NDA/NA 2022-I MATHEMATICS
Questions + Solutions (Clean Copy-Paste Ready)

1.
Δ1=∣1pq1qr1rp∣\Delta_1=\begin{vmatrix}1&p&q\\1&q&r\\1&r&p\end{vmatrix}Δ1​=​111​pqr​qrp​​, Δ2=∣111qrprpq∣\Delta_2=\begin{vmatrix}1&1&1\\q&r&p\\r&p&q\end{vmatrix}Δ2​=​1qr​1rp​1pq​​, p≠q≠rp\neq q\neq rp=q=r
Δ1+Δ2=\Delta_1+\Delta_2=Δ1​+Δ2​=
(a) 0 (b) always positive (c) always negative (d) positive if p,q,rp,q,rp,q,r positive else negative

Ans: (c)
Δ1=Δ2\Delta_1=\Delta_2Δ1​=Δ2​ (by column interchange)
Δ1+Δ2=2Δ2=−[(p−q)2+(q−r)2+(r−p)2]<0\Delta_1+\Delta_2=2\Delta_2=-[(p-q)^2+(q-r)^2+(r-p)^2]<0Δ1​+Δ2​=2Δ2​=−[(p−q)2+(q−r)2+(r−p)2]<0

2.
(a−b)(b−c)(c−a)=2(a-b)(b-c)(c-a)=2(a−b)(b−c)(c−a)=2, abc=6abc=6abc=6
Value of ∣abca2b2c2a3b3c3∣\begin{vmatrix}a&b&c\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}​aa2a3​bb2b3​cc2c3​​
(a) 3 (b) 12 (c) 14 (d) 15

Ans: (b)
=abc(a−b)(b−c)(c−a)=6×2=12=abc(a-b)(b-c)(c-a)=6\times2=12=abc(a−b)(b−c)(c−a)=6×2=12

3.
Determinant ∣abcbcacab∣\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}​abc​bca​cab​​ vanishes if

1. a+b+c=0a+b+c=0a+b+c=0
2. a3+b3+c3=3abca^3+b^3+c^3=3abca3+b3+c3=3abc
3. a2+b2+c2−ab−bc−ca=0a^2+b^2+c^2-ab-bc-ca=0a2+b2+c2−ab−bc−ca=0
   (a) 1,2 only (b) 2,3 only (c) 1,3 only (d) 1,2,3

Ans: (d)
All three conditions make the determinant zero.

4.
A=[m n]A=[m\ n]A=[m n], B=[−n −m]B=[-n\ -m]B=[−n −m], C=[m−m]C=\begin{bmatrix}m\\-m\end{bmatrix}C=[m−m​]

1. CA=CBCA=CBCA=CB
2. AC=BCAC=BCAC=BC
3. C(A+B)=CA+CBC(A+B)=CA+CBC(A+B)=CA+CB
   (a) 1 only (b) 2 only (c) 2 and 3 (d) 1 and 2

Ans: (c)
CA≠CBCA\neq CBCA=CB, but AC=BCAC=BCAC=BC and C(A+B)=CA+CBC(A+B)=CA+CBC(A+B)=CA+CB.

5.
A=[2sin⁡θcos⁡θ0−2cos⁡θsin⁡θ0−111]A=\begin{bmatrix}2\sin\theta&\cos\theta&0\\-2\cos\theta&\sin\theta&0\\-1&1&1\end{bmatrix}A=​2sinθ−2cosθ−1​cosθsinθ1​001​​
A(adj⁡A)=A(\operatorname{adj}A)=A(adjA)=
(a) null (b) −I-I−I (c) III (d) 2I2I2I

Ans: (d)
∣A∣=2|A|=2∣A∣=2 ⇒A(adj⁡A)=∣A∣I=2I\Rightarrow A(\operatorname{adj}A)=|A|I=2I⇒A(adjA)=∣A∣I=2I

6.
Matrix [2cos⁡2θ2cos⁡2θ61−2sin⁡2θ2cos⁡2θ−13k2k1]\begin{bmatrix}2\cos2\theta&2\cos2\theta&6\\1-2\sin^2\theta&2\cos^2\theta-1&3\\k&2k&1\end{bmatrix}​2cos2θ1−2sin2θk​2cos2θ2cos2θ−12k​631​​ singular for
(a) 0 only (b) 1 only (c) 2 only (d) any real kkk

Ans: (d)
Rows 1 and 2 are identical ⇒\Rightarrow⇒ singular for every real kkk.

7.
AAA non-singular, B=adj⁡AB=\operatorname{adj}AB=adjA

1. AB=BAAB=BAAB=BA
2. ABABAB scalar matrix
3. ABABAB can be null
   (a) 1 only (b) 1 and 2 only (c) 2 only (d) 1,2,3

Ans: (b)
AB=BA=∣A∣IAB=BA=|A|IAB=BA=∣A∣I (scalar, non-null).

8.
Square matrices A,BA,BA,B same order

1. AB=O⇒AB=O\RightarrowAB=O⇒ at least one is null
2. AB=I⇒BA=ABAB=I\Rightarrow BA=ABAB=I⇒BA=AB
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (b)
Counter-example shows 1 false; 2 true because B=A−1B=A^{-1}B=A−1.

