online user counter

MATHMAA

1Q) Prove that for every prime P>7, $$P^{6}-1$$ is divisible by 504.

2Q) a) Determine the smallest positive integer x, whose last digit is 6 and if we erase this 6 and put it in left most of the number so obtained, the number becomes 4x.

b) For any real numbers a and b, prove that:

$$3a^{4}-4a^{3}b+b^{4} \geq 0$$

3Q) If squares of the roots of $$x^{4}+bx^{2}+cx+d=0$$ are $$\alpha ,\beta ,\gamma ,\delta$$, then prove that:

$$64 \alpha \beta \gamma \delta -[4\sum\alpha\beta -(\sum\alpha)^{2}]^{2} = 0$$.

4Q) a) Let a, b,c be the length of the sides of a triangle and r be the in-radius. Show that:

r< $$\frac{a^{2}+b^{2}+c^{2}}{3(a+b+c)}$$.

b) A family consists of a grand father, 6 sons & daughters and 4 grand children. They are to be seated in a row for a dinner. The grand children wish to occupy the two seats at each end and the grand father refuses to have a grand child on either side of him. Determine the number of ways in which they can be seated for the dinner.

5Q) A semi-circle is drawn outwardly on chord AB of the circle with centre O and unit radius. The perpendicular from O to AB, meets the semi-circle on AB at C. Determine the measure of $$\angle AOB$$ and length AB so that OC has maximum length.

6Q) If $$cos \alpha + cos \beta + cos \gamma = sin \alpha + sin \beta + sin \gamma = 0$$, Prove that

$$cos (2\alpha) + cos (2\beta) + cos (2\gamma)$$ =  $$sin (2\alpha) + sin (2\beta) + sin (2\gamma) =0$$.

7Q) A point 'A' is randomly chosen in a square of side length 1 unit. Find the probability that the distance from A to the centre of the square does not exceed x.

8Q) (a) Two parallel chords in a circle have lengths 10cm an 14cm and distance between them is 