Ethiopian University Entrance Examination (EUEE) Mathematics
for Natural Science Ginbot 2011/June 2019
BOOKLET CODE: 17 SUBJECT CODE:02
Number Of Items:65 Time Allowed:3 HOURS
DIRECTIONS: For each of the following problems. Choose the best answer from
the given alternative and carefully blacken the letter of your best choice on the
separate answer sheet provided.
1. A man wants to fence two identical area of each enclosure that he can make with 192
mater fencing material if the side along the river does not need a fence?
What is the maximum area of each enclosure that he can make with 192 meter fencing material if
the side along the river does not need a fence?
(A) 1530 m2
(B) 1564 m2
(C) 1536 m2
(D) 1664 m2
Let x and y be the width and depth of rectangle. A inters of x and y becomes
A= xy
The total length of the fencing is 192 m
3y= 192
X= 96 โ3
2y
A (y) = (96- 3
2 ๐ฆ) ๐ฆ = 96โ
3
2๐ฆ2 = ; 0โค ๐ฆ โค 64
A (y)= 96y -3
2๐ฆ2 is contain [0,64] and diff. on (0,64)
Aโ(y)= 0 โน 96-3y=0
โนy=32m
Hence, y=32 is a critical number to get the max area at y=32
A(0)=0 A=64 and A (32)= 1536 ๐2
Ans. C
2. If ๐(๐ฅ)= ๐ฅ3 -2 ๐ฅ+1, then which one of the following is NOT true?
(A) ๐(๐)= 1
4 for some โ โ [โ1, 0].
(B) ๐(๐)= 1
2 for some โ โ [0, 1].
(C) ๐(๐)= 0 for some โ โ [โ2, 0].
(D) ๐(๐)= 3 for some โ โ [1, 2].
3. If ) ๐(๐ฅ) = In (2tan ๐ฅ), then what is the value of ๐โฒ(0)?
(A) 1
(B) -2ln 2
(C) ln 2
2
(D) ln2
4. If ๐(๐ฅ) = ๐ผ๐ฅ โ ๐ and ๐โ1 (๐ฅ + 1)=1
2๐ฅ+2, for each ๐ฅ ๐, then what must be the value of
๐ผ and ๐?
(A) ๐ผ = ยฝ, ๐= -2
(B) ๐ผ =2, ๐ =3
Let CโR such that
๐(๐ฅ) = ๐ฅ2 โ 2๐ฅ + 1
Ans. A is using intermediate value theorem
If C โ [โ1,0]
๐(โ1) = 2 ๐๐๐ ๐(0) = 1
Hence, 1
4 is not included b/n that values
๐(๐ฅ) = ln(2tan ๐ฅ)
๐(๐ฅ) = ๐(โ(๐ฅ))
Where, g(x) - ln ๐ฅ, ๐โฒ(๐ฅ) =1
๐ฅ
h(x) = 2tan ๐ฅ , hโฒ(x) = 2tan ๐ฅ , ๐ก๐ ๐ 2 ๐ ๐๐2๐ฅ
๐โฒ(๐ฅ) = 1
2tan ๐ฅ . ln 2 ๐ ๐๐2 ๐ฅ
๐โฒ(0) = 1
2tan 0. ln 2 ๐ ๐๐2 ๐ฅ
= ๐ฅ๐ง 2
Ans. D
(C) ๐ผ =1, ๐=1
(D) ๐ผ =2, ๐=2
(E)
5. If 2(2๐ฅ ๐ฅโ5 โ3
) -1 = (3 2
โ5 โ4), then what is the value of ๐ฅ?
(A) 2
(B) -2
(C) 1
2
(D) - 1
2
6. What is the area of the region bounded by the lines ๐ฅ = 0, ๐ฅ = 2 and ๐ฆ = 1 and the
curve ๐ฆ = ๐2๐ฅ?
(A) ๐2
2 -
1
2
(B) ๐4
2 -
5
2
(C) ๐3
3 -
1
3
(D) - ๐4
2 +
5
2
๐(๐ฅ) = ๐๐ฅ โ ๐
๐โ1(๐ฅ) =๐ฅ + ๐
๐
๐โ1(๐ฅ + 1) =๐ฅ + 1 + ๐
๐ , ๐๐ข๐ก ๐โ1(๐ฅ + 1) =
1
2๐ฅ + 2 =
๐ฅ + 4
2
๐ฅ + 1 + ๐
๐=
๐ฅ + 4
2
โด ๐ = 2 ๐๐๐ 1 + ๐ = 4 ๐ = 3
Ans. B
2(2๐ฅ ๐ฅ โ5 โ3
)โ1
= (3 2โ5 โ4
) , ๐ดโ1 =1
2 ( 3 2
โ5 โ4)
Let ๐ดโ1 =1
2( 1 2
โ5 โ4 ) ๐ดโ1 =
1
2 (3 2
โ5 โ4)
โ1
A=( 4 2โ5 โ3
) , 2x = 4
X= 2
Ans, A
7. If ๐ if the greatest integer function and g is the absolute value function, the what is the
value of (๐ ๐ g) (3
2) + (g ๐ ๐) (โ
4
3) ?
(A) 1
(B) 2
(C) -1
(D) 3
๐ = ๐๐๐
The area b/n the line x=0, x=2 and y=1 is
๐4
y = 1
y = 0 x = 2
A โซ ๐2๐ฅ2
0 ๐๐ฅ , let u = 2๐ฅ , ๐ฅ = 0, u =
๐ฅ = 2, ๐ข = 4
๐๐ = 2 ๐๐ฅ
1
2 ๐๐ = ๐๐ฅ
A1
2 โซ ๐44
0 ๐๐ฆ
= 1
2[๐4]4
=1
2 [๐4 โ ๐0] =
1
2 [๐4 โ 1]
๐4
2โ
1
2 Area of rectangle
๐4
2โ
1
2โ 2 =
๐4
2โ
5
2
Ans, B
8. Which one of following is true about the graph of ๐(๐ฅ)๐ฅ2+5๐ฅ+6
๐ฅ2โ4 +3?
(A) A horizontal asymptote of the graph of the graph is ๐ฆ =4.
(B) The vertical asymptotes of the graph are ๐ฅ = -2 and ๐ฅ =2.
(C) The graph has a hole at ๐ฅ =2.
(D) The graph has ๐ฆ- intercept at (0, โ3
2)
9. What is the solution of โซ 4๐ฅ (โ๐ ๐ฅ +1
๐ฅ2)d๐ฅ ?