9.
A=I3A=I_3A=I3​, B=ATB=A^TB=AT, C=A+BC=A+BC=A+B
∣C∣=|C|=∣C∣=
(a) 1 (b) 2 (c) 4 (d) 8

Ans: (d)
C=2IC=2IC=2I ⇒∣C∣=8\Rightarrow|C|=8⇒∣C∣=8

10.
AB=AAB=AAB=A, BA=BBA=BBA=B, both non-singular

1. A2=AA^2=AA2=A
2. AB2=A2BAB^2=A^2BAB2=A2B
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (c)
Both follow from A=B=IA=B=IA=B=I.

11.
Number of terms in (1+2x)9(1−2x)9\Bigl(1+\frac2x\Bigr)^9\Bigl(1-\frac2x\Bigr)^9(1+x2​)9(1−x2​)9
(a) 9 (b) 10 (c) 19 (d) 20

Ans: (b)
=(1−4x2)9=\bigl(1-\frac4{x^2}\bigr)^9=(1−x24​)9 has 10 terms.

12.
(x+y)10(x+y)^{10}(x+y)10

1. Coefficient of 6th term is highest
2. Coefficient of 3rd term = coefficient of 9th term
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (c)
Middle term (6th) has highest coefficient; binomial coefficients are symmetric.

13.
3nC2n=3nC2n−7^{3n}C_{2n}=^{3n}C_{2n-7}3nC2n​=3nC2n−7​
Value of nCn−5^nC_{n-5}nCn−5​
(a) 42 (b) 35 (c) 28 (d) 21

Ans: (d)
n=7n=7n=7 ⇒7C2=21\Rightarrow^7C_2=21⇒7C2​=21

14.
51C21−51C22+⋯−51C30^{51}C_{21}-^{51}C_{22}+\dots-^{51}C_{30}51C21​−51C22​+⋯−51C30​
(a) 51C25^{51}C_{25}51C25​ (b) 51C27^{51}C_{27}51C27​ (c) 51C51−51C0^{51}C_{51}-^{51}C_051C51​−51C0​ (d) 51C25−51C27^{51}C_{25}-^{51}C_{27}51C25​−51C27​

Ans: (a)
The alternating sum equals 51C25^{51}C_{25}51C25​.

15.
Odd numbers between 300 and 400 with no repeated digits
(a) 32 (b) 36 (c) 40 (d) 45

Ans: (a)
Hundreds fixed as 3; units 4 choices; tens 8 choices ⇒1×4×8=32\Rightarrow1\times4\times8=32⇒1×4×8=32

16.
Permutations of TIGER in which vowels do not occupy even places
(a) 72 (b) 36 (c) 18 (d) 12

Ans: (b)
Vowels in odd places: 2P2×3!=36^2P_2\times3!=362P2​×3!=36

17.
α,β\alpha,\betaα,β roots of x2+px+q=0x^2+px+q=0x2+px+q=0; α3,β3\alpha^3,\beta^3α3,β3 roots of x2+mx+n=0x^2+mx+n=0x2+mx+n=0
m+n=m+n=m+n=
(a) p3+q3+pqp^3+q^3+pqp3+q3+pq (b) p3+q3−pqp^3+q^3-pqp3+q3−pq (c) p3+q3+3pqp^3+q^3+3pqp3+q3+3pq (d) p3+q3−3pqp^3+q^3-3pqp3+q3−3pq

Ans: (d)
m=p3−3pqm=p^3-3pqm=p3−3pq, n=q3n=q^3n=q3 ⇒m+n=p3+q3−3pq\Rightarrow m+n=p^3+q^3-3pq⇒m+n=p3+q3−3pq

18.
α,β\alpha,\betaα,β roots of x2−(a+b)x+ab−c=0x^2-(a+b)x+ab-c=0x2−(a+b)x+ab−c=0
Quadratic with roots a,ba,ba,b
(a) x2−αx−βx+αβ+c=0x^2-\alpha x-\beta x+\alpha\beta+c=0x2−αx−βx+αβ+c=0 (b) x2−αx−βx+αβ−c=0x^2-\alpha x-\beta x+\alpha\beta-c=0x2−αx−βx+αβ−c=0
(c) x2+αx+βx+αβ+c=0x^2+\alpha x+\beta x+\alpha\beta+c=0x2+αx+βx+αβ+c=0 (d) x2+αx+βx+αβ−c=0x^2+\alpha x+\beta x+\alpha\beta-c=0x2+αx+βx+αβ−c=0

Ans: (a)

19.
Equal roots of x2−ax−bx−cx+bc+ca=0x^2-ax-bx-cx+bc+ca=0x2−ax−bx−cx+bc+ca=0
(a) a+b+c=0a+b+c=0a+b+c=0 (b) a−b+c=0a-b+c=0a−b+c=0 (c) a+b−c=0a+b-c=0a+b−c=0 (d) −a+b+c=0-a+b+c=0−a+b+c=0

Ans: (c)

20.
α>β\alpha>\betaα>β roots of x2−8x+q=0x^2-8x+q=0x2−8x+q=0, α2−β2=16\alpha^2-\beta^2=16α2−β2=16
q=q=q=
(a) –15 (b) –10 (c) 10 (d) 15

Ans: (d)

21.
Highest power of 5 dividing 30!+35!30!+35!30!+35!
(a) 4 (b) 6 (c) 7 (d) 8

Ans: (c)
30!(1+35⋅34⋅33⋅32⋅31)30!(1+35\cdot34\cdot33\cdot32\cdot31)30!(1+35⋅34⋅33⋅32⋅31); the term in parentheses is not divisible by 5, so exponent is that of 30!=730!=730!=7.

