Part A

Problem 1

In the addition shown, each of the letters $\mathrm{T}, \mathrm{H}, \mathrm{I}, \mathrm{S}$ represents a non zero digit. What is $\mathrm{T}+\mathrm{H}+\mathrm{I}+\mathrm{S}$ ?

(A) 34
(B) 32
(C) 24
(D) 22

Problem 2

We have four sets $S_1, S_2, S_3, S_4$ each containing a number of parallel lines. The set $S_1$ contains $i+1$ parallel lines $i=1,2,3,4$. A line in $S_i$ is not parallel to lines in $S_j$ when $i \neq j$. In how many points do these lines intersect?

(A) 54
(B) 63
(C) 71
(D) 95

Problem 3

An old tanker is $100 \mathrm{~km}$ due north of a cruise liner. The tanker sails Southeast at a speed of 20 kilometers per hour and the liner sail Northwest at a speed of 10 kilometres per hour. What is the shortest distance between the two boats during the subsequent motion?

(A) $50 \sqrt{2} \mathrm{~km}$
(B) $60 \mathrm{~km}$
(C) $80 \mathrm{~km}$
(D) $100 \mathrm{~km}$

Problem 4

Volume A equals one fourth of the sum of the volumes B and C, while volume B equals one sixth of the sum of the volumes $A$ and $C$. The ratio of volume $C$ to the sum of volumes of $A$ and $B$ is

(A) $2: 3$
(B) $9: 10$
(C) $7: 12$
(D) $12: 23$

Problem 5

In the ninety-nine shop every item costs some whole number of rupees plus 99 paise. Rhea spent sixty five rupees and seventy six paise in buying some items from the shop. How many items did she buy?

(A) 23
(B) 24
(C) 65
(D) 66

Problem 6

The diagram shows a rectangle $A B C D$ where $A B: A D=1: 2$. Point $E$ on $A C$ is such that $D E$ is perpendicular to $A C$. What is the ratio of the area of the triangle DCE to the rectangle ABCD?

(A) $1: 4 \sqrt{2}$
(B) $1: 6$
(C) $1: 8$
(D) $1: 10$

Problem 7

The numbers $2,3,12,14,15,20,21$ may be divided into two sets so that the product of the numbers in each set is the same. What is this product?

(A) 420
(B) 1260
(C) 2520
(D) 6720

Problem 8

$A B C D$ is a trapezium with $A D=D C=C D=10$ units and $A B=22$ units. Semi circles are drawn as shown in the figure. The area of the region bounded by these semi circles in square units is

(A) $128+48 \pi$
(B) $128+24 \pi$
(C) $116+48 \pi$
(D) $116+24 \pi$

Problem 9

Consider the number of ways in which five girls and five boys sit in ten seats that are equally spaced around a circle. The proportion of the seating arrangements in which no two girls sit at the ends of a diameter is

(A) $\frac{1}{2}$
(B) $\frac{8}{63}$
(C) $\frac{55}{63}$
(D) None of the above

Problem 10

Let $A=1^{-4}+2^{-4}+3^{-4}+\ldots \ldots \ldots \ldots$, the sum of reciprocals of fourth powers of integers and $\mathrm{B}=1^{-4}+3^{-4}+5^{-4}+\ldots \ldots \ldots \ldots$, the sum of reciprocals of fourth powers of odd positive integers. The value of $\mathrm{A} / \mathrm{B}$ as a fraction is

(A) $\frac{16}{15}$
(B) $\frac{32}{31}$
(C) $\frac{64}{63}$
(D) $\frac{128}{127}$

Problem 11

The number $5^{\left(6^7\right)}$ is written on the board (in base 10). Gia takes two of the digits at a time, erases them but appends the sum of those digits at the end. She repeats this till she ends up with one digit on the board. What is the digit that remains on the board?

(A) 1
(B) 5
(C) 6
(D) 7

Problem 12

Seven points are marked on the circumference of a circle and all pairs of points are joined by straight lines. No three of these lines have a common point and any two intersect at a point inside the circle. Into how many regions is the interior of the circle divided by these lines?

