Mathematics NDA 2 2017 Solution- Sept 10th - Part 1 - Intresting questions
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NDA 2 2017 Solution- Sept 10th - Part 1 - Intresting questions

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NDA 2 Paper 1

Selected problem
Q 3250491314

The points `(a,b),(0,0),(-a,-b)` and `(ab, b^2)` are
NDA Paper 1 2017
(A)

the vertices of a parallelogram

(B)

the vertices of a rectangle

(C)

the vertices of a square

(D)

collinear

Solution:

All given points lie on the line of equation
`ay = bx`

`=>` All points are collinear.
Correct Answer is `=>` (D) collinear
Q 3240191913

The sum of all real roots of the equation `| x - 3|^2 + | x -3 | -2 = 0` is
NDA Paper 1 2017
(A)

2

(B)

3

(C)

4

(D)

6

Solution:

`|x-3|^2+ |x-3|-2=0`

Let us qssume `|x-3|=y`

`y^2+y-2=0`

`(y+2)(y-1)=0`

`:. y= |x-3|=-2` [ can't possible ]

`y=|x-3|=1`

`x-3= pm 1`

`x=4,2`

sum of real roots `= 2+4=6`
Correct Answer is `=>` (D) 6
Q 3210391219

Given that the arithmetic mean and standard deviation of a sample of `15` observations are `24` and `0` respectively. Then which one of the following is the arithmetic mean of the smallest five observations in the data?
NDA Paper 1 2017
(A)

`0`

(B)

`8`

(C)

`16`

(D)

`24`

Solution:

As standard deviation is `‘0’`, therefore all observations will be equal to `24`.

`=>` Average of any five observations `= 24`.
Correct Answer is `=>` (D) `24`
Q 3200391218

The function `f(x) = |x| - x^3` is
NDA Paper 1 2017
(A)

odd

(B)

even

(C)

both even and odd

(D)

neither even nor odd

Solution:

` f(x) = { tt (( x - x^3 , x >= 0),( - x - x^3 , x < 0))`

` f(-x) = { tt ((x - x^3 , x >= 0 = f(x) ),( x + x^3 , x < 0 = - f(x) ))`

Neither even nor odd
Correct Answer is `=>` (D) neither even nor odd
Q 3250391214

The left - hand derivative of
` f(x) = [x] sin (pi x)` at ` x = k`
Where ` k` is an integer and `[x]` is the greatest function is
NDA Paper 1 2017
(A)

`(-1)^k (k -1) pi`

(B)

`(-1)^(k-1) (k -1) pi`

(C)

`(-1)^k k pi`

(D)

`(-1)^(k-1) k pi`

Solution:

L.H.D

` = lim_(h -> 0) ( [ k - h] sin (k- h) - [k] sin k pi)/(-h)`

` = lim_(h -> 0) ( ( k - 1) sin (k pi - pi h))/(-h)`

` = lim_(h -> 0) ( pm ( k -1) sin (pi h))/(-h)`

` = pm (k -1) pi`

` = (-1)^k (k -1) pi`
Correct Answer is `=>` (A) `(-1)^k (k -1) pi`
Q 3230391212

The area bounded by the curve ` |x| + |y| = 1` is
NDA Paper 1 2017
(A)

`1` square unit

(B)

`2 sqrt2` square unit

(C)

`2` square unit

(D)

`2 sqrt3` square unit

Solution:

The area formed by `|x| + |y| = 1` is square shown as below,
Here `4` cases will be

CASE `I x>0 ` and `y>0`

CASE `II x>0` and `y<0`

CASE `III x<0` and `y>0`

CASE `IV x<0` and `y<0`

Following graph shows all the straight lines for all the cases.

`∴` Area of square `= (sqrt2)^2 = 2` sq unit

The correct answer is: `2` sq unit
Correct Answer is `=>` (B) `2 sqrt2` square unit
Q 3210491319

Match List - I with List - II and select the correct answer using the code given below the lists :






Column I(Function)Column I(Maximum value)
(A) `sin x + cos x` (P) `sqrt(10)`
(B) `3 sin x + 4 cos x` (Q) `sqrt2`
(C) `2 sin x + cos x` (R) `5`
(D) `sin x + 3 cos x` (S) `sqrt5`

NDA Paper 1 2017
(A)

2 3 1 4

(B)

2 3 4 1

(C)

3 2 1 4

(D)

3 2 4 1

Solution:

`f(x) = sin x + cos x`

`=>` maximum value `=sqrt 2`

`(A) -> (2)`

`f(x) = 3 sin x + 4 cos x`

`=>` maximum value `= sqrt(3^2 + 4^2) = 5`

`(B) -> (3)`

`f(x) = 2 sin + cos x`

`=>` maximum value `= sqrt(4 + 1) =sqrt 5`

`(c) -> (4)`

`f(x) = sin + 3 cos x`

maximum value `= sqrt(1 + 9) = sqrt(10)`

`(D) -> (1)`
Correct Answer is `=>` (B) 2 3 4 1
Q 3260291115

If `a,b,c` are non-zero real numbers, then the inverse of the matrix `A=[(a,0,0),(0,b,0),(0,0,c)]` is equal to
NDA Paper 1 2017
(A)

`[(a^(-1), 0, 0),(0, b^(-1), 0),(0,0,c^(-1))]`

(B)

`1/(abc)[(a^(-1), 0, 0),(0, b^(-1), 0),(0,0,c^(-1))]`

(C)

`1/(abc)[(1, 0, 0),(0, 1, 0),(0,0,1)]`

(D)

`1/(abc)[(a, 0, 0),(0, b, 0),(0,0,c)]`

Solution:

`A=[(a,0,0),(0,b,0),(0,0,c)] a = ` diag `[a, b, c]`

`A^(–1) = d i a g^(–1) [a, b, c] = d i a g [a^(–1), b^(–1), c^(–1) ]`

` = [(a^(-1), 0, 0),(0, b^(-1), 0),(0,0,c^(-1))]`
Correct Answer is `=>` (A) `[(a^(-1), 0, 0),(0, b^(-1), 0),(0,0,c^(-1))]`
Q 3260491315

Which one of the Following graphs represent the function ` f(x) = x/x , x ne 0` ?
NDA Paper 1 2017
(D)

None of the above

Solution:

`f(x) = x/x , != 0`

` => y = 1, x != 0`
Correct Answer is `=>` (D) None of the above

Some more Problems

Some More Problems
Q 3200591418

The value of

`sin^-1 (3/5) + tan ^-1 (1/7)`

is equal to
NDA Paper 1 2017
(A)

0

(B)

`pi/4`

(C)

`pi/3`

(D)

`pi/2`

Solution:

`sin^-1 (3/5) + tan ^-1 (1/7)`
`=tan^(-1)3/4 + tan ^-1 (1/7)`
`= tan^(-1) [(3/4 + 1/7)/(1-3/28)]`
`=tan^(-1)(25/25) = tan^(-1)1 = pi/4`
Correct Answer is `=>` (B) `pi/4`
Q 3250691514

`sqrt(1 + sinA) = - ( sin\ \A/2 + cos \ \ A/2)` is true if
NDA Paper 1 2017
(A)

`(3 pi)/2 < A < (5 pi)/2` only

(B)

`pi/2 < A < (3pi )/2 ` only

(C)

`(3 pi)/2 < A < (7 pi)/2`

(D)

`0 < A < (3 pi)/2`

Solution:

`sqrt(1 + sinA) = sqrt[( sin^2 \ \A/2 + cos^2 \ \ A/2 + 2 sin \ \ A/2 cos \ \
A/2)^2] `


` =sqrt( ( sin\ \A/2 + cos \ \ A/2)^2) `

` = | sin \ \ A/2 + cos \ \ A/2 |`


To be - ve of molecules ` sin\ \A/2 + cos \ \ A/2 <0`

` sqrt2 [ sin 45^o sin A/2 + cos A/2 cos 45^o ] < 0`

` sqrt2 sin ( A/2 + pi/4 ) < 0`

` sin ( A/2 + pi/4 ) < 0`

To be + ve of

` sin ( A/2 + pi/4 )`

` pi < A/2 + pi/4 < 2 pi`

` (3 pi)/2 < A < (5 pi)/2`
Correct Answer is `=>` (A) `(3 pi)/2 < A < (5 pi)/2` only
Q 3230791612

The principle value of `sin^-1 x ` lies in the interval
NDA Paper 1 2017
(A)

`(-pi/2 , pi/2)`

(B)

`[-pi/2, pi/2]`

(C)

`[0,pi/2]`

(D)

`[0,pi]`

Solution:

Principal value `[-pi/2 , pi/2]`
Correct Answer is `=>` (B) `[-pi/2, pi/2]`
Q 3250791614

If `alpha, beta ` and `gamma` are the angles which the vector `vec(OP)` ( O being the origin ) makes with positive direction of the coordinates axes , then which of the following are correct ?

1. `cos^2 alpha + cos^2 beta = sin^2 gamma`

2. ` sin^2 alpha + sin^2 beta = cos^2 gamma`

3. `sin^2 alpha + sin^2 beta + sin^2 gamma =2`

Select the correct answer using the code given below
NDA Paper 1 2017
(A)

1 and 2 only

(B)

2 and 3 only

(C)

1 and 3 only

(D)

1, 2 and

Solution:

We know

`cos^2 alpha + cos^2 beta+ cos^2 gamma=1`................(I)

`cos^2 alpha+ cos^2 beta+ 1- sin^2 gamma=1`

`cos^2 alpha+ cos^2 beta= sin^2 gamma`..................(II)

by (I)

`1- sin^2 alpha+ 1- sin^2 beta+1- sin^2 gamma=1`

`sin^2 alpha+ sin^2 beta+ sin^2 gamma=2`..........(III)
Correct Answer is `=>` (C) 1 and 3 only
Q 3270791616

The angle between the lines `x + y - 3 = 0` and ` x - y + 3 = 0` is `alpha` and the acute angle between the lines ` x - sqrt 3 y + 2 sqrt3 = 0` and `sqrt3x - y + 1 = 0` is `beta`. Which one of the following is correct ?
NDA Paper 1 2017
(A)

`alpha = beta`

(B)

`alpha > beta`

(C)

`alpha < beta`

(D)

`alpha = 2 beta`

Solution:

`=> angle` between `x + y – 3 = 0` & `x – y + 3 = 0` is `90°`

` => alpha = 90°`

As, `beta` is acute, therefore ` alpha > beta`
Correct Answer is `=>` (B) `alpha > beta`
Q 3200791618