(A) 4๐ฅ2(โ๐ ๐ฅ + 1)-2๐ฅ2+c
(B) ๐ฅ2(2โ๐ ๐ฅ โ 1)+4 ๐๐ ๐ฅ +c
(C) ๐ฅ2(2โ๐ ๐ฅ + 1)+4 ๐๐ ๐ฅ +c
(D) ๐ฅ2(4โ๐ ๐ฅ โ 1)+2 ๐๐ ๐ฅ +c
Give โf is the greatest integer function and g is the absolute value function , then (fog) (3/2)+ (gof)(-
4/3)= 1+2=3
(๐๐๐) (3/2) = ๐(๐ (3/2))
= ๐(3/2) = 1
(๐๐๐) (โ4/3) = ๐(๐(โ4/3))
= ๐(โ2)
= 2
Ans. D
๐(๐) ๐๐+๐๐+๐
๐๐โ๐ +3
=๐๐+๐๐+๐+๐(๐๐โ๐)
๐๐โ๐ =
๐
โ๐+ ๐
=๐๐+๐๐+๐+๐๐ ๐โ๐๐
๐๐โ๐ =
โ๐
๐+ ๐ โ
๐
๐
= ๐๐๐+๐๐โ๐
๐๐โ๐
Hence ๐ = ๐, โ ๐ = ๐ is a H.A
๐ฟ = ๐ is a V.A
Its y intercept is ( ๐,๐
๐)
Makes a hole at ๐ = โ๐
Ans. A
10. The earthโs orbit has a semi-major axis ๐ผ โ 149.6 Gm (gigameters) and an eccentricity
of ๐ โ 0.017. What is the approximate value of the semi-minor axis?
(A) 149.58 Gm
(B) 145.32 Gm
(C) 149.06 Gm
(D) 152.14 Gm
โซ 4๐ฅ (โ๐ ๐ฅ +1
๐ฅ2 ) ๐๐ฅ
= โซ 4๐ฅ โ๐ ๐ฅ +4
๐ฅ ๐๐ฅ
= โซ 4๐ฅ โ๐ ๐ฅ ๐๐ฅ + โซ4
๐ฅ ๐๐ฅ
= 4 [โซ ๐ฅ โ๐ ๐ฅ ๐๐ฅ + โซ1
๐ฅ ๐๐ฅ]
= 4 [๐ฅ2โ๐ ๐ฅ
2โ
๐ฅ2
4+ โ๐ ๐ฅ] + ๐
=2๐ฅ2โ๐ ๐ฅ โ ๐ฅ2 + 4 โ๐ ๐ฅ + ๐
= ๐ฅ2(2โ๐ ๐ฅ โ 1) + 4 โ๐ ๐ฅ + ๐
Ans. B
๐ โ 149 ๐บ๐
๐ โ 0.017
๐๐
๐
โ ๐ = ๐๐
= 0.017 x149.6 G.M
= 2.5432 G.M
From the relation
๐2 = ๐2+๐2
๐2 = ๐2-๐2
=(149.6)2 โ (2.5432)2
๐2 = 22,373.69 G.M2
๐ = 149.58๐
Ans. A
11. A water tank is a rectangular parallelepiped with has length 3 m, width 2 m and height 2.5
m. If water is flowing into the tank at the rate of 0.12 m3/sec, then how fast does the level of
water rise up in the tank?
(A) 0.04 m/sec
(B) 0.03m/sec
(C) 0.02 m/sec
(D) 0.06 m/sec
12. What is the sum of the infinite series โ 5โ๐=0
2๐ + 5๐
10 ๐
(A) 15
(B) 25
4
(C) 65
4
(D) 5
L= 3m.w = 2m h=25m rate 0.12m3 /sec
V= Ab h , Ab= Lw=6m2
๐๐ฃ
๐๐ก= (๐ด๐โ)
๐โ
๐๐ก ,
๐๐ฃ
๐๐ก= 0.12๐2/๐ ๐๐
๐๐ฃ
๐๐ก= ๐ด๐
๐โ
๐๐ก
โด๐โ
๐๐ก=
๐๐ฃ
๐๐ก ,
1
๐ด๐
=0.12๐3/๐ ๐๐
6๐2= 0.02๐/๐ ๐๐
Ans. C
Sum of โ 5 (2๐+5๐
10๐ )โ๐ฟ=0
โ 5 [โ2๐
10๐
โ
๐=0
+ โ5๐
10๐
โ
๐=0
]
โ 5 [5
4+ 2]
โ 5 [13
4]
= 65
4
Ans. C
13. The marks of 50 students is given below:
Marks 0-10 10-20 20-30 30-40 40-50
No. of students 5 8 ๐1 10 ๐2
If the median of the data is 26, what are the values of ๐1๐๐๐ ๐2 ?
(A) ๐1 = 20, ๐2 = 7
(B) ๐1 = 12, ๐2 = 15
(C) ๐1 = 15, ๐2 = 12
(D) ๐1 = 7, ๐2 = 20
14. There are three children in a room with, ages four, five, and six. If a five-year-old child enters
the room, then which of the following statement is correct?
(A) Mean age will stay the same but the standard deviation will increase.
(B) Mean age will stay the same but the standard deviation will decrease.
(C) Mean age and standard deviation will increase.
(D) Mean age and standard deviation will stay the same.
Data, 4, 5, 6, ๏ฟฝฬ ๏ฟฝ = 5
New data, 4, 5, 5, 6, ๏ฟฝฬ ๏ฟฝ = 5 mean stays
S.D (data) =2
3= 0.67
S.D (new data)= 2
4= 0.5 It decreases
Ans. B
Given ๐ = 50, ๐ฟ๐ถ๐ต = 19.5 ๐ถ + ๐ = 11
๐๐ = ๐1 , ๐๐๐๐๐๐ = 26
Median = LCB + (๐
2 โ๐ถ๐๐
๐๐) ๐
26 = 19.5 + (25 โ 13
๐1) 11
6.5 =12.11
๐1
๐1 =132
6.5= 20 ๐2 = 50 โ 5 + 8 + 20 + 10 = 7
Ans.A
15. For real numbers x and y, which one of the following is NOT true?
(A) (โ๐ฅ)(โ๐ฆ) (๐ฆ2 + ๐ฅ2 โฅ โ1)
(B) (โ๐ฅ)(โ๐ฆ) (๐ฆ โฅ ๐ฅ2 + 1)
(C) (โ๐ฅ)(โ๐ฆ) (๐ฆ โฅ ๐ฅ2 + 1)
(D) (โ๐ฅ)(โ๐ฆ) (๐ฆ โฅ ๐ฅ2 + 1)
16. If the second and fifth terms of geometric progression are - 1
2 and
1
16 ,
(A) 128
85
(B) 255
256
(C) 85
128
(D) 256
255
17. Which one of the following is a valid argument?
(A) If I donโt change my oil regularly, my engine will die. My engine died, Thus, I didnโt
change my oil regularly.
