PART - A

Problem 1

Let $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}=\frac{m}{n}$, where $m$ and $n$ are positive integers with no common divisors other than 1 . The highest power of 7 that divides $m$ is

A. 0
B. 1
C. 2
D. 3

Problem 2

Five spherical balls of diameter 10 cm each fit inside a closed cylindrical tin with internal diameter 16 cms . What is the smallest height of the tin can be?

A. 39
B. 42
C. 45
D. 48

Problem 3

The expression $\frac{7 n+18}{2 n+3}$ takes integer values for certain integer values of $n$. The sum of all such values of the expression is

A. 14
B. 21
C. 24
D. 28

Problem 4

$P$ is a point inside $A B C D$ such that $P A=2, P B=4, P C=5$ and $P D=6$. The maximum area of the quadrilateral $A B C D$ is

A. 30
B. 33
C. 35
D. 38

Problem 5

The value of

$$ \left(3^{4 / 3}-3^{1 / 3}\right)^3+\left(3^{5 / 3}-3^{2 / 3}\right)^3+\left(3^{6 / 3}-3^{3 / 3}\right)^3+\cdots+\left(3^{10 / 3}-3^{7 / 3}\right)^3$$ is

A. $12\left(3^7-1\right)$
B. $12\left(3^7+1\right)$
C. $6\left(3^7-1\right)$
D. $6\left(3^7+1\right)$

Problem 6

One hundred people are standing in a line and they are required to count off in fives as "one, two, three, four, five" and so on from the first person in the line. Anyone who counts "five" walks out of the line. Those remaining repeat this procedure until only four people remain in the line. What was the original position in the line of the last person to leave?

A. 94
B. 96
C. 97
D. 98

Problem 7

The number of values of positive integers $n$ for which $1!+2!+\cdots+n$ ! is a perfect square is

A. 0
B. 1
C. 2
D. Infinitely many

Problem 8

When $2025^{2026}-2025$ is divided by $2025^2+2026$, the remainder is

A. 0
B. 2025
C. 2026
D. None of these

Problem 9

For a real number $x$, let $\lfloor x\rfloor$ denote the largest integer $\leq x$. For example, $\lfloor 3.4\rfloor=2$ and $\lfloor 4.9\rfloor=4$. Let $N=\left\lfloor(\sqrt{27}+\sqrt{23})^6\right\rfloor$. The remainder when $N$ is divided by 1000 is

A. 799
B. 599
C. 399
D. 199

Problem 10

$A B C$ is a triangle. $D$ lies on $A C$ such that $A B=B D=C D$. All the angles in the diagram are a positive whole number of degrees. The largest possible size, in degrees of $\angle A B C$ is

A. 171
B. 173
C. 175
D. 177

Problem 11

The side lengths of a right angled triangle are in geometric progression and the smallest side has length 2 units. The length of the hypotenuse is

A. $1+\sqrt{5}$
B. $\sqrt{10}$
C. $3 \sqrt{2}-1$
D. $\sqrt{11}$

Problem 12

In a regular polygon there are two diagonals that intersect inside the polygon at an angle $50^{\circ}$. The least number of sides of the polygon for which this is possible is

A. 12
B. 18
C. 24
D. 36

Problem 13

In triangle $P Q R, \angle R=2 \angle P, P R=5$ and $Q R=4$. The length of $P Q$ is

A. $2 \sqrt{10}$
B. 6
C. 7
D. $2 \sqrt{7}$

Problem 14

Each of ten people around a circle chooses a number and tells it to the neighbor on each side. Thus each person gives out one number and receives two numbers. The players then announce the average of the two numbers they received. The announced numbers, in order around the circle were $1,2,3,4,5,6,7,8,9,10$. The number chosen by the person who announced the number 6 is

A. 7
B. 5
C. 3
D. 1

Problem 15

A regular octagon is formed by cutting off four equal isosceles right angled triangles from the corners of a square of side length 1 . The area of the octagon is

A. $2(\sqrt{2}-1)$
B. $4 \sqrt{2}-3$
C. $\sqrt{2}-1$
D. $3 \sqrt{2}-2$

PART - B

Problem 16

The diagram shows the net of a cube, that is, we can fold along the edges of the squares to make a cube from this net. On each face there is an integer written - $1, a, b, c, d, 2026$. If each of the four numbers $a, b, c, d$ equals the average of the numbers on the four faces of the cube adjacent to it, the value of $a$ is $\rule{2cm}{0.2mm}$