Let `vec alpha = hat i + 2 hat j - hat k , vec beta = 2 hat i - hat j + 3 hat k ` and ` gamma = 2 hat i + hat j + 6 hat k ` be three vectors . If `vec alpha` and `vec beta` are both perpendicular to the vector `delta ` and ` vec delta * vec gamma = 10` ,, then what is the magnitude of `vec delta` ?
NDA Paper 1 2017
(A)

`sqrt3` unit

(B)

`2 sqrt3` unit

(C)

`sqrt3/2` unit

(D)

`1/sqrt3` unit

Solution:

Let ` vec delta = a hat i + b hat j + c hat k`

` vec alpha . vec delta => a + 2b – c = 0` ...(i)

` vec beta . vec delta = 0 => 2a – b + 3c = 0` ...(ii)
from (i) and (ii)

`a/5 = b/(-5) = c/(-5)`

` => a/1 = b/(-1) = c/(-1) = lamda ` (say)

` => a = lamda , b = - lamda , c =- lamda`

Again, `vec delta . vec gamma = 10`

`=> 2a + b + 6c = 2 lamda - lamda - 6 lamda = - 5 lamda = 10`

` => lamda =- 2`

` :. vec delta = - 2 hat i + 2 hat j + 2 hat k => | vec delta| = sqrt(12) = 2 sqrt3`
Correct Answer is `=>` (B) `2 sqrt3` unit
Q 3210591410

If `S_n= nP+(n(n-1)Q)/2` where `S_n` denotes the sum of the first `n` terms of AP then the common difference is
NDA Paper 1 2017
(A)

`P+Q`

(B)

`2P+3Q`

(C)

`2Q`

(D)

`Q`

Solution:

`S_n = nP+(n(n-1) Q)/2`

`S_1=T_1=P`

`S_2=T_2+T_1= 2P+ Q`

`d=T_2-T_1=(T_2+T_1)-2T_1= 2P+ Q - 2P=Q`
Correct Answer is `=>` (D) `Q`
Q 3200691518

The roots of the equation `(q-r)x^2 +(r-p) x +(p-q)-0` are
NDA Paper 1 2017
(A)

`(r-p)//(q-r),1//2`

(B)

`(p-q)//(q-r),1`

(C)

`(q-r)//(p-q),1`

(D)

`(r-p)//(p-q),1//2`

Solution:

Let's put `x=1` in equation `(q-r)x^2 +(r-p) x +(p-q) = 0`

Hence `x=> 1` is root of equation

Another is is a root = multiplication of roots `= (p-q)(q-r)`
Correct Answer is `=>` (B) `(p-q)//(q-r),1`
Q 3200891718

If E is the universal set and `A = B uu C` then the set `E - ( E - ( E - ( E - ( E -A ))))` is same as the set
NDA Paper 1 2017
(A)

`B' uu C'`

(B)

`B uu C`

(C)

`B' nn C'`

(D)

` B nn C`

Solution:

If ` E` : universal set

` E - ( E - (E - (E - (E - A))))`

` = E - ( E - (E - (E - A')))`

` = A'`

` = ( B cup C)'`

` = B' cap C'`
Correct Answer is `=>` (C) `B' nn C'`
Q 3220091811

If `A =` {`x : x ` is a multple of 2 } , ` B =` { `x : x` is a multiple of 5 } and `C = { x : x ` is a multiple of 10 } then `A nn ( B nn C ) ` is equal to
NDA Paper 1 2017
(A)

A

(B)

B

(C)

C

(D)

{ `x : x` is a multiple of 100 }

Solution:

Here, `C subset A` and `C subset B`

`C = A cap B = A cap (B cap C)`

`= A cap C = C`
Correct Answer is `=>` (C) C
Q 3260191915

It is given that the roots of the equation `x^2 - 4 x - log_3 P = 0` are real . For this . the minimum value of P is
NDA Paper 1 2017
(A)

`1/27`

(B)

`1/64`

(C)

`1/81`

(D)

1

Solution:

If Roots are real of `x^2 - 4 x - log_3 P = 0`

`Dge0`
`D= 16+4log_3Pge0`
`log_3P le-4`
`Ple3^(-4)`
`Ple1/81`
Correct Answer is `=>` (C) `1/81`
Q 3271101026

For any vector `vec a`

`| vec a xx hat i |^2 + | vec a xx hat j | ^2 + | vec a xx hat k |^2`

is equal to
NDA Paper 1 2017
(A)

`|vec a|^2`

(B)

`2| vec a|^2`

(C)

`3| vec a |^2`

(D)

`4|vec a|^2`

Solution:

Let `vec a = ahati + bhatj + chatk`

`=| vec a xx hat i |^2 + | vec a xx hat j | ^2 + | vec a xx hat k |^2 `

`=| (ahati + bhatj + chatk) xx hat i |^2 + | (ahati + bhatj + chatk)xx hat j | ^2 + | (ahati + bhatj + chatk)xx hat k |^2 `


`=|chatj-bhatk|^2 +|-chati+ahatk|^2 + |bhati - ahatj|^2`

`=2(a^2+ b^2+c^2) = 2|veca|`
Correct Answer is `=>` (B) `2| vec a|^2`
Q 3221201121

The distance of the points `(1,3)` from the line `2 x + 3 y = 6` measured parallel to the line `4 x + y = 4` , is
NDA Paper 1 2017
(A)

`5/sqrt13` units

(B)

`3/sqrt17` units

(C)

`sqrt17` units

(D)

`sqrt17/2` units

Solution:

Equation of a line through the point `A (1, 3)` and parallel to

the line `4x + y = 4` is `4x+y = 7`...........(1)

Intersection of this line(1) and given line `2 x + 3 y = 6` is `(3/2,1)`

Now Required distance `AB = sqrt[(3/2-1)^2 + (1-3)^2] = sqrt17/2` units
Q 3250191914

Which one of the following is correct in respect of the function
` f(x) = x(x - 1) (x + 1)` ?
NDA Paper 1 2017
(A)

The local maximum value in large than local maximum value

(B)

The local maximum value in smaller than local maximum value

(C)

The function has no local maximum

(D)

The function has no local minimum

Solution:

`f(x) = x(x - 1) (x + 1) = x(x^2-1) = x^3 -x`

`f'(x) = 3x^2 - 1`

Fom min & max `f'(x) = 0`

`x = pm 1/sqrt3`

Now `f''(x) = 6x`

So Point of maxima `f''(x)<0` (for `x = -1/sqrt3`

Point of minima `f''(x)>0` (for `x =1/sqrt3)`

`f_(max)(x) = x(x^2-1) = (-1/sqrt3)(1/3 -1) = -1/sqrt3 (-2/3) = 2/(3sqrt3)`


`f_(min)(x) = x(x^2-1) = (1/sqrt3)(1/3 -1) = 1/sqrt3 (-2/3) = -2/(3sqrt3)`

The local maximum value in large than local maximum value
Correct Answer is `=>` (A) The local maximum value in large than local maximum value

Some more Problems

Some more Problems
Q 3240591413

The Value of

`tan 9° - tan 27° - tan 63° + tan 81°`

is equal to
NDA Paper 1 2017
(A)

-1

(B)

0

(C)

1

(D)

4

Solution:

`tan 9 - tan 27 - tan 63 + tan 81`
`= tan 9 - tan 27 - tan (90-27) + tan (90-9)`
`= tan 9 - tan 27 - cot 27 + cot 9`
`=[tan 9 + cot 9] - [ tan 27 + cot 27]`
`=[sin 9/ cos 9 + cos 9/ sin 9] - [ sin 27/cos 27 + cos 27/sin 27]`
`= 2/sin 18 - 2/sin 54`
`= 2[ 1/sin 18 - 1/ sin 54]`
`=2 [ (sin 54 - sin 18)/ ( sin 18 sin 54) ]`
`= 2 [ ( 2 cos 36 sin 18)/( sin 18 sin 54)]`
`= 4 cos 36 / sin 54`
`= 4 cos(90 - 54) / sin 54`
`= 4 sin 54 / sin 54`
`= 4`
Correct Answer is `=>` (D) 4
Q 3250591414

The Value of `sqrt3 cosec 20° - sec 20°` is equal to
NDA Paper 1 2017
(A)

4

(B)

2

(C)

1

(D)

-4

Solution:

`= (sqrt(3) cos 20 - sin 20)/ ( sin 20 cos 20) `
`= 2( sqrt(3)/2 cos 20 - (1/2) sin 20)/ ( sin 20 cos 20) `
`= 4 ( cos 30 cos 20 - sin 30 sin 20)/ ( 2 sin 20 cos 20) `
`= 4 (cos (30+20))/ sin 40 `
`= 4 cos 50/cos 50 `
`= 4`
Correct Answer is `=>` (A) 4
Q 3280591417

Angle `alpha` is divided into two parts A and B such that `A -B = x ....` and `tan A : tan B = p :q`. The Value of `sin x` is equal to
NDA Paper 1 2017
(A)

`((p+q) sin alpha)/(p-q)`

(B)

`(p sin alpha )/(p + q)`

(C)

`(p sin alpha) /( p-q)`

(D)

`((p -q) sin alpha ) / ( p +q)`

Solution:

If `alpha` is divided in `A` & `B`

`A+B= alpha`

`A-B= x` (given)

`(tan A)/(tan B)= p/q`

`(sin A)/(cos A) * (cos B)/(sin B)= p/q`

by `C` & `D` Theorem

`(sin A cos B - sin B cos B)/(sin A cos B+ sin B cos A)=(p-q)/(p+q)`

`(sin (A-B))/(sin (A+B)) =(p-q)/(p+q)`

Put the value of `A-B` & `A+B`

`(sin x)/(sin alpha) =(p-q)/(p+q)`

`sin x =((p-q)/(p+q)) sin alpha`
Correct Answer is `=>` (D) `((p -q) sin alpha ) / ( p +q)`
Q 3210891710

If `x + log_10 ( 1 +2 ^x) = x log_10 5 + log_10 6`

then x is equal to
NDA Paper 1 2017
(A)

`2,-3`

(B)

`2` only

(C)

`1`

(D)

`3`

Solution:

