of side QR. If \(PQ = 11\), \(PR = 17, PS = 13\), find QR.
14. Prove that, tangent segments drawn from an external point to the circle are congruent.
15. Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.
16. A metal cuboid of measures \(16 \text{ cm} \times 11 \text{ cm} \times 10 \text{ cm}\) was melted to make coins. How many coins were made, if the thickness and diameter of each coin was 2 mm and 2 cm respectively? (\(\pi = 3.14\))
17. In \(\Delta ABC\), PQ is a line segment intersecting AB at P and AC at Q such that seg \(PQ \parallel\) seg BC. If PQ divides \(\Delta ABC\) into two equal parts having equal areas, find \(\frac{BP}{AB}\).
18. Draw a circle of radius 2.7 cm and draw a chord PQ of length 4.5 cm. Draw tangents at points P and Q without using centre.
19. In the figure given above \(\square ABCD\) is a square of side 50 m. Points P,Q,R,S are midpoints of side AB, side BC, side CD, side AD respectively. Find area of shaded region.
20. Circles with centres A, B and C touch each other externally. If AB = 3cm, BC = 3cm, CA=4cm, then find the radii of each circle.
21. If \(\sin\theta + \sin^2\theta = 1\), show that: \(\cos^2\theta + \cos^4\theta = 1\).
22. In \(\Delta ABC\), \(\angle ABC = 90^\circ\), \(\angle BAC = \angle BCA = 45^\circ\). If \(AC = 9\sqrt{2}\), then find the value of AB.
23. Chord AB and chord CD of a circle with centre O are congruent. If \(m(\text{arc}AB) = 120^\circ\), then find the \(m(\text{arc}CD)\).
24. Find the Y-co-ordinate of the centroid of a triangle whose vertices are \((4,-3), (7,5)\) and \((-2,1)\).
25. If \(\sin\theta = \cos\theta\), then what will be the measure of angle \(\theta\)?
26. In the above figure, seg AC and seg BD intersect each other in point P. If \(\frac{AP}{CP} = \frac{BP}{DP}\), then complete the following activity to prove \(\Delta ABP \sim \Delta CDP\).
27. In the above figure, \(\square ABCD\) is a rectangle. If \(AB = 5\), \(AC = 13\), then complete the following activity to find BC.
28. Complete the following activity to prove : \(\cot\theta + \tan\theta = \text{cosec}\theta \times \sec\theta\)
29. If \(\Delta ABC \sim \Delta PQR, AB : PQ = 4:5\) and \(A(\Delta PQR) = 125 \text{ cm}^2\), then find \(A(\Delta ABC)\).
30. In the above figure, \(m(\text{arcDXE}) = 105^\circ, m(\text{arcAYC}) = 47^\circ\), then find the measure of \(\angle DBE\).
31. Draw a circle of radius 3.2 cm and centre 'O'. Take any point P on it. Draw tangent to the circle through point P using the centre of the circle.
32. If \(\sin\theta = \frac{11}{61}\), then find the value of \(\cos\theta\) using trigonometric identity.
33. In \(\Delta ABC, AB = 9 \text{ cm}, BC = 40 \text{ cm}, AC = 41 \text{ cm}\). State whether \(\Delta ABC\) is a right-angled triangle or not? Write reason.
34. In the above figure, chord PQ and chord RS intersect each other at point T. If \(\angle STQ = 58^\circ\) and \(\angle PSR = 24^\circ\), then complete the following activity to verify : \(\angle STQ = \frac{1}{2} [m(\text{arcPR}) + m(\text{arcSQ})]\)
35. Complete the following activity to find the co-ordinates of point P which divides seg AB in the ratio 3:1 where \(A(4, -3)\) and \(B(8,5)\).
36. In \(\Delta ABC\), seg \(XY \parallel\) side AC. If \(2AX = 3BX\) and \(XY = 9\), then find the value of AC.
37. Prove that "Opposite angles of cyclic quadrilateral are supplementary."
38. \(\Delta ABC \sim \Delta PQR\). In \(\Delta ABC, AB = 5.4 \text{ cm}, BC = 4.2 \text{ cm}, AC = 6.0 \text{ cm}, AB : PQ = 3 : 2\), then construct \(\Delta ABC\) and \(\Delta PQR\).