Problem 17

Let $S={1,2,3, \ldots, 15}$. The number of subsets $A$ of $S$ containing four elements such that any two elements of $A$ differ by at least 2 is $\rule{2cm}{0.2mm}$

Problem 18

Given a deck of 52 cards with numbers $1,2, \ldots, 52$ written on them, one number per card. The deck is shuffled and 13 cards are chosen at random form the shuffled deck and thrown away, without noting the numbers on them. From the remaining 39 cards, one card is chosen at random. If the probability that the number on this card is a multiple of 13 is $\frac{m}{n}$, where $m, n$ are integers with no common divisor other than 1 , then $m+n$ equals $\rule{2cm}{0.2mm}$

Problem 19

Suppose 10 objects are placed along a circle at equal distances. The number of ways can three objects be chosen from among them so that no two of the chosen objects are adjacent or diametrically opposite is $\rule{2cm}{0.2mm}$

Problem 20

For a positive integer $n$, let $n \bmod 13$ denote the remainder $r, 0 \leq r<13$ when divided by 13. If $a, b, c$ are integers such that

$$
\begin{array}{rr}
4 a+5 b+6 c=1 & \bmod 13 \
a-b-7 c=3 & \bmod 13 \
3 a-4 b+5 c=9 & \bmod 13
\end{array}
$$

then $a+b+c \bmod 13$ is $\rule{2cm}{0.2mm}$

Problem 21

The sum of all positive integers $N$ less than 2024 such that $N$ equals 13 times the sum of digits of $N$ (when $N$ is written in base 10 ) is $\rule{2cm}{0.2mm}$

Problem 22

Six identical regular hexagons are arranged inside a larger hexagon as shown in the Figure. The outer hexagon has area 900 square units. The area of the shaded region (the total area contained in the smaller hexagons) is $\rule{2cm}{0.2mm}$ square units.

Problem 23

A positive integer is said to be special if the sum of the remainders obtained when it is divided by five consecutive positive integers is 32 . For example, 24 is special since when divided by $11,12,13,14,15$ the remainders are $2,0,11,10,9$ and these add up to 32 . The smallest positive integer that is special is $\rule{2cm}{0.2mm}$

Problem 24

The largest three digit number with the property that the number is equal to the sum of the hundreds digit, the square of its tens digit and the cube of its units digit is $\rule{2cm}{0.2mm}$

Problem 25

It is a surprising fact that $1 \times 2 \times 3 \times 4 \times 5 \times 6=8 \times 9 \times 10$. It is more surprising that $$8 \times 9 \times 10 \times 11 \times 12 \times 13 \times 14$$ can be written as a product of consecutive positive integers. The smallest number in the product is $\rule{2cm}{0.2mm}$

Problem 26

There are 100 points $P_1, P_2, \ldots, P_{100}$ placed on a line such that the distance between $P_i$ and $P_{i+1}$ is $\frac{1}{i}$ for $1 \leq i \leq 99$. The sum of the distances between every pair of these points is $\rule{2cm}{0.2mm}$

Problem 27

For any positive integer $n$, let $d(n)$ denote the number of divisors of $n$. For example, $d(4)=3$, since the divisors of 4 are $1,2,4$. The smallest positive integer $n$ for which $d(n-2)+d(n)+d(n+2)=21$ is $\rule{2cm}{0.2mm}$

Problem 28

Two bugs sit at the vertices $A$ and $H$ of a cube $A B C D E F G H$ with edge length $4 \sqrt{110}$ units. The bugs start moving simultaneously along $A C$ and $H F$ with the speed of the first bug twice that of the other one. The shortest distance between the bugs is $\rule{2cm}{0.2mm}$

Problem 29

The smallest positive integer $n$ for which it is possible to draw an $n$-gon whose vertex angles all measure $163^{\circ}$ or $171^{\circ}$ is $\rule{2cm}{0.2mm}$