`x + log10 (1 + 2x) = x log_(10) 5 + log_(10) 6`

`=> x(1 – log_(10) 5) = log_(10) 6 – log_(10)
(1+2x)`

` => x (log_(10) 10 – log_(10) 5) = log_(10) ( 6/( 1 + 2^x))`

` => x log_(10) 2 = log 10 ( 6/( 1 + 2^x)) => x = 1`
Correct Answer is `=>` (C) `1`
Q 3250891714

The matrix A has x rows and `x +5` columns. The matrix B has y rows and 11-y columns. Both AB and BA exist. What are the values of `x` and `y` respectively ?
NDA Paper 1 2017
(A)

8 and 3

(B)

3 and 4

(C)

3 and 8

(D)

8 and 8

Solution:

order of `[A]=(x)` row `xx (x+5)` column

order of `[B] =(y)` row `xx (11-y)` column

If `AB` exist

Column of `[A]=` Row of `[B]`

`x+5=4`.........(i)

If `BA` exist

column of `[B]=` Row of `[A]`

by solving (I) & (II)

`x=8, y=3`
Correct Answer is `=>` (A) 8 and 3
Q 3220091811

If `A =` {`x : x ` is a multple of 2 } , ` B =` { `x : x` is a multiple of 5 } and `C = { x : x ` is a multiple of 10 } then `A nn ( B nn C ) ` is equal to
NDA Paper 1 2017
(A)

A

(B)

B

(C)

C

(D)

{ `x : x` is a multiple of 100 }

Solution:

Here, `C subset A` and `C subset B`

`C = A cap B = A cap (B cap C)`

`= A cap C = C`
Correct Answer is `=>` (C) C
Q 3210191910

IF `A = [ (4 i - 6 , 10 i ) , ( 14 i, 6 + 4 i ) ]` and `k ==1/(2i)` , where ` i = sqrt(-1)` , then `k A ` is equal to
NDA Paper 1 2017
(A)

`[(2 +3i, 5 ), ( 7 ,2 -3 i)]`

(B)

`[(2 -3i, 5 ), ( 7 , 2 +3 i)]`

(C)

`[(2- 3 i, 7 ), ( 5 , 2 + 3 i )]`

(D)

`[(2 + 3 i, 5 ) , ( 7 , 2 + 3 i)]`

Solution:

`A=[(4i-6, 10i),(14 i , 6+4i)] `

`KA= [((4i-6)/(2i) , (10 i)/(2i)),((14 i)/(2i) , (6+4i)/(2i))]`

`=[(2+3i , 5),(7, 2-3i)]`
Correct Answer is `=>` (A) `[(2 +3i, 5 ), ( 7 ,2 -3 i)]`
Q 3271101026

For any vector `vec a`

`| vec a xx hat i |^2 + | vec a xx hat j | ^2 + | vec a xx hat k |^2`

is equal to
NDA Paper 1 2017
(A)

`|vec a|^2`

(B)

`2| vec a|^2`

(C)

`3| vec a |^2`

(D)

`4|vec a|^2`

Solution:

Let `vec a = ahati + bhatj + chatk`

`=| vec a xx hat i |^2 + | vec a xx hat j | ^2 + | vec a xx hat k |^2 `

`=| (ahati + bhatj + chatk) xx hat i |^2 + | (ahati + bhatj + chatk)xx hat j | ^2 + | (ahati + bhatj + chatk)xx hat k |^2 `


`=|chatj-bhatk|^2 +|-chati+ahatk|^2 + |bhati - ahatj|^2`

`=2(a^2+ b^2+c^2) = 2|veca|`
Correct Answer is `=>` (B) `2| vec a|^2`
Q 3251301224

The equation of a straight line which cuts off an intercept of 5 units on negative direction of y- axis and makes an angle 120° with positive direction of x - axis is
NDA Paper 1 2017
(A)

`y + sqrt3x + 5 = 0`

(B)

`y - sqrt 3 x + 5 = 0`

(C)

`y + sqrt 3 x - 5 = 0`

(D)

`y - sqrt 3x - 5 = 0`

Solution:

`C= - 5`

`theta=120`

So, `y=mx+c`

`y= tan 120 x +(-5)`

`y+sqrt(3) x +5 =0`
Correct Answer is `=>` (A) `y + sqrt3x + 5 = 0`
Q 3271301226

The equation of the the line passing through the points `(2,3)` and the point of intersect of line `2x -3 y + 7 = 0` and `7 x +4 y + 2 = 0` is
NDA Paper 1 2017
(A)

`21x + 46y - 180 = 0`

(B)

`21 x - 46 y + 96 = 0`

(C)

`46 x + 21y - 155 = 0`

(D)

`46 x - 21 y -29 = 0`

Solution:

Line through two line is `l_1 +lambdal_2 = 0`

Here `l_1 & l_2 : 2x -3y +7=0 & 7x +4y + 2=0`

`(2x -3y +7) +lambda (7x +4y + 2)=0`.......(i)

Now this line also passes throgh(2,3)

`4-9+7+lambda(14+12+2)=0`

`2+lambda(28) = 0`

`lamda = -1/14`

Now put in (i)

`21 x - 46 y + 96 = 0`

`color{blue}bbul{"Alternatively"}`

Point of intersection of `l_1 & l_2 : 2x -3y +7=0 & 7x +4y + 2=0`

`P(-(34)/(29) , (45)/(29))`

`=>` Required Equation line is

`y-3= (45/29 -3)/(- 34/29 -2) ( x-2)`

`y-3= (-42)/(-92) (x-2)`

`21 x -46 y +96 =0`
Correct Answer is `=>` (B) `21 x - 46 y + 96 = 0`
Q 3261401325

The value of the determinant

`| (cos^2""theta/2 , sin^2""theta/2 ), ( sin^2""theta/2 , cos ^2""theta/2 ) |`

For all values of `theta`, is
NDA Paper 1 2017
(A)

`1`

(B)

`cos theta`

(C)

`sin theta`

(D)

`cos 2 theta`

Solution:

`| (cos^2""theta/2 , sin^2""theta/2 ), ( sin^2""theta/2 , cos ^2""theta/2 ) |`

` = ( cos^4 \ \ theta/2 - sin^4 \ \ theta/2 )`

` = ( cos^2 \ \ theta/2 + sin^2 \ \ theta/2 ) ( cos^2 \ \ theta/2 - sin^2 \ \ theta/2 ) = cos theta`
Correct Answer is `=>` (B) `cos theta`

Some more Problems

Some more Problems
Q 3280191917

If `A` is a square matrix, then the value of `adj A^T - (adj A )^T` is equal to
NDA Paper 1 2017
(A)

A

(B)

`2 |A| I` , where I is the identity matrix

(C)

null matrix whose order is same a that of A

(D)

Unit matrix whose order is same as that of A

Solution:

If `A` is square matrix of order ` m xx m`

`[Adj A^T]_(m xx m) -[(adj A)^T]_(m xx m) `

`=[0]_(m xx m) ` {As we know by property of `Adj` since `Adj A^T= (adj A)^T` }
Correct Answer is `=>` (C) null matrix whose order is same a that of A
Q 3231101022

If `hat a ` and `hat b` are two unit vectors, then the vector `(hat a + hat b ) xx ( hat a xx hat b )` is parallel to
NDA Paper 1 2017
(A)

`(hat a - hat b)`

(B)

`( hat a + hat b)`

(C)

`(2 hat a - hat b )`

(D)

`(2 hat a + hat b)`

Solution:

`(hat a + hat b) xx (hat a xx hat b)= hat a xx (hat a xx hat b) + hat b xx (hat a xx hat b) `

` = (hat a * hat b) hat a - (hat a * hat a) hat b + (hat b * hat b)hat a - (hat b * hat a) hat b`

` = k hat a - hat b + hat a - k hat b`

` = (k + 1) (hat a - hat b)`
Correct Answer is `=>` (A) `(hat a - hat b)`
Q 3270391216

If ` f(x) = x/ 2 - 1`, then one the interval ` [0 , pi]` which one of the following is correct ?
NDA Paper 1 2017
(A)

`tan [f(x) ]` where `[.]` is the greatest integer function , and ` 1/(f(x))` are both continuous

(B)

`tan [f(x) ]` where `[.]` is the greatest integer function , and ` f^(-1) (x)` are both continuous

(C)

`tan [f(x) ]` where `[.]` is the greatest integer function , and ` 1/(f(x))` are both discontinuous

(D)

`tan [f(x) ]` where `[.]` is the greatest integer function is discontinous but ` 1/(f(x))` is continuous

Solution:

` f(x) = x/2 - 1 , [ 0 , pi]`

`tan [f(x)] = tan [ x/2 -1 ]`

` 1/(f(x)) = 1/( x/2 - 1) ` is discontinuous at `x = 2`

Also, `tan [f(x)]` is discontinuous for `x = 2` in `[0, pi ]`
Q 3251201124

If the vectors `a hat i + hatj + hat k, hat i + b hat j + hat k` and ` hat i + hat j + c hat k ( a, b , c != 1 )` are coplanar, then the value of

` 1 /( 1 - a ) + 1/ (1 - b ) + 1/ (1 -c)` is equal to
NDA Paper 1 2017
(A)

`0`

(B)

`1`

(C)

`a +b + c`

(D)

`abc`

Solution:

` | (a ,1,1),(1 , b ,1),(1,1,c) | = 0`

`=> C_2 -> C_2 – C_1 , C_3 -> C_3 – C_1`

` | ( a, 1-a, 1-a),(1, b-1, 0),(1,0 ,c-1) | = 0`

`=> a(b – 1) (c – 1) – (1 – a) (c – 1) – (1 – a) (b – 1) = 0`

Dividing by `(1 – a) (1 – b) (1 – c)`, we get

` a/(1 -a) + 1/(1 - b) + 1/(1 - c) = 0`

` => 1/(1-b) + 1/( 1 - c) = - a/(1 -a)`

Adding `1/(1 -a)` on both sides

` 1/(1 -a) + 1/(1 - b) + 1/(1 - c) = 1`
Correct Answer is `=>` (B) `1`
Q 3281201127

The point of intersection of the line joining the points `( -3 ,4 ,- 8)` and ` (5, -6 ,4 )` with the `XY` - plane is
NDA Paper 1 2017
(A)

`(7/3, -8/3 , 0 )`

(B)

`(-7/3 , -8/3 , 0)`

(C)

`(-7/3, 8/3, 0 )`

(D)

`(7/3 , 8/3 , 0 )`

Solution:

