MathDB

1

Distance the bigger wheel have to roll so that the points will be togather

In the following figure, the bigger wheel has circumference

12

m and the inscribed wheel has circumference

8

m.

P_{1}

denotes a point on the bigger wheel and

P_{2}

denotes a point on the smaller wheel. Initially

P_{1}

and

P_{2}

coincide as in the figure. Now we roll the wheels on a smooth surface and the smaller wheel also rolls in the bigger wheel smoothly. What distance does the bigger wheel have to roll so that the points will be together again?

SAQ: P 5

1

Prove that LCM of n numbers is n×min{n numbers} and HCF is max{n numbers}/n

Let

a

be the smallest and

A

the largest of

n

distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to

n a

and that the greatest common divisor is less than or equal to

\frac{A}{n}

SAQ: P 4

1

Find all the real numbers satisfying the equiation

Find all real numbers

x_{1}, x_{2}, \cdots, x_{n}

satisfying,

\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{1}{2}\left(x_{1}+x_{2}+\cdots+x_{n}\right)

SAQ: P 3

1

It is possible to traverse to any given square.

Show that, in a chessboard, it is possible to traverse to any given square from another given square using a knight. (A knight can move in a chessboard by going two steps in one direction and one step in a perpendicular direction as shown in the given figure)

SAQ: P 2

1

If log_{x}a + log_{y}a= 4 log_{xy}a then prove x=y

Let

x, y

be numbers in the interval (0,1) such that for some

a>0, a \neq 1

\log _{x} a+\log _{y} a=4 \log _{x y} a

Prove that

x=y

SAQ: P 1

1

How many people were involved in the entire process?

In a room there is a series of bulbs on a wall and corresponding switches on the opposite wall. If you put on the

n

-th switch the

n

-th bulb will light up. There is a group of men who are operating the switches according to the following rule: they go in one by one and starts flipping the switches starting from the first switch until he has to turn on a bulb; as soon as he turns a bulb on, he leaves the room. For example the first person goes in, turns the first switch on and leaves. Then the second man goes in, seeing that the first switch on turns it off and then lights the second bulb. Then the third person goes in, finds the first switch off and turns it on and leaves the room. Then the fourth person enters and switches off the first and second bulbs and switches on the third. The process continues in this way. Finally we find out that first 10 bulbs are off and the 11 -th bulb is on. Then how many people were involved in the entire process?

MCQ: P 15

1

Find out the number of real solutions of $x^2e^{\sin x}=1$

Find out the number of real solutions of

x^2e^{\sin x}=1

[*] 0 [*] 1 [*] 2 [*] 3

MCQ: P 14

1

Find the locus of the point (x, y) in the plane

z=x+i y

where

x

and

y

are two real numbers. Find the locus of the point

(x, y)

in the plane, for which

\frac{z+i}{z-i}

is purely imaginary (that is, it is of the form

i b

where

b

is a real number). [Here,

i=\sqrt{-1}

[*] A straight line [*] A circle [*] A parabole [*] None of these

MCQ: P 13

1

Find at how many points f(x) is not differentiable

Define

f(x)=\max \{\sin x, \cos x\} .

Find at how many points in

(-2 \pi, 2 \pi), f(x)

is not differentiable?
[*] 0 [*] 2 [*] 4 [*]

\infty

MCQ: P 12

1

Maximum value of $\sin^4\theta +\cos^6\theta $

Maximum value of

\sin^4\theta +\cos^6\theta

will be ?
[*]

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

[*]

\frac{1}{2}

[*]

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

[*] 1

MCQ: P 11

1

Find out the number of unordered pairs of (A, B)

S=\{1,2, \ldots, 6\} .

Then find out the number of unordered pairs of

(A, B)

such that

A, B \subseteq S

and

A \cap B=\phi

[*] 360 [*] 364 [*] 365 [*] 366

MCQ: P 10

1

Find $\sum_{i=1}^{n} \alpha_{i}$

If

\sum_{i=1}^{n} \cos ^{-1}\left(\alpha_{i}\right)=0,

then find

\sum_{i=1}^{n} \alpha_{i}

[*]

\frac{n}{2}

[*]

n

[*]

n\pi

[*]

\frac{n\pi}{2}

MCQ: P 9

1

How many 5×5 grid is possible

How many

5 \times 5

grids are possible such that each element is either 1 or 0 and each row sum and column sum is

4 ?

[*] 64 [*] 32 [*] 120 [*] 96

MCQ: P 8

1

Find the relation between BC^3 and BD^3+BE^3

In

\triangle A B C

,

A B=A C

and

D

is foot of the perpendicular from

C

to

A B

and

E

the foot of the perpendicular from

B

to

A C,

then
[*]

BC^3>BD^3+BE^3

[*]

BC^3 <BD^3+BE^3

[*]

BC^3=BD^3+BE^3

[*] None of these

MCQ: P 7

1

How many x are there such that x,[x],{x} are in harmonic progression

How many

x

are there such that

x,[x],\{x\}

are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here

[x]

is the largest integer less than equal to

x

and

\{x\}=x-[ x]

)
[*] 0 [*] 1 [*] 2 [*] 3

MCQ: P 6

1

Find the area of Triangle BDF

Let

A C

and

C E

be perpendicular line segments, each of length

18 .