Problem 30

Let $P(x)=a x^3+b x^2+c x+d$ be a cubic polynomial such that $P(2)=7, P(3)=13$ and $P(5)=7$. If the sum of the three roots of $P(x)=0$ is 40 , the value of $P(35)$ is $\rule{2cm}{0.2mm}$

Problem 1

Saket wanted to add two 2-digit numbers. But he multiplied them and got 629 as the answer. The sum of the two 2-digit numbers is

a)56
b) 52
c) 54
d) 46

Problem 2

The sum of three integers is 1 . Their product is 36 . The greatest of these three numbers is

a) 12
b) 8
c) 4
d) 6

Problem 3

The sum of five consecutive even numbers is 150 . When written in ascending order, the fourth number is

a) 34
b) 32
c) 36
d) 38

Problem 4

The price of a cell phone is decreased by $25 \%$. What percentage increase must be done in the delivered price to get back the original price?

a) $25 \%$
b) $271 / 2 \%$
c) $30 \%$
d) $331 \frac{1}{3} \%$

Problem 5

A rectangular carpet is placed in $8 m \times 8 \mathrm{~m}$ room, as shown in the diagram. What fraction of the floor is not covered?

a) $\frac{1}{4}$
b) $\frac{5}{11}$
c) $\frac{5}{8}$
d) $\frac{13}{24}$

Problem 6

$p$ and $p+1$ are two prime numbers. Then the value of $\frac{p(p+1)}{2 p+1}$ lies between

a) $\frac{4}{5}$ and 1
b) 1 and $\frac{7}{5}$
c) $\frac{6}{5}$ and $\frac{7}{5}$
d) $\frac{7}{5}$ and $\frac{8}{5}$

Problem 7

$a, b, c, d$ are real numbers such that $a-2023=b+2024=c-2025=d+2026$. Then the greatest among $a, b, c, d$ is

b) $a$
b) $b$
c) $c$
d) $d$

Problem 8

In the adjoining figure, $A D=A E$. Then measure of $\angle E A D$ is

a) $100^{\circ}$
b) $105^{\circ}$
c) $106^{\circ}$
d) $108^{\circ}$

Problem 9

The largest 3-digit number which is exactly divisible by the H.C.F. of 24 and 36 is $n$. Then $n+4$ is equal to

a) 994
b) 996
c) 998
d) 1000

Problem 10

In the given figure, $\mathrm{AB} / / \mathrm{HG} / / \mathrm{CD} / / \mathrm{FE}$.

$$\mathrm{AB}=6, \mathrm{GH}=4, \mathrm{CD}=5, \mathrm{FE}=9 \text { and } \mathrm{BC}=8 \text {. }$$ Distance between the pair of parallel lines $(\mathrm{AB}, \mathrm{HG}),(\mathrm{BC}, \mathrm{GF})$ and $(\mathrm{CD}, \mathrm{FE})$ is the same and equal to 3 . The area of the total figure is

a) 64
b) 60
c) 45
d) 65

Problem 11

The number of pairs of two digit square numbers, the sum or difference of which are also squares is

a) 0
b) 1
c) 2
d) 3

Problem 12

There are 20 people around a table. Each of them shakes hands with the people to his (or her) immediate left and immediate right. The total number of handshakes that takes place is

a) 40
b) 30
c) 32
d) 20

Problem 13

In the adjoining figure, $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are the vertices of a square of side 3 units. All the semi-circles are equal. Then the area of the shaded region is (in sq.units)

a) $8+\pi$
b) $6+\pi$
c) $12+\pi$
d) $7+\pi$

Problem 14

There are two boxes $A$ and $B$ which can hold 38 candles and 20 candles respectively.
288 candles have to be placed to the maximum capacity of the boxes. If we require $m$ number of A-type boxes and $n$ number of B-type boxes, then the value of $\frac{m}{n}$ is

a) 1
b) 2
c) 3
d) 4

Problem 15

The divisors of 6 are $1,2,3,6$. Leaving 1 and 6 , the divisors are 2 and 3 . Let us denote \([6]=2+3=5\). Then the value of \([[[12]]]\) is $\qquad$

a) 6
b) 8
c) 5
d) 12

Section B (Fill in the blanks

Problem 16

The fraction $\frac{(2 \times 3 \times 4)+(4 \times 6 \times 8)+(6 \times 9 \times 12)+\ldots+(20 \times 30 \times 40)}{(1 \times 2 \times 3)+(2 \times 4 \times 6)+(3 \times 6 \times 9)+\ldots+(10 \times 20 \times 30)}$ reduces to $\rule{2cm}{0.2mm}$