(B) If you do very problem in the book, then you will learn the subject. You learned the
subject. Thus , you did every problem in the book,
(C) If I am literate, then I can read and write. I can read but I canโt write. Thus I am not
literate.
(D) If it rains or snows, then my roof leaks. My roof is leaking. Thus, it is raining and
snowing.
Ans. B
๐ฎ๐ = โ๐๐โ , ๐ฎ๐ = ๐
๐๐โ
๐ฎ๐
๐ฎ๐
=๐๐ฎ๐
๐๐ ๐ฎ๐
, ๐๐ =โ๐
๐โ๐
๐๐โโ =
โ๐
๐
โด ๐ฎ๐ = ๐ ๐ =โ๐
๐
๐บ๐ =๐ฎ๐(๐โ๐๐)
๐ โ ๐= ๐๐
๐๐๐โ
Ans. C
18. What is the value of โซ๐ฅ๐โ1 + ๐๐ฅโ1
๐ฅ๐ + ๐๐ฅ d๐ฅ?
(A) 1
๐ ln(๐ฅ๐ + ๐๐ฅ+1)+ c
(B) 1
๐ ln(๐ฅ๐+1 + ๐๐ฅ+1)+ c
(C) 1
๐ ln((๐ฅ + 1) ๐ + ๐๐ฅ+1)+ c
(D) 1
๐ ln(๐ฅ ๐ + ๐๐ฅ)+ c
(E)
19. Suppose 2,500 items are produced by a machine and 2 % of the product are randomly
selected and tested. If 5 of the tested items have a defect, then what is the probability that an
item produced by the machine No defect?
(A) 0.80
(B) 0.90
(C) 0.85
(D) 095
(E)
AnS. C โ ๐ โ ๐โ๐ P: I am L
๐โงยฌ๐
ยฌ๐ q: I can read
r- I can write
โซ (๐ฅ๐โ1 + ๐๐ฅโ1
๐ฅ๐ + ๐๐ฅ ) ๐๐ฅ
(๐ฅ๐ + ๐๐ฅ )1 = ๐๐ฅ๐โ1 + ๐๐ฅ
Multiply both sides by 1 ๐โ ๐ค๐ ๐๐๐ก
(๐ฅ๐โ1 + ๐๐ฅโ1) 1๐โ
โซ๐ฅ๐โ1 + ๐๐ฅโ1
๐ฅ๐ + ๐๐ฅ๐๐ฅ =
1
๐โ๐ |๐ฅ๐ + ๐๐ฅ| + ๐
Ans. D
Total item 2500
Test 2% of 2500=50
Defect=5
Non defect=45
P(E)= 4550
=0.9
Ans. B
20. If ๐(๐ฅ) = 1
๐ฅ, then what is the value of ๐(๐)(๐ฅ)?
(A) ๐(๐)(๐ฅ)= (โ1) ๐ ๐!
๐ฅ๐
(B) ๐(๐)(๐ฅ)= (โ1) ๐+1 ( ๐+1)!
๐ฅ๐+1
(C) ๐(๐)(๐ฅ)= (โ1) ๐ ๐!
๐ฅ๐+1
(D) ๐(๐)(๐ฅ)= (โ1) ๐ ๐!
๐ฅ๐โ1
21. If the region enclosed by the graphs of ๐(๐ฅ)= ๐ฅ2 and ๐(๐ฅ)= ๐ฅ3 from ๐ฅ =0 to ๐ฅ =1 rotates
about the ๐ฅ- axis, what is the volume of the solid of revolution?
(A) 2๐
35 cubic units
(B) 2๐
25 cubic units
(C) 2๐
5 cubic units
(D) 2๐
27 cubic units
(E)
(F)
๐(๐ฅ) =1
๐ฅ , ๐โฒ(๐ฅ) =
โ1
๐ฅ2 , ๐โฒโฒ(๐ฅ) =
2
๐ฅ3
๐โฒโฒโฒ(๐ฅ) =โ6
๐ฅ4
โด ๐๐(๐ฅ) =(โ1)๐. ๐:
๐ฅ๐+1
Ans. C
๐ = ๐ โซ (๐ฅ2)21
0
โ (๐ฅ3)2๐๐ฅ
= ๐ โซ ๐ฅ4 โ ๐ฅ6๐๐ฅ1
0
= ๐ [๐ฅ5
5โ
๐ฅ7
7]
= ๐ [1
5โ
1
7]
=2๐
35
Ans. A
22. What is the limit of sequence 1, 2
22โ12 ,
3
32โ22 ,
4
42โ32 ,โฆ?
(A) 0
(B) 1
2
(C) 1
(D) 3
2
(E)
23. At the value of x does the function ๐(๐ฅ) = 4๐ฅ2
3 - ๐ฅ4 attains maximum value?
(A) -1
(B) 0
(C) 4
3
(D) 1
2
22 โ 12 ,
3
32 โ 22 ,
4
42 โ 32 , โฆ
โ๐
๐2 โ (๐ โ 1)2
โ๐
๐2 โ (๐2 โ 2๐ + 1 =
๐
๐2 โ ๐2 + 2๐ โ 1
โ๐
2๐ โ 1 =
1
2 โ1๐
, ๐ โ โ
lim๐โโ
1
2 โ1๐
=1
2
Ans.B
๐(๐ฅ) =4๐ฅ3
3โ ๐ฅ4
๐โฒ(๐ฅ) = 4๐ฅ2 โ 4๐ฅ3
= 4๐ฅ2(1 โ ๐ฅ)
๐โฒ(๐ฅ) = 0, ๐ฅ = 0,1 ๐๐๐ ๐๐๐๐ก๐๐๐๐ ๐๐๐๐๐ก
0 1
4๐ฅ2 โ โ โ โ โ โ + + + + + + + + +
1 โ ๐ฅ + + + + + โ โ โ โ โ โ โ
(4๐ฅ)2(1 โ ๐ฅ) โ โ โ โ0 + +0 โ โ โ โ โ โ โ
๐โฒ(๐ฅ)๐โ๐๐๐๐๐ ๐ ๐๐๐ ๐๐๐๐ โ ๐ฃ๐ ๐ก๐ + ๐ฃ๐ โ ๐๐ก ๐ฅ = 0
๐ โ๐๐ ๐ ๐๐๐๐๐ ๐๐๐๐๐๐ข๐
๐โฒ(๐ฅ)๐โ๐๐๐๐๐ ๐ ๐๐๐ ๐๐๐๐ + ๐ฃ๐ ๐ก๐ โ ๐ฃ๐ ๐๐ก ๐ฅ = 1
๐ โ๐๐ ๐๐๐๐๐๐ ๐๐๐ฅ๐๐๐ข๐
Ans. D
24. If โ(๐ฅ) =โ1 + โ๐ฅ , then which of the following is equal to โโฒ(๐ฅ)?
(A) 1
4โ๐ฅ+๐ฅโ๐ฅ
(B) 1
2โ1+โ๐ฅ
(C) ๐ฅ
2โ1+โ๐ฅ
(D) ๐ฅ
4โ๐ฅ+๐ฅโ๐ฅ
25. Fatuma can solve 90% of the problems given in a book and Mesfin can solve 70%. What is the
probability that at least one of them will solve the problem, selected at random from the book?