(A) 64
(B) 63
(C) 57
(D) 56

Problem 13

The diagram below shows a regular hexagon with side length 1 , insceibed in a square. Two of the vertices lie on the diagonal of the square and the remaining vertices lie on its sides. What is the area of the square?

(A) $\frac{7}{2}$
(B) 4
(C) $2+\sqrt{3}$
(D) $3+\sqrt{2}$

Problem 14

$\mathrm{AB}$ is a diameter of a semicircle of centre $\mathrm{O}$. C is the midpoint of the arc $\mathrm{AB}$. $\mathrm{AC}$ and the tangent at $B$ to the semicircle meet at P. D is the midpoint of $B P$. If $A C D O$ is a parallelogram and $\angle P A D=\theta$, then $\sin \theta$ is

(A) $\frac{1}{\sqrt{5}}$
(B) $\frac{1}{\sqrt{10}}$
(C) $\frac{2}{\sqrt{10}}$
(D) $\frac{3}{\sqrt{10}}$

Problem 15

The real valued function $f(x)$ satisfies the equation $2 f(1-x)+1=x f(x)$ for all $x$. Then $\left(x^2-x+4\right)$ $f(x)$ equals

(A) $x-1$
(B) $x$
(C) $x+1$
(D) $x-3$

Part B

Problem 16

The number of ways in which 26 identical chocolates be distributed between Amy, Bob, Cathy and Daniel so that each receives at least one chocolate and Amy receives more chocolates than Bob is $\rule{2cm}{0.15mm}$

Problem 17

A set $\mathrm{S}$ contains 11 numbers. The average of the numbers in $\mathrm{S}$ is 302 . The average of the six smallest numbers of $S$ is 100 and the average of the six largest of the numbers is 300 . What is the median of the numbers in $\mathrm{S}$ $\rule{2cm}{0.15mm}$

Problem 18

The sum of the angles $1,2,3,4,5,6,7,8$ in degrees shows in the following figure is $\rule{2cm}{0.15mm}$

Problem 19

The number of positive integers less than 2018 that are divisible by 6 but are not divisible by at least one of the numbers 4 or 9 is $\rule{2cm}{0.15mm}$

Problem 20

\[x(x+1)(x+2) \ldots \ldots(x+23)=\sum_{n=1}^{24} a_n x^n\] the number of coefficients $a_n$ that are multiples of 3 is $\rule{2cm}{0.15mm}$

Problem 21

A square is cut into 37 squares of which 36 have area 1 square $\mathrm{cms}$. The length of the side of the original square is $\rule{2cm}{0.15mm}$

Problem 22

There are 4 coins in a row and all are showing heads to start with. The coins can be flipped with the following rules :
(a) The fourth coin (from the left) can be flipped any time
(b) An intermediate coin can be changed to tail only if its immediate neighbor on the right is heads and all other coins (if any) to its right are tails.
(c) Only one coin can be flipped in one step.

The minimum number of steps required to bring all coins to show tails is $\rule{2cm}{0.15mm}$

Problem 23

A poet met a tortoise sitting under a tree. When the tortoise was the poet's age, the poet was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If all the ages are in whole number of years, and the sum of their ages is now 264 , the age of the tree in years is $\rule{2cm}{0.15mm}$

Problem 24

The sum of all real value of $x$ satisfying $\left(x+\frac{1}{x}-17\right)^2=x+\frac{1}{x}+17$ is $\rule{2cm}{0.15mm}$

Problem 25

On the inside of a square with side length 6 , construct four congruent isosceles triangles each with base 6 and height 5 , and each having one side coinciding with a different side of the square. The area of the octagonal region common to the interiors of all four triangles is $\rule{2cm}{0.15mm}$

Problem 26

In a triangle with integer side lengths, one side is thrice the other. The third side is $15 \mathrm{~cm}$. The greatest possible perimeter of the triangle is (in $\mathrm{cm}$ ) $\rule{2cm}{0.15mm}$