Problem 17

In the adjoining figure, $A B C D$ is a rectangle. $A E$ and $C F$ are quadrants. The length of the rectangle is twice its breadth. Taking $\pi=\frac{22}{7}$, the area of the shaded region is $21 \mathrm{~cm}^2$. Then the area of the rectangle is $\rule{2cm}{0.2mm}$

Problem 18

The number $m$ has factors 2,5 and 6 . The number $n$ has factors 4 and 8 . The smallest value of $m+n$ is $\rule{2cm}{0.2mm}$

Problem 19

There are two bus stops on opposite sides of a road. Bus route $X$ has a frequency of 15 minutes at one stop. Bus route $Y$ has a frequency of 40 minutes in the opposite bus stop. Currently both buses arrived in the opposite stops. Again two buses will simultaneously arrive at opposite stops after $\rule{2cm}{0.2mm}$ hours.

Problem 20

In the adjoining figure, $\angle A B C=60^{\circ}$ and $\angle A C B=80^{\circ}$. AD is the bisector of $\angle A$. Through C, a line making $\frac{\angle A}{2}$ with BC is drawn. This line cuts the bisector at D and the perpendicular from B to AD at E. Then the measure of $x$ (in degrees) is $\rule{2cm}{0.2mm}$

Problem 21

For two real numbers $a$ and $b$, we have $$ a * b=\left(a+\frac{b}{2}\right) \times\left(b+\frac{a}{2}\right) $$ Then the value of $(2 * 8) * 2$ is $\rule{2cm}{0.2mm}$

Problem 22

A 2-digit number has repeated digits. The number of such numbers having exactly 4 divisors is $\rule{2cm}{0.2mm}$

Problem 23

The salaries of Peter and Ali are in the ratio 3:2. Their expenditures are in the ratio 5:3 in that order. If each saves Rs. 5000 , then Peter's income (in Rs) is $\rule{2cm}{0.2mm}$

Problem 24

In the given figure, $\angle \mathrm{A}: \angle \mathrm{B}: \angle \mathrm{C}=14: 3: 1$. A line BE through B making an angle $\frac{\angle B}{3}$ with BC is drawn. A line through A, making an angle $\frac{1}{4} \angle C A D$ with $A C$ is drawn. They cut at G. Then the measure of $\angle \mathrm{EGF}$ is $\rule{2cm}{0.2mm}$ degrees

Problem 25

Ramaswamy, Krishnaswamy, Rangawamy, Gopalaswamy and Kumaraswamy have different amounts of money in rupees in their pockets, each an odd number and less than Rs. 100. The largest possible total sum of money in rupees is $\rule{2cm}{0.2mm}$

Problem 26

The cost price of 10 articles is equal to the selling price of 9 articles. The profit percent is $11 \frac{1}{a}$. Then $a=$ $\rule{2cm}{0.2mm}$

Problem 27

When $2 \frac{6}{11}$ of $1 \frac{2}{7}$ is divided by $3 \frac{3}{11}$, we get $\rule{2cm}{0.2mm}$

Problem 28

Two cell phones were sold at the same price. If there is $10 \%$ gain on the one and $10 \%$ loss on the other, then the total percent of loss is $\rule{2cm}{0.2mm}$

Problem 29

An office staff works for 4 days consecutively, then has the next day off; he works for 4 more days and has a day off on the next day; and so on. Today is his day-off and it is a Sunday. The minimum number of days the staff must work to have his off-day as Sunday is $\rule{2cm}{0.2mm}$

Problem 30

In the adjoining figure, triangle $B C D$ is equilateral. If $\angle \mathrm{AFB}=90^{\circ}$ and AH is the bisector of $\angle F A E$, then the measure of $\angle \mathrm{HGE}$ (in degrees) is $\rule{2cm}{0.2mm}$

