NDA Previous Year Question Paper 2016 | Free PDF Download 

NDA previous Year Question paper

1. Suppose ωω is a cube root of unity with ω≠1ω=1 . Suppose PP and QQ are the points on the complex plane defined by ωω and ω2ω2 . If OO is the origin, then what is the angle between OPOP and OQOQ ?

(a) 60∘60∘
(b) 90∘90∘
(c) 120∘120∘
(d) 150∘150∘

1. Suppose there is a relation ∗∗ between the positive numbers xx and yy given by x∗yx∗y if and only if x≤y2x≤y2 . Then which one of the following is correct?

(a) ∗∗ is reflexive but not transitive and symmetric
(b) ∗∗ is transitive but not reflexive and symmetric
(c) ∗∗ is symmetric and reflexive but not transitive
(d) ∗∗ is symmetric but not reflexive and transitive

1. If x2−px+4>0x2−px+4>0 for all real values of xx then which one of the following is correct?

(a) ∣p∣<4∣p∣<4
(b) ∣p∣≤4∣p∣≤4
(c) ∣p∣>4∣p∣>4
(d) ∣p∣≥4∣p∣≥4

1. If z=x+iy=(12−i2)−25z=x+iy=(2​1​−2​i​)−25 , where i=−1i=−1​ , then what is the fundamental amplitude of z−2z−i2z−i2​z−2​​ ?

(a) ππ
(b) π22π​
(c) π33π​
(d) π44π​

1. If

f(x1)−f(x2)=f(x1−x21−x1x2)f(x1​)−f(x2​)=f(1−x1​x2​x1​−x2​​)

for x1,x2∈(−1,1)x1​,x2​∈(−1,1) , then what is f(x)f(x) equal to?

(a) ln⁡(1−x1+x)ln(1+x1−x​)
(b) ln⁡(2+x1−x)ln(1−x2+x​)
(c) tan⁡−1(1−x1+x)tan−1(1+x1−x​)
(d) tan⁡−1(1+x1−x)tan−1(1−x1+x​)

1. What is the range of the function

y=x21+x2y=1+x2x2​

where x∈Rx∈R ?

(a) [0, 1]
(b) [0, 1]
(c) (0, 1)
(d) (0, 1)

NDA previous Year Question paper

1. मान लीजिए कि ωω एक (यूिनिट) का घनमूल है और ω≠1ω=1 है। मान लीजिए PP और QQ, ωω तथा ω2ω2 द्वारा परिभाषित सम्मिश्र समतल पर बिंदुएँ हैं। यदि OO मूलबिंदु है, तो OPOP और OQOQ के बीच का कोण क्या है?

(a) 60°
(b) 90°
(c) 120°
(d) 150°

1. मान लीजिए कि धनात्मक संख्याओं xx और yy के बीच एक संबंध ∗∗ इस प्रकार दिया गया है कि x∗yx∗y, यदि और केवल यदि x≤y2x≤y2 है। तो निम्नलिखित में से कौन-सा एक सही है?

(a) * स्वतुल्य है लेकिन संक्रामक और सममित नहीं
(b) * संक्रामक है लेकिन स्वतुल्य और सममित नहीं
(c) * सममित और स्वतुल्य है लेकिन संक्रामक नहीं
(d) * सममित है लेकिन स्वतुल्य और संक्रामक नहीं

1. यदि xx के सभी वास्तविक मानों के लिए x2−px+4>0x2−px+4>0 है, तो निम्नलिखित में से कौन-सा एक सही है?

(a) ∣p∣<4∣p∣<4
(b) ∣p∣≤4∣p∣≤4
(c) ∣p∣>4∣p∣>4
(d) ∣p∣≥4∣p∣≥4

1. यदि z=x+iy=(12−i2)−25z=x+iy=(2​1​−2​i​)−25, जहाँ i=−1i=−1​ है, तो z−2z−i2z−i2​z−2​​ का मूल आयाम क्या है?

(a) π
(b) π22π​
(c) π33π​
(d) π44π​

1. यदि x1,x2∈(−1,1)x1​,x2​∈(−1,1) के लिए

f(x1)−f(x2)=f(x1−x21−x1x2)f(x1​)−f(x2​)=f(1−x1​x2​x1​−x2​​)

है, तो f(x)f(x) किसके बराबर है?