Problem 27

A cube has edge length $x$ (an integer). three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If exactly 343 of these cubes have no faces painted blue, then the value of $x$ is $\rule{2cm}{0.15mm}$

Problem 28

If $f(x)=a x^4-b x^2+x+5$ and $f(3)=8$, the value of $f(-3)$ is $\rule{2cm}{0.15mm}$

Problem 29

Archana has to choose a three-digit code for her bike lock. The digits can be chosen from 1 to 9 . To help her remember them, she decides to choose three different digits in increasing order, for example 278 . The number of such codes she can choose is $\rule{2cm}{0.15mm}$

Problem 30

Let $\mathrm{S}$ be a set of five different positive integers, the largest of which is $\mathrm{n}$. It is impossible to construct a quadrilateral with non-zero area, whose side-lengths are all distinct elements of $\mathrm{S}$. The smallest possible value of $n$ is $\rule{2cm}{0.15mm}$

Part A

Problem 1

The value of $\frac{3+\sqrt{6}}{8 \sqrt{3}-2 \sqrt{12}-\sqrt{32}+\sqrt{50}-\sqrt{27}}$ is
(A) $\sqrt{2}$
(B) $\sqrt{3}$
(C) $\sqrt{6}$
(D) $\sqrt{18}$

Problem 2

A train moving with a constant speed crosses a stationary pole in 4 seconds and a platform $75 \mathrm{~m}$ long in 9 seconds. The length of the train is (in meters)
(A) 56
(B) 58
(C) 60
(D) 62

Problem 3

One of the factors of $9 x^2-4 z^2-24 x y+16 y^2+20 y-15 x+10$ is

(A) $3 x-4 y-2 z$
(B) $3 x+4 y-2 z$
(C) $3 x+4 y+2 z$
(D) $3 x-4 y+2 z$

Problem 4

The natural number which is subtracted from each of the four numbers $17,31,25,47$ to give four numbers in proportion is
(A) 1
(B) 2
(C*) 3
(D) 4

Problem 5

The solution to the equation $5\left(3^x\right)+3\left(5^x\right)=510$ is

(A) 2
(B) 4
(C) 5
(D) No solution

Problem 6

If $(x+1)^2=x$, the value of $11 x^3+8 x^2+8 x-2$ is
(A) 1
(B) 2
(C) 3
(D) 4

Problem 7

There are two values of $m$ for which the equation $4 x^2+m x+8 x+9=0$ has only one solution for $x$. The sum of these two value of $m$ is

(A) 1
(B) 2
(C) 3
(D) 4

Problem 8

The number of zeros in the product of the first 100 natural numbers is
(A) 12
(B) 15
(C) 18
(D) 24

Problem 9

The length of each side of a triangle in increased by $20 \%$ then the percentage increase of area is
(A) $60 \%$
(B) $120 \%$
(C) $80 \%$
(D) $44 \%$

Problem 10

The number of pairs of relatively prime positive integers $(a, b)$ such that $\frac{a}{b}+\frac{15 b}{4 a}$ is an integer is
(A) 1
(B) 2
(C) 3
(D) 4

Problem 11

The four digit number $8 a b 9$ is a perfect square. The value of $a^2+b^2$ is
(A) 52
(B) 62
(C) 54
(D) 68

Problem 12

$a, b$ are positive real numbers such that $\frac{1}{a}+\frac{9}{b}=1$. The smallest value of $a+b$ is
(A) 15
(B) 16
(C) 17
(D) 18

Problem 13

$a, b$ real numbers. The least value of $a^2+a b+b^2-a-2 b$ is
(A) 1
(B) 0
(C) -1
(D) 2

Problem 14

I is the incenter of a triangle $\mathrm{ABC}$ in which $\angle \mathrm{A}=80^{\circ} . \angle \mathrm{BIC}=$
(A) $120^{\circ}$
(B) $110^{\circ}$
(C) $125^{\circ}$
(D) $130^{\circ}$