(A) 0.77
(B) 0.97
(C) 0.87
(D) 0.67
Fatuma answers 90% โ ๐(๐น) = 0.9
Mesfin answers 70%โ ๐(๐) = 0.7
โด ๐(๐น๐๐) = ๐(๐น) + ๐(๐) โ ๐(๐น๐๐)
= ๐(๐น) + ๐(๐) โ [๐(๐น) โ ๐(๐)]
= 0.9 + 0.7 โ [0.9 โ 0.7]
= 1.6 โ 0.63
= 0.97
Ans. B
โ(๐ฅ) = โ1 + โ๐ฅ
Let โ(๐ฅ) = ๐(๐(๐ฅ))๐คโ๐๐๐
๐(๐ฅ) = โ๐ฅ , ๐โฒ(๐ฅ) = ๐โฒ(๐ฅ) =1
2(๐ฅ
12โ ) =
1
2โ๐ฅ
๐(๐ฅ) = 1 + โ๐ฅ =1
2โ๐ฅ
โโฒ(๐ฅ) = ๐โฒ(๐(๐ฅ)), ๐(๐ฅ) =1
2โ1 + โ๐ฅ ,
1
2โ๐ฅ
=1
4โ๐ฅ(1 + โ๐ฅ
= 1
4โx+xโx
Ans. A
26. What is the polar form of 7โ๐
3โ4๐ ?
(A) โ2(๐๐๐ ๐
2+ ๐ ๐ ๐๐
๐
2)
(B) 2(๐๐๐ ๐
4+ ๐ ๐ ๐๐
๐
4)
(C) 2(๐๐๐ ๐
2+ ๐ ๐ ๐๐
๐
2)
(D) โ2(๐๐๐ ๐
4+ ๐ ๐ ๐๐
๐
4)
27. ๐ผ๐ ยฌ๐ โ ๐ is False and p q is True , then which of the following is True?
(A) ๐ v (ยฌq ^r)
(B) ยฌ ๐ โ (q v r)
(C) ยฌ ๐ ^(q โ r)
(D) P (ยฌq v r)
Polar form of
7 โ ๐
3 โ 4๐ ,
3 + 4๐
3 + 4๐=
7(3 + 4๐) โ ๐(3 + 4๐)
3(3 + 4๐) โ 4๐(3 + 4๐)
=21 + 28๐ โ 3๐ + 4
9 + 12๐ โ 12๐ + 16
=25 + 25๐
25
= 1 + ๐ โ ๐ฅ = 1, ๐ฆ = 1
๐ก๐๐ ๐ =๐ฆ
๐ฅ= 01, ๐ = ๐
4โ , ๐ = โ๐ฅ2 + ๐ฆ2 = โ2 =
โด ๐(cos ๐ + ๐ ๐ ๐๐ ๐)
โ2 (๐๐๐ ๐4โ + ๐ ๐ ๐๐ ๐
4โ )
Ans. D
ยฌpโ ๐ ๐๐ ๐๐๐๐ ๐
ยฌ๐ is T and F is false
๐ ๐๐ ๐น
๐ โ ๐ ๐๐ ๐๐๐ข๐
โด ๐ ๐๐ ๐น๐๐๐ ๐
Truth value of p, q and r respectively
F,F,F
Ans.C ยฌ ๐ โง (๐ โ ๐)
T โง (๐น โ ๐น)
T โง ๐ = ๐
28. Which one of the following is the set of critical numbers of ๐(๐ฅ) =3
8 ๐ฅ8/3- 6๐ฅ2/3?
(A) {โ2, 2}
(B) {โ1, 1}
(C) {โ2, 0, 2}
(D) {โ1, 0, 1}
29. The center of a circle on the line y= 2x and the line x=1 is tangent to the circle at (1,6). How
long is the radius of the circle?
(A) 5
(B) 4
(C) 2
(D) 3
30. If ๐1 =2, ๐2=6, ๐3=10, ๐4= 14,โฆ. then ๐=1
100๐๐ =
(A) 20,020
(B) 20,000
(C) 20,200
(D) 22,200
๐(๐ฅ) =๐ฅ
8 ๐ฅ
83โ โ 6๐ฅ
23โ
๐โฒ(๐ฅ)3
8,8
3๐ฅ
53โ โ 6.
2
3 ๐ฅ
โ13โ
=๐ฅ5/3 โ4
๐ฅ1
3โ
๐โฒ(๐ฅ) =๐ฅ
53
+13โ โ 4
๐ฅ1
3โ=
๐ฅ2 โ 4
๐ฅ1
3โ
๐โฒ(๐ฅ) = 0, ๐ฅ2 โ 4 = 0, ๐ฅ = ยฑ 2, 0
Ans.C
Tangent at (1,6) and center at (h, k) here K=6, and y= 2x is on the center .
Hence, 2h=6โ โ = 3
Distance b/n (1, 6) to (3,6) is r
๐ = โ(1 โ 3)2 + (6 โ 6)2 = 42
Ans. C
31. Let ๐(๐ฅ) = 2๐๐ฅ-๐ sin ๐ฅ+1. If the equation of the tangent line to the graph of ๐ at (0, 3) is ๐ฆ=
5x+3, then what is the value of ๐?
(A) -3
(B) 3
(C) -5
(D) 2
32. Among 2000 students who took a regional exam, the percentile of certain studentโs score is 90.
Which of the following is correct about the studentโs score?
(A) The students have answered 90% of the questions correctly.
(B) The score of the student is as good as that of 90% of the students.
(C) The studentโs score is the same as the top 10% of the score.
(D) The studentโs score is greater than or equal to that of 1800 students.
๐1=2 ๐2 = 6, ๐3=10 โด ๐ = ๐3โ ๐2= ๐2โ ๐1= 4
Sn=โ ๐ด๐๐๐=0 =
๐
2(2๐ด1 + (๐ โ 1)๐)
โ ๐ด๐= 100
2 (2.2 + (100 โ 1)4)
100
๐=1
= 50(4 + 99.4)
= 50(4 + 100) = 20,000
Ans. B
๐(๐ฅ) = 2๐๐ฅ โ k sin x + 1
Tangent at (0,3) the line is y=5x+3
๐โฒ(๐ฅ) = 2๐๐ฅ โ ๐ cos ๐ฅ
๐โฒ(0) = ๐ ๐๐๐๐ ๐๐ ๐ก๐๐๐๐๐๐ก ๐๐๐๐
โ 5 = 2 โ ๐ โ 5 โ 2 = โ๐ โ ๐ = โ3
Ans. D
Ans. D
33. What is the value of lim๐โโ
(๐ฅ3โ8
๐ฅ2โ4)?