22.
2⋅2!+3⋅3!+⋯+9⋅9!+2=2\cdot2!+3\cdot3!+\dots+9\cdot9!+2=2⋅2!+3⋅3!+⋯+9⋅9!+2=
(a) 11! (b) 10! (c) 10+10!10+10!10+10! (d) 11+10!11+10!11+10!

Ans: (b)
Telescopes to 10!10!10!.

23.
A={(1,2,3)}A=\{(1,2,3)\}A={(1,2,3)}, number of elements in power set of AAA
(a) 1 (b) 2 (c) 4 (d) 8

Ans: (b)
n(A)=1⇒21=2n(A)=1\Rightarrow2^1=2n(A)=1⇒21=2

24.
a,b,c>0a,b,c>0a,b,c>0 in GP

1. a2,b2,c2a^2,b^2,c^2a2,b2,c2 in GP
2. 1/a,1/b,1/c1/a,1/b,1/c1/a,1/b,1/c in GP
3. a,b,c\sqrt a,\sqrt b,\sqrt ca​,b​,c​ in GP
   (a) 1,2 only (b) 2,3 only (c) 1,3 only (d) 1,2,3

Ans: (d)

25.
a+b2,b+c2\dfrac{a+b}2,\dfrac{b+c}22a+b​,2b+c​ in HP ⇒\Rightarrow⇒
(a) a,b,ca,b,ca,b,c AP (b) a,b,ca,b,ca,b,c GP (c) a+b,b+c,c+aa+b,b+c,c+aa+b,b+c,c+a GP (d) a+b,b+c,c+aa+b,b+c,c+aa+b,b+c,c+a AP

Ans: (b)

26.
cot⁡215∘+tan⁡215∘=\cot^215^\circ+\tan^215^\circ=cot215∘+tan215∘=
(a) 12 (b) 14 (c) 838\sqrt383​ (d) 4

Ans: (b)
=csc⁡215∘+sec⁡215∘−2=14= \csc^215^\circ+\sec^215^\circ-2=14=csc215∘+sec215∘−2=14

27.
In △ABC\triangle ABC△ABC, sin⁡A=cos⁡B+cos⁡C\sin A=\cos B+\cos CsinA=cosB+cosC
B=B=B=
(a) π/6\pi/6π/6 (b) π/4\pi/4π/4 (c) π/3\pi/3π/3 (d) π/2\pi/2π/2

Ans: (d)

28.
α+β=π/4\alpha+\beta=\pi/4α+β=π/4, 2tan⁡α=12\tan\alpha=12tanα=1
tan⁡2β=\tan2\beta=tan2β=
(a) 1/31/31/3 (b) 2/32/32/3 (c) 3/43/43/4 (d) 3/53/53/5

Ans: (c)

29.
tan⁡(45∘+θ)=1+sin⁡2θ\tan(45^\circ+\theta)=1+\sin2\thetatan(45∘+θ)=1+sin2θ, −π/4<θ<π/4-\pi/4<\theta<\pi/4−π/4<θ<π/4
cos⁡2θ=\cos2\theta=cos2θ=
(a) 0 (b) 1/21/21/2 (c) 1 (d) 2

Ans: (c)
θ=0⇒cos⁡2θ=1\theta=0\Rightarrow\cos2\theta=1θ=0⇒cos2θ=1

30.
sin⁡2θ=cos⁡3θ\sin2\theta=\cos3\thetasin2θ=cos3θ (θ\thetaθ acute)
1+4sin⁡θ=1+4\sin\theta=1+4sinθ=
(a) 3\sqrt33​ (b) 2 (c) 5\sqrt55​ (d) 3

Ans: (c)
θ=18∘⇒1+4sin⁡18∘=5\theta=18^\circ\Rightarrow1+4\sin18^\circ=\sqrt5θ=18∘⇒1+4sin18∘=5​

31.
tan⁡θ=−5/12\tan\theta=-5/12tanθ=−5/12
Possible value of sin⁡θ\sin\thetasinθ
(a) 5/135/135/13 only (b) −5/13-5/13−5/13 only (c) both (d) none

Ans: (c)

32.
cos⁡4(7π/8)+cos⁡4(5π/8)=\cos^4(7\pi/8)+\cos^4(5\pi/8)=cos4(7π/8)+cos4(5π/8)=
(a) 3/23/23/2 (b) 3/43/43/4 (c) 3/83/83/8 (d) 3/163/163/16

Ans: (b)

33.
sin⁡2(π/4+θ)−sin⁡2(π/4−θ)=\sin^2(\pi/4+\theta)-\sin^2(\pi/4-\theta)=sin2(π/4+θ)−sin2(π/4−θ)=
(a) sin⁡2θ\sin2\thetasin2θ (b) cos⁡2θ\cos2\thetacos2θ (c) 2sin⁡θ2\sin\theta2sinθ (d) 2cos⁡θ2\cos\theta2cosθ