Problem 15

In the adjoining figure $A B C D$ is a square and DFEB is a rhombus $\angle C D F=$

(A) $15^{\circ}$
(B) $18^{\circ}$
(C) $20^{\circ}$
(D) $30^{\circ}$

Part B

Problem 16

$A B C D$ is a square $E, F$ are point on $B C, C D$ respectively and $E A F=45^{\circ}$. The value of $\frac{E F}{B E+D F}$ is $\rule{1cm}{0.15mm}$

Problem 17

The average of 5 consecutive natural numbers is 10 . The sum of the second and fourth of these numbers is $\rule{1cm}{0.15mm}$

Problem 18

The number of natural number $n$ for which $n^2+96$ is a perfect square is $\rule{1cm}{0.15mm}$

Problem 19

$n$ is an integer and $\sqrt{\frac{3 n-5}{n+1}}$ is also an integer. The sum of all such $n$ is $\rule{1cm}{0.15mm}$

Problem 20

$\frac{a}{b}$ is a fraction where $a, b$ have no common factors other 1 . b exceeds a by 3 . If the numerator is increased by 7 , the fraction is increased by unity. The value of $a+b$ $\rule{1cm}{0.15mm}$

Problem 21

If $x=\sqrt[3]{2}+\frac{1}{\sqrt[3]{2}}$ then the value of $2 x^3-6 x$ is $\rule{1cm}{0.15mm}$

Problem 22

The angle of a heptagon are $160^{\circ}, 135^{\circ}, 185^{\circ}, 140^{\circ}, 125^{\circ}, x^{\circ}, x^{\circ}$. The value of $x$ is $\rule{1cm}{0.15mm}$

Problem 23

$A B C$ is a triangle and $A D$ is its altitude. If $B D=5 D C$, then the value of $\frac{3\left(A B^2-A C^2\right)}{B C^2}$ is $\rule{1cm}{0.15mm}$

Problem 24

As sphere is inscribed in a cube that has surface area of $24 \mathrm{~cm}^2$. A second cube is then inscribed within the sphere. The surface area of the inner cube $\left(\right.$ in $\left.\mathrm{cm}^2\right)$ is $\rule{1cm}{0.15mm}$

Problem 25

A positive integer $n$ is multiple of 7 . If $\sqrt{n}$ lies between 15 and 16 , the number of possible values (s) of n is $\rule{1cm}{0.15mm}$

Problem 26

The value of $x$ which satisfies the equation $\frac{\sqrt{x+5}+\sqrt{x-16}}{\sqrt{x+5}-\sqrt{x-16}}=\frac{7}{3}$ is $\rule{1cm}{0.15mm}$

Problem 27

$\mathrm{M}$ man do a work in $\mathrm{m}$ days. If there had been $\mathrm{N}$ men more, the work would have been finished $\mathrm{n}$ days earlier, then the value of $\frac{m}{n}-\frac{M}{N}$ is $\rule{1cm}{0.15mm}$

Problem 28

The sum of the digit of a two number is 15 . If the digits of the given number are reversed, the number is increased by the square of 3 . The original number is $\rule{1cm}{0.15mm}$

Problem 29

When expanded the units place of $(3127)^{173}$ is $\rule{1cm}{0.15mm}$

Problem 30

If $a:(b+c)=1: 3$ and $c:(a+b)=5: 7$, then $b:(c+a)$ is $\rule{1cm}{0.15mm}$

Part A

Problem 1

The fraction greater than $8 \frac{4}{9}$ is
(A) $8 \frac{1}{3}$
(B) $\frac{150}{18}$
(C) $8 \frac{2}{3}$
(D) $\frac{216}{27}$

Problem 2

A car is slowly driven in a road full of fog. The car passes a man who was walking at the rate of 3 $\mathrm{km}$ an hour in the same direction. He could see the car for 4 minutes and was visible for up to a distance of 100 meters. The speed of the car is (in $\mathrm{km}$ per hours)
(A) $4 \frac{1}{2}$
(B) $4$
(C) $3 \frac{1}{2}$
(D) $3$