Line going to given point

`(x+3)/8 =(y-4)/(-10)= (z+8)/12 =r` & `xy ` plane ` z=0`

`(x+3)/8 =(y-4)/(-10) = 8/12 " " {z=0}`

on solving `x= 7/3 , y= (-8)/3 , z=0`
Correct Answer is `=>` (A) `(7/3, -8/3 , 0 )`
Q 3231301222

The position of the point `(1,2)` relative to the ellipse `2 x^2 + 7 y^2 = 20` is
NDA Paper 1 2017
(A)

outside the ellipse

(B)

inside the ellipse but not at the focus

(C)

on the ellipse

(D)

at the focus

Solution:

`=> S(1,2)`

`=> 2(1)^2+ 7(2)^2 -20`

`=> 30 -20`

`=> 10 > 0`

Here point lies outside the ellipse.
Correct Answer is `=>` (A) outside the ellipse
Q 3221401321

The equation of the circle which passes through the points `(1,0) , (0, -6 )` and `(3,4)` is
NDA Paper 1 2017
(A)

`4 x^2 + 4 y^2 + 142 x + 47 y + 140 = 0`

(B)

`4 x^2 + 4 y^2 - 142 x - 47 y + 138 = 0`

(C)

`4x^2 + 4y^2 -142 x + 47 y + 138 = 0`

(D)

`4x^2 + 4 y^2 + 150 x - 49y + 138 = 0`

Solution:

Let equation of circle is

`x^2 – x + y^2 + 6y + lamda (–y – 6 + 6x) = 0`

Putting `(3, 4)`, we get

`9 – 3 + 16 + 24 + lamda (–4 – 6 + 18) = 0`

`=> 46 + 8 lamda = 0 => lamda = (- 23)/4`

` :. x^2 – x + y^2 + 6y – (23)/4 (–y – 6 + 6x) = 0`

`=> 4x^2 + 4y^2 – 4x + 24y + 23y + 138 – 138x = 0`

`=> 4x^2 + 4y^2 – 142x + 47y + 138 = 0`
Correct Answer is `=>` (C) `4x^2 + 4y^2 -142 x + 47 y + 138 = 0`
Q 3241401323

A variable plane passes through a fixed point `(a,b,c)` and cuts the axes in `A,B` and `C` respectively. The locus of the centre of the sphere `OABC`, O being the origin, is
NDA Paper 1 2017
(A)

`x/a + y/b + z /c = 1`

(B)

`a/x + b / y +c/z = 1`

(C)

`a/x + b/y + c/z = 1`

(D)

`a/x + b/y + c/z = 2`

Solution:

Equation of plane is `x/p + y/q + z/r = 1`

It passes through `(a, b, c) => a/p + b/q + c/r = 1`

Equation of sphere is given by

`x^2 + y^2 + z^2 – px– qy – rz = 0`

with its centre at (`x_c , y_c , z_c`) such that

`x_c = p/2 , y_c = q/2 , z_c = r/2`

` => p = 2 x_c , q = 2 y_c , r = 2 z_c`

`:. ` locus of centre

` => a/x + b/y + c/z = 2`
Correct Answer is `=>` (D) `a/x + b/y + c/z = 2`
Q 3251401324

The equations of the plane passing through the line of intersect of the plnes `x + y + z =1, 2x + 3y +4z = 7` and perpendicular to the plane `x - 5 y + 3 z =5` is given by
NDA Paper 1 2017
(A)

`x +2 y + 3z - 6 = 0`

(B)

`x +2 y + 3z + 6 = 0`

(C)

`3x +4 y + 5 y + 5 z - 8 = 0`

(D)

`3x + 4 y + 5 z + 8 = 0`

Solution:

Let `P_1 : x + y + z – 1 = 0`

`P_2 : 2x + 3y + 4z – 7 = 0`

Equation of plane passing through the line of

intersection of `P_1` and `P_2` is given by

`x + y + z – 1 + lamda (2x + 3y + 4z – 7) = 0`

`=> x(1 + 2 lamda ) + y(1 + 3 lamda ) + z(1 + 4 lamda ) – 1 – 7 lamda = 0`

This is perpendicular to `x –5y + 3z –5 = 0`

`=> 1(1 + 2 lamda ) – 5(1 + 3 lamda ) + 3(1 + 4 lamda ) = 0`

`=> 1 + 2 lamda –5 – 15 lamda + 3 + 12 lamda = 0`

`=> – lamda – 1 = 0 => lamda = – 1`

`:.` Equation of plane is

`– x – 2y – 3z – 1 + 7 = 0 => x + 2y + 3z – 6 = 0`
Correct Answer is `=>` (A) `x +2 y + 3z - 6 = 0`
Q 3211201129

If the angle between the line whose direction ratios are `(: 2 , -1 , 2 :)` and ` (:x, 3 ,5 :)` is `pi/4 ` , then the smaller value of x is
NDA Paper 1 2017
(A)

`52`

(B)

`4`

(C)

`2`

(D)

`1`

Solution:

`2x –3 + 10 = 3 sqrt((34 + x^2)/2)`

`4x^2 + 49 + 28x = ( 9(34 + x^2))/2`

` 8x^2 + 98 + 56x = 306 + 9x^2`

`x^2 – 56x + 208 = 0`

` x = ( + 56 pm sqrt ( 3136 -832))/2 = 28 pm 24 = 4, 52`
Correct Answer is `=>` (A)

Some more Problems

Some more Problems
Q 3281301227

The number of terms in the expansion of `(x+a)^100+(x-a)^100` after simplification is
NDA Paper 1 2017
(A)

`202`

(B)

`101`

(C)

`51`

(D)

`50`

Solution:

`(x + a)^(100) + (x – a)^(100)`

Number of terms `= 101 – 50 = 51`
Correct Answer is `=>` (C) `51`
Q 3241301223

In the expansion of `(1+x)^50` the sum of the coefficient of odd power of `x` is
NDA Paper 1 2017
(A)

`2^26`

(B)

`2^49`

(C)

`2^50`

(D)

`2^51`

Solution:

In `(1 + x)^(50)`

`C_1 + C_3 + C_5 + .... = 1/2 xx 2^(50) = 2^(49)`

` [ ∵ C_0 + C_1 + C_2 + .... C_n = 2^n`
Correct Answer is `=>` (B) `2^49`
Q 3241301223

In the expansion of `(1+x)^50` the sum of the coefficient of odd power of `x` is
NDA Paper 1 2017
(A)

`2^26`

(B)

`2^49`

(C)

`2^50`

(D)

`2^51`

Solution:

In `(1 + x)^(50)`

`C_1 + C_3 + C_5 + .... = 1/2 xx 2^(50) = 2^(49)`

` [ ∵ C_0 + C_1 + C_2 + .... C_n = 2^n`
Correct Answer is `=>` (B) `2^49`
Q 3210391210

The smallest positive integer `n` for which `((1+i)/(1-i))^n=1`, is
NDA Paper 1 2017
(A)

`1`

(B)

`4`

(C)

`8`

(D)

`16`

Solution:

`((1+i)/(1-i))^n=1 = i^n = 1 => n = 4`
Correct Answer is `=>` (B) `4`
Q 3240391213

If we define a relation `R` on the set `N xx N` as `(a,b) R (c,d) ⇔a+d=b+c` for all `(a,b),(c,d) in (N xx N)`, then the relation is
NDA Paper 1 2017
(A)

symmetric only

(B)

symmetric and transitive only

(C)

equivalane relation

(D)

reflexive only

Solution:

`(a, b) R (c, d) < = > a + b = b + c`

`a + a = a + a`

`=> (a, a) R (a, a) => R` is reflexive.

Next, Let `(a, b) R (c, d) => a + b = b + c`

`=> c + b = d + a => (c, d) R (a, b)`

`=> R` is symmetric.

Next, Let `(a, b) R (c, d)` and `(c, d) R (e, f)`

`=> a + b = b + c` and `c + f = d + e`

`=> a + d + c + f = b + c + d + e`

`=> a + f = b + e => (a, b) R (e, f)`

`=> R` is transitive `=> R` is an equivalence relation.
Correct Answer is `=>` (C) equivalane relation
Q 3260391215

If `y=x+x^2+x^3+.....` up to infinite terms where `x<1` ,then which one of the following is correct ?
NDA Paper 1 2017
(A)

`x= y/(1+y)`

(B)

`x=y/(1-y)`

(C)

`x= (1+y)/y`

(D)

`x= (1-y)/y`

Solution:

` y = x/(1 - x) => x = y/(1 + y)`
Correct Answer is `=>` (A) `x= y/(1+y)`
Q 3280391217

If `alpha` and `beta` are the roots of the equation `3x^2+2x+1=0` then the equation whose roots are `alpha+beta^(-1)` and `beta+alpha^(-1)` is
NDA Paper 1 2017
(A)

`3x^2+8x+16=0`

(B)

`3x^2-8x-16=0`

(C)

`3x^2+8x-16=0`

(D)

`x^2+8x+16=0`

Solution:

`3x^2 + 2x + 1 = 0 => alpha + beta = - 2/3 , alpha beta = 1/3`

`S = alpha + beta + 1/alpha + 1/beta = - 2/3 - 2 = (-8)/3`

`P = (alpha + beta^(-1) ) ( beta + alpha^(-1) ) = alpha beta + 2 + 1/(alpha beta) = (16)/3`

Required equation is `x^2 – Sx + P = 0`

` x^2 + 8/3 x + (16)/3 = 0 => 3x^2 + 8x + 16 = 0`
Correct Answer is `=>` (A) `3x^2+8x+16=0`
Q 3221712621

The System of equations `kx +y +z = 1,x +ky +z = k` and `x +y +kz = k^2` has no solution if k equals
NDA Paper 1 2017
(A)

`0`

(B)

`1`

(C)

`-1`

(D)

`-2`

Solution:

` |( k ,1 , 1),(1 , k ,1),( 1 ,1 , k) | = 0`

` => k(k^2 – 1) – (k–1) + (1 – k) = 0`

`=> k(k + 1) – 1 – 1 ] = 0`

`=> k^2 + k – 2 = 0 => k = 1, –2`

For `k = 1`, first two equations will become same.