Ans: (a)

34.
Tower height hhh, angles θ\thetaθ and 2θ2\theta2θ
Height of tower =
(a) hcos⁡θh\cos\thetahcosθ (b) hsin⁡θh\sin\thetahsinθ (c) hcos⁡2θh\cos2\thetahcos2θ (d) hsin⁡2θh\sin2\thetahsin2θ

Ans: (c)

35.
Shadow increases by xxx when elevation falls from 60∘60^\circ60∘ to θ\thetaθ; tower height 3x\sqrt3x3​x
(a) 0<θ<30∘0<\theta<30^\circ0<θ<30∘ (b) 30∘<θ<45∘30^\circ<\theta<45^\circ30∘<θ<45∘ (c) 45∘<θ<60∘45^\circ<\theta<60^\circ45∘<θ<60∘ (d) 60∘<θ<90∘60^\circ<\theta<90^\circ60∘<θ<90∘

Ans: (b)

36.
tan⁡−1(1/2)+tan⁡−1(x/3)=π/4\tan^{-1}(1/2)+\tan^{-1}(x/3)=\pi/4tan−1(1/2)+tan−1(x/3)=π/4, 0<x<60<x<60<x<6
x=x=x=
(a) 1 (b) 2 (c) 3 (d) 5

Ans: (a)

37.
3sin⁡−1x+cos⁡−1x=π3\sin^{-1}x+\cos^{-1}x=\pi3sin−1x+cos−1x=π
x=x=x=
(a) 0 (b) 1/21/21/2 (c) 1/21/\sqrt21/2​ (d) 1/31/\sqrt31/3​

Ans: (c)

38.
tan⁡α+tan⁡β=1−tan⁡αtan⁡β\tan\alpha+\tan\beta=1-\tan\alpha\tan\betatanα+tanβ=1−tanαtanβ
One value of α+β\alpha+\betaα+β
(a) π/6\pi/6π/6 (b) π/4\pi/4π/4 (c) π/3\pi/3π/3 (d) π/2\pi/2π/2

Ans: (b)

39.
(1+tan⁡θ)(1+tan⁡9θ)=2(1+\tan\theta)(1+\tan9\theta)=2(1+tanθ)(1+tan9θ)=2
tan⁡10θ=\tan10\theta=tan10θ=
(a) 0 (b) 1 (c) 2 (d) infinite

Ans: (b)

40.
sin⁡0∘+sin⁡10∘+⋯+sin⁡360∘=\sin0^\circ+\sin10^\circ+\dots+\sin360^\circ=sin0∘+sin10∘+⋯+sin360∘=
(a) –1 (b) 0 (c) 1 (d) 2

Ans: (b)

41.
Number of supersets of {4}\{4\}{4} among subsets of {1,2,3,4}\{1,2,3,4\}{1,2,3,4}
(a) 6 (b) 7 (c) 8 (d) 9

Ans: (c)

42.

1. x∉A∪B⇒x∉Ax\notin A\cup B\Rightarrow x\notin Ax∈/A∪B⇒x∈/A or x∉Bx\notin Bx∈/B
2. x∉A∩B⇒x∉Ax\notin A\cap B\Rightarrow x\notin Ax∈/A∩B⇒x∈/A and x∉Bx\notin Bx∈/B
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (d)

43.

1. A∪B=A∩B⇔A=BA\cup B=A\cap B\Leftrightarrow A=BA∪B=A∩B⇔A=B
2. A−B=∅⇔A=BA-B=\emptyset\Leftrightarrow A=BA−B=∅⇔A=B
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (c)

44.
xRy⇔x2−5xy+4y2=0xRy\Leftrightarrow x^2-5xy+4y^2=0xRy⇔x2−5xy+4y2=0 on N\mathbb NN

1. reflexive 2. symmetric 3. transitive
   (a) 1 only (b) 2 only (c) 1,2 only (d) 1,2,3

Ans: (a)

45.
For any relation RRR

1. reflexive ⇒R−1\Rightarrow R^{-1}⇒R−1 reflexive
2. symmetric ⇒R−1\Rightarrow R^{-1}⇒R−1 symmetric
3. transitive ⇒R−1\Rightarrow R^{-1}⇒R−1 transitive
   (a) 1,2 only (b) 2,3 only (c) 1,3 only (d) 1,2,3

Ans: (d)

46.
Principal argument of 1/(1+i)1/(1+i)1/(1+i)
(a) −3π/4-3\pi/4−3π/4 (b) −π/4-\pi/4−π/4 (c) π/4\pi/4π/4 (d) 3π/43\pi/43π/4

Ans: (b)

47.
Modulus of (3/2−1/2)200\bigl(\sqrt3/2-1/2\bigr)^{200}(3​/2−1/2)200
(a) 1/41/41/4 (b) 1/21/21/2 (c) 1 (d) 22002^{200}2200

Ans: (c)

48.