Problem 3

Kiran sells pens at a profit of $20 \%$ for Rs. 60 . But due to lack of demand he reduced its price to Rs. 55. Then
(A) He gets a profit of $10 \%$
(B) He gets a profit of $12 \%$
(C) He incurs a loss of $10 \%$
(D) He incurs a loss of $8 \%$

Problem 4

If $40 \%$ of a number is added to another number then it becomes $125 \%$ of itself. The ratio of the second to the first number is
(A) $5: 8$
(B) $7: 5$
(C) $8: 5$
(D) None of these

Problem 5

The length of a rectangular sheet of paper is $33 \mathrm{~cm}$. It is rolled along its length into a cylinder so that width becomes height of the cylinder. The volume is 1386 cubic cms. The width of the rectangular sheet (in $\mathrm{cm}$ ) is
(A) 14
(B) 15
(C) 16
(D) 18

Problem 6

If $\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\ldots \ldots . .+\frac{1}{\mathrm{n} \times(\mathrm{n}+1)}=\frac{19}{20}$ then $\mathrm{n}=$
(A) 18
(B) 19
(C) 20
(D) 25

Problem 7

$a, b$ are natural numbers. If $9 a^2=12 a+96$ and $b^2=2 b+3$, the value of $2018(a+b)$ is
(A) 14226
(B) 14128
(C) 14126
(D) 14246

Problem 8

Shanti has three daughters. The average age of them is 15 years. Their ages are in the ratio $3: 5$ : 7. The age of the youngest daughter is (in years)
(A) 8
(B) 9
(C) 10
(D) 12

Problem 9

In the adjoining figure, $\mathrm{ABCD}$ is a quadrilateral. The bisectors of $\angle \mathrm{B}$ and the exterior angle at $\mathrm{D}$ meet at $\mathrm{P}$. Given $\angle \mathrm{C}=80^{\circ}, \angle \mathrm{ADC}=\frac{1}{2} \angle \mathrm{A}$ and $\angle \mathrm{A}=\angle \mathrm{C}+40^{\circ}$. Then $\angle \mathrm{DPB}$ is

(A) $50^{\circ}$
(B) $60^{\circ}$
(C) $70^{\circ}$
(D) $80^{\circ}$

Problem 10

The number of 3-digit number which contain 6 and 7 is
(A) 52
(B) 60
(C) 62
(D) 64

Problem 11

The difference between the biggest and the smallest three digit number each of which has different digits is
(A) 864
(B) 875
(C) 885
(D) 895

Problem 12

If $3 x+1=2 y-1=5 z+3=7 w+1=15$, the value of $6 x-3 y+5 z-8 w$ is
(A) 1
(B) 2
(C) 3
(D) None of these

Problem 13

Five years ago the average age of Aruna, Roy, David and salman is 45 years. Sita joins them now,. The average age of all the five now is 49 years. The present age of sita is (in years)
(A) 45
(B) 43
(C) 51
(D) 48

Problem 14

The fraction $\frac{B}{3 x-1}$ is subtracted from the fraction $\frac{A}{2 x+3}$. The resulting fraction is $\frac{-11}{(2 x+3)(3 x-1)}$. Then $A+B=$
(A) 11
(B) -11
(C) 5
(D) -5

Problem 15

There are some cows and ducks. The total number of legs is equal to 14 more than twice the number of heads. The number of cows is
(A) 5
(B) 6
(C) 7
(D) 8

Problem 16

The sum of $5 \%$ of a number and $9 \%$ another number is equal to sum of the $8 \%$ first number and $7 \%$ of the second number. The ratio between the numbers is
(A) $3: 2$
(B) $5: 7$
(C) $7: 9$
(D) $2: 3$