(a) ln⁡(1−x1+x)ln(1+x1−x​)
(b) ln⁡(2+x1−x)ln(1−x2+x​)
(c) tan⁡−1(1−x1+x)tan−1(1+x1−x​)
(d) tan⁡−1(1+x1−x)tan−1(1−x1+x​)

1. फलन y=x21+x2y=1+x2x2​ का परास क्या है, जहाँ x∈Rx∈R है?

(a) [0,1)[0,1)
(b) [0,1][0,1]
(c) (0,1)(0,1)
(d) (0,1](0,1]

NDA previous Year Question paper

1. A straight line intersects xx and yy axes at PP and QQ respectively. If (3, 5) is the middle point of PQPQ , then what is the area of the triangle OPQOPQ ?

(a) 12 square units
(b) 15 square units
(c) 20 square units
(d) 30 square units

1. If a circle of radius bb units with centre at (0,b)(0,b) touches the line y=x−2y=x−2​ , then what is the value of bb ?

(a) 2+22+2​
(b) 2−22−2​
(c) 2222​
(d) 22​

For the next three (3) items that follow :

Consider the function

f(θ)=4(sin⁡2θ+cos⁡4θ)f(θ)=4(sin2θ+cos4θ)

1. What is the maximum value of the function f(θ)f(θ) ?

(a) 1
(b) 2
(c) 3
(d) 4

1. What is the minimum value of the function f(θ)f(θ) ?

(a) 0
(b) 1
(c) 2
(d) 3

1. Consider the following statements :

2. f(θ)=2f(θ)=2 has no solution.

3. f(θ)=72f(θ)=27​ has a solution.

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

For the next two (2) items that follow :

Consider the curves

1. Where do the curves intersect?

(a) At (2, 3) only
(b) At (- 1, - 2) only
(c) At (2, 3) and (- 1, - 2)
(d) Neither at (2, 3) nor at (- 1, - 2)

NDA previous Year Question paper

1. What is the area bounded by the curves?

(a) 176617​ square units
(b) 8338​ square units
(c) 2 square units
(d) 1331​ square unit

For the next two (2) items that follow :

Consider the function

f(x)=27(x2/3−x)4f(x)=427(x2/3−x)​

1. How many solutions does the function f(x)=1f(x)=1 have?

(a) One
(b) Two
(c) Three
(d) Four

1. How many solutions does the function f(x)=−1f(x)=−1 have?

(a) One
(b) Two
(c) Three
(d) Four

For the next two (2) items that follow :

Consider the functions

f(x)=xg(x) and g(x)=[1x]f(x)=xg(x) and g(x)=[x1​]

where [⋅][⋅] is the greatest integer function.

1. What is ∫1312g(x)dx∫31​21​​g(x)dx equal to?

(a) 1661​
(b) 1331​
(c) 518185​
(d) 536365​

1. What is ∫131f(x)dx∫31​1​f(x)dx equal to?

(a) 37727237​
(b) 2332​
(c) 17727217​
(d) 3714414437​

For the next five (5) items that follow :

Consider the function

f(x)=∣x−1∣+x2f(x)=∣x−1∣+x2

where x∈Rx∈R

1. Which one of the following statements is correct?

(a) f(x)f(x) is continuous but not differentiable at x=0x=0
(b) f(x)f(x) is continuous but not differentiable at x=1x=1
(c) f(x)f(x) is differentiable at x=1x=1
(d) f(x)f(x) is not differentiable at x=0x=0 and x=1x=1

NDA previous Year Question paper

1. Which one of the following statements is correct?

(a) f(x)f(x) is increasing in (−∞,12)(−∞,21​) and decreasing in (12,∞)(21​,∞)
(b) f(x)f(x) is decreasing in (−∞,12)(−∞,21​) and increasing in (12,∞)(21​,∞)
(c) f(x)f(x) is increasing in (−∞,1)(−∞,1) and decreasing in (1,∞)(1,∞)
(d) f(x)f(x) is decreasing in (−∞,1)(−∞,1) and increasing in (1,∞)(1,∞)

1. Which one of the following statements is correct?

(a) f(x)f(x) has local minima at more than one point in (−∞,∞)(−∞,∞)
(b) f(x)f(x) has local maxima at more than one point in (−∞,∞)(−∞,∞)
(c) f(x)f(x) has local minimum at one point only in (−∞,∞)(−∞,∞)
(d) f(x)f(x) has neither maxima nor minima in (−∞,∞)(−∞,∞)

1. What is the area of the region bounded by xx -axis, the curve y=f(x)y=f(x) and the two ordinates x=12x=21​ and x=1x=1 ?