1. n!/3!n!/3!n!/3! divisible by 6 (n>3n>3n>3)
2. n!/3!+3n!/3!+3n!/3!+3 divisible by 7 (n>3n>3n>3)
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (d)

49.
Ways to choose 5 players out of 9 excluding two particular ones
(a) 14 (b) 21 (c) 35 (d) 42

Ans: (b)
7C5=21^7C_5=217C5​=21

50.
(n+1)(n+1)(n+1)th term from end in (x+1/x)2n\bigl(x+1/x\bigr)^{2n}(x+1/x)2n
(a) 2nCnx^{2n}C_n x2nCn​x (b) 2nCn−1x^{2n}C_{n-1}x2nCn−1​x (c) 2nCn^{2n}C_n2nCn​ (d) 2nCn−1^{2n}C_{n-1}2nCn−1​

Ans: (c)

51.
S9=S11S_9=S_{11}S9​=S11​ of an AP
S20=S_{20}=S20​=
(a) 20 (b) 10 (c) 2 (d) 0

Ans: (d)

52.
5th term =1/10=1/10=1/10, 10th term =1/5=1/5=1/5
S50=S_{50}=S50​=
(a) 25 (b) 25.5 (c) 26 (d) 26.5

Ans: (b)

53.
(1110011)2÷(10111)2=(1110011)_2\div(10111)_2=(1110011)2​÷(10111)2​=
(a) (101)2(101)_2(101)2​ (b) (1001)2(1001)_2(1001)2​ (c) (111)2(111)_2(111)2​ (d) (1011)2(1011)_2(1011)2​

Ans: (a)

54.
x2+y2=(100010111)2x^2+y^2=(100010111)_2x2+y2=(100010111)2​, x+y=(11111)2x+y=(11111)_2x+y=(11111)2​
(x−y)2+xy=(x-y)^2+xy=(x−y)2+xy=
(a) (1101)2(1101)_2(1101)2​ (b) (1001)2(1001)_2(1001)2​ (c) (1011)2(1011)_2(1011)2​ (d) (1111)2(1111)_2(1111)2​

Ans: (b)

55.
Region 5x−4y+12<05x-4y+12<05x−4y+12<0, x+y<2x+y<2x+y<2, x<0x<0x<0, y>0y>0y>0
Point inside
(a) (0,0)(0,0)(0,0) (b) (−2,4)(-2,4)(−2,4) (c) (−1,4)(-1,4)(−1,4) (d) (−1,2)(-1,2)(−1,2)

Ans: (d)

56.
y=[x]y=[x]y=[x], x∈(−1,1)x\in(-1,1)x∈(−1,1)

1. derivative 0 at x=0.5x=0.5x=0.5
2. continuous at x=0x=0x=0
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (a)

57.
Degree of 1+(y′)2=(y′′)4/31+(y')^2=(y'')^{4/3}1+(y′)2=(y′′)4/3
(a) 4/34/34/3 (b) 2 (c) 3 (d) 4

Ans: (d)

58.
Half-life 100 yr; decay constant
(a) ln⁡2/100\ln2/100ln2/100 (b) ln⁡5/100\ln5/100ln5/100 (c) ln⁡10/100\ln10/100ln10/100 (d) 2ln⁡2/1002\ln2/1002ln2/100

Ans: (a)

59.
Domain of 1−(x−1)2\sqrt{1-(x-1)^2}1−(x−1)2​
(a) (0,1)(0,1)(0,1) (b) [−1,1][-1,1][−1,1] (c) (0,2)(0,2)(0,2) (d) [0,2][0,2][0,2]

Ans: (d)

60.
Area of parabola y2=4kxy^2=4kxy2=4kx and latus rectum = 24
k=k=k=
(a) 1 (b) 2 (c) 3 (d) 4

Ans: (c)

61.
∫0π/4dx(sin⁡x+cos⁡x)2=\displaystyle\int_0^{\pi/4}\frac{dx}{(\sin x+\cos x)^2}=∫0π/4​(sinx+cosx)2dx​=
(a) −1/2-1/2−1/2 (b) 1/21/21/2 (c) 1 (d) 3/23/23/2

Ans: (b)

62.
∫(sin⁡x)−1/2(cos⁡x)−3/2 dx=\displaystyle\int(\sin x)^{-1/2}(\cos x)^{-3/2}\,dx=∫(sinx)−1/2(cosx)−3/2dx=
(a) tan⁡x+c\sqrt{\tan x}+ctanx​+c (b) 2tan⁡x+c2\sqrt{\tan x}+c2tanx​+c (c) cot⁡x+c\sqrt{\cot x}+ccotx​+c (d) 2tan⁡x+c\sqrt{2\tan x}+c2tanx​+c

Ans: (b)

63.
I1=∫exex+e−xdxI_1=\int\frac{e^x}{e^x+e^{-x}}dxI1​=∫ex+e−xex​dx, I2=∫dxe2x+1I_2=\int\frac{dx}{e^{2x}+1}I2​=∫e2x+1dx​
I1+I2=I_1+I_2=I1​+I2​=
(a) x/2+cx/2+cx/2+c (b) x+cx+cx+c (c) ln⁡(ex+e−x)+c\ln(e^x+e^{-x})+cln(ex+e−x)+c (d) ln⁡(ex−e−x)+c\ln(e^x-e^{-x})+cln(ex−e−x)+c

Ans: (b)

