KEAM 2018

KEAM Paper 2 Answer Key 2018: Mathematics

A

$\frac{5}{\sqrt{2}} (1 + i)$

B

$\frac{10}{\sqrt{2}} (1 + i)$

C

$\frac{10}{\sqrt{2}} ( 1 - i ) $

D

$\frac{5}{\sqrt{2}} (1 - i)$

E

$\frac{1}{\sqrt{2}} (1 + i)$

A

$\frac{1}{i - 1} $

B

$\frac{1}{i + 1} $

C

$\frac{-1}{i - 1} $

D

$\frac{ - 1}{i + 1} $

E

$\frac{1}{i}$

A

$\frac{1 + i}{2}$

B

$ 2 + 2i $

C

$\frac{1 - i }{2} $

D

$\sqrt{2} - \sqrt{2} i $

E

$ 2 - 2 i$

A

2

B

$\sqrt{2}$

C

4

D

8

E

10

A

-2

B

-1

C

-i

D

-2i

E

0

A

$ i + 3 $

B

$ 1 + 3i$

C

$ 1 - 3i$

D

0

E

$- i - 3 $

A

$3x^2 -19x -3=0$

B

$3x^2 +19x -3=0$

C

$x^2 +19x +3=0$

D

$3x^2 -19x-19=0$

E

$3x^2 -19x +3=0$

A

$\left( \frac{3}{4} , 2 \right)$

B

$\left( \frac{3}{4} , - 2 \right)$

C

$\left( 2 , \frac{3}{4} \right)$

D

$\left( \frac{-3}{4} , 2 \right)$

E

$\left( 2 , \frac{- 3}{4} \right)$

A

$-f [(1)$

B

$f(1) - f '(1)$

C

$2 f(1) - f ' (1)$

D

$2 f (1) + f '(1) $

E

$f(1) + f '(1)$

A

$\frac{\sqrt{3}}{2}$

B

$\frac{\sqrt{3}}{4}$

C

$\frac{\sqrt{3}}{\sqrt{2}}$

D

$\frac{\sqrt{3}}{8}$

E

$\frac{\sqrt{3}}{16}$

A

(–4, –1)

B

(–1, –4)

C

(1, 4)

D

(4, 1)

E

(–1, 4)