(A) 3
(B) 2
(C) -2
(D) The limit does not exist.
34. What is the greatest lower bound of sequence {(โ1)๐ 1
๐+1}
๐=0
โ?
(A) -1
(B) 0
(C) -1/2
(D) ยฝ
35. What is the value of lim๐โโ
(3๐+ 2โฒโฒ
6โฒโฒ )?
(A) 1
(B) โ
(C) 5
6
(D) 0
๐๐๐๐โ๐ ๐๐ โ ๐
๐๐โ๐= ๐๐๐๐โ๐
(๐ โ ๐)(๐๐ + ๐๐ + ๐
(๐ โ ๐)(๐ + ๐)
=4+4+4
4=
12
4= 3
Ans. A
Ans. B
lim๐โโ
(3๐ + 2๐
6๐)
= lim๐โโ
3๐
6๐+ lim
๐โโ
2๐
6๐
= lim๐โโ
(1
2)
๐+ lim
๐โโ (
1
3)
๐
= lim ๐โโ
1
2๐+ lim
๐โโ
1
3๐
= 0+0
= 0
Ans. D
36. What is the value of limโโ0
8 (1
2+ โ) โ 8 (
1
2)
2
?
โ
(A) 8
(B) 4
(C) 0
(D) The limit does not exist
37. Which one of the following is the multiplicative inverse of ๐ง = 3+4๐
4โ5๐
(A) 8
25โ
31
25๐
(B) 8
25โ
25
31๐
(C) 8
25+
25
31๐
(D) 8
25+
31
25๐
38. If A is a 3x3 matrix and det (A)= 5, then how much is det (2A A) ?
(A) 20
(B) 100
(C) 50
(D) 200
8 (
12 + h)
2
+ 8 (12)
2
h
lim โโ0
8h2 + 8h + 2 โ 2
h
lim 8โโ0
โ + 8 = 8
Ans. A
๐ง =3 + 4๐
4 โ 5๐
โด 1
๐ง=
13 + 4๐4 โ 5๐
= 4 โ 5๐
3 + 4๐ ,
3 โ 4๐
3 โ 4๐=
โ8 โ 3 + ๐
2.5
Ans, B
39. Which one of the following is true about the pair of line: 3x +9y-24=0 and 4x+12y+32=0 ?
(A) parallel and distinct line
(B) intersecting line
(C) perpendicular line
(D) representing the same lines
40. If F(x)=โซ ๐โ1๐ฅ
0 dt , then what is the value of Fโ (x) ?
(A) โ๐โ1-1
(B) ๐โ๐ฅ
(C) ๐โ๐ฅ+1
โ๐ฅ+1
(D) โ๐โ๐ฅ
โ1 โถ ๐ฆ = โ13โ ๐ฅ + 8
3โ
โ2 โถ ๐ฆ = โ13โ ๐ฅ โ 8
3โ
Having the same slope
Ans. A
det (A) =5
det(2 ๐ด ๐๐ด)
Since T+ is 3x3 max
det (2 ๐ด ๐๐ด) = 23๐ฅ5๐ฅ5 = 200
Ans.D
๐น(๐ฅ) = โซ ๐โ๐ก
๐ฅ
0
๐๐ก
๐นโฒ(๐ฅ) = โ[๐โ๐ก]๐ฅ
= ๐น(๐ฅ) โ ๐น(0)
= โ๐โ๐ฅ + 1, ๐กโ๐๐ ๐นโฒ(๐ฅ) = (โ๐โ๐ฅ + 1)1 = ๐โ๐ฅ
Ans.B
41. If there are two children in a family, what is the probability that there is at least one girl in
the family?
(A) 1
2
(B) 1
4
(C) 2
3
(D) 3
4
42. Every day a person saves 5 cents more than the amount he saved on the previous day. His
target is to save the total amount of 3225 cents by the end of 30 days. How much must be
the starting saving to meet the target?
(A) 50 cents
(B) 25 cents
(C) 35 cents
(D) 60 cents
43. What is the value of โซ ln ๐ฅ
2 1
๐ฅ2 dx ?
(A) - ๐๐ 2
2 +
1
2
(๐ต) ๐๐ 2
2 -
1
2
A. - ๐๐ 2
2 -
1
2
(๐ท) ๐๐ 2
2 +
1
2
๐โ๐๐ ๐ ๐๐ฅ๐๐ ๐ก๐ 2 ๐โ๐๐๐๐๐๐, ๐ ๐ก๐๐๐ก <๐บ<๐บ
๐ต๐<๐บ
๐ต = BB, BG,GB,GG
๐(๐บ)=3
4
Ans. D
๐30 = 3225 ๐ = 5 , ๐ = 30 , ๐ด1 =?
๐๐ =๐
2(2๐ด1 + (๐ โ 1)๐) โ ๐30 =
30
2(2๐ด1 + (30 โ 1)5)
3225 = 15(๐ด1 + 29๐ฅ5)
215 = 2๐ด1 + 145
70 = 2๐ด1 โ ๐ด1 = 35 ๐๐๐๐ก๐
Ans. C
44. If the circle passing through the point (-1, 0) touches the y-axis at (0,2), then what is the
equation of the circle ?
(A) ๐ฅ2 + ๐ฆ2 + 5๐ฅ + 4๐ฆ + 4 = 0
(B) ๐ฅ2 + ๐ฆ2 โ 5๐ฅ + 4๐ฆ + 4 = 0
(C) ๐ฅ2 + ๐ฆ2 + 5๐ฅ โ 4๐ฆ + 4 = 0
(D) ๐ฅ2 + ๐ฆ2 โ 5๐ฅ โ 4๐ฆ + 4 = 0
45. Let ศค be a complex number and ๐ค = 3 + 4๐ . if ๐ง2+1
๐ง+๐ = |๐ค|ศค โ
1
๐ ๏ฟฝฬ ๏ฟฝ , then what is the value of ศค ?