64.
∫−2−1x∣x∣ dx=\displaystyle\int_{-2}^{-1}\frac x{|x|}\,dx=∫−2−1​∣x∣x​dx=
(a) –2 (b) –1 (c) 1 (d) 2

Ans: (b)

65.
Number of extreme values of sin⁡4x+2x\sin4x+2xsin4x+2x in (0,π/2)(0,\pi/2)(0,π/2)
(a) 1 (b) 2 (c) 4 (d) 8

Ans: (b)

66.
Maximum of 1tan⁡x+cot⁡x\dfrac1{\tan x+\cot x}tanx+cotx1​ in (0,π/2)(0,\pi/2)(0,π/2)
(a) 1/41/41/4 (b) 1/21/21/2 (c) 1 (d) 2

Ans: (b)

67.
4f(x)−f(1/x)=(2x+1/x)(2x−1/x)4f(x)-f(1/x)=(2x+1/x)(2x-1/x)4f(x)−f(1/x)=(2x+1/x)(2x−1/x)
f(2)=f(2)=f(2)=
(a) 0 (b) 1 (c) 2 (d) 4

Ans: (d)

68.
f(x)=4x+3f(x)=4x+3f(x)=4x+3
f∘f∘f(−1)=f\circ f\circ f(-1)=f∘f∘f(−1)=
(a) –1 (b) 0 (c) 1 (d) 2

Ans: (a)

69.
x3y3=1x^3y^3=1x3y3=1, dydx\dfrac{dy}{dx}dxdy​ at (1,1)(1,1)(1,1)
(a) –1 (b) 0 (c) 1 (d) 4

Ans: (a)

70.
y=(xx)xy=(x^x)^xy=(xx)x, dydx\dfrac{dy}{dx}dxdy​ at x=1x=1x=1
(a) 1/21/21/2 (b) 1 (c) 2 (d) 4

Ans: (b)

71.
y=[x+1]y=[x+1]y=[x+1], −4<x<−3-4<x<-3−4<x<−3
Derivative at x=−3.5x=-3.5x=−3.5
(a) –4 (b) –3.5 (c) –3 (d) 0

Ans: (d)

72.
dydx=(ln⁡5)y\dfrac{dy}{dx}=(\ln5)ydxdy​=(ln5)y, y(0)=ln⁡5y(0)=\ln5y(0)=ln5
y(1)=y(1)=y(1)=
(a) 0 (b) 5 (c) 2ln⁡52\ln52ln5 (d) 5ln⁡55\ln55ln5

Ans: (d)

73.
f(x)=10f(x)=10f(x)=10

1. domain (−∞,∞)(-\infty,\infty)(−∞,∞)
2. continuous
3. differentiable at 0
   (a) 1,2 only (b) 2,3 only (c) 1,3 only (d) 1,2,3

Ans: (d)

74.
lim⁡x→0x3(csc⁡x)2=\lim_{x\to0}x^3(\csc x)^2=limx→0​x3(cscx)2=
(a) 0 (b) 1/21/21/2 (c) 1 (d) DNE

Ans: (a)

75.
lim⁡x→1x3−1x−1=\lim_{x\to1}\dfrac{x^3-1}{\sqrt x-1}=limx→1​x​−1x3−1​=
(a) 0 (b) 3 (c) 6 (d) DNE

Ans: (c)

76.
f(x)=x33−7x22+6x+5f(x)=\dfrac{x^3}3-\dfrac{7x^2}2+6x+5f(x)=3x3​−27x2​+6x+5 decreasing on
(a) (−∞,1)(-\infty,1)(−∞,1) only (b) (1,6)(1,6)(1,6) (c) (6,∞)(6,\infty)(6,∞) only (d) (−∞,1)∪(6,∞)(-\infty,1)\cup(6,\infty)(−∞,1)∪(6,∞)

Ans: (b)

77.
f′(2)=0f'(2)=0f′(2)=0 for f(x)=mx+2nx+1f(x)=\dfrac mx+2nx+1f(x)=xm​+2nx+1
m+8n=m+8n=m+8n=
(a) –2 (b) 0 (c) 2 (d) cannot be determined

Ans: (d)

78.
Area in first quadrant between y=xy=xy=x and y=x2y=x^2y=x2
(a) 1/81/81/8 (b) 1/41/41/4 (c) 1/21/21/2 (d) 1

Ans: (b)

79.
xy=4225xy=4225xy=4225, x,y∈Nx,y\in\mathbb Nx,y∈N, minimum of x+yx+yx+y
(a) 130 (b) 260 (c) 2113 (d) 4226

Ans: (a)

80.
xdydx−2y=0x\dfrac{dy}{dx}-2y=0xdxdy​−2y=0 represents
(a) family of straight lines (b) circles (c) parabolas (d) ellipses

Ans: (c)

81.
Points (−5,0)(-5,0)(−5,0), (5p2,10p)(5p^2,10p)(5p2,10p), (5q2,10q)(5q^2,10q)(5q2,10q) collinear, p≠qp\neq qp=q
pq=pq=pq=
(a) –2 (b) –1 (c) 1 (d) 2

Ans: (c)