(a) 512125​ square unit
(b) 5665​ square unit
(c) 7667​ square units
(d) 2 square units

1. What is the area of the region bounded by xx -axis, the curve y=f(x)y=f(x) and the two ordinates x=1x=1 and x=32x=23​ ?

(a) 512125​ square unit
(b) 712127​ square unit
(c) 2332​ square unit
(d) 11121211​ square unit

For the next two (2) items that follow :

Given that

an=∫0πsin⁡2{(n+1)x}sin⁡2xdxan​=∫0π​sin2xsin2{(n+1)x}​dx

1. Consider the following statements :

2. The sequence {a2n}{a2n​} is in AP with common difference zero.

3. The sequence {a2n+1}{a2n+1​} is in AP with common difference zero.

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

1. What is an−1−an−4an−1​−an−4​ equal to?

(a) -1
(b) 0
(c) 1
(d) 2

NDA previous Year Question paper

1. The centroid of the triangle is at which one of the following points?

(a) (3, 6)
(b) (32,6)(23​,6)
(c) (3, 3)
(d) (32,9)(23​,9)

For the next two (2) items that follow :

Consider the function

f(x)=(x−1)2(x+1)(x−2)3f(x)=(x−1)2(x+1)(x−2)3

1. What is the number of points of local minima of the function f(x)f(x) ?

(a) None
(b) One
(c) Two
(d) Three

1. What is the number of points of local maxima of the function f(x)f(x) ?

(a) None
(b) One
(c) Two
(d) Three

NDA previous Year Question paper

1. Let f(x)f(x) and g(x)g(x) be twice differentiable functions on [0,2][0,2] satisfying f′′(x)=g′′(x),f′(1)=4,g′(1)=6,f′′(x)=g′′(x),f′(1)=4,g′(1)=6, f(2)=3f(2)=3 and g(2)=9g(2)=9 . Then what is f(x)−g(x)f(x)−g(x) at x=4x=4 equal to?

(a) -10
(b) -6
(c) -4
(d) 2

For the next two (2) items that follow :

Consider the curves

y=∣x−1∣and∣x∣=2y=∣x−1∣and∣x∣=2

1. What is/are the point(s) of intersection of the curves?

(a) (-2,3) only
(b) (2,1) only
(c) (-2,3) and (2,1)
(d) Neither (-2,3) nor (2,1)

1. What is the area of the region bounded by the curves and xx -axis?

(a) 3 square units
(b) 4 square units
(c) 5 square units
(d) 6 square units

For the next two (2) items that follow :

Consider the function

where pp is a constant.

1. What is the value of f′(0)f′(0) ?

(a) p3p3
(b) 3p33p3
(c) 6p36p3
(d) −6p3−6p3

1. What is the value of pp for which f′′(0)=0f′′(0)=0 ?

(a) −16−61​ or 0
(b) -1 or 0
(c) −16−61​ or 1
(d) -1 or 1

For the next two (2) items that follow :

Consider a triangle ABC in which

cos⁡A+cos⁡B+cos⁡C=3sin⁡π3cosA+cosB+cosC=3​sin3π​

1. What is the value of sin⁡A2sin⁡B2sin⁡C2sin2A​sin2B​sin2C​ ?

(a) 1221​
(b) 1441​
(c) 1881​
(d) 116161​

NDA previous Year Question paper

1. What is the value of

cos⁡(A+B2)cos⁡(B+C2)cos⁡(C+A2)?cos(2A+B​)cos(2B+C​)cos(2C+A​)?

(a) 1441​
(b) 1221​
(c) 116161​
(d) None of the above

For the next two (2) items that follow :

Given that tan⁡αtanα and tan⁡βtanβ are the roots of the equation x2+bx+c=0x2+bx+c=0 with b≠0b=0 .

