CDS Maths(1) Questions and answers

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Exams

These are the list of questions asked in the previous year CDS Maths Exams. You can practice these questions as real time exam in this link.

61). what is (1 + cot x - cosec x)(1 + tan x + cot x) equal to?

A). Sin x + cos x

B). Sin x - cos x

C). 2

D). 1

View Answer

Correct Answer: Sin x - cos x

(1 + cotx - cosecx)( 1 + tanx + secx)

Let x = 30 deg then

\( \Large =(1 + \sqrt{3} - 2) \left(1 + \frac{1}{3} + \frac{2}{\sqrt{3}}\right) \)

\( \Large = (\sqrt{3} - 1) \left( \frac{3 + \sqrt{3}}{\sqrt{3}} \right) \)

\( \Large (\sqrt{3} - 1)(\sqrt{3} + 1) \)

= 2

Option (B) is correct.

62). What is (cosec x - sin x)(sec x - cos x)(tan x + cot x)equal to?

A). Sin x + cos x

B). Sin x - cos x

C). 2

D). 1

View Answer

Correct Answer: 1

(cosecx - sinx) (secx - cosx)(tanx + cotx)

\( \Large \left(\frac{1}{sinx} - sinx\right) \left(\frac{1}{cosx} - cosx\right) \left( \frac{sinx}{cosx} + \frac{cosx}{sinx}\right) \)

\( \Large = \frac{cos^{2}x}{sinx} \times \frac{sin^{2}x}{cosx} \times \frac{1}{sinxcosx} = 1\)

63). Consider the following statements:
1. \(Sin 1^{\circ} > Sin 1\)
2. \(Cos 1^{\circ} < Cos 1\)
Which of the above statements is/are correct?

A). 1 only

B). 2 only

C). Both 1 and 2

D). Neither 1 nor 2

View Answer

64). If sin x + cosec x = 2, then what is \(sin^{9}x + cosec^{9}x\) equal to ?

A). 2

B). 18

C). 512

D). 1024

View Answer

65). If sin x + cos x = p and \( sin^{3}x + cos^{3}x = q\), then what is p^{3} - 3p equal to ?

A). 0

B). -2q

C). 2q

D). 4q

View Answer

Correct Answer: -2q

Similarly,

\( \Large sin^{3}x + cos^{3}x = q \)

sinx + cosx = p ... (i)

squaring on both side

So,\( \Large 1 + 2sinxcosx = p^{2} \)

\( \Large sin x cos x = \frac{p^{2} - 1}{2} \)... (ii)

Cubing on both side of equation (i) we get

\( \Large (sinx + cos x)^{3} = p^{3} \)

\( \Large sin^{3} + cos^{3}x + 3 sinx cosx(sinx + cosx) = p^{3} \)

\( \Large q + 3 \left( \frac{p^{2} - 1}{2}\right)(p) = p^{3} \)

\( \Large 2q + 3(p^{2} - 1)p = 2p^{3}\)

\( \Large 2q + 3p^{3} - 3p = 2p^{3} \)

\( \Large p^{3} - 3p = -2q \)

66). What is the number of pairs of perpendicular planes in cuboid?

A). 4

B). 8

C). 12

D). None of these

View Answer

67). How many equilateral triangles can be formed by joining any three vertices of a cube?

A). 0

B). 4

C). 8

D). None of these

View Answer

68). For the next two (2) items that follow :
ABCD is a trapezium in which AB is parallel to CD. Let M be the midpoint of BC.
68. Consider the following statements :
1. 'Area of triangle ADM + Area of triangle DCM' is equal to three-fourth of the area of trapezium ABCD, if AB = CD.
2. 'Area of triangle DCM + Area of triangle ABM' is always greater than half of the area of trapezium ABCD.
Which of the above statements is/are correct?

A). 1 only

B). 2 only

C). Both 1 and 2

D). Neither 1 nor 2

View Answer

Correct Answer: 2 only

Let M be the midpoint of BC and let AD = 2 cm and BC = 4 cm

Now area of \( \Large \vartriangle ADM = 1cm^{2} \)

and area of \( \Large \vartriangle DCM = \frac{1}{2} \times 2 \times 1 = 1 cm^{2} \)

Area of trapezium = \( \Large \frac{1}{2} x 1 x (2 + 4) = 3 cm^{2}\)

Now, \( \Large Area of \vartriangle ADM + Area of \vartriangle DCM = 26 cm^{2}\) Area of trapezium ABCD = \( \Large 3 cm^{2} \)

Statement (1) is wrong

For Statement 2

\( \Large Area of \vartriangle DCM + Area of \vartriangle ABM = \frac{1}{2} \times 2 \times 1 + \frac{1}{2} \times 2 \times 1 \)

= 1 + 1

=2 \( \Large cm^{2} \)

Now \( \Large \frac{1}{2} \times Area of trapezium = \frac{3}{2} cm^{2} \)

\( \Large = 1.5 cm^{2} \)

2 > 1.5

Statement (2) is correct.

Option (B)is correct.

69). Consider the following statements :
1. 'Area of triangle ADM Area of triangle ABM' is always equal to area of triangle DCM, if AB = CD.
2. Half of area of triangle ABM is equal to one-eighth of area of trapezium ABCD. if AB = CD
Which of the above statements is/are correct?

A). 1 only

B). 2 only

C). Both 1 and 2

D). Neither 1 nor 2

View Answer

70). ABCD is a parallelogram. P and R are the midpoints of DC and BC respectively. The line PR intersects the diagonal AC at Q. The distance CQ will be equal to

A). \( \Large \frac{AC}{4} \)

B). \( \Large \frac{BD}{3} \)

C). \( \Large \frac{BD}{4} \)

D). \( \Large \frac{AC}{3} \)

View Answer

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