Geometry MCQs

21

Geometry • Polygons

निम्नलिखित में से किस ज्यामितीय आकृति की भुजाओं की संख्या सबसे अधिक है? Which of the following geometrical shapes has the greatest number of sides?

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A. सप्तभुज Heptagon B. त्रिभुज Triangle C. अष्टभुज Octagon D. वर्ग Square

सही उत्तर: C Correct Answer: C

अष्टभुज की 8 भुजाएँ होती हैं। सप्तभुज की 7, वर्ग की 4 और त्रिभुज की 3 भुजाएँ होती हैं। इसलिए अष्टभुज की भुजाएँ सबसे अधिक हैं।

An octagon has eight sides, a heptagon has seven, a square has four and a triangle has three. Therefore, the octagon has the greatest number of sides.

22

Geometry • Triangles

किसी त्रिभुज के कोणों की माप पूर्णांक में है। निम्नलिखित में से कौन-सा अनुपात उसके कोणों का अनुपात नहीं हो सकता? The angle measures of a triangle are integers. Which of the following cannot be the ratio of its angles?

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A. 2:3:4 2:3:4 B. 2:5:7 2:5:7 C. 3:4:5 3:4:5 D. 4:5:6 4:5:6

सही उत्तर: B Correct Answer: B

2:5:7 के पदों का योग 14 है। प्रत्येक अनुपात इकाई 180/14 = 90/7 होगी, जिससे कोण पूर्णांक नहीं बनेंगे। अन्य विकल्पों में अनुपात के पदों का योग 180 को पूर्णतः विभाजित करता है।

The terms of 2:5:7 add to 14. Each ratio unit would be 180/14 = 90/7, which does not produce integer-valued angles. The sums in the other options divide 180 exactly.

23

Geometry • Coordinate Geometry

Find the rectangular coordinates of the point with cylindrical coordinates (2, 2pi/3, 1). Find the rectangular coordinates of the point with cylindrical coordinates (2, 2pi/3, 1).

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A. (-1, sqrt(3), 1) (-1, sqrt(3), 1) B. (2, sqrt(3), -2) (2, sqrt(3), -2) C. (1, sqrt(3), 0) (1, sqrt(3), 0) D. (1, sqrt(3), -1) (1, sqrt(3), -1)

सही उत्तर: A Correct Answer: A

Using x = r cos(theta), y = r sin(theta), z = z, we get x = 2 cos(2pi/3) = -1, y = 2 sin(2pi/3) = sqrt(3), and z = 1.

24

Geometry • Conic Sections

For what value of m is the line y = mx + 5 tangent to the ellipse represented by 4x^2/9 + y^2/16 = 1? For what value of m is the line y = mx + 5 tangent to the ellipse represented by 4x^2/9 + y^2/16 = 1?

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A. 2 2 B. plus or minus 2 plus or minus 2 C. plus or minus 3 plus or minus 3 D. 3 3

सही उत्तर: B Correct Answer: B

For y = mx + c to be tangent to x^2/a^2 + y^2/b^2 = 1, c^2 = a^2m^2 + b^2. Here a^2 = 9/4, b^2 = 16 and c = 5, giving m^2 = 4, so m = plus or minus 2.

25

Geometry • Coordinate Geometry

The Cartesian equation of the surface x = mu cos(nu), y = mu sin(nu), z = mu cot(nu) is: The Cartesian equation of the surface x = mu cos(nu), y = mu sin(nu), z = mu cot(nu) is:

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A. x^2 + z^2 = y^2 tan(nu) x^2 + z^2 = y^2 tan(nu) B. x^2 + y^2 = z^2 cot(nu) x^2 + y^2 = z^2 cot(nu) C. x^2 + y^2 = z^2 tan^2(nu) x^2 + y^2 = z^2 tan^2(nu) D. x^2 + z^2 = y^2 cot(nu) x^2 + z^2 = y^2 cot(nu)

सही उत्तर: C Correct Answer: C

From x^2 + y^2 = mu^2 and z = mu cot(nu), we have mu = z tan(nu). Therefore x^2 + y^2 = z^2 tan^2(nu).

26

Geometry • Solid Geometry

Revolving a circular area of radius R through 360 degrees about the x-axis generates a complete torus. The distance between the centre of the circle and the x-axis is d. The surface area of the torus is: Revolving a circular area of radius R through 360 degrees about the x-axis generates a complete torus. The distance between the centre of the circle and the x-axis is d. The surface area of the torus is:

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A. 3pi^2Rd 3pi^2Rd B. 4pi^2Rd 4pi^2Rd C. pi^2Rd pi^2Rd D. 2pi^2Rd 2pi^2Rd

सही उत्तर: B Correct Answer: B

By Pappus' centroid theorem, surface area = circumference of the generating circle x distance travelled by its centroid = 2pi R x 2pi d = 4pi^2Rd.

27

Geometry • Conic Sections

Find the focus of the conic x^2 + 19 - 8x - y = 0. Find the focus of the conic x^2 + 19 - 8x - y = 0.

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A. (4, 3.25) (4, 3.25) B. (2.75, 4) (2.75, 4) C. (3.25, 4) (3.25, 4) D. (4, 2.75) (4, 2.75)

सही उत्तर: A Correct Answer: A

Rewriting gives (x - 4)^2 = y - 3. Comparing with (x-h)^2 = 4a(y-k), we get a = 1/4, so the focus is (4, 3 + 1/4) = (4, 3.25).

28

Geometry • Solid Geometry

Find the volume of the solid bounded by the ellipsoid x^2/4 + y^2/9 + z^2/49 = 1. Find the volume of the solid bounded by the ellipsoid x^2/4 + y^2/9 + z^2/49 = 1.

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A. 124 124 B. 176 176 C. 248 248 D. 312 312

सही उत्तर: B Correct Answer: B

The semiaxes are 2, 3 and 7. Thus volume = (4/3)pi(2)(3)(7) = 56pi, approximately 176.

29

Geometry • Conic Sections

For the conic r = 1 + sin(omega), the slope of the tangent to the conic at omega = pi/3 is: For the conic r = 1 + sin(omega), the slope of the tangent to the conic at omega = pi/3 is:

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A. -1/2 -1/2 B. 1/2 1/2 C. -1 -1 D. 1 1

सही उत्तर: C Correct Answer: C

For a polar curve, dy/dx = (r' sin(omega) + r cos(omega))/(r' cos(omega) - r sin(omega)), where r' here denotes dr/domega. Substituting r = 1 + sin(omega) at omega = pi/3 gives -1.

30

Geometry • Coordinate Geometry

The equation x^2 - 6x + 6y + y^2 + 2 = 0 represents: The equation x^2 - 6x + 6y + y^2 + 2 = 0 represents:

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A. An ellipse with latus rectum 4 and focus at (-3,3) An ellipse with latus rectum 4 and focus at (-3,3) B. A circle with radius 4 and centre at (-3,3) A circle with radius 4 and centre at (-3,3) C. A circle with radius 4 and centre at (3,-3) A circle with radius 4 and centre at (3,-3) D. An ellipse with latus rectum 4 and focus at (3,-3) An ellipse with latus rectum 4 and focus at (3,-3)

सही उत्तर: C Correct Answer: C

Completing squares gives (x-3)^2 + (y+3)^2 = 16. This is a circle with centre (3,-3) and radius 4.

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