Question 01

If $x^2+x=1$, then the value of $\frac{x^7+34}{x+2}$ is equal to

a) 7
b) 1
c) 13
d) 17

Question 02

The angle between the hour hand and the minute hand of a clock at the time $9: 38 \mathrm{pm}$ is

a) $60^{\circ}$
b) $61^{\circ}$
c) $59^{\circ}$
d) $62^{\circ}$

Question 03

In the adjoining figure, $A O B$ is a diameter of the circle with centre O. PC and PD are two tangents. Then the measure of $\angle E P D$ is $\qquad$

a) $15^{\circ}$
b) $10^{\circ}$
c) $12^{\circ}$
d) $20^{\circ}$

Question 04

The value of $x$ satisfying $4^x-3^{x-1 / 2}=3^{x+1 / 2}-2^{2 x-1}$ is of the form $\frac{a}{b}$ where $\operatorname{gcd}(a, b)=1$. Then the value of $\left(\frac{a+b}{a-b}\right)$ is equal to

a) 7
b) -5
c) 4
d) 5

Question 05

The number of polynomials of the form $\left(x^3+a x^2+b x+c\right)$ which are divisible by $x^2+1$ where $a, b, c \in{1,2,3,4, \ldots, 12}$ is

a) $12^3$
b) $12^2$
c) 12
d) 1

Question 06

The number of real solutions of the equation $\frac{(x+2)(x+3)(x+4)(x+5)}{(x-2)(x-3)(x-4)(x-5)}=1$ is

a) 1
b) 2
c) 3
d) 0

Question 07

If $a=\sqrt{23 a+b}, b=\sqrt{23 b+a}, a \neq b$, then the value of $\sqrt{a^2+b^2+48}$ is

a) 30
b) 25
c) 24
d) 23

Question 08

In the adjoining figure, PA and PB are tangents to the circle.
$A C$ is parallel to $P B$.
Then measure of $\angle C D A$ is

a) $118^{\circ}$
b) $108^{\circ}$
c) $98^{\circ}$
d) $88^{\circ}$

Question 09

If $\sqrt{\frac{19^8+19^x}{19^x+1}}=361$, then $x$ satisfies the equation

a) $4 x^2-7 x-15=0$
b) $2 x^2-9 x-5=0$
c) $3 x^2+11 x-4=0$
d) $3 x^2-11 x-4=0$

Question 10

If $S=4^2+2.5^2+3.6^2+\ldots \ldots \ldots+25.28^2$, then the value of $\frac{S}{325}$ is equal to

a) 436
b) 326
c) 346
d) 324

Question 11

A sequence $\{a_n\}, n \geq 1$ with $a_1=\frac{1}{2}$ and $a_n=\frac{a_{n-1}}{2 n a_{n-1}+1}$ is given. Then the value of $a_1+a_2+a_3+\ldots \ldots \ldots+a_{2024}$ is equal to

a) $\frac{2025}{2024}$
b) $\frac{2024}{2025}$
c) 2025
d) $\frac{1}{2025}$

Question 12

If $\alpha$ and $\beta(\alpha>\beta)$ satisfy the equation $x^{1+\log _{10} x}=10 x$ then the value of $\alpha+\frac{1}{\beta}$ is equal to

a) 100
b) 20
c) 10
d) $\frac{1}{100}$

Question 13

In the adjoining figure, four successively touching circles are placed in the interior of $\angle A O B$. The first (smallest) has a radius 7 cm . The third circle has a radius 28 cm . Then the radius of the largest circle (in cm ) is

a) 42
b) 48
c) 52
d) 56

Question 14

The coefficient of $x$ in the equation $x^2+p x+q=0$ was taken as 17 , in place of 13 and its roots were found to be -2 and -15 . If $\alpha, \beta$ are the roots of the original equation, then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is

a) $30 x^2+109 x+30=0$
b) $20 x^2-107 x+20=0$
c) $30 x^2-109 x+30=0$
d) $20 x^2+107 x+20=0$

Question 15

If $(1+x y+x+y)^2-(1-x y+x-y)^2=k y(1+x)^2$, then $k$ equals to

a) 1
b) 2
c) 3
d) 4

Section B (Fill in the blanks)