1. What is tan⁡(α+β)tan(α+β) equal to?

(a) b(c−1)b(c−1)
(b) c(b−1)c(b−1)
(c) c(b−1)−1c(b−1)−1
(d) b(c−1)−1b(c−1)−1

1. What is sin⁡(α+β)sin(α+β) sec αα sec ββ equal to?

(a) bb
(b) −b−b
(c) cc
(d) −c−c

For the next two (2) items that follow :

Consider the two circles (x−1)2+(y−3)2=r2(x−1)2+(y−3)2=r2 and x2+y2−8x+2y+8=0x2+y2−8x+2y+8=0

1. What is the distance between the centres of the two circles?

(a) 5 units
(b) 6 units
(c) 8 units
(d) 10 units

1. If the circles intersect at two distinct points, then which one of the following is correct?

(a) r=1r=1
(b) 1<r<21<r<2
(c) r=2r=2
(d) 2<r<82<r<8

For the next two (2) items that follow :

Consider the two lines

x+y+1=0and3x+2y+1=0x+y+1=0and3x+2y+1=0

1. What is the equation of the line passing through the point of intersection of the given lines and parallel to xx -axis?

(a) y+1=0y+1=0
(b) y−1=0y−1=0
(c) y−2=0y−2=0
(d) y+2=0y+2=0

NDA previous Year Question paper

1. What is the equation of the line passing through the point of intersection of the given lines and parallel to yy -axis?

(a) x+1=0x+1=0
(b) x−1=0x−1=0
(c) x−2=0x−2=0
(d) x+2=0x+2=0

For the next two (2) items that follow :

Consider the equation

ksin⁡x+cos⁡2x=2k−7ksinx+cos2x=2k−7

1. If the equation possesses solution, then what is the minimum value of kk ?

(a) 1
(b) 2
(c) 4
(d) 6

1. If the equation possesses solution, then what is the maximum value of kk ?

(a) 1
(b) 2
(c) 4
(d) 6

For the next two (2) items that follow :

Consider the function

f(x)=a∣x∣+x−1∣x∣+xf(x)=∣x∣+xa∣x∣+x−1​

where [·] denotes the greatest integer function.

1. What is lim⁡x→0+f(x)limx→0+​f(x) equal to?

(a) 1
(b) ln⁡alna
(c) 1−a−11−a−1
(d) Limit does not exist

1. What is lim⁡x→0−f(x)limx→0−​f(x) equal to?

(a) 0
(b) ln⁡alna
(c) 1−a−11−a−1
(d) Limit does not exist

For the next two (2) items that follow :

Let z1z1​ , z2z2​ and z3z3​ be non- zero complex numbers satisfying z2=izˉz2=izˉ , where i=−1i=−1​ .

1. What is z1+z2+z3z1​+z2​+z3​ equal to?

(a) ii
(b) −i−i
(c) 0
(d) 1

NDA previous Year Question paper

1. Consider the following statements :

2. z1z2z3z1​z2​z3​ is purely imaginary.

3. z1z2+z2z3+z3z1z1​z2​+z2​z3​+z3​z1​ is purely real.

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

For the next two (2) items that follow :

Given that log⁡xylogx​y, log⁡zxlogz​x, log⁡yzlogy​z are in GP, xyz=64xyz=64 and x3,y3,z3x3,y3,z3 are in AP.

1. Which one of the following is correct?

x,y and z are

(a) in AP only
(b) in GP only
(c) in both AP and GP
(d) neither in AP nor in GP

1. Which one of the following is correct?

xy, yz and zx are

(a) in AP only
(b) in GP only
(c) in both AP and GP
(d) neither in AP nor in GP

For the next two (2) items that follow :

Let zz be a complex number satisfying

∣z−4z−8∣=1 and ∣zz−2∣=32​z−8z−4​​=1 and ​z−2z​​=23​

1. What is ∣z∣∣z∣ equal to?

(a) 6
(b) 12
(c) 18
(d) 36

1. What is ∣z−6z+6∣​z+6z−6​​ equal to?

(a) 3
(b) 2
(c) 1
(d) 0

NDA previous Year Question paper

1. Consider the following statements :

2. The function f(x)f(x) is continuous at x=0x=0 .

3. The function f(x)f(x) is continuous at x=π2x=2π​ .

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

1. Consider the following statements :

2. The function f(x)f(x) is differentiable at x=0x=0 .

3. The function f(x)f(x) is differentiable at x=π2x=2π​ .

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

For the next two (2) items that follow :

Let αα and ββ (α<β)(α<β) be the roots of the equation x2+bx+c=0x2+bx+c=0 , where b>0b>0 and c<0c<0 .