39. Show that:
\(\frac{\tan A}{(1 + \tan^2 A)^2} + \frac{\cot A}{(1 + \cot^2 A)^2} = \sin A \times \cos A\)
40. \(\square ABCD\) is a parallelogram. Point P is the midpoint of side CD. seg BP intersects diagonal AC at point X, then prove that : \(3AX = 2AC\)
41. In the above figure, seg AB and seg AD are tangent segments drawn to a circle with centre C from exterior point A, then prove that :
\(\angle A = \frac{1}{2} [m(\text{arcBYD}) - m(\text{arcBXD})]\)
42. Find the co-ordinates of centroid of a triangle if points \(D(-7,6), E(8,5)\) and \(F(2,-2)\) are the mid-points of the sides of that triangle.
43. If a and b are natural numbers and \(a > b\). If \((a^2 + b^2), (a^2 - b^2)\) and \(2ab\) are the sides of the triangle, then prove that the triangle is right angled. Find out two Pythagorean triplets by taking suitable values of a and b.
44. Construct two concentric circles with centre O with radii 3 cm and 5 cm. Construct tangent to a smaller circle from any point A on the larger circle. Measure and write the length of tangent segment. Calculate the length of tangent segment using Pythagoras theorem.
45. If \(\Delta ABC \sim \Delta PQR\) and \(\frac{A(\Delta ABC)}{A(\Delta PQR)} = \frac{16}{25}\), then find \(AB : PQ\).
46. In \(\Delta RST, \angle S = 90^\circ, \angle T = 30^\circ, RT = 12 \text{ cm}\), then find RS.
47. If radius of a circle is 5 cm, then find the length of longest chord of a circle.
48. Find the distance between the points \(O(0,0)\) and \(P(3,4)\).
49. In the figure, \(\angle L = 35^\circ\), find: i. m(arc MN) ii. m(arcMLN)
50. Show that, \(\cot\theta + \tan\theta = \text{cosec}\theta \times \sec\theta\).
51. Find the surface area of a sphere of radius 7 cm.
52. In trapezium ABCD, side AB \(\parallel\) side PQ \(\parallel\) side DC, \(AP = 15, PD = 12, QC = 14\), Find BQ.
53. Find the length diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.
54. In the given figure, points G,D,E, and F are concyclic points of a circle with centre C. \(\angle ECF = 70^\circ, m(\text{arcDGF}) = 200^\circ\). Find m(arcDE) and m(arc DEF).
55. Show that points \(A(-1,-1), B(0,1), C(1,3)\) are collinear.
56. A person is standing at a distance of 50 m from a temple looking at its top. The angle of elevation is \(45^\circ\). Find the height of the temple.
57. In \(\Delta PQR\) seg PM is a median. Angle bisectors of \(\angle PMQ\) and \(\angle PMR\) intersect side PQ and side PR in points X and Y respectively. Prove that \(XY \parallel QR\). Complete the proof by filling in the boxes.
58. Find the coordinates of point P where P is the midpoint of a line segment AB with A(–4, 2) and B(6, 2).
59. In \(\Delta ABC\), seg AP is a median. If \(BC = 18, AB^2 + AC^2 = 260\), Find AP.
60. Prove the following theorem: Angles inscribed in the same arc are congruent.
61. Draw a circle with a radius of 3.3 cm. Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at points P and Q. Write your observation about the tangents.
62. The radii of the ends of a frustum are 14 cm and 6 cm respectively and its height is 6 cm. Find its curved surface area.
63. In \(\Delta ABC\), seg \(DE \parallel\) side BC. If \(2A(\Delta ADE) = A(\square DBCE)\), find \(AB : AD\) and show that \(BC = \sqrt{3}DE\).
64. \(\Delta SHR \sim \Delta SVU\). In \(\Delta SHR, SH = 4.5 \text{ cm}, HR = 5.2 \text{ cm}, SR = 5.8 \text{ cm}\) and \(\frac{SH}{SV} = \frac{3}{5}\). Construct \(\Delta SVU\).