`=> k = –2`
Correct Answer is `=>` (D) `-2`
Q 3270891716

If `1.3 + 2.3^2 +3.3^3 +.....+ n.3^n=((2n-1)3^a+b)/4` then `a` and `b` are respectively.
NDA Paper 1 2017
(A)

`n,2`

(B)

`n,3`

(C)

`n+1,2`

(D)

`n+1,3`

Solution:

`S_n = (ab)/(1 - r) + ( dbr ( 1 - r^(n -1)))/(1 - r)^2`

`1.3 + 2.3^2 + ....n. 3^n`

`= ( {(2n - 1) 3^(n + 1) + 3})/4`

` ( (2n -1) 3^a + b)/4 = ( ( 2n - 1) 3^(n +1) + 3)/4`

` a = n + 1, b = 3`
Correct Answer is `=>` (D) `n+1,3`
Q 3210891719

In `Delta PQR, angle R=pi/2` . If `tan (P/2)` and `tan (Q/2)` are the roots of the equation `ax^2+bx+c=0` , Then which one of the following is correct?
NDA Paper 1 2017
(A)

`a+b+c`

(B)

`b=c+a`

(C)

`c=a+b`

(D)

`b-c`

Solution:

`P/2 + Q/2 = 45^o`

` tan ( P/2 + Q/2) = ( tan (P/2) +tan( Q/2))/(1 - tan P/2 tan Q/2) = 1`

` = (-b)/a / (1 - c/a) => a + b = c`
Correct Answer is `=>` (C) `c=a+b`
Q 3230091812

If `|z-4/z|=2`, then the maximum value of `|z|` is equal to
NDA Paper 1 2017
(A)

`1+ sqrt 3`

(B)

`1+ sqrt 5`

(C)

`1- sqrt 5`

(D)

`sqrt 5-1`

Solution:

` | z - 4/z | >= |z| - | 4/z| => 2>= |z| - 4/(|z|)`

`=> |z|^2 – 2|z| – 4 <= 0`

`=> |z| = ( 2 + 2 sqrt5)/2 = 1 + sqrt5` (neglecting –ve value)
Correct Answer is `=>` (B) `1+ sqrt 5`
Q 3280091817

let `g` be the greatest integer function. Then the function `f(x)=(g(x))^2-g(x^2)` is discontinuous at
NDA Paper 1 2017
(A)

all integers

(B)

all integers except `0` and `1`

(C)

all integers except `0`

(D)

all integers except `1`

Solution:

`g(x) = [x]`

`f(x) = [x]^2 – [x]`

`f(x)` is discontinuous at every integers except `x = 1`.
Correct Answer is `=>` (D) all integers except `1`
Q 3230191912

The differential equation of minimum order by eliminating the arbitrary constant A and C in the equation `y= A[sin (x+C)+ cos (x+C)]` is
NDA Paper 1 2017
(A)

`y''+(sin x+ cos x) y'=1`

(B)

`y''= (sin x+ cos x)y'`

(C)

`y''=(y')^2 + sin x cos x`

(D)

`y''+y=0`

Solution:

`y = A[sin(x + c) + cos(x + c)]`

` (dy)/(dx) = A[cos(x + c) – sin(x + c)]`

` (d^2 y)/(dx^2 ) = –A[sin (x + c) + cos (x + c)] = – y`

` => (d^2 y)/(dx^2 ) + y =0`
Q 3271401326

consider the following statement :
statement `I` :
`x > sin x ` for all ` x > 0`
statement `II` :
`f(x) = x - sin x ` is an increasing function for all ` x > 0`
which one of the following is correct in respect of the above statement ?


NDA Paper 1 2017
(A)

Both Statement `I` and Statement `II` are true and Statement `II` is the correct explanation of Statement `I`

(B)

Both Statement `I` and Statement `II` are true and Statement `II` is not the correct explanation of Statement `I`

(C)

Statement `I` is the true but Statement `II` is false

(D)

Statement `I` is the False but Statement `II` is true

Solution:

Both statements are correct but statement `2` is not
the correct explanation of statement `1`.
Correct Answer is `=>` (B) Both Statement `I` and Statement `II` are true and Statement `II` is not the correct explanation of Statement `I`
Q 3200291118

A person is to count `4500` notes. let `a_n` denote the number of notes he count in the `n^(th)` minute. If `a_1=a_2=a_3=.....a_10=150` and `a_10, a_11, a_12,.......` are in A.P with the common difference `-2` then the time taken by him to count all the notes is
NDA Paper 1 2017
(A)

`24` minutes

(B)

`34` minute

(C)

`125` minutes

(D)

`135` minutes

Solution:

Let us assume that total minutes `= x`

For first nine minutes `= 150 xx 9`

Let the remaining minutes `= y = x – 9`

Now `(150 xx 9) + y/2 [2 xx 150 + (y – 1) (–2)] = 4500`

`y/2 (302 – 2y) = 3150 => y^2 – 151 y + 3150 = 0`

`=> (y – 126) (y – 25) = 0 => y = 25` or `126` (Rejected)

So, `x = y + 9 = 25 + 9 = 34` minutes.
Correct Answer is `=>` (B) `34` minute

Some more Problems

Some more Problems
Q 3200291118

A person is to count `4500` notes. let `a_n` denote the number of notes he count in the `n^(th)` minute. If `a_1=a_2=a_3=.....a_10=150` and `a_10, a_11, a_12,.......` are in A.P with the common difference `-2` then the time taken by him to count all the notes is
NDA Paper 1 2017
(A)

`24` minutes

(B)

`34` minute

(C)

`125` minutes

(D)

`135` minutes

Solution:

Let us assume that total minutes `= x`

For first nine minutes `= 150 xx 9`

Let the remaining minutes `= y = x – 9`

Now `(150 xx 9) + y/2 [2 xx 150 + (y – 1) (–2)] = 4500`

`y/2 (302 – 2y) = 3150 => y^2 – 151 y + 3150 = 0`

`=> (y – 126) (y – 25) = 0 => y = 25` or `126` (Rejected)

So, `x = y + 9 = 25 + 9 = 34` minutes.
Correct Answer is `=>` (B) `34` minute
Q 3200191918

The solution of the differential equation `(dy)/(dx)=(y phi ' (x) -y^2)/(phi (x))` is
NDA Paper 1 2017
(A)

`y= x/(phi(x)+c)`

(B)

`y= (phi (x))/x+ c`

(C)

`y= (phi(x)+ c)/x`

(D)

`y= (phi(x))/(x+c)`

Solution:

` (dy)/(dx) - ( y phi ' (x))/(phi (x)) = (-y^2)/(phi (x)) `

` => 1/y^2 . ( (dy)/(dx) ) - 1/y . ( phi ' (x))/(phi (x)) = - 1/(phi (x))`

Put, `t = 1/y`

we get ` (dt)/(dx) = - 1/y^2 ( (dy)/(dx) ) `

`=> (dt)/(dx) + ( phi ' (x))/(phi (x)) t = 1/( phi (x))`

I.F ` = e ^( int ( phi' (x))/( phi (x)) dx ) = e^(log phi (x)) = phi (x)`
Solution of differential equation is

` t . phi (x) = int 1/(phi (x)) xx phi (x) dx`

` => 1/y phi' (x) = x+c`

`y= (phi(x))/(x+c)`
Correct Answer is `=>` (B) `y= (phi (x))/x+ c`
Q 3211101020

If `f(x)=(4x+x^4)/(1+ 4x^3)` and `g(x)=ln ((1+x)/(1-x))` then what is the value of `f * g ((e-1)/(e+1))` equal to ?
NDA Paper 1 2017
(A)

`2`

(B)

`1`

(C)

`0`

(D)

`1/2`

Solution:

`f(x) = ( 4x + x^2)/(1 + 4 x^3)`

` g(x) = In ( (1 +x)/(1 -x))`

` g ((e -1)/(e +1)) = In [ ( e + 1 + e - 1)/( e+ 1 -e +1) ] = In ((2e)/2) = 1`

` fog ((e -1)/(e +1)) = f(1) = (4 +1)/(1 +4) =1`
Correct Answer is `=>` (B) `1`
Q 3261101025

The value of the determinant `|(1- alpha, alpha-alpha^2, alpha^2),(1- beta , beta- beta^2, beta^2),(1- gamma, gamma-gamma^2, gamma^2)|` is equal to
NDA Paper 1 2017
(A)

`( alpha- beta)(beta- gamma)(alpha- gamma)`

(B)

`(alpha- beta)(beta- gamma)(gamma- alpha)`

(C)

`(alpha- beta)(beta- gamma)(gamma- alpha)(alpha+ beta+ gamma)`

(D)

`0`

Solution:

`|(1- alpha, alpha-alpha^2, alpha^2),(1- beta , beta- beta^2, beta^2),(1- gamma, gamma-gamma^2, gamma^2)|`

`C_1 -> C_1 + C_2 + C_3`

` = |(1, alpha-alpha^2, alpha^2),(1 , beta- beta^2, beta^2),(1, gamma-gamma^2, gamma^2)|`

`C_2 -> C_2 + C_3`

` = |(1, alpha, alpha^2),(1 , beta, beta^2),(1, gamma, gamma^2)|`

` = ( alpha – beta )(beta – gamma )( gamma – alpha )`
Correct Answer is `=>` (B) `(alpha- beta)(beta- gamma)(gamma- alpha)`
Q 3201101028

The adjoint of the matrix `A= [(1,0,2),(2,1,0),(0,3,1)]` is
NDA Paper 1 2017
(A)

`[(-1,6,2),(-2,1,-4),(6,3,1)]`

(B)

`[(1,6,-2),(-2,1,4),(6,-3,1)]`

(C)

`[(6,1,2),(4,-1,2),(6,3,-1)]`

(D)

`[(-6,2,1),(4,-2,1),(3,1,-6)]`

Solution:

` A = [ (1,0 ,2),( 2 , 1 ,0),(0 ,3,1) ]`

`C_(11) = 1 quad C_(12) = –2 C_(13) = 6`

`C_(21) = 6 quad C_(22) = 1 quad C_(23) = –3`

`C_(31) = –2 quad C_(32) = 4 quad C_(33) = 1`

`:. Adj A = [(1,6,-2),(-2,1,4),(6,-3,1)]`
Correct Answer is `=>` (B) `[(1,6,-2),(-2,1,4),(6,-3,1)]`
Q 3231201122

If `A=[(-2,2),(2,-2)]` then which one of the following is correct ?
NDA Paper 1 2017
(A)

`A^2=-2A`

(B)

`A^2=-4A`

(C)

`A^2=-3A`

(D)

`A^2=4A`

Solution:

`A^2 = [(-2, 2),(2, -2)] [(-2, 2),(2, -2)] = [(8, -8),(-8, 8)] `

` = -4 [(-2, 2),(2, -2)] = - 4A`
Correct Answer is `=>` (B) `A^2=-4A`
Q 3241201123