1. Consider the following :

2. β<−αβ<−α

3. β<∣α∣β<∣α∣

Which of the above is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

1. Consider the following :

2. α+β+αβ>0α+β+αβ>0

3. α2β+β2α>0α2β+β2α>0

Which of the above is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

NDA previous Year Question paper

Consider a parallelogram whose vertices are A(1,2)A(1,2) , B(4,y)B(4,y) , C(x,6)C(x,6) and D(3,5)D(3,5) taken in order.

1. What is the value of AC2−BD2AC2−BD2 ?

(a) 25
(b) 30
(c) 36
(d) 40

1. What is the point of intersection of the diagonals?

(a) (72,4)(27​,4)
(b) (3, 4)
(c) (72,5)(27​,5)
(d) (3, 5)

1. What is the area of the parallelogram?

(a) 7227​ square units
(b) 4 square units
(c) 112211​ square units
(d) 7 square units

For the next four (4) items that follow :

Let f:R→Rf:R→R be a function such that

f(x)=x3+x2f′(1)+xf′′(2)+f′′′(3)f(x)=x3+x2f′(1)+xf′′(2)+f′′′(3)

for x∈Rx∈R .

1. What is f(1)f(1) equal to?

(a) -2
(b) -1
(c) 0
(d) 4

1. What is f′(1)f′(1) equal to?

(a) -6
(b) -5
(c) 1
(d) 0

1. What is f′′′(10)f′′′(10) equal to?

(a) 1
(b) 5
(c) 6
(d) 8

NDA previous Year Question paper

1. Consider the following :

2. f(2)=f(1)−f(0)f(2)=f(1)−f(0)

3. f′′(2)−2f′(1)=12f′′(2)−2f′(1)=12 .

Which of the above is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

For the next three (3) items that follow :

A plane PP passes through the line of intersection of the planes 2x−y+3z=22x−y+3z=2 , x+y−z=1x+y−z=1 and the point (1, 0, 1).

1. What are the direction ratios of the line of intersection of the given planes?

(a) ⟨2,−5,−3⟩⟨2,−5,−3⟩
(b) ⟨1,−5,−3⟩⟨1,−5,−3⟩
(c) ⟨2,5,3⟩⟨2,5,3⟩
(d) ⟨1,3,5⟩⟨1,3,5⟩

1. What is the equation of the plane PP ?

(a) 2x+5y−2=02x+5y−2=0
(b) 5x+2y−5=05x+2y−5=0
(c) x+z−2=0x+z−2=0
(d) 2x−y−2z=02x−y−2z=0

1. If the plane PP touches the sphere x2+y2+z2=r2x2+y2+z2=r2 , then what is rr equal to?

(a) 22929​2​
(b) 42929​4​
(c) 52929​5​
(d) 1

For the next two (2) items that follow :

Consider the function

f(x)=∣x2−5x+6∣f(x)=∣x2−5x+6∣

1. What is f′(4)f′(4) equal to?

(a) -4
(b) -3
(c) 3
(d) 2

NDA previous Year Question paper

1. What is f′′(2.5)f′′(2.5) equal to?

(a) -3
(b) -2
(c) 0
(d) 2

For the next two (2) items that follow :

Let f(x)f(x) be the greatest integer function and g(x)g(x) be the modulus function.

1. What is (g∘f)(−53)−(f∘g)(−53)(g∘f)(−35​)−(f∘g)(−35​) equal to?

(a) -1
(b) 0
(c) 1
(d) 2

1. What is (f∘f)(−95)+(g∘g)(−2)(f∘f)(−59​)+(g∘g)(−2) equal to?

(a) -1
(b) 0
(c) 1
(d) 2

For the next two (2) items that follow :

Consider a circle passing through the origin and the points (a,b)(a,b) and (−b,−a)(−b,−a) .