Question 16

When $x^{10}+1$ is divided by $x^2+1$, we get

$$
a x^8+b x^7+c x^6+d x^5+e x^4+f x^3+g x^2+h x+k
$$

as quotient. Then the value of $a^{2024}+b^{2024}+c^{2024}+d^{2024}+e^{2024}+f^{2024}+g^{2024}+h^{2024}+k^{2024}$ is $\rule{2cm}{0.2mm}$

Question 17

The equation $x^4-4 x^3+a x^2+b x+1=0$ has 4 positive roots. Then $a+b=$ $\rule{2cm}{0.2mm}$

Question 18

In the adjoining figure, $B O C$ is the diameter of the semicircle with centre O.
DE is the tangent at D .
If $\mathrm{AB}=k(\mathrm{AE})$, then the numerical value of $k$ is $\rule{2cm}{0.2mm}$

Question 19

In triangle $A B C$,
$\tan A: \tan B: \tan C=1: 2: 3$.
If $\frac{A C}{A B}=\frac{p \sqrt{q}}{r}$, where $q$ is Square free and $\operatorname{gcd}(p, r)=1$ then the value of $p+q+r$ is $\rule{2cm}{0.2mm}$

Question 20

Simon was given a number and asked to divide it by 120. He divided the number by 5,6 and 7 and got 3,2 and 2 as remainders respectively. The remainder when the number is divided by 120 is $\rule{2cm}{0.2mm}$ .

Question 21

The greatest number that leaves the same remainder when it divides 30,53 and 99 is$\rule{2cm}{0.2mm}$ ـ.

Question 22

If $f(x+1)=x^2-3 x+2$ and if the roots of the equation $f(x)=0$ are $\alpha$ and $\beta$, then the value of $\alpha^2+\beta^2$ is $\rule{2cm}{0.2mm}$ .

Question 23

The maximum volume of a cylinder is cut from a cube of edge $a$. The volume of the remaining solid is $k a^3$, where $k=\frac{p}{q}, \operatorname{gcd}(p, q)=1$. Taking $\pi=\frac{22}{7}$, the value of $p+q$ is $\rule{2cm}{0.2mm}$ .

Question 24

If the irreducible quadratic factor of $5 x^4+9 x^3-2 x^2-4 x-8$ is $a x^2+b x+c$, then the value of $a^2+b^2-c^2$ is $\rule{2cm}{0.2mm}$ .

Question 25

In the adjoining figure, POQ is the diameter of the semicircle with centre O.
OABC is a square whose area is $36 \mathrm{~cm}^2$. If $\mathrm{QD}=x \mathrm{~cm}$, the value of $x \sqrt{3}$ is $\rule{2cm}{0.2mm}$ .

Question 26

If $a=\sqrt{2024}, b=\sqrt{2025}$, the value of $2(a b)^{1 / 2}(a+b)^{-1}\left\{1+\frac{1}{4}\left(\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}\right)^2\right\}^{1 / 2}$ is $\rule{2cm}{0.2mm}$

Question 27

In a decreasing geometric progression, the $2^{\text {nd }}$ term is 6. The sum of all infinite terms of the progression is one-eighth of the sum to infinity of the squares of the terms. The sum of the $1^{\text {st }}$ and the $4^{\text {th }}$ terms is $\frac{p}{q}$ where $p, q$ are relatively prime to each other. Then the value of $\left[\frac{p}{q}\right]$, where $[x]$ represents the greatest integer not exceeding $x$ is $\rule{2cm}{0.2mm}$

Question 28

The value of $\left(\frac{\sqrt{10}}{10}\right)^{\left(\log _{10} 9\right)-2}$ is of the form $\frac{a}{b}$, where $a, b$ are relatively prime to each other. Then $a-b$ is equal to ـ.$\rule{2cm}{0.2mm}$

Question 29

ABCD is a square. BE is the tangent to the semicircle on AD as diameter. The area of the triangle BCE is $216 \mathrm{~cm}^2$. The radius of the semicircle (in cm ) is $\rule{2cm}{0.2mm}$

Question 30

$a, b, c, d$ are real constants in a $f(x)=a x^{2025}+b x^{2023}+c x^{2021}+d x^{2019}$ and $f(-4)=18$. Then the maximum value of $|f(4)|+|2 \cos x|$ is $\rule{2cm}{0.2mm}$ .