Suppose

B

and

D

are the midpoints of

A C

and

C E

respectively. If

F

be the point of intersection of

E B

and

A D,

then the area of

\triangle B D F

is?
[*]

27\sqrt{2}

[*]

18\sqrt{2}

[*] 13.5 [*] 18

MCQ: P 5

1

Integral solutions of x^5-31x+2015

How many integral solutions are there of the equation

x^5 -31x+2015=0

?
[*] 2 [*] 4 [*] 1 [*] None of these

MCQ: P 4

1

Find the greatest number of X's that can be put on the grid

Let

n

be an odd integer. Placing no more than one

X

in each cell of a

n \times n

grid, what is the greatest number of

X

's that can be put on the grid without getting

n

X

's together vertically, horizontally or diagonally?
[*]

2{{n}\choose {2}}

[*]

{{n}\choose {2}}

[*]

2n

[*]

2{{n}\choose {2}}-1

MCQ: P 3

1

Find $\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)$

If

A_i=\frac{x-a_i}{|x-a_i|}

,

i=1,2,\cdots,n

for

n

numbers

a_1<a_2<\cdots<a_m<\cdots<a_n,

then

\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)=?

[*]

(-1)^{m-1}

[*]

(-1)^m

[*]

1

[*] None of these

MCQ: P 2

1

Find the value of $\Big(f_n(3)\Big)^2+f_n(2)$

Let

f_n(x)=\underbrace{xx\cdots x}_{n\ \text{times}}

that is,

f_n(x)

is a

n

digit number with all digits

x

, where

x\in \{1,2,\cdots,9\}

. Then which of the following is

\Big(f_n(3)\Big)^2+f_n(2)

?
[*]

f_n(5)

[*]

f_{2n}(9)

[*]

f_{2n}(1)

[*] None of these

MCQ: P 1

1

How many distinct arrangements are possible for wearing five different rings

How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand? (We can wear multiple rings in one finger)
[*]

\frac{10!}{5!}

[*]

5^5

[*]

\frac{9!}{4!}

[*] None of these

2015 Mathematical Talent Reward Programme

Subcontests

Distance the bigger wheel have to roll so that the points will be togather

Prove that LCM of n numbers is n×min{n numbers} and HCF is max{n numbers}/n

Find all the real numbers satisfying the equiation

It is possible to traverse to any given square.

If log_{x}a + log_{y}a= 4 log_{xy}a then prove x=y

How many people were involved in the entire process?

Find out the number of real solutions of $x^2e^{\sin x}=1$

Find the locus of the point (x, y) in the plane

Find at how many points f(x) is not differentiable

Maximum value of $\sin^4\theta +\cos^6\theta $

Find out the number of unordered pairs of (A, B)

Find $\sum_{i=1}^{n} \alpha_{i}$

How many 5×5 grid is possible

Find the relation between BC^3 and BD^3+BE^3

How many x are there such that x,[x],{x} are in harmonic progression

Find the area of Triangle BDF

Integral solutions of x^5-31x+2015

Find the greatest number of X's that can be put on the grid

Find $\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)$

Find the value of $\Big(f_n(3)\Big)^2+f_n(2)$

How many distinct arrangements are possible for wearing five different rings

2015 Mathematical Talent Reward Programme

Subcontests

Distance the bigger wheel have to roll so that the points will be togather

Prove that LCM of n numbers is n&times;min{n numbers} and HCF is max{n numbers}/n

Find all the real numbers satisfying the equiation

It is possible to traverse to any given square.

If log_{x}a + log_{y}a= 4 log_{xy}a then prove x=y

How many people were involved in the entire process?

Find out the number of real solutions of $x^2e^{\sin x}=1$

Find the locus of the point (x, y) in the plane

Find at how many points f(x) is not differentiable

Maximum value of $\sin^4\theta +\cos^6\theta $

Find out the number of unordered pairs of (A, B)

Find $\sum_{i=1}^{n} \alpha_{i}$

How many 5&times;5 grid is possible

Find the relation between BC^3 and BD^3+BE^3

How many x are there such that x,[x],{x} are in harmonic progression

Find the area of Triangle BDF

Integral solutions of x^5-31x+2015

Find the greatest number of X's that can be put on the grid

Find $\lim \limits_{x\to a_m}\Big(A_1A_2\cdots A_n\Big)$

Find the value of $\Big(f_n(3)\Big)^2+f_n(2)$

How many distinct arrangements are possible for wearing five different rings

Prove that LCM of n numbers is n×min{n numbers} and HCF is max{n numbers}/n

How many 5×5 grid is possible