1. On which line does the centre of the circle lie?

(a) x+y=0x+y=0
(b) x−y=0x−y=0
(c) x+y=a+bx+y=a+b
(d) x−y=a2−b2x−y=a2−b2

1. What is the sum of the squares of the intercepts cut off by the circle on the axes?

(a) (a2+b2a2−b2)2(a2−b2a2+b2​)2
(b) 2(a2+b2a−b)22(a−ba2+b2​)2
(c) 4(a2+b2a−b)24(a−ba2+b2​)2
(d) None of the above

NDA previous Year Question paper

1. What is cos⁡(θ2)cos(2θ​) equal to?

(a) ∣a^−b^∣22∣a^−b^∣​
(b) ∣a^+b^∣22∣a^+b^∣​
(c) ∣a^−b^∣44∣a^−b^∣​
(d) ∣a^+b^∣44∣a^+b^∣​

1. What is sin⁡(θ2)sin(2θ​) equal to?

(a) ∣a^−b^∣22∣a^−b^∣​
(b) ∣a^+b^∣22∣a^+b^∣​
(c) ∣a^−b^∣44∣a^−b^∣​
(d) ∣a^+b^∣44∣a^+b^∣​

1. Consider the following statements :

2. There exists θ∈(−π2,π2)θ∈(−2π​,2π​) for which tan⁡−1(tan⁡θ)≠θtan−1(tanθ)=θ .

3. sin⁡−1(13)−sin⁡−1(15)=sin⁡−1(22(3−1)15)sin−1(31​)−sin−1(51​)=sin−1(1522​(3​−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

1. Consider the following statements :

2. tan⁡−1x+tan⁡−1(1x)=πtan−1x+tan−1(x1​)=π

3. There exist x,y∈[−1,1]x,y∈[−1,1] , where x≠yx=y such that sin⁡−1x+cos⁡−1y=π2sin−1x+cos−1y=2π​ .

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

NDA previous Year Question paper

1. What are the order and degree respectively of the differential equation whose solution is y=cx+c2−3c3/2+2y=cx+c2−3c3/2+2 , where cc is a parameter?

(a) 1, 2
(b) 2, 2
(c) 1, 3
(d) 1, 4

1. What is

∫−22xdx−∫−22[x]dx∫−22​xdx−∫−22​[x]dx

equal to, where [⋅][⋅] is the greatest integer function?

(a) 0
(b) 1
(c) 2
(d) 4

1. If

∫−25f(x)dx=4 and ∫05{1+f(x)}dx=7∫−25​f(x)dx=4 and ∫05​{1+f(x)}dx=7

then what is ∫−20f(x)dx∫−20​f(x)dx equal to?

(a) -3
(b) 2
(c) 3
(d) 5

1. If lim⁡x→0ϕ(x)=a2limx→0​ϕ(x)=a2 , where a≠0a=0 , then what is lim⁡x→0ϕ(xa)limx→0​ϕ(ax​) equal to?

(a) a2a2
(b) a−2a−2
(c) −a2−a2
(d) −a−a

1. What is lim⁡x→0e−1x2limx→0​e−x21​ equal to?

(a) 0
(b) 1
(c) -1
(d) Limit does not exist

1. If AA is a square matrix, then what is adj⁡(A−1)−(adj⁡A)−1adj(A−1)−(adjA)−1 equal to?

(a) 2∣A∣2∣A∣
(b) Null matrix
(c) Unit matrix
(d) None of the above

1. What is the binary equivalent of the decimal number 0-3125?

(a) 0-0111
(b) 0-1010
(c) 0-0101
(d) 0-1101

1. Let RR be a relation on the set NN of natural numbers defined by ∀nRm⇔n∀nRm⇔n is a factor of m′m′ . Then which one of the following is correct?