Geometrically `Re (z^2-i)=2`, where `i= sqrt(-1)` and `Re` is the real part, represents
NDA Paper 1 2017
(A)

Circle

(B)

ellipse

(C)

rectangular hyperbola

(D)

parabola

Solution:

Let `z = x + iy`

`z^2 – i = x^2 – y^2 + 2xyi – i`

`= x^2 – y^2 + (2xy – 1) i`

`Re (z^2 – i) = 2`

`=> x^2 – y^2 = 2`

This equation represents rectangular hyperbola.
Correct Answer is `=>` (C) rectangular hyperbola
Q 3271201126

If `p+q+r = a+b+c=0` then the determinant `|(pa, qb , rc),(qc, ra, pb),(rb, pc, qa)|` equals
NDA Paper 1 2017
(A)

`0`

(B)

`1`

(C)

`pa+qb=rc`

(D)

`pa+qb+rc+a+b+c`

Solution:

`p + q + r = a + b + c = 0`

`|(pa, qb , rc),(qc, ra, pb),(rb, pc, qa)|`

`= pqr (a^3 + b^3 + c^3) – abc (p^3 + q^3 + r^3)`

`= pqr (3abc) – abc (3pqr)`

`= 0 ( ∵ a^3 + b^3 + c^3 = 3abc p^3 + q^3 + r^3 = 3pqr )`
Correct Answer is `=>` (A) `0`
Q 3201201128

A committee of two persons is selected from two men and two women . The probability that the committee will have exactly one woman is
NDA Paper 1 2017
(A)

`1/6`

(B)

`2/3`

(C)

`1/3`

(D)

`1/2`

Solution:

`n(s) = text()^4C_2 = (4 xx 3)/2 = 6`

`n(E) = text()^2C_1 xx text()^2C_1 = 2 xx 2 = 4`

`P(E) = 4/6 = 2/3`
Correct Answer is `=>` (B) `2/3`
Q 3221301221

let a die be loaded in such a way that even faces are twice likely to occur as the odd faces. what is the probability that a prime number will show up when the die is tossed ?
NDA Paper 1 2017
(A)

`1/3`

(B)

`2/3`

(C)

`4/9`

(D)

`5/9`

Solution:

Possible primes are `2, 3, 5`.

` 2/3 xx 1/3 + 1/3 xx 2/3 = 4/9`
Correct Answer is `=>` (C) `4/9`
Q 3231401322

Let the sample space consist of non - negative integers up to `50` ,`X` denote the numbers which are multiples of `3` and `Y` denote the odd numbers , which of the following is/are correct ?
1. `P(X) = 8/(25)`
2. ` P(Y) = 1/2`

select the correct ans. using the code given below .
NDA Paper 1 2017
(A)

`1` only

(B)

`2` only

(C)

Both `1` and `2`

(D)

neither `1` nor `2`

Solution:

`n(X) = 16, n(Y) = 25` and `S = 51`

`=> P(X) = (16)/(51) , P(Y) = (25)/(51)`
Q 3211401320

For two events A and B , let `P(A) = 1/2 , P(A cup B ) = 2/3` and ` P(A cap B ) = 1/6` what is ` P( bar A cap B)` equal to?
NDA Paper 1 2017
(A)

`1/6`

(B)

`1/4`

(C)

`1/3`

(D)

`1/2`

Solution:

`P(A cup B) = P(A) + P(B) – P(A cap B)`

` P(B) = 2/3 - 1/2 + 1/6 = 1/3`

`:. P( bar A cap B ) = 1/3 - 1/6 = 1/6`
Correct Answer is `=>` (A) `1/6`
Q 3201301228

Consider the following statement :
1. Cofficient of variation depends on the unit of measurement of the variable.
2. Range is a measure of dispersion.
3.mean deviation is least when measured about median.
which of the above statement are correct ?
NDA Paper 1 2017
(A)

`1` and `2` only

(B)

`2` and `3` only

(C)

`1` and `3` only

(D)

`1 , 2 ` and `3`

Solution:

Statement `1` and statement `2` are correct.
Mean derivation is least when measured about
mean, therefore statement `3` is wrong.
Q 3261301225

The inverse of the function ` y = 5^(log x)` is
NDA Paper 1 2017
(A)

` x = y^(1/(log 5)) , y> 0`

(B)

` x = y^(log 5) , y> 0`

(C)

` x = y^(1/(log 5)) , y < 0`

(D)

` x = 5 In , y , y> 0`

Solution:

`y = 5^(log x)`

`=> log x = (log y)/(log 5) = log y^(1/ (log 5)) => x = y^(1/(log 5)) , y> 0`
Correct Answer is `=>` (A) ` x = y^(1/(log 5)) , y> 0`
Q 3211301220

A function is defined as follow :
` f(x) = { tt (( - x/sqrt(x^2) , x != 0) , ( 0 , x = 0))`
Which one of the following is correct in respect of the above function ?

NDA Paper 1 2017
(A)

`f(x) ` is continuous at ` x = 0` but not differentiable at ` x = 0 `

(B)

`f(x) ` is continuous at as well as differentiable at ` x = 0 `

(C)

`f(x) ` is discontinuous at ` x = 0`

(D)

None of the above

Solution:

` f(x) = { tt (( - x/(|x|) , x != 0) , ( 0 , x = 0))`

` = { tt ( (-1 , x > 0),( 1 , x < 0),( 0 , x = 0))`

`f(x)` is discontinuous at `x = 0`
Correct Answer is `=>` (C) `f(x) ` is discontinuous at ` x = 0`
Q 3261201125

If ` y = (cos x)^ ((cos x)^((cos x)^((oo) ) ) ) ` , then `(dy)/(dx)` is equal to
NDA Paper 1 2017
(A)

` - (y^2 tan x) /(1 - y log ( cos x))`

(B)

` (y^2 tan x) /(1 + y log ( cos x))`

(C)

` (y^2 tan x) /(1 - y log ( sin x))`

(D)

` (y^2 sin x) /(1 + y log ( sin x))`

Solution:

`y = (cos x)y`

`=> log y = y log cos x`

Differentiating both sides,

`1/y (dy )/(dx) = y .( tanx) + log cos x . (dy)/(dx)`

` => (1/y - log cos x) (dy)/(dx) = – y tan x`

` => (dy)/(dx) = ( -y^2 tan x)/(1 - y log ( cos x))`
Correct Answer is `=>` (A) ` - (y^2 tan x) /(1 - y log ( cos x))`
Q 3211201120

Consider the following
1. ` x + x^2` is continuous at ` x = 0`
2. ` x + cos \ 1/x` is discontinuous at ` x = 0`
3 . ` x^2 + cos \ 1/x` is continuous at ` x = 0`
Which of the above are correct?


NDA Paper 1 2017
(A)

`1` and `2` only

(B)

`2` and `3` only

(C)

`1` and `3` only

(D)

`1 , 2` and `3`

Solution:

Both statements are correct.
Correct Answer is `=>` (A) `1` and `2` only
Q 3281101027

Consider the following statement:
1. `(dy)/(dx)` at a point on the curve gives slope of the tangent at the point .
2. If `a(t)` denotes acceleration of a particle , then `int a(t) dt + c` gives velocity of the particle.
3 . If ` s(t)` gives displacement of a particle at time `t` , then `(ds)/( dt)` gives its acceleration at that instant.
NDA Paper 1 2017
(A)

` 1` and `2` only

(B)

`2` only

(C)

`1` only

(D)

`1 , 2 ` and `3`

Solution:

Statement `1` and `2` are correct
Q 3241101023

If ` y = sec^(-1) ( (x +1)/(x -1)) + sin^(-1) ( (x - 1)/(x +1))` , then `(dy)/(dx)` is equal to

NDA Paper 1 2017
(A)

`0`

(B)

`1`

(C)

` (x - 1)/(x +1)`

(D)

` (x + 1)/(x -1)`

Solution:

`cos^(–1) (x-1)/(x +1) + sin^(-1) (x-1)/(x +1) = pi/2`

` => (dy)/(dx) = 0`
Correct Answer is `=>` (A) `0`
Q 3231712622

What is `int tan^-1 (sec x + tan x ) dx ` equal to ?
NDA Paper 1 2017
(A)

`(pix)/4 + x^2/4 +c`

(B)

`(pix)/2 + x^2/4 +c`

(C)

`(pi)/4 + (pix^2)/4 +c`

(D)

`(pix)/4 - x^2/4 +c`

Solution:

` int tan^(-1) (sec x + tan x ) dx = int tan^(-1) { tan ( pi/4 - x/2) } dx`

` = int ( pi/4 - x/2) dx = (pix)/4 - x^2 /4 + c`
Correct Answer is `=>` (D) `(pix)/4 - x^2/4 +c`
Q 3270191916

A function of defined in `(0 , oo)` by
` f (x) = { tt (( 1 - x^2 , f o r , 0 < x <= 1),( I n x , f o r , 1 < x <= 2) , ( I n 2 - 1 + 0.5 x , f o r , 2 < x < oo ))`
Which one of the following is correct in respect of the derivation of the function , i.e `f'(x)` ?
NDA Paper 1 2017
(A)

`f'(x) = 2x ` for ` 0 < x <= 1`

(B)

`f'(x) = -2x ` for ` 0 < x <= 1`

(C)

`f'(x) = -2x ` for ` 0 < x < 1`

(D)

`f'(x) = 0 ` for ` 0 < x < oo`

Solution:

` f (x) = { tt (( 1 - x^2 , f o r , 0 < x <= 1),( I n x , f o r , 1 < x <= 2) , ( I n 2 - 1 + 0.5 x , f o r , 2 < x < oo ))`

`f '(x) = – 2x, 0 < x <= 1`
Correct Answer is `=>` (B) `f'(x) = -2x ` for ` 0 < x <= 1`
Q 3220191911

Consider the following statements :
1 . Derivation of ` f(x)` may not exist at some point.
2 . Derivation of ` f(x)` may exist finitely at some point.
3 . Derivation of ` f(x)` may exist infinite (geometrically) at some point.
which of the above statement are correct ?
NDA Paper 1 2017
(A)

`1` and `2` only

(B)

`2` and `3` only

(C)

`1` and `3` only

(D)

`1 , 2 ` and `3`

Solution:

All statements are correct.
Correct Answer is `=>` (D) `1 , 2 ` and `3`
Q 3200091818