Problem 17

The length of two sides of an isosceles triangle are $8 \mathrm{~cm}$ and $14 \mathrm{~cm}$. The perimeter of the triangle (in $\mathrm{cm}$ ) is
(A) 30
(B) 36
(C) 19
(D) 30 or 36

Problem 18

There are three cell phones A, B, C. A is $50 \%$ costlier than C and B is $25 \%$ costlier than C.A is a $\%$ costlier than $\mathrm{B}$. Then $a=$
(A) 25
(B) 20
(C) 15
(D) 10

Problem 19

Sushant wrote a two digit number. He added 5 to the tens digit and subtracted 3 from the unit digit of the number and got a number equal to twice the original number. The original number is
(A) 47
(B) 74
(C) 37
(D) 73

Problem 20

The units digit of $5^{2018}-3^{2018}$ is
(A) 5
(B) 6
(C) 7
(D) 4

Part B

Problem 21

The smallest natural number that has to be added to 803642 to get a number which is divisible by 9 is $\rule{2cm}{0.15mm}$.

Problem 22

The greatest two digit number that will divided 398, 436, and 542 leaving respectively 7, 11 and 15 as remainders is $\rule{2cm}{0.15mm}$.

Problem 23

$\frac{2}{3}$ is $\rule{2cm}{0.15mm}$ of $\frac{1}{3}$.

Problem 24

The sum of 5 positive integers is 280. The average of the first 2 number is 40. The average of the third and fourth number is 60. The fifth number is $\rule{2cm}{0.15mm}$.

Problem 25

If $a: b=3: 4$ and $\frac{p}{q}=\frac{a^2+b^2+a b}{a^2+b^2-a b}$, where $p$, $q$ have no common divisors other than $1, p+q$ is $\rule{2cm}{0.15mm}$.

Problem 26

$a$ is a natural number such that a has exactly two divisors and $(a+1)$ has exactly three divisors. The number of divisors of $a+2$ is $\rule{2cm}{0.15mm}$.

Problem 27

The first term of a series is $\frac{2}{5}$. If $x$ is a term of this series, the next term is $\frac{1-x}{1+x}$. If $t_n$ denotes the $n$ th term and $t_{2018}-t_{2017}=\frac{p}{q}$, where $p$, $q$ are integers having no common factors other than $1, p+q$ is $\rule{2cm}{0.15mm}$.

Problem 28

In the adjoining figure, the side of the square is $\sqrt{\frac{2018}{\pi}} \mathrm{cm}$. The area of the unshaded region is $\left(\frac{\pi-2}{\pi}\right)$ A sq. cms. The value of $A$ is $\rule{2cm}{0.15mm}$.

Problem 29

$n$ is a natural number. The square root of the sum of the square of $n$ and 19 is equal to the next natural number to $n$. The value of $n$ is $\rule{2cm}{0.15mm}$.

Problem 30

Using only the digits 1,2, 4, 5, two- digit numbers are formed. The digits of the two digit number may be the same or different. The number of such two-digit number is $\rule{2cm}{0.15mm}$.

Part A

Problem 1

Observe the following sequence. What is the 100th term?
$$
7,8,1,0,0,1,0,1,1,0,2,1,0,3, \ldots \ldots \ldots
$$
(A) 1
(B) 0
(C) 2
(D) 3

Problem 2

A number is multiplied by 2 then by $\frac{1}{3}$, then by 4 , then by $\frac{1}{5}$ then by 6 and finally by $\frac{1}{7}$. The answer is 16 . Then the number is
(A) odd
(B) even
(C) Square
(D) a cube

Problem 3

Samrud bought a t- shirt for Rs.250. His friend Shlok wanted by buy it. Samrud wants to have a $10 \%$ profit on that. The selling price is (in rupees)
(A) 280
(B) 278
(C) 276
(D) 275

Problem 4

The value of $1+21+4161+81-11-31-51-71-91$ is
(A) -50
(B) 50
(C) 100
(D) -100

Problem 5

In the adjoining figure what portion of the figure is shaded ?