A

$x^2 - 4x + 16y + 17 = 0$

B

$4x^2 + 4y2 - 16x + 4y -60 = 0$

C

$x^2 + 2y^2 + 4x + 2y -27 = 0$

D

$x^2 - y^2 + 3x - 2y -43 = 0$

E

$x^2 + 4x + 6y - 2 = 0$

A

$2^{29}$

B

$2^{59}$

C

$2^{58}$

D

$2^{59} - 2^{29}$

E

$2^{60}$

A

$20 \sqrt{6}$

B

$30 \sqrt{6}$

C

$5 \sqrt{10}$

D

$40 \sqrt{6}$

E

$10 \sqrt{6}$

A

$\frac{\sqrt{3}}{2}$

B

$\frac{1}{\sqrt{2}}$

C

$\frac{ - \sqrt{3}}{2}$

D

$\frac{-1 }{\sqrt{2}}$

E

$\frac{1}{2}$

A

$(1121)^2$

B

$(1101)^2$

C

$(1001)^2$

D

$(1021)^2$

E

$(1011)^2$

A

$2 \ \cos^2 \ x $

B

$2 \ \cos^3 \ x $

C

$ - \cos - 2x $

D

$\cos \ 2x $

E

$ 3 \ \cos \ x $

A

$\left( \frac{24 \pm 5 \sqrt{145}}{12} , 1 \right)$

B

$\left( \frac{21 \pm 5 \sqrt{145}}{12} , 1 \right)$

C

$\left(1 , \frac{24 \pm 5 \sqrt{145}}{2} \right)$

D

$\left(1 , \frac{21 \pm 5 \sqrt{145}}{2} \right)$

E

$\left( \frac{21 \pm 5 \sqrt{145}}{2} , - 1 \right)$

A

2.23

B

3.23

C

2.57

D

3.57

E

1.03

A

$\frac{1}{2^8}$

B

$\frac{1}{2^{12}}$

C

$\frac{1}{2^{6}}$

D

$\frac{1}{2^{4}}$

E

$\frac{1}{2^{5}}$

A

90

B

45

C

60

D

70

E

10

A

0

B

$\frac{1 }{\sqrt{2}}$

C

$\frac{1 }{2}$

D

$\sqrt{2}$

E

1

A

$a^2 - b^2 + 2ac = 0 $

B

$(a - c)^2 =b^2 +c^2$

C

$a^2 +b^2 -2ac =0$

D

$a^2 +b^2 + 2ac =0$

E

$a + b + c = 0$

A

$\frac{\pi}{4}$

B

$\frac{\pi}{4} \sqrt{7}$

C

$\frac{4}{15}$

D

$\frac{4}{15} \sqrt{7}$

E

$\frac{15}{4} \sqrt{7}$

A

3

B

5

C

4

D

0

E

1

A

$\frac{2}{9}$

B

$\frac{1}{2}$

C

$\frac{-9}{2}$

D

$\frac{3}{4}$

E

$\frac{9}{2}$

A

$\frac{1}{2}$

B

$\frac{3}{2}$

C

1

D

0

E

2

A

$x =- 2, y = -3$

B

$x = 2, y = - 3$

C

$x =2, y = 3$

D

$x =4, y = 3$

E

$x =3, y = 4$

A

12 x

B

x + 4

C

$6x^5 + 6^z \ \log (6)$

D

$6x^5 + x6^{x - 1} $

E

$x^6$

A

$\frac{\sqrt{52}}{7}$

B

$\frac{52}{7}$

C

$\frac{\sqrt{53}}{7}$

D

$\frac{53}{7}$

E

6

A

$\frac{m^2}{n^2}$

B

$\frac{n^2}{m^2}$

C

$\infty$

D

$ - \infty$

E

0

A

0

B

-1

C

$\frac{1}{2}$

D

1

E

$\frac{-1}{2}$

A

1

B

$\frac{1}{\sqrt{2}}$

C

2

D

4

E

0

A

$\frac{4}{45} e^{-2} $

B

$\frac{1}{45} e^{-1} $

C

$\frac{1}{9} e^{-2} $

D

$\frac{1}{4} e^{-2} $

E

$\frac{1}{45} e^{-2} $

A

$\frac{\pi}{3}$

B

$\frac{3 \pi}{4}$

C

$\frac{\pi}{2}$

D

0

E

$\pi$

A

1, 1024

B

–1, 1024

C

1, 1031

D

–1024, 1

E

–1,1031

A

$\frac{7}{36}$

B

$\frac{2}{9}$

C

$\frac{2}{3}$

D

$\frac{5}{9}$

E

$\frac{6}{7}$

A

$\frac{2^9}{10!}$

B

$\frac{2^{10}}{8!}$

C

$\frac{2^{11}}{9!}$

D

$\frac{2^{10}}{7!}$

E

$\frac{2^8}{9!}$

A

3 and 2

B

1 and 2

C

2 and 3

D

1 and 4

E

3 and 5

A

0

B

1

C

2

D

4

E

$\frac{1}{2}$

A

0

B

$\frac{\pi}{4}$

C

$\frac{- \pi}{4}$

D

$\frac{\pi}{2}$

E

$\frac{ - \pi}{2}$

A

$\frac{e^x}{ 1 +x} + c $

B

$\frac{e^x}{ 1 + e^x} + c $

C

$\frac{e^{2x}}{ 1 + e^x} + c $

D

$\frac{e^{-x}}{ 1 +x} + c $

E

$\frac{e^{- 2 x}}{ 1 +x} + c $

A

1

B

2

C

8

D

12

E

4

A

1542

B

1512

C

2512

D

12

E

4

A

$|z|$

B

$|\bar{z}|^2$

C

$\frac{1}{2} |z|^2$

D

$\frac{1}{2} |z + iz |^2$

E

$ |z + iz |^2$

A

(16, 18)