(a) RR is reflexive, symmetric but not transitive
(b) RR is transitive, symmetric but not reflexive
(c) RR is reflexive, transitive but not symmetric
(d) RR is an equivalence relation

NDA previous Year Question paper

1. What is ∫04π∣cos⁡x∣dx∫04π​∣cosx∣dx equal to?

(a) 0
(b) 2
(c) 4
(d) 8

1. What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?

(a) 860
(b) 854
(c) 840
(d) 824

1. (a,2b)(a,2b) is the mid- point of the line segment joining the points (10, - 6) and (k,4)(k,4) . If a−2b=7a−2b=7 , then what is the value of kk ?

(a) 2
(b) 3
(c) 4
(d) 5

1. Consider the following statements :

2. If ABC is an equilateral triangle, then 3tan⁡(A+B)tan⁡C=13tan(A+B)tanC=1 .

3. If ABC is a triangle in which A=78∘A=78∘ , B=66∘B=66∘ , then

tan⁡(A2+C)<tan⁡Atan(2A​+C)<tanA

1. If ABC is any triangle, then

tan⁡(A+B2)sin⁡(C2)<cos⁡(C2)tan(2A+B​)sin(2C​)<cos(2C​)

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) 1 and 2
(d) 2 and 3

1. If A=(cos⁡12∘−cos⁡36∘)(sin⁡96∘+sin⁡24∘)A=(cos12∘−cos36∘)(sin96∘+sin24∘) and B=(sin⁡60∘−sin⁡12∘)(cos⁡48∘−cos⁡72∘)B=(sin60∘−sin12∘)(cos48∘−cos72∘) , then what is ABBA​ equal to?

(a) -1
(b) 0
(c) 1
(d) 2

NDA previous Year Question paper

1. What is the mean deviation from the mean of the numbers 10, 9, 21, 16, 24 ?

(a) 5.2
(b) 5.0
(c) 4.5
(d) 4.0

1. Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?

(a) 17181817​
(b) 53545453​
(c) 103108108103​
(d) 215216216215​

1. Two independent events AA and BB have P(A)=13P(A)=31​ and P(B)=34P(B)=43​ . What is the probability that exactly one of the two events AA or BB occurs?

(a) 1441​
(b) 5665​
(c) 512125​
(d) 712127​

1. A coin is tossed three times. What is the probability of getting head and tail alternately?

(a) 1881​
(b) 1441​
(c) 1221​
(d) 3443​

1. If the total number of observations is 20, ∑xi=1000∑xi​=1000 and ∑xi2=84000∑xi2​=84000 , then what is the variance of the distribution?

(a) 1500
(b) 1600
(c) 1700
(d) 1800

1. A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is queen of spade?

(a) 152521​
(b) 113131​
(c) 1441​
(d) 1881​

NDA previous Year Question paper

1. If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?

(a) 13181813​
(b) 5665​
(c) 11121211​
(d) 35363635​

1. A certain type of missile hits the target with probability p=0.3p=0.3 . What is the least number of missiles should be fired so that there is at least an 80%80% probability that the target is hit?

(a) 5
(b) 6
(c) 7
(d) None of the above

1. For two mutually exclusive events A and B, P(A)=0.2P(A)=0.2 and P(A‾∩B)=0.3P(A∩B)=0.3 . What is P(A∣(A∪B))P(A∣(A∪B)) equal to?

(a) 1221​
(b) 2552​
(c) 2772​
(d) 2332​

1. What is the probability of 5 Sundays in the month of December?

(a) 1771​
(b) 2772​
(c) 3773​
(d) None of the above

1. If mm is the geometric mean of

(yz)log⁡(yz),(zx)log⁡(zx)and(xy)log⁡(xy).(zy​)log(yz),(xz​)log(zx)and(yx​)log(xy).

then what is the value of mm ?

(a) 1
(b) 3
(c) 6
(d) 9

1. A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?

(a) 2332​
(b) 1331​
(c) 1441​
(d) 2552​

NDA previous Year Question paper

1. The mean of the series x1,x2,…,xnx1​,x2​,…,xn​ is X‾X . If x2x2​ is replaced by λλ , then what is the new mean?

(a) X‾−x2+λnnX−x2​+λ​
(b) X‾−x2−λnnX−x2​−λ​
(c) X‾−x2+λnnX−x2​+λ​
(d) nX‾−x2+λnnnX−x2​+λ​

1. For the data

3,5,1,6,5,9,5,2,8,63,5,1,6,5,9,5,2,8,6

the mean, median and mode are xx , yy and zz respectively. Which one of the following is correct?