(A) -4- 2i
(B) โ4 โ 2๐
(C) โ1 + ๐
(D) โ2๐
โซโ๐ ๐ฅ
๐ฅ2
2
1
๐๐ฅ
Let u=โ๐ ๐ฅ ๐๐ฃ =1
๐ฅ2 ๐๐ฅ
๐๐ =1
2 ๐๐ฅ ๐ =
โ1
๐ฅ
โซโ๐ ๐ฅ
๐ฅ2
2
1
= [โโ๐ ๐ฅ
๐ฅโ
1
2]
1
2
=โโ๐ 2
2+
1
2
Ans. A
let (h, k) be the center and touches at (0,12) hence (h,2) be the centers at p(-1, 0)
(๐ฅ โ โ)2 + (๐ฆ โ ๐)2 = ๐2
(โ1 โ โ)2 + (โ2)2 = (โ โ 0)2 + (2 โ 2)2
โ2 โ 2โ + 1 + 4 = โ2
โ2โ = โ5
โ = 52โ
โด (๐ฅ โ 52โ )2 + (๐ฆ โ 2)2 = โ2 =
25
4
โด ๐ฅ2 + ๐ฆ2 + 5๐ฅ โ 4๐ฆ + 4 = 0
Ans. C
46. What is the value of the left-hand side limit, lim๐ฅโ0โ
sin ๐ฅ
๐ฅ2 โ4๐ฅ2 + 9๐ฅ2 ?
(A) Does not exist
(B) 3
(C) 2
(D) -3
47. If โ1 1 23 2 ๐ฅ2 4 1
= โ๐ฅ 3 22 2 31 โ1 โ1
, then what is the value of x
(๐ด) โ2
5
(๐ต) 2
5
(๐ถ) 2
3
(๐ท) โ2
3
lim๐ฅโ0ฬ
sin ๐ฅ
๐ฅ2 โ4๐ฅ3 + 9๐ฅ2
โ lim๐ฅโ0ฬ
sin ๐ฅ
๐ฅ โ
4๐ฅ3 + 9๐ฅ2
๐ฅ2
โ lim๐ฅโ0ฬ
sin ๐ฅ
๐ฅ , lim
๐ฅโ0ฬ โ4๐ฅ + 9
1,-3
=-3
Ans. D
๐ง2 + 1 = (๐ง + ๐)(๐ง โ ๐)
[๐ค] = โ32 + 42 = 5
๐ค ฬ ฬ ฬ = 3 โ 4๐
โด๐ง2 + 1
๐ง + ๐= |๐ค|๐ง โ
1
๐ ๏ฟฝฬ ๏ฟฝ
(๐ง + ๐)(๐ง โ ๐)
2 + ๐= 5๐ง + ๐(3 โ 4๐)
๐ง โ ๐ = 5๐ง + 3๐ + 4
๐ง โ 5๐ง = 3๐ + ๐ + 4
โ4๐ง = 4๐ + 4
๐ง =4๐ + 4
โ4
๐ง = โ๐ โ 1
Ans. B
48. Consider the following system of equations: {
๐ฅ โ 2๐ฆ + ๐ง = 1โ๐ฅ + ๐ฆ + ๐ง = 33๐ฅ โ 5๐ฆ + ๐ง = ๐
How much must be the value of k so that the system has a solution?
(A) 7
(B) 1
(C) -1
(D) 0
+ - + + - +
๐ด = |โ1 1 23 2 ๐ฅ2 4 1
| = ๐ต |โ๐ฅ 3 22 2 31 โ1 โ2
|
โ โ1 |2 ๐ฅ4 1
| โ |3 ๐ฅ2 1
| + 2 |3 32 4
| = โ1(2 โ 4๐ฅ) โ 1(3 โ 2๐ฅ) + 2(12 โ 4)
โ โ2 + 4๐ฅ โ 3 + 2๐ฅ + 16
โ 6๐ฅ + 11
โ โ๐ฅ | 2 3
โ1 โ 2| โ 3 |
2 31 โ 2
| + 2 |2 2
1 โ 1| = โ๐ฅ(โ4 + 3) โ 3(โ4 โ 3) + 2(โ2 โ 2)
โ โ๐ฅ (โ1) โ 3(โ7) + 2(โ4)
โ ๐ฅ + 21 โ 8
โ ๐ฅ โ 13
โด 6๐ฅ + 11 = ๐ฅ + 13
6๐ฅ โ ๐ฅ = 13 โ 11
5๐ฅ = 2
๐ฅ = 25โ
Ans. B
Using Augment matrix and by applying elementary raw operations to write in row echelon from then
solve for k. so that k = -1 .
Ans. C
49. What is the simplified form of ๐โ1 ๐โ1
๐โ3โ ๐โ3 ?
(๐ด) ๐2๐2
๐2 โ ๐2
(๐ต) ๐3 โ ๐3
๐ โ ๐
(๐ถ) ๐3 โ ๐3
๐๐
(๐ท) ๐2 ๐2
๐3 โ ๐3
50. If ๐(๐ฅ) = ๐๐ฅ3 +๐
๐ฅ +5 has local minimum at (2, -3) , what are the values of a and b?
(A) ๐ = +1
4 , b=12
(B) ๐ =1
4 , b=12
(C) ๐ = โ1
4 , b=-12
(D) ๐ =1
4 , b=-12
๐(๐ฅ) = ๐๐ฅ3 +๐
๐ฅ+ 5 โ๐๐ ๐๐๐๐๐๐ข๐ ๐๐ก (2, โ3)
๐(2) = โ3
16๐ + ๐ = โ16 โฆ โฆ (๐ฅ)
๐๐๐๐๐ โ 2 ๐๐ ๐ ๐๐๐๐ก๐๐๐๐ ๐๐ข๐๐๐๐
๐โฒ(๐ฅ) = 0, ๐ฅ = 2
๐โฒ(๐ฅ) = 3๐๐ฅ2 โ๐
๐ฅ2
๐โฒ(2) = 0
48๐ โ ๐ = 0 โ โ โ (๐ฅ๐ฅ)
๐๐๐๐ (โ)๐๐๐ (โโ)๐ค๐ ๐๐๐ก
๐ =โ1
4 ๐๐๐ ๐ = โ12
Ans. C
๐โ1 ๐โ1
๐โ3 โ ๐โ3=
1๐๐
1๐3 โ
1๐3
=๐2๐2
๐3 โ ๐3
Ans. D
51. Which one of the following is equal to (๐ฅ) = โ(๐ฅ + 4)2 , for every ๐ฅ ๐ ?
(A) ๐(๐ฅ) = |๐ฅ + 4|
(๐) ๐(๐ฅ) = ๐ฅ + 2
(๐ถ) ๐ (๐ฅ) = ๐ฅ + 4
(๐ท) ๐(๐ฅ) = |๐ฅ| +4
52. Ship A and B depart from the point at the same time on the course N600 E and N400 E,
respectively. If the speed of ship A is 20 km per hour and the speed of ship B is 30 km per hour,
what is the distance between the two ships just after 30 minutes of their departure?
[๐ฆ๐๐ข ๐๐๐ฆ ๐ก๐๐๐: cos(400) = 0.77, cos(200) = 0.94, sin(200) = 0.34 ]
(A) โ43 km
(B) โ40 km
(C) โ50 km
(D) โ53 km
53. Which the plane is rotated 450 about the point (1,-2), then what would be the image of the point
(2,4)?