B

(18, 16)

C

$\left( \frac{19}{3} , \frac{8}{3} \right)$

D

$\left( \frac{8}{3} , \frac{19}{3}\right)$

E

$( 7 , 3 ) $

A

abc

B

$a^2 + b^2 + c^2$

C

$ab + bc + ca$

D

0

E

$a(ab + bc + ca)$

A

ax + by = 0

B

ax – by = 0

C

bx + ay = 0

D

bx – ay = 0

E

x = y

A

$\frac{x}{a} + \frac{y}{b} = 3 $

B

$\frac{x}{a} + \frac{y}{b} = 2 $

C

$\frac{x}{a} + \frac{y}{b} = 0 $

D

$\frac{x}{a} + \frac{y}{b} + 2 = 0 $

E

$\frac{x}{a} + \frac{y}{b} = 4 $

A

(4, 4)

B

(8, 8)

C

(–4, –4)

D

$(4 \sqrt{2}, 4 \sqrt{2} )$

E

(6, 6)

A

$\left( \frac{ -3}{2} , \frac{-3}{2}\right) $

B

$\left( \frac{3}{4} , \frac{-3}{2}\right) $

C

$\left( \frac{7}{2} , \frac{13}{2}\right) $

D

$\left( \frac{ -1}{4} , \frac{1}{2}\right) $

E

$\left( \frac{3}{2} , \frac{3}{2}\right) $

A

$\frac{9}{16}$

B

$\frac{16}{25}$

C

$\frac{9}{25}$

D

$\frac{21}{25}$

E

$\frac{25}{9}$

A

x = 1, y = 2, z = 3

B

x = 2, y = 3, z = 1

C

x = 3, y = 1, z = 2

D

x = 1, y = 3, z = 2

E

x = 2, y = 2, z = 3

A

$\frac{\sqrt{3} - 1 }{2 \sqrt{2}}$

B

$\frac{\sqrt{3} + 1 }{2 \sqrt{2}}$

C

$\frac{ 1- \sqrt{3}}{2 \sqrt{2}}$

D

$\frac{ 1 + \sqrt{3}}{ \sqrt{2}}$

E

$\frac{- ( 1 + \sqrt{3})}{2 \sqrt{2}}$

A

$3 (\hat{i} + \hat{j} + \hat{k} )$

B

$ \hat{i} + 2\hat{j} + 2 \hat{k} $

C

$2 \hat{i} + 2\hat{j} + 2 \hat{k} $

D

$ \hat{i} + 3 \hat{j} + 3 \hat{k} $

E

$ \hat{i} -3 \hat{j} + 2 \hat{k} $

A

30

B

$\frac{30}{25} \sqrt{233} $

C

$\frac{30}{33} \sqrt{193} $

D

$\frac{65}{23} \sqrt{493} $

E

$\frac{65}{13} \sqrt{133} $

A

5

B

$ 5 \sqrt{5}$

C

$\sqrt{5}$

D

$\frac{1}{4}$

E

$\frac{3}{\sqrt{5}}$

A

$2^8 - 2 $

B

$2^7 - 1 $

C

$2^7$

D

$2^8 - 1 $

E

$2^7 - 2 $

A

$x^2 + y^2 – 4x + 4y – 77 = 0$

B

$x^2 + y^2 – 4x – 4y – 5 = 0$

C

$x^2 + y^2 + 2x + 2y – 59 = 0$

D

$x^2 + y^2 – 2x – 2y – 23 = 0$

E

$x^2 + y^2 + 4x – 2y – 26 = 0$

A

(1, 2, 3)