(a) x=y≠zx=y=z
(b) x≠y=zx=y=z
(c) x≠y≠zx=y=z
(d) x=y=zx=y=z

1. Consider the following statements in respect of a histogram :

2. The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and the xx -axis.

3. When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency.

Which of the above statements is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

1. A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?

(a) 1221​
(b) 3883​
(c) 1441​
(d) 1881​

1. What is the number of ways in which 3 holiday travel tickets are to be given to 10 employees of an organization, if each employee is eligible for any one or more of the tickets?

(a) 60
(b) 120
(c) 500
(d) 1000

1. If one root of the equation

(l−m)x2+λx+1=0(l−m)x2+λx+1=0

is double the other and ll is real, then what is the greatest value of mm ?

(a) −98−89​
(b) 9889​
(c) −89−98​
(d) 8998​

1. What is the number of four-digit decimal numbers (<1)(<1) in which no digit is repeated?

(a) 3024
(b) 4536
(c) 5040
(d) None of the above

NDA previous Year Question paper

1. What is a vector of unit length orthogonal to both the vectors i⃗+j⃗+k⃗i+j​+k and 2i⃗+3j⃗−k⃗2i+3j​−k ?

(a) −4i⃗+3j⃗−k⃗2626​−4i+3j​−k​
(b) −4i⃗+3j⃗+k⃗2626​−4i+3j​+k​
(c) −3i⃗+2j⃗−k⃗1414​−3i+2j​−k​
(d) −3i⃗+2j⃗+k⃗1414​−3i+2j​+k​

1. If a⃗,b⃗a,b and c⃗c are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?

(a) a⃗+b⃗+c⃗=0⃗a+b+c=0
(b) a⃗+b⃗+c⃗=unit vectora+b+c=unit vector
(c) a⃗+b⃗=c⃗a+b=c
(d) a⃗=b⃗+c⃗a=b+c

1. What is the area of the parallelogram having diagonals 3i⃗+j⃗−2k⃗3i+j​−2k and i⃗−3j⃗+4k⃗i−3j​+4k ?

(a) 5555​ square units
(b) 4545​ square units
(c) 5353​ square units
(d) 152152​ square units

1. Consider the following in respect of the matrix A=(−111−1)A=(−11​1−1​) :

2. A2=−AA2=−A

3. A3=4AA3=4A

Which of the above is/are correct?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

1. Which of the following determinants have value 'zero'?

Select the correct answer using the code given below.

(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3

NDA previous Year Question paper

1. What is the acute angle between the lines represented by the equations y−3x−5=0y−3​x−5=0 and 3y−x+6=03​y−x+6=0 ?

(a) 30∘30∘
(b) 45∘45∘
(c) 60∘60∘
(d) 75∘75∘

1. The system of linear equations kx+y+z=1kx+y+z=1 , x+ky+z=1x+ky+z=1 and x+y+kz=1x+y+kz=1 has a unique solution under which one of the following conditions?

(a) k≠1k=1 and k≠−2k=−2
(b) k≠1k=1 and k≠2k=2
(c) k≠−1k=−1 and k≠−2k=−2
(d) k≠−1k=−1 and k≠2k=2

1. What is the number of different messages that can be represented by three O's and two 1's?

(a) 10
(b) 9
(c) 8
(d) 7

1. If log⁡a(ab)=xloga​(ab)=x , then what is log⁡b(ab)logb​(ab) equal to?

(a) 1xx1​
(b) xx+1x+1x​
(c) x1−x1−xx​
(d) xx−1x−1x​

1. If

y=log⁡10x+log⁡x10+log⁡xx+log⁡1010y=log10​x+logx​10+logx​x+log10​10

then what is

(dydx)x=10(dxdy​)x=10​

equal to?

(a) 10
(b) 2
(c) 1
(d) 0

1. Suppose ω1ω1​ and ω2ω2​ are two distinct cube roots of unity different from 1. Then what is (ω1−ω2)2(ω1​−ω2​)2 equal to?

(a) 3
(b) 1
(c) -1
(d) -3

NDA previous Year Question paper

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