The maximum value of ` (In x)/x` is
NDA Paper 1 2017
(A)

`e`

(B)

`1/e`

(C)

`2/e`

(D)

`1`

Solution:

` f(x) = ( In x)/x`

` f' (x) = ((1/x) x - In x)/x^2 = (1 - In x)/x^2`

` f'' (x) = ( - 1/x . x^2 - 2x (1 - In x))/x^4`

` = ( - x - 2x (1 - In x ))/x^4 = ( - [ 1 + 2 - 2 In x ])/x^3`

` = ( - (3 - 2 In x))/x^3`

At `f' (x) = 0 => ln x = 1 => x = e`

` f'' (e) = ( - (3 -2))/e^3 = - 1/e^3 < 0`

`f(x)_(max) = f(e) = (In e)/e = 1/e`
Correct Answer is `=>` (B) `1/e`

Some more Problems

Some more Problems
Q 3220491311

The general solution of
` (dy)/(dx) = (ax +h)/(by + k)`
represent a circle only when
NDA Paper 1 2017
(A)

` a = b = 0`

(B)

` a = - b != 0`

(C)

` a = b != 0 , h = k`

(D)

` a = b != 0`

Solution:

` (dy)/(dx) = (ax +h)/(by + k)`

`int (b y + k ) dy = int (ax + h) dx`

` (b y^2)/2 + ky = (ax^2)/2 + hx + c`

` (ax^2)/2 - (b y^2)/2 + hx – ky + c = 0`

`a = – b != 0`
Correct Answer is `=>` (B) ` a = - b != 0`
Q 3240491313

If ` lim_(x -> pi/2) (sin x)/x = l` and ` lim_(x -> oo) (cos x)/x = m` then , which one of the following is correct ?
NDA Paper 1 2017
(A)

`l = 1 , m = 1`

(B)

`l = 2/pi , m = oo`

(C)

`l = 2/pi , m = 0`

(D)

`l = 1 , m = oo`

Solution:

` lim_(x -> pi/2) (sin x)/x = L => L = 2//pi`

` m = 0`
Correct Answer is `=>` (C) `l = 2/pi , m = 0`
Q 3230491312

which one of the following can be considered as appropriate pair of values of regression coefficient of `y` on `x` and regression coefficient of `x` on `y` ?
NDA Paper 1 2017
(A)

`(1,1)`

(B)

`(-1,1)`

(C)

`(-1/2,2)`

(D)

`(1/3 , 10/3)`

Solution:

regression coefficient of `y` on `x =` regression coefficient of `x` on `y`

`=> (x, y)` lies on `(y = x)` line.
Correct Answer is `=>` (A) `(1,1)`
Q 3270491316

Let `A` and `B` be two events with `P(A)=1/3` `P(B)=1/6` and `P(A cap B)=1/12` what is `P(B| bar(A))` is equal to ?
NDA Paper 1 2017
(A)

`1/5`

(B)

`1/7`

(C)

`1/8`

(D)

`1/10`

Solution:

`P(A) = 1/3 , P(B)= 1/6 , P(A cap B) = 1/12`

`P(B | bar A) =(P(B cap bar A))/(P(bar A))`

`P(B cap bar A) = (P( B) - P (A cap B)) =1/12`

`P( bar A) = 2/3`

`:. P(B // bar A)=(1//12)/(2//3) = 1/8`
Correct Answer is `=>` (C) `1/8`
Q 3230291112

In a binomial distribution , the mean is `2/3 ` and the variance is `5/9` what is the probability that `X = 2 ?`
NDA Paper 1 2017
(A)

`5/(36)`

(B)

`(25)/(36)`

(C)

`(25)/(216)`

(D)

`(25)/(54)`

Solution:

`np = 2/3 , npq = 5/9`

` => q = 5/9 xx 3/2 = 5/6`

` => p = 1/6, n = 4`

` p (x = 2) = text()^4C_2 (1/6)^2 xx (5/6)^2`

` = 6 xx 1/(36) xx (25)/(36) = (25)/(216)`
Correct Answer is `=>` (C) `(25)/(216)`
Q 3280891717

The probability that a ship safely reaches a port is ` 1/3`. The probability that out of `5` ships at least ` 4` ships would arrive safely is

NDA Paper 1 2017
(A)

`1/(243)`

(B)

`(10)/(243)`

(C)

`(11)/(243)`

(D)

`(13)/(243)`

Solution:

`"p(all reach safely)" = (1/3)^5`

`"p(4 reach safely)" = 5 xx (1/3)^4 (2/3)`

`"p(at least 4 reach safely)" = 11/243`
Correct Answer is `=>` (C) `(11)/(243)`
Q 3260891715

What is the probability that at least two persons out of the group of three persons were born in the same month (disregard year) ?
NDA Paper 1 2017
(A)

`(33)/(144)`

(B)

`(17)/(72)`

(C)

`1/(144)`

(D)

`2/9`

Solution:

`p`(none born in same month) `=(12 xx 11 xx 10)/(12 xx 12 xx 12)`

`p`(at least two born in same month)

`= 1- (12 xx 11 xx 10)/(12 xx 12 xx 12) =(144 -110)/144 =17/72`
Correct Answer is `=>` (B) `(17)/(72)`
Q 3230891712

It is given that ` bar X = 10 , bar Y = 90 , sigma_x = 3 , sigma_y = 12` and ` r_(xy) = 0.8` the regression equation of `X` on `Y` is
NDA Paper 1 2017
(A)

` Y = 3 .2X + 58`

(B)

` X = 3 .2Y + 58`

(C)

` X = -8 + 0 .2 y`

(D)

` Y = - 8 + 0.2 X`

Solution:

`bar x = 10 , bar y =90`

`sigma x =3 , sigma y=12`

`r_(xy) =0.8`

Regression equation `x` on `y` is

`=> x -10 = r (sigma x )/(sigma y) (y-90)`

`=> x - 10 = 0.8 xx 3/12 (y-90)`

`=> x -10 =0.2 (y-90)`

`=> x =-8 + 0.2 y`
Correct Answer is `=>` (C) ` X = -8 + 0 .2 y`
Q 3210791619

If ` P(B) = 3/4 , P (A cap B cap bar C ) = 1/3` and ` P ( bar A cap B cap bar B ) = 1/3 ` then what is ` P ( B cap C)` equal to ?


NDA Paper 1 2017
(A)

`1/(12)`

(B)

`3/4`

(C)

`1/(15)`

(D)

`1/9`

Solution:

`P(B cap bar C) = P(A cap B cap bar C)+ P(bar A cap B cap bar C)`

`=1/3 + 1/3 =2/3`

`p(B) =P(B cap C) + P(B cap bar C)`

`=> P(B cap C) =P(B) -P(B cap bar C)`

`=3/4 -2/3 =1/12`
Correct Answer is `=>` (A) `1/(12)`
Q 3280791617

The following table given the monthly expenditure of two families :
is contructing a pie diagram to the above data , the radii of the circle are to
be chosen by which one of the following ratio ?
NDA Paper 1 2017
(A)

`1:1`

(B)

`10 : 9`

(C)

` 100 : 91`

(D)

`5 : 4`

Solution:

Total expenditure of `A = 10,000`

Total expenditure of `B = 8,100`

So area of `A :` area of `B = 10,000 : 8100 = 100 : 81`

`=>` radii of `A :` radii of `B = sqrt(100) : sqrt(81) = 10 : 9`
Correct Answer is `=>` (B) `10 : 9`
Q 3260791615

If a variable takes values ` 0 , 1,2 ,3 ........ , n` with frequencies
` 1 , C (n , 1) , C (n , 2) , C (n , 3) , ......... , C ( n , n)`
respectively , then the arithmetic mean is
NDA Paper 1 2017
(A)

` 2n`

(B)

`n + 1`

(C)

`n`

(D)

`n/2`

Solution:

The arithmetic mean will always be between minimum and maximum value so out of the given option, `‘n//2’` is possible value.
Correct Answer is `=>` (D) `n/2`
Q 3240791613

In a multiple - choice test an examinee either know the correct ans. with probability ` p`, or guesses with probability ` 1 - p` . the probability of answering a question correctly is `1/m` , if he or see merely guesses correctly the probability that he or she really knows the answer is
NDA Paper 1 2017
(A)

`(mp)/( 1 + mp) `

(B)

`(mp)/( 1 + (m -1) p)`

(C)

` ((m-1)p)/(1 + ( m -1) p)`

(D)

`((m-1) p)/( 1 + mp)`

Solution:

`P`(to know correct answer) `= p`

`P`(to guess correct answer) `= (1 – p) xx (1//m)`

`P`(to answer correctly) `=p+ (1-p)/m`

So, required probability `= p/(p+ (1-p)/m) = (mp)/(1+p(m-1))`
Correct Answer is `=>` (B) `(mp)/( 1 + (m -1) p)`
Q 3210691519

If ` x_1` and `x_2` are positive quantities , then the condition for the difference b/t the arithmetic mean and the geometric mean to be greater `1` is
NDA Paper 1 2017
(A)

` x_1 + x_2 > 2 sqrt (x_1 x_2)`

(B)

` sqrt (x_1 ) + sqrt (x_2) > sqrt2`

(C)

`| sqrt (x_1 ) - sqrt (x_2) | > sqrt2`

(D)

` x_1 + x_2 < 2 (sqrt (x_1 x_2) +1)`

Solution:

`(x_1+x_2)/2 - sqrt(x_1x_2) > 1`

`=> (x_1+x_2)/2 > sqrt(x_1 x_2) +1`

`=> x_1 +x_2 > 2 sqrt(x_1 x_2) > 2`

`=> ( sqrt (x_1) - sqrt(x_2))^2 > 2`

`=> | sqrt (x_1) - sqrt(x_2)| > sqrt 2`
Correct Answer is `=>` (C) `| sqrt (x_1 ) - sqrt (x_2) | > sqrt2`
Q 3260691515

Consider of the following statement :
1 . Variance is unaffected by change of origin and change of scale .
2 . Coefficient of variance is independent of the unit of the observation
Which of the statement given above is/are correct ?
NDA Paper 1 2017
(A)

`1` only

(B)

`2` only

(C)

Both `1 ` and `2`

(D)

Neither `1` nor `2`

Solution:

Variance is independent of change of origin but not
scale. So, Statement 1 is incorrect Statement 2 is
correct.
Correct Answer is `=>` (A) `1` only
Q 3230691512