B

(–3, 2, 0)

C

(3, –2, 0)

D

(3, 2, 0)

E

(3, 2, 1)

A

-1

B

1

C

0

D

2

E

4

A

c

B

$\frac{\pi}{2} + c $

C

$c + 1 $

D

$2 \pi + c $

E

$c + 2 $

A

$ \pi ( \sqrt{2} - 1) $

B

$ \pi ( \sqrt{2} + 1) $

C

$ 2 \pi ( \sqrt{2} - 1) $

D

$2 \pi ( \sqrt{2} + 1) $

E

$\frac{\pi}{\sqrt{2} + 1 } $

A

2

B

$\pi $

C

$\frac{\pi}{4} $

D

$2 \pi $

E

0

A

$\frac{2}{3}$

B

$\frac{2}{9}$

C

$\frac{1}{3}$

D

0

E

$\frac{1}{6}$

A

$\frac{\pi}{2}$

B

$\frac{\pi}{4}$

C

$\frac{\pi}{8}$

D

$\frac{\pi}{16}$

E

$2 \pi $

A

$y'' -2y' + 2y = 0$

B

$y'' + 2y' - 2y = 0$

C

$y'' + y'^2 + 2y = 0$

D

$y'' + 2y' - 2y = 0$

E

$y'' - 2y' - 2y = 0$

A

$2^{-10} $

B

$2^{12} $

C

$ 2^{-12} $

D

$ - 2^{- 12} $

E

$2^{10}$

A

$\frac{1}{2}$

B

$\frac{ - 1}{2}$

C

1

D

-1

E

0

A

$\frac{\sin x + \cos x}{\sin2x } + c$

B

$\frac{\sin x - \cos x}{\sin2x} +c $

C

$\frac{\sin \ x}{\sin x + \cos x} +c $

D

$\frac{\sin \ x}{\sin x - \cos x} + c$

E

$\frac{\sin \ x - \cos \ x }{\sin x + \cos x} + c$

A

$\vec{r} ( 2 \hat{i} + \hat{j} - 2 \hat{k} ) = 15 $

B

$\vec{r} ( 2 \hat{i} + \hat{j} - \hat{k} ) = 15 $

C

$\vec{r} ( 2 \hat{i} + \hat{j} + 2 \hat{k} ) = 15 $

D

$\vec{r} ( \hat{i} + \hat{j} + 2 \hat{k} ) = 15 $

E

$\vec{r} ( 2 \hat{i} - \hat{j} + 2 \hat{k} ) = 15 $

A

$\sin \left( \frac{A + B}{2} \right)$

B

$2 \ \tan (A + B ) $

C

$\cot \left( \frac{A - B}{2} \right)$

D

$\tan \left( \frac{A - B}{2} \right)$

E

$2 \ \cot(A + B ) $

A

$x$

B

$ - 16 x $

C

$15 x $

D

$16 x $

E

$ - 15 x $

A

$\frac{2^n}{ n +1 } $

B

$\frac{2^n}{n} $

C

$\frac{2^{n - 1}}{n + 1 } $

D

$\frac{2^{n - 1}}{n } $

E

$\frac{2^{n + 1}}{n } $

A

$\frac{399}{2} $

B

$\frac{379}{12} $

C

$\frac{133}{2} $

D

$\frac{133}{4} $

E

$\frac{169}{2} $

A

$\frac{1}{6}$

B

$\frac{1}{12}$

C

$\frac{1}{9}$

D

$\frac{11}{12}$

E

$\frac{1}{11}$

A

1

B

0

C

(1-a) (1-b)(1- c)

D

a + b + c

E

2(a +b +c)

A

5x + y +12z -23 = 0

B

5x + 6y + 2z = 23

C

5x + 6y + 2z = 23

D

x + y + z =13

E

2x + 6y +5z = 7

A

50(507)

B

51(506)

C

50(506)

D

51(507)

E

52(506)

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