Question 01

There is a 6-digit number in which the first and the fourth digit from the first are the same, the second and the fifth digit from the first are the same and the third and the sixth digit from the first are the same. Then the number is always

a) A square number
b) Divisible by 5
c) Divisible by 11
d) An odd number.

Question 02

Starting from the number 1, Ritu generates a series of numbers as

$$
1,3,6,11,18,29,42, \ldots
$$

such that the differences of the consecutive numbers from the beginning give consecutive primes. In this series she came across a perfect square for the first time. The Square root of this perfect square is

a) 14
b) 19
c) 23
d) 21

Question 03

The expression $\frac{x\left(\frac{\sqrt{x}+\sqrt{y}}{2 y \sqrt{x}}\right)^{-1}+y\left(\frac{\sqrt{x}+\sqrt{y}}{2 x \sqrt{y}}\right)^{-1}}{\left(\frac{x+\sqrt{x y}}{2 x y}\right)^{-1}+\left(\frac{y+\sqrt{x y}}{2 x y}\right)^{-1}}$ reduces to

a) $\sqrt{x y}$
b) $\frac{\sqrt{x}+\sqrt{y}}{2}$
c) $\frac{2}{\sqrt{x}+\sqrt{y}}$
d) $\frac{\sqrt{x y}}{\sqrt{x}+\sqrt{y}}$

Question 04

The sum of the digits of a two-digit number is multiplied by 8 and the result is found to be 13 more than the number. Then the two digit number is

a) A prime number
b) An even number
c) Such that the difference of its digits is 2 .
d) Such that the sum of its digits is a composite number.

Question 05

A water tank is fitted with four different taps as outlets. If the tank is full, it takes 1 hour to empty the tank when the first tap alone is opened; it takes 2 hours to empty the tank when the second tap alone is opened; it takes 3 hours to empty the tank when the third tap alone is opened; it takes 4 hours to empty the tank when the fourth tap alone is opened. When all the taps are opened simultaneously, the full tank will be emptied in

a) More than 29 minutes
b) Between 28 and 29 minutes
c) Between 29 and 30 minutes
d) Less than 28 minutes.

Question 06

Two primes $p, q$ are such that $p+q$ is odd and $q-10 p=23$. Then $q-20 p$ equals to

a) 1
b) 3
c) 5
d) 7

Question 07

Which one of the following is a false statement?

a) Diagonals of a square bisect each other at right angles.
b) Diagonals of a rectangle bisect each other.
c) Diagonals of a rhombus bisect each other at right angles.
d) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a rectangle.

Question 08

Soham has his $23^{\text {rd }}$ birthday on $1^{\text {st }}$ January 2024 and he noticed that 2024 is divisible by 23. If he lives till 100 years of age, how many times other than the above, his age would be a divisor of the then year?

a) 2
b) 3
c) 4
d) 5

Question 09

Consider the two figures shown here $\mathrm{AB}=16 \mathrm{~cm}$ in both the figures.

Points $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ divide AB in equal lengths in fig. 1 Similarly $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{T}, \mathrm{L}, \mathrm{M}$ divide AB in equal length in fig 2

All the curves are semi-circles.
If $[a]$ and $[b]$ are the areas of
the shaded figures respectively in fig 1 and fig 2 , then

a) [a] - [b] is a non-zero number.
b) $[\mathrm{a}]=\frac{5}{4}[\mathrm{~b}]$
c) $\quad[\mathrm{a}]=\frac{4}{5}[\mathrm{~b}]$
d) $[\mathrm{a}]=[\mathrm{b}]$

Question 10

The sum of 11 consecutive natural numbers is 121 . The sum of the next three numbers is

a) 54
b) 55
c) 53
d) 57

Question 11

A big ship wrecked and 1000 people landed in a remote island. The food material was available for them for 60 days. After 16 days another small ship, which had no food stock, wrecked and 100 people landed in the same island. The number of days the food material for all of them available is

a) 42
b) 35
c) 40
d) 41

Question 12

Two numbers are respectively $28 \%$ and $70 \%$ of a third number. The percentage of the first number to the second is

a) 40
b) 36
c) 45
d) 50

Question 13

The sum of two natural numbers is 150 . Their HCF is 15 . The number of pairs of such numbers is