(A) (1 โ โ2
2, โ2 +
โ2
2)
(B) (1 โ โ 2
5
2, โ2 +
โ27
2)
๐(๐ฅ)โ(๐ฅ + 4)2, ๐ ๐๐๐๐ ๐กโ๐ ๐๐๐๐๐ฅ ๐๐ ๐๐ฃ๐๐
๐(๐ฅ) = |๐ฅ + 4|
Ans. A
Speed of ship A= 20 km/hr
Speed of shiep B= 30 km/hr
Both goes with in 30 min=0.5hr
The distance covered by sheep A= 10 km and, B= 15km using the relation S= Vt and angle b/n two
ships is ๐ = 200 using cosine low the distance b/n them is ๐2 = 102 + 152 โ 2 โ 10 โ 15 cos 20
๐ = โ43 ๐๐
Ans. A
(C) (โ 2
5
2,
โ27
2)
(D) (โ2
2,
โ2
2)
54. If u= 2 ๐โ โ ๐โ and v=๐โ โ 8๐โ + 3๐โ, then what is the unit vector in the
direction of 5u+v?
(A) ๐โ + 2๐โ โ 2๐โ
(B) 2
3๐โ +
1
3๐โ โ
2
3๐โ
(C) 1
3๐โ +
2
3๐โ +
2
3๐โ
(D) 1
3๐โ +
2
3๐โ โ
2
3๐โ
๐๐ข๐๐ ๐ก๐๐ก๐ข๐ก๐ (๐ฅ0, ๐ฆ0) = (1, โ2), ๐ = 450 ๐๐๐ (๐ฅ, ๐ฆ) = (2,4)
๐ฅ1 = ๐ฅ0 + (๐ฅ โ ๐ฅ0)๐๐๐ ๐ โ (๐ฆ โ ๐ฆ0) ๐ ๐๐ ๐
= 1 + (2, โ1) ๐๐๐ 45โ (4 โ (โ2)) ๐ ๐๐ 450
= 1 +โ2
2โ
6โ2
2= 1 โ
5โ2
2
๐ฆโฒ = ๐ฆ00 + (๐ฅ โ ๐ฅ0) ๐ ๐๐ ๐ + (๐ฆ โ ๐ฆ0) ๐๐๐ ๐
= โ2 +โ2
2+
6โ2
2= โ2 +
7โ2
2
Ans. B
๐ข = 2๐ โ ๐ ๐๐๐ ๐ฃ = ๐ โ 8๐ + 3๐
5๐ข = 10๐ โ 5๐
5๐ข + ๐ฃ = ๐ + 2๐ โ 2๐
|5๐ข + ๐ฃ| = โ12 + 22 + (22) = โ9 =3
The unit rector in the direction of 5u+v is
=5u+v
|5u+v|
1๐
3+
2
3๐ โ
2
3๐
Ans. D
55. The diagram below is a representation of a 25m vertical observation tower TB and two cars K
and L on a road. The angle of depression from T to car L is300. The angle of elevation from car
K to the of the tower is 600. B, K and L lie in a straight line lie on the same horizontal plane as
the base of the tower.
T
300
600
B K L
What is the distance between the two cars?
(A) 50โ3
2 m
(B) 50โ3 m
(C) 50โ3
3 m
(D) 50+โ3 m
25 m
30
25 B=60
๐ = 60
x y
tan ๐ = 25
๐ฅ โ ๐ฅ =
25
tan 60=
25
โ3=
25โ3
3
tan ๐ = ๐ฅ + ๐ฆ
25 โ ๐ฅ + ๐ฆ = 25 tan ๐ต = 25โ3
Now we need to calculate y to determine the distance b/n the two ships
๐ฆ = 25โ3 โ ๐ฅ = 25โ3 โ25โ3
3
๐ฆ = 50โ3
3m
Ans. C
56. Which of the following is a correct assertion that can be proved by the principle of
mathematical induction?
(A) 1
๐+1 โค 1, for each real number ๐ โฅ 1.
(B) ๐! โฅ 2๐, for each integer ๐ โฅ 4.
(C) 2๐ โ 1, is prime for each prime integer ๐.
(D) ๐! โค 4๐, for each positive integer ๐.
57. If the between the vectors ๐ดโ=(2,-1,1) and ๐ตโ=(1,1, ๐) is ๐
3, then what is the value of ๐ ?
(A) 2
(B) -1
(C) 1
(D) -2
Ans. B
๐ด(2, โ1,1) ๐๐๐ ๏ฟฝโโ๏ฟฝ = (1,1, ๐), ๐ =๐
3
๐ด. ๐ต = |๐ด| |๐ต| cos ๐
(2)(1) + (โ1)(1) + (1)(๐) = โ22 + 12 + 12, โ12 + 12 + ๐2 cos 600
1 + ๐ = โ6 , โ2 + ๐2.1
2
2 + 2๐ = โ12 + 6๐2
(2 + 2๐)2 = 12 + 6๐2
4๐2 + 8๐ + 4 = 12 + 6๐2
โ2๐2 + 8๐ โ 8 = 0
๐2 โ 4๐ + 4 = 0
(๐ โ 2)2 = 0
๐ โ 2 = 0
๐ = 2
Ans. A
58. Let L be the line by the vector equation (๐ฅ, ๐ฆ) = (1,1) + ๐ก(โ3.1 ),๐ก๐.. What is the equation of
the image of L after being rotated 150 about (1,1) and then translated by vector u= (โ1,1)
(A) ๐ฅ โ ๐ฆ = 2
(B) โ ๐ฅ + ๐ฆ = 2
(C) โ3 ๐ฅ โ ๐ฆ = 2
(D) โ+โ3 y=1
59. What is the solution set of ๐ ๐๐2 ๐ฅ โ ๐ ๐๐ ๐ฅ ๐๐๐ ๐ฅ = 0 over [0,2๐]?
(A) {0, ๐,5๐
4, 2๐}
(B) {0,๐
4, ๐, ,2๐}
(C) {0,๐
4, ๐,
5๐
4}
(D) {0,๐
4, ๐,
5๐
4, 2๐}
L:(๐ฅ, ๐ฆ) = (1,1) + ๐ก(โ3 ,1) , ๐ก๐ธ๐
๐ฅ = 1 + โ3 ๐ก , ๐ฆ = 1 + ๐ก
โ๐ฅ โ 1
โ3= ๐ฆ โ 1
๐ฅ โ 1 = โ3 ๐ฆ โ โ3
๐ฆ =1
โ3 ๐ฅ +
โ3 โ 1
โ3
Here, slope m=1
โ3 , tan ๐ =
1
โ3
๐ = 300
(1,1) ๐๐(โ1,1), (0,2)
Slope of image of โ = 450 , tan 450 = 1
โด ๐๐๐๐๐๐ ๐๐๐๐๐ก (0,2)๐๐๐ ๐ ๐๐๐๐ ๐ = 1
๐ฆ โ 2 = 1(๐ฅ โ 0)
๐ฆ โ 2 = ๐ฅ
๐ฆ = ๐ฅ + 2 โ ๐ฆ โ ๐ฅ = 2
Ans. B
60. Consider a rectangle ABCD with base vertices A= (0,3) and B= (4,0) and the other vertices, C
and D, in the first-quadrant of the coordinate plane. If its height BC is half of the length of its
base, then which of the following indicates the coordinates of vertex C?