82.
Line through (1,−2)(1,-2)(1,−2) with equal intercepts
(a) x+y−1=0x+y-1=0x+y−1=0 (b) x−y−1=0x-y-1=0x−y−1=0 (c) x+y+1=0x+y+1=0x+y+1=0 (d) x−y−2=0x-y-2=0x−y−2=0

Ans: (c)

83.
Circle touching both axes in first quadrant and line y−2=0y-2=0y−2=0
(a) x2+y2−2x−2y−1=0x^2+y^2-2x-2y-1=0x2+y2−2x−2y−1=0 (b) x2+y2+2x+2y+1=0x^2+y^2+2x+2y+1=0x2+y2+2x+2y+1=0
(c) x2+y2−2x−2y+1=0x^2+y^2-2x-2y+1=0x2+y2−2x−2y+1=0 (d) x2+y2−4x−4y+4=0x^2+y^2-4x-4y+4=0x2+y2−4x−4y+4=0

Ans: (c)

84.
Parabola focus (−3,0)(-3,0)(−3,0), directrix x−3=0x-3=0x−3=0
(a) y2=3xy^2=3xy2=3x (b) x2=12yx^2=12yx2=12y (c) y2=12xy^2=12xy2=12x (d) y2=−12xy^2=-12xy2=−12x

Ans: (d)

85.
Distance between foci of x2+2y2=1x^2+2y^2=1x2+2y2=1
(a) 1 (b) 2\sqrt22​ (c) 2 (d) 222\sqrt222​

Ans: (b)

86.
Sides a,b,ca,b,ca,b,c, perimeter ppp, area qqq
p(p−2a)tan⁡(A/2)=p(p-2a)\tan(A/2)=p(p−2a)tan(A/2)=
(a) qqq (b) 2q2q2q (c) 3q3q3q (d) 4q4q4q

Ans: (d)

87.
Line through intersection of x+2y+2=0x+2y+2=0x+2y+2=0 and 2x−3y−3=02x-3y-3=02x−3y−3=0, equal intercepts in 4th quadrant
Sum of absolute intercepts
(a) 2 (b) 3 (c) 4 (d) 6

Ans: (a)

88.
Lines ax+by+c=0ax+by+c=0ax+by+c=0 and bx+ay+c=0bx+ay+c=0bx+ay+c=0 parallel iff
(a) a−b=0a-b=0a−b=0 only (b) a+b=0a+b=0a+b=0 only (c) a2−b2=0a^2-b^2=0a2−b2=0 (d) ab+1=0ab+1=0ab+1=0

Ans: (c)

89.
Locus of mid-point of intercept of x+y=px+y=px+y=p
(a) x−y=0x-y=0x−y=0 (b) x+y=0x+y=0x+y=0 (c) x−y=px-y=px−y=p (d) x+y=px+y=px+y=p

Ans: (a)

90.
Point equidistant from (2a,0)(2a,0)(2a,0) and (0,3a)(0,3a)(0,3a)
(a) 2x−3y=02x-3y=02x−3y=0 (b) 3x−2y=03x-2y=03x−2y=0 (c) 4x−6y+5a=04x-6y+5a=04x−6y+5a=0 (d) 4x−6y−5a=04x-6y-5a=04x−6y−5a=0

Ans: (c)

91–93 (plane 6x+ky+3z=126x+ky+3z=126x+ky+3z=12, sphere x2+y2+z2−2x−3y−4z=0x^2+y^2+z^2-2x-3y-4z=0x2+y2+z2−2x−3y−4z=0)
91. k=k=k= (a) 3 (b) 4 (c) 6 (d) 12 Ans: (b)
92. Perp. distance ppp from centre to plane
(a) 0<p<0.50<p<0.50<p<0.5 (b) 0.5<p<10.5<p<10.5<p<1 (c) 1<p<1.51<p<1.51<p<1.5 (d) p>1.5p>1.5p>1.5 Ans: (b)
93. Line through origin and centre
(a) x−y=zx-y=zx−y=z (b) 2x−3y=4z2x-3y=4z2x−3y=4z (c) 6x=3y=4z6x=3y=4z6x=3y=4z (d) 6x=4y=3z6x=4y=3z6x=4y=3z Ans: (d)

94–95 (plane 2xk+2y3+z3=2\frac{2x}k+\frac{2y}3+\frac z3=2k2x​+32y​+3z​=2 through (2,3,−6)(2,3,-6)(2,3,−6))
94. Direction ratios of normal
(a) ⟨3,2,1⟩\langle3,2,1\rangle⟨3,2,1⟩ (b) ⟨2,3,6⟩\langle2,3,6\rangle⟨2,3,6⟩ (c) ⟨6,3,2⟩\langle6,3,2\rangle⟨6,3,2⟩ (d) ⟨1,2,3⟩\langle1,2,3\rangle⟨1,2,3⟩ Ans: (a)
95. p+q+r=p+q+r=p+q+r=
(a) 10 (b) 11 (c) 12 (d) 13 Ans: (b)

96. 4i^+j^−3k^4\hat i+\hat j-3\hat k4i^+j^​−3k^ and pi^+qj^−2k^p\hat i+q\hat j-2\hat kpi^+qj^​−2k^ collinear
Possible (p,q)(p,q)(p,q)
(a) 4,1 (b) 1,4 (c) 8/3,2/38/3,2/38/3,2/3 (d) 2/3,8/32/3,8/32/3,8/3

Ans: (c)

97. Position vector of centroid GGG
(a) a⃗+b⃗+c⃗3\frac{\vec a+\vec b+\vec c}33a+b+c​ (b) 2a⃗−b⃗−c⃗3\frac{2\vec a-\vec b-\vec c}332a−b−c​ (c) b⃗+c⃗−2a⃗3\frac{\vec b+\vec c-2\vec a}33b+c−2a​ (d) a⃗−2b⃗−2c⃗3\frac{\vec a-2\vec b-2\vec c}33a−2b−2c​

Ans: (a)

98.