a) 1
b) 2
c) 3
d) 4

Question 14

ABC and ADE are isosceles triangles.
If $\angle B F D=156^{\circ}$, then $\angle A=$

a) $68^{\circ}$
b) $70^{\circ}$
c) $66^{\circ}$
d) $70^{\circ}$

Question 15

Some students are made to stand in rows of equal number, one behind the other. Saket is in the $3^{\text {rd }}$ row from the front and $5^{\text {th }}$ row from the back. He is $4^{\text {th }}$ from the left and $6^{\text {th }}$ from right. The total number of students is

a) 45
b) 72
c) 63
d) 81

FILL IN THE BLANKS

Question 16

In the adjoining figure, ABCD is a rectangle. Then,
$\angle \mathrm{EBD}=$ $\rule{2cm}{0.2mm}$ degrees.

Question 17

An infinite sequence of positive numbers $x_1, x_2, x_3, \ldots, x_n, x_{n+1}, \ldots$ satisfies $x_n^2=(3 n+7)+(n-3) x_{n+1}$, where $x_n$ is the $n^{\text {th }}$ term of the sequence. Then the numerical value of $x_1$ is $\rule{2cm}{0.2mm}$

Question 18

For $n \geq 2$ and $n \in Z$, the smallest positive integer $n$ for which none of the fractions $\frac{17}{n+17}, \frac{18}{n+18}, \frac{19}{n+19}, \ldots \ldots, \frac{100}{n+100}$ can be simplified is $\rule{2cm}{0.2mm}$ .

Question 19

In triangle $\mathrm{ABC}, \mathrm{AB}=15 \mathrm{~cm}, \mathrm{BC}=20 \mathrm{~cm}$ and $\mathrm{CA}=25 \mathrm{~cm}$. Then the length of the shortest altitude of the triangle (in cm ) is $\rule{2cm}{0.2mm}$

Question 20

The units digit of $19^{2025}+999^{2023}$ is $\rule{2cm}{0.2mm}$

Question 21

$N$ is a 2-digit number. When 6 is added to the tens digit and 2 is subtracted from the units digit, we get a two digit number which is equal to $3 N$. Then $N$ is $\rule{2cm}{0.2mm}$

Question 22

$A B C D$ is a quadrilateral. $A B$ is parallel to $C D$ and $A B>C D$. If $A D=A B=B C$ and $\angle \mathrm{ADC}=140^{\circ}$, then the measure of $\angle \mathrm{CAB}$ is $\rule{2cm}{0.2mm}$ degrees.

Question 23

The product of two positive numbers $x$ and $y$ is 4 times their Sum and the same product is 8 times their difference. If $x \geq y$, then $x=$ $\rule{2cm}{0.2mm}$

Question 24

In the adjoining figure, $A B C D E F G H$ is a regular Octagon. The measure of $\angle \mathrm{ADG}$ (in degrees) is $\rule{2cm}{0.2mm}$ .

Question 25

If $2^{3 a+2}=4^{b+7}$ and $3^{a+10}=27^{2 b+10}$ then the value of $a^2+b^2$ is $\rule{2cm}{0.2mm}$

Question 26

ABCD is a rectangle. $\mathrm{AB}=6$ and $\mathrm{AD}=10$.
E is a point on BC such that $\mathrm{AE}=10$.
Then area of $\triangle \mathrm{ADE}$ (in square units) is $\rule{2cm}{0.2mm}$

Question 27

The numbers $1,4,7,10$ and 13 are placed in each box of the figure, such that the sum of the numbers in the horizontal or vertical boxes are the same. The largest possible value of the horizontal or vertical sum is $\rule{2cm}{0.2mm}$

Question 28

The number of integer pairs $(m, n)$ such that $m\left(n^2+1\right)=48$ is $\rule{2cm}{0.2mm}$

Question 29

In the adjoining figure, $\triangle \mathrm{ABD}$ and $\triangle \mathrm{BCE}$ are equilateral triangles.

The measure of $\angle \mathrm{AFC}=$ $\rule{2cm}{0.2mm}$ degrees.

Question 30

The value of $\frac{\sqrt[4]{27 \cdot \sqrt[3]{9}}}{\sqrt[6]{9 \cdot 3^3 \cdot \sqrt{3}}}$ is $\rule{2cm}{0.2mm}$