(A) (4,5/2)
(B) (11/2,2)
(C) (5/2,2)
(D) (6,3/2)
61. โ๐ โN, 3๐ โ 2 prime that can be proved or disproved by which one of the following
mathematical proofs?
(A) Disprove by counter example
(B) Proof by exhaustion
(C) Direct proof
(D) Proof by contradiction
๐ด๐ต๐ถ๐ท ๐๐ ๐๐๐๐ก๐๐๐๐๐
๐ด๐ต = โ32 + 42 = 5
๐ต๐ถ =1
2 ๐ด๐ต =
5
2
(๐ฅ โ 4)2 + ๐ฆ2 =25
4โ ๐๐ ๐ ๐๐๐๐๐๐ ๐ค๐๐กโ ๐๐๐๐ก๐๐ ๐(4,0)๐๐๐ ๐ =
5
2
โด ๐ ๐๐ ๐ ๐๐๐๐๐ก ๐๐ ๐กโ๐ ๐ถ๐๐๐๐๐
๐ถ(6, 32โ )
Ans. D
๐ ๐๐2๐ฅ โ sin ๐ฅ cos ๐ฅ = 0
sin ๐ฅ (sin ๐ฅ โ cos ๐ฅ) = 0
sin ๐ฅ = 0 or sin ๐ฅ โ cos ๐ฅ = 0
sin ๐ฅ = 0, ๐ฅ = 0 , ๐, 2๐
sin ๐ฅ โ cos ๐ฅ = 0 โ sin ๐ฅ = cos ๐ฅ, ๐ฅ =๐
4 ,
5๐
4
๐ . ๐ = {0,๐
4 , ๐,
5๐
4, 2๐}
Ans. D
62. If the point (๐, 0, 3) is on the sphere centered at (1, 2, 3) with radius 2, what is the value of ๐?
(A) 0
(B) 2
(C) 1
(D) -3
(E)
63. What is the image of the circle ๐ฅ2+๐ฆ2 + 4๐ฅ โ 6๐ฆ + 11 = 0, when the origin is shifted to the
point (1,1) after translation of axes ?
(A) ๐ฅ2+๐ฆ2 โ 4๐ฅ โ 2๐ฆ + 3 = 0
(B) ๐ฅ2+๐ฆ2 โ 4๐ฅ โ 6๐ฆ + 3 = 0
(C) ๐ฅ2+๐ฆ2 โ 6๐ฅ + 8๐ฆ โ 23 = 0
(D) ๐ฅ2+๐ฆ2 โ 6๐ฅ โ 8๐ฆ + 23 = 0
Let ๐ฅ0 , ๐ฆ0 ๐๐๐ ๐ง0 be the center of the sphere
โด (๐ฅ โ ๐ฅ0)2 + (๐ฆ โ ๐ฆ0)2 + (๐ง โ ๐ง0)2 = ๐2, ๐คโ๐๐๐ ๐ฅ, ๐ฆ ๐๐๐ ๐ง ๐๐๐ ๐๐๐๐๐ก๐ ๐๐ ๐กโ๐ ๐ ๐โ๐๐๐.
(๐ โ 1)2 + (0 โ 2)2 + (3 โ 3)2 = 22
(๐ โ 1)2 + 4 + 0 = 4
(๐ โ 1)2 = 0 โ ๐ = 1
Ans.C
โ๐ โ ๐, 3๐ โ 2 ๐๐ ๐๐๐๐๐
๐ฟ๐๐ก ๐ = 1
3๐ โ 2 = 31 โ 2 = 1 โ ๐๐๐๐กโ๐๐๐ ๐๐๐๐๐ ๐๐๐ ๐๐๐๐๐๐ ๐๐ก๐
Disprove by a counter example.
Ans. A
Give a circle with equation
๐ฅ2 + ๐ฆ2 โ 4๐ฅ โ 6๐ฆ + 11 = 0, ๐๐๐ค ๐๐ ๐ค๐ ๐ โ๐๐๐ก
(1.1)units the image becomes
(๐ฅ โ 1)2 + (๐ฆ โ 1)2 โ 4(๐ฅ โ 1) โ 6(๐ฆ โ 1) + 11 = 0
๐ฅ2 โ 2๐ฅ + 1 + ๐ฆ2 โ 2๐ฆ + 1 โ 4๐ฅ + 4 โ 6๐ฆ + 6 + 11 = 0
๐ฅ2 + ๐ฆ2 โ 2๐ฅ โ 4๐ฅ โ 2๐ฆ โ 6๐ฆ + 1 + 1 + 4 + 6 + 11 = 0
๐ฅ2 + ๐ฆ2 โ 6๐ฅ โ 8๐ฆ + 23 = 0
Ans. D
64. What is the value of cot 2700+2cos 900+4๐ ๐๐2 + 1800 ?
(A) 4
(B) 8
(C) -2
(D) 7
65. Which one of the following is a valid assertion that can be proved by the principle of
mathematical induction?
(A) 3๐ + 25 < 3๐, for every integer ๐ โฅ 3.
(B) 2๐ > ๐ + 20, for every integer ๐ โฅ 4.
(C) ๐3 โ ๐ ๐๐ ๐๐๐ฃ๐๐ ๐๐๐๐ ๐๐ฆ 6, for every integer ๐ โฅ 1.
(D) ๐2 โค 2๐, for every integer ๐ โฅ 1.
THE END
๐โ๐ ๐ฃ๐๐๐ข๐ ๐๐ cot 2700 + 2 ๐๐๐ 900 + 4 ๐ ๐๐2 1800
cot 2700 =1
tan 270=
๐๐๐ 2700
๐ ๐๐2700=
0
โ1= 0
2 cos 900 = ๐ฅ โ 0 = 0
4 ๐ ๐๐2 1800 = 4(โ1)2 = 4
โด 0 + 0 + 4 = 4
Ans. A
Ans. C , sine ๐3 โ ๐ ๐๐๐ ๐ โฅ 1 ๐๐ ๐๐๐ค๐๐ฆ๐ ๐๐๐ฃ๐๐ ๐๐๐๐ ๐๐ฆ 6