Five sticks of length ` 1,3,5,7` and `9` feet are given . three of these sticks are selected at random , What is the probability that the selected sticks can from a triangle ?
NDA Paper 1 2017
(A)

`0.5`

(B)

`0.4`

(C)

`0.3`

(D)

`0`

Solution:

`n(S) = text()^5C_3 = 10`

`n(E) = text()^4C_3 – 1 = 3`

`P(E) = 3//10 = 0.3`
Correct Answer is `=>` (C) `0.3`
Q 3210691510

The cofficient of correlation when cofficient of regression are ` 0.2` and `1.8` is
NDA Paper 1 2017
(A)

`0.36`

(B)

`0.2`

(C)

`0.6`

(D)

`0.9`

Solution:

Coefficient of correlation

`= sqrt(0.2 xx 1.8) =0.6`
Correct Answer is `=>` (A) `0.36`
Q 3220391211

What is ` int_0^(2pi) sqrt(1 + sin \ x/2) dx` equal to
NDA Paper 1 2017
(A)

`8`

(B)

`4`

(C)

`2`

(D)

`0`

Solution:

`int_0^(2 pi) sqrt(1+ sin \ x/2) dx`

`= int_0^(2 pi) | sin \ x/4 + cos \ x/4 | dx`

`=4 [ sin \ x/4 - cos \ x/4 ]_0^(2 pi)`

`=8`
Correct Answer is `=>` (A) `8`
Q 3210591419

If `x` is any real number , then ` x^2/(1 + x^4)` belongs to which one of the following intervals?
NDA Paper 1 2017
(A)

`(0 ,1)`

(B)

`( 0 , 1/2]`

(C)

`( 0 , 1/2)`

(D)

None of these

Solution:

`y=(x^2)/(1+x^4)`

`=> y ge 0`

Also, `y= (x^2)/(1+x^4) = 1/(x^2+ 1/x^2) => y le 1/2`

`=> y in [0, 1/2]`
Correct Answer is `=>` (D) None of these
Q 3270591416

The order and degree of the differential equation
` [ 1 + ((dy)/(dx))^2 ]^3 = p^2 [ (d^2y)/(dx^2) ]^2`
are respectively


NDA Paper 1 2017
(A)

`3` and `2`

(B)

`2` and `2`

(C)

`2` and `3`

(D)

`1` and `3`

Solution:

` [ 1 + ((dy)/(dx))^2 ]^3 = p^2 [ (d^2y)/(dx^2) ]^2`

Order `= 2`

Degree `= 2`
Correct Answer is `=>` (B) `2` and `2`
Q 3210291119

If ` y = cos^(-1) ((2x)/(1 + x^2) )` then ` (dy)/(dx)` is equal to
NDA Paper 1 2017
(A)

` - (2)/(1 + x^2)` for all ` |x | < 1`

(B)

` - (2)/(1 + x^2)` for all ` |x | > 1`

(C)

` (2)/(1 + x^2)` for all ` |x | < 1`

(D)

None of the above

Solution:

`y = cos^(–1) ( (2x)/(1 + x^2))`

` = pi/2 - sin^(-1) , ( (2x)/(1 + x^2))`

` = pi/ 2 - 2 tan^(-1) x , | x | < 1`
Correct Answer is `=>` (D) None of the above
Q 3220691511

The angles of elevation of he top of the tower from the top and foot of a pole ......? respectively 30° and 45° . If `h_T` is the height of the tower and `h_p` is the height of the pole, then which of the following are correct ?

1. `(2 h_P h_T)/(3 + sqrt3) = h_p^2`

2. `(h_T- h _P)/ ( sqrt3 + 1 ) = h_P/2`

3. `2 ( h_P + h_T)/h_P = 4 + sqrt3`

The select the correct answer using the cod given below.
NDA Paper 1 2017
(A)

1 and 3 only

(B)

2 and 3 only

(C)

1 and 2 only

(D)

1,2 and 3

Solution:

Let the distance between pole & tower is ‘b’.

Now , ` h_T/b = tan 45º = 1 => h_T = b`

` ( h_T - h_P)/b = tan 45º = 1/sqrt3 => ( h_T - h_P)/h_P = 1/sqrt3`

` => ( h_T - h_P)/( h_T - ( h_T - h_P) ) = 1/(sqrt3 - 1) => ( h_T - h_P)/h_P = ( sqrt3 + 1)/2`

`=>` Statement ‘`2`’ is correct,

` ( h_T - h_P + 2 h_p)/h_P = ( sqrt3 + 1 + 4)/2 => ( h_T +h_P)/h_P = (5 + sqrt3)/2`

`=>` Statement ‘`3`’ is incorrect.

`:. ` Option ‘`c`’ is right choice.
Correct Answer is `=>` (C) 1 and 2 only
Q 3230591412

The set of all points , where the function ` f(x) = sqrt ( 1 - e^(-x^2) ) ` is differentiable , is
NDA Paper 1 2017
(A)

`(0 ,oo)`

(B)

`( -oo ,oo)`

(C)

`( -oo , 0) cup ( 0 , oo)`

(D)

`(-1 , oo)`

Solution:

` f'(x) = ( - 2x ( -e^(-x^2)))/(2 sqrt ( 1 - e^(-x^2)) )= ( xe^(-x^2))/sqrt( 1 - e^(-x^2))`

Which is defined `AA x in R`, except `x = 0`

`=> f(x)` is differentiable on ` (- oo , 0) cup ( 0 , oo)`
Correct Answer is `=>` (C) `( -oo , 0) cup ( 0 , oo)`
Q 3280491317

If ` f(x) = x ( sqrtx - sqrt(x +1) )` then `f(x)` is
NDA Paper 1 2017
(A)

continuous but not differentiable at ` x = 0`

(B)

differentiable at ` x = 0`

(C)

not continuous at ` x = 0`

(D)

None of the above

Solution:

L. H. Lt = R. H. Lt `= f(0) = 0`

`=> f(x)` is continuous at `x = 0`

L. H. D = R. H. D `= –1`

`=> f(x)` is differentiable at `x = 0`
Q 3270291116

` int (ln x)^(-1) dx - int (ln x)^(-2) dx` is equal to
NDA Paper 1 2017
(A)

` x (In x)^(-1) + c`

(B)

` x (In x)^(-2) + c`

(C)

` x (In x) + c`

(D)

` x (In x)^(2) + c`

Solution:

` int [ 1/(log x) - 1/(log x)^2 ] dx`

Putting `log x = t`

` int e^t [ 1/t - 1/t^2 ] dt = e^t/t + c = x/(log x) + c`
Q 3250291114

A cylindrical jar a lid has to be constructed using a given surface area of a metal sheet . if the capacity of the jar is a maximum , then the diameter of the jar must be K times the height of the jar . the value of K is
NDA Paper 1 2017
(A)

`1`

(B)

`2`

(C)

`3`

(D)

`4`

Solution:

The height and the radius of the base of an open cylinder of given surface area and maximum volume are equal. i.e., radius `=` height.

`=>` Diameter `= 2 xx` height

`=> k = 2`
Correct Answer is `=>` (B) `2`
Q 3240291113

The value of ` int_0^(pi/4) sqrt( tan x) dx + int_0^(pi/4) sqrt(cot x) dx` is equal to


NDA Paper 1 2017
(A)

`pi/4`

(B)

`pi/2`

(C)

`pi/(2 sqrt2)`

(D)

`pi/sqrt2`

Solution:

` int_0^(pi/4) sqrt( tan x) dx + int_0^(pi/4) sqrt(cot x) dx`

` = int_0^(pi/4) ( sqrt( tan x) + sqrt(cot x) ) dx`

` = int_0^(pi/4) ( sinx + cos x)/sqrt( sinx cos x) dx`

` = sqrt2 int_0^(pi/4) ( sinx + cos x)/sqrt ( 1 - (sinx - cos x)^2) dx`

Putting `sin x – cos x = t`

`=> dt = (sin x + cos x) dx`

when `x = 0, t = –1`

and `x = pi/4 , t = 0`

` = sqrt2 int_(-1)^0 1/sqrt(1 -t^2) dt = sqrt2 int_(-1)^0 [ sin^(-1) t ]_(-1)^0`

` = sqrt 2 [ 0 - ( - pi //2) ] = pi/sqrt2`
Correct Answer is `=>` (D) `pi/sqrt2`
Q 3220591411

The angle Of elevation of a stationary cloud from a point 25 m above a lake is 15° and the angle of depression of its image in the lake is 45° . The Height of the cloud above the lake level is
NDA Paper 1 2017
(A)

`25` m

(B)

`25 sqrt3 ` m

(C)

`50` m

(D)

`50 sqrt3` m

Solution:

`tan 15° = 2 – sqrt3 = x/(x + 50)`

`=> (2 – sqrt 3 –1 ) x = – 50(2 – sqrt3 )`

` => x = ( – 50(2 – sqrt3))/(1 - sqrt3)`

` => x + 25 = ( – 50(2 – sqrt3) + 25 (1 - sqrt3))/(1 - sqrt3)`

` = ( - 100 + 50 sqrt3 + 25 - 25 sqrt3)/(1 - sqrt3)`

` = ( - 75 + 25 sqrt3)/(1 - sqrt3) = ( 25(3 - sqrt3))/(sqrt3 - 1) = 25 sqrt3`
Correct Answer is `=>` (B) `25 sqrt3 ` m
Q 3270091816

If `|a |` denotes the absolute value of an integer, then which of the following are correct ?

1. `| ab | = | a | | b |`

2. `| a + b | <= | a| + |b |`

3. `|a - b | >= ||a | = | b | |`

Select the correct answer using the code given below .
NDA Paper 1 2017
(A)

1 and 2 only

(B)

2 and 3 only

(C)

1 and 3 only

(D)

1, 2 and 3

Solution:

(i) `|ab|=|a| |b|` correct

(ii) `|a+b| le |a| +|b| ` (By triangular equality) correct

(iii) `|a-b| > ||a| -|b||` (By `Delta` inequality) correct
Correct Answer is `=>` (D) 1, 2 and 3
Q 3210091819

How many different permutation can be made out of the letters of the word `"PERMUTATION"` ?
NDA Paper 1 2017
(A)

`19958400`

(B)

`19954800`

(C)

`19952400`

(D)

`39916800`

Solution:

‘T’ is repeated twice.

So Number of permutations

` = ((11)!)/(2!) = 19958400`
Correct Answer is `=>` (A) `19958400`

 
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