1. Dot product distributive over addition
2. Cross product distributive over addition
3. Cross product associative
   (a) 1 only (b) 2 only (c) 1 and 2 only (d) 1,2,3

Ans: (c)

99. a⃗×b⃗=c⃗\vec a\times\vec b=\vec ca×b=c

1. a⃗\vec aa unique if b⃗,c⃗\vec b,\vec cb,c given
2. c⃗\vec cc unique if a⃗,b⃗\vec a,\vec ba,b given
   (a) 1 only (b) 2 only (c) both (d) neither

Ans: (b)

100. Unit vectors, ∣a⃗−b⃗∣<2|\vec a-\vec b|<2∣a−b∣<2, 2θ=2\theta=2θ= angle
(a) 0<sin⁡θ<10<\sin\theta<10<sinθ<1 only (b) −1/2<sin⁡θ<1/2-1/2<\sin\theta<1/2−1/2<sinθ<1/2 only
(c) −1<sin⁡θ<0-1<\sin\theta<0−1<sinθ<0 only (d) −1<sin⁡θ<1-1<\sin\theta<1−1<sinθ<1

Ans: (d)

101. Two digits from 1–5 multiplied; probability last digit 0
(a) 1/101/101/10 (b) 1/51/51/5 (c) 2/52/52/5 (d) 4/54/54/5

Ans: (b)

102. Left-skewed unimodal frequency curve
(a) Mean > Median > Mode (b) Mean > Mode > Median
(c) Median > Mean > Mode (d) Mode > Median > Mean

Ans: (d)

103. Variance of five positive numbers = 3.6; four are 2,2,4,5
Fifth number
(a) 4 (b) 5 (c) 7 (d) 9

Ans: (c)

104. Mean of 50 terms of AP with first term 4, common difference 4
(a) 50 (b) 51 (c) 100 (d) 102

Ans: (d)

105. Coefficient of mean deviation of 21,34,23,39,26,37,40,20,33,27
(a) 0.11 (b) 0.22 (c) 0.33 (d) 0.44

Ans: (b)

106–108 (deviations from 100 sum to –20; from 92 sum to 140)
106. Mean
(a) 91 (b) 96 (c) 98 (d) 99 Ans: (d)
107. Sum of deviations from 99
(a) 0 (b) 10 (c) 20 (d) 40 Ans: (a)
108. Value of yyy if sum of deviations from yyy is 180
(a) 80 (b) 85 (c) 90 (d) 95 Ans: (c)

109–111 (marks of 51 students in AP, first term 4, common difference 3)
109. Mean
(a) 67 (b) 71 (c) 75 (d) 79 Ans: (d)
110. Median
(a) 79.5 (b) 79 (c) 78.5 (d) 77 Ans: (b)
111. Sum of deviations from median
(a) –1 (b) 0 (c) 1 (d) 2 Ans: (b)

112–114 (90 applicants table)
112. P(G∩Tˉ)=P(G\cap\bar T)=P(G∩Tˉ)=
(a) 1/51/51/5 (b) 2/52/52/5 (c) 3/53/53/5 (d) 4/54/54/5 Ans: (b)
113. P(G∣Tˉ)=P(G|\bar T)=P(G∣Tˉ)=
(a) 2/72/72/7 (b) 3/73/73/7 (c) 4/74/74/7 (d) 5/75/75/7 Ans: (c)
114. P(Tˉ∣Gˉ)=P(\bar T|\bar G)=P(Tˉ∣Gˉ)=
(a) 1/41/41/4 (b) 1/31/31/3 (c) 3/53/53/5 (d) 3/43/43/4 Ans: (d)

115–117 (disease probability 1/31/31/3)
115. Exactly 3 out of 6
(a) 80/72980/72980/729 (b) 10/8110/8110/81 (c) 10/24310/24310/243 (d) 160/729160/729160/729 Ans: (d)
116. None out of 6
(a) 665/729665/729665/729 (b) 64/72964/72964/729 (c) 4/2434/2434/243 (d) 1/7291/7291/729 Ans: (b)
117. At least one out of 6
(a) 728/729728/729728/729 (b) 665/729665/729665/729 (c) 653/729653/729653/729 (d) 596/729596/729596/729 Ans: (b)

118–120 (frequency distribution, total 120, mean 50)
118. p=p=p=
(a) 25 (b) 26 (c) 27 (d) 28 Ans: (c)
119. q=q=q=
(a) 1 (b) 2 (c) 3 (d) 4 Ans: (a)
120. If every frequency doubled, new mean
(a) 25 (b) 50 (c) 75 (d) 100 Ans: (b