65. An ice cream pot has a right circular cylindrical shape. The radius of the base is 12cm and the height is 7 cm. This pot is completely filled with ice cream. The entire ice cream is given to the students in the form of right circular ice cream cones, having a diameter of 4 cm and a height is 3.5 cm. If each student is given one cone, how many students can be served?
66. A circle touches side BC at point P of the \(\Delta ABC\), from outside of the triangle. Further extended lines AC and AB are tangents to the circle at N and M respectively. Prove that:
\(AM = \frac{1}{2} (\text{Perimeter of } \Delta ABC)\)
67. Eliminate \(\theta\) if \(x = r\cos\theta\) and \(y = r\sin\theta\).
68. If \(\Delta ABC \sim \Delta PQR\) and \(AB : PQ = 2:3\), then find the value of \(\frac{A(\Delta ABC)}{A(\Delta PQR)}\).
69. Two circles of radii 5 cm and 3 cm touch each other externally. Find the distance between their centres.
70. Find the side of a square whose diagonal is \(10\sqrt{2} \text{ cm}\).
71. Angle made by the line with the positive direction of X-axis is given. Find the slope of the line. \(45^\circ\)
72. In the above figure, \(\angle ABC\) is inscribed in arc ABC. If \(\angle ABC = 60^\circ\). find \(m\angle AOC\).
73. Find the value of \(\sin^2\theta + \cos^2\theta\)
74. In the figure given above, \(\square ABCD\) is a square and a circle is inscribed in it. All sides of a square touch the circle. If \(AB = 14 \text{ cm}\), find the area of shaded region.
75. The radius of a sector of a circle is 3.5 cm and length of its arc is 2.2 cm. Find the area of the sector.
76. Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.
77. In the given figure, \(m(\text{arcNS}) = 125^\circ\), \(m(\text{arcEF}) = 37^\circ\), find the measure \(\angle NMS\).
78. Find the slope of the line passing through the points \(A(2,3)\) and \(B(4,7)\).
79. Find the surface area of a sphere of radius 7 cm.
80. In \(\Delta ABC\), ray BD bisects \(\angle ABC\), \(A-D-C\), seg \(DE \parallel\) side BC, \(A-E-B\), then for showing \(\frac{AB}{BC} = \frac{AE}{EB}\), complete the following activity:
81. Determine whether the points are collinear. \(A(1, -3), B(2, -5), C(-4,7)\)
82. \(\Delta ABC \sim \Delta LMN\). In \(\Delta ABC, AB = 5.5 \text{ cm}, BC = 6 \text{ cm}, CA = 4.5 \text{ cm}\). Construct \(\Delta ABC\) and \(\Delta LMN\) such that \(\frac{BC}{MN} = \frac{5}{4}\).
83. In \(\Delta PQR\), seg PM is a median, \(PM = 9\) and \(PQ^2 + PR^2 = 290\). Find the length of QR.
84. Prove that, if a line parallel to a side of a triangle intersects the other sides in two district points, then the line divides those sides in proportion.
85. \(\frac{1}{\sin^2\theta} - \frac{1}{\cos^2\theta} - \frac{1}{\tan^2\theta} - \frac{1}{\cot^2\theta} - \frac{1}{\sec^2\theta} - \frac{1}{\text{cosec}^2\theta} = -3\), then find the value of \(\theta\).
86. A cylinder of radius 12 cm contains water up to the height 20 cm. A spherical iron ball is dropped into the cylinder and thus water level raised by 6.75 cm. What is the radius of iron ball?
87. Draw a circle with centre O having radius 3 cm. Draw tangent segments PA and PB through the point P outside the circle such that \(\angle APB = 70^\circ\).
88. \(\square ABCD\) is trapezium, \(AB \parallel CD\) diagonals of trapezium intersects in point P.
Write the answers of the following questions:
a. Draw the figure using the given information.
b. Write any one pair of alternate angles and opposite angles.
c. Write the names of similar triangles with the test of similarity.
89. AB is a chord of a circle with centre O. AOC is diameter of circle, AT is a tangent at A.
Write answers of the following questions:
a. Draw the figure using the given information.
b. Find the measures of \(\angle CAT\) and \(\angle ABC\) with reasons.
c. Whether \(\angle CAT\) and \(\angle ABC\) are congruent? Justify your answer.