(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) $\frac{3}{10}$

Problem 6

The sum of the numbers in the three brackets ( ) is
$$\frac{()}{24}=\frac{20}{()}=\frac{24}{18}=\frac{4}{()}$$
(A) 60
(B) 55
(C) 50
(D) 45

Problem 7

A is the smallest three digit number which leaves a remainder 2 when divided by $17 . B$ is the smallest three digit number which leaves remainder 7 . When divided by 12 . Then $A+B$ is
(A) 205
(B) 312
(C) 215
(D) 207

Problem 8

A square of side 3 cm in cut into 9 equal squares. Another square of side 4 cm is cut into 16 equal squares. Saket made a bigger square using all the smaller square bits. The length of the side of the bigger square is (in cm)

(A) 7
(B) 6
(C) 5
(D) 8

Problem 9

A contractor constructed a big hall, rectangular in shape, with length 32 meters and breadth 18 meters. He wanted to buy 1 meter by 1 meter tiles. But in the shop 3 meter by 2 meter tiles only were available. How many tiles he has to buy for tilting the floor?

(A) 48
(B) 96
(C) 120
(D) 126

Problem 10

The fraction to be added to $2 \frac{1}{3}$ to get the fraction $4 \frac{4}{7}$ is
(A) $2 \frac{1}{21}$
(B) $2 \frac{4}{21}$
(C) $2 \frac{5}{21}$
(D) $2 \frac{6}{21}$

Part B

Problem 11

In the adjoining figure $\angle \mathrm{BAD}=\angle \mathrm{DAF}=\angle \mathrm{FAC}$. GE is parallel to $\mathrm{DF}$, and $\angle \mathrm{EGA}=90^{\circ}$. If $\angle \mathrm{ACE}=70^{\circ}$, the measure of $\angle \mathrm{FDE}$ is $\rule{1cm}{0.15mm}$.

Problem 12

$A B C$ is a triangle in which the angles are in the ratio $3: 4: 5$. PQR is a triangle in which the angles are in the ratio $5: 6: 7$. The difference between the least angle of $A B C$ and the least angle of PQR is $a^{\circ}$. Then $a=$ $\rule{1cm}{0.15mm}$.

Problem 13

Samrud had to multiply a number by 35 . By mistake he multiplied by 53 and got a result 720 more. The new product is $\rule{1cm}{0.15mm}$.

Problem 14

Vishva plays football every 4 th day. He played on a Tuesday. He plays football on a Tuesday again in is $\rule{1cm}{0.15mm}$ days.

Problem 15

In an elementary school $26 \%$ of the students are girls. If there are 240 less girls than boys, then the strength of the school is $\rule{1cm}{0.15mm}$.

Problem 16

There are three concentric circles as shown in the figure. The radii of them are $2 \mathrm{~cm}, 4 \mathrm{~cm}$ and 6 $\mathrm{cm}$. The ratio of the area of the shaded region to the area of the dotted region is $\frac{a}{b}$ where $a, b$ are integers and have no common factor other than 1. Then $a+b=$ $\rule{1cm}{0.15mm}$.

Problem 17

The value of $\left(1+\frac{1}{9}\right)\left(1+\frac{1}{8}\right)\left(1+\frac{1}{7}\right)\left(1+\frac{1}{6}\right)\left(1+\frac{1}{5}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{2}\right)$ is $\rule{1cm}{0.15mm}$.

Problem 18

When a two digit number divides 265 , the remainder is 5 . The number of such two digit numbers is $\rule{1cm}{0.15mm}$.

Problem 19

If $A \# B=\frac{A \times B}{A+B}$, the value of $\frac{12 \# 8}{8 \# 4}+\frac{10 \# 6}{6 \# 2}$ is $\rule{1cm}{0.15mm}$.

Problem 20

When water becomes ice, its volume increases by $10 \%$. When ice melts into water its volume decreases by $a \%$. Then $a=$ $\rule{1cm}{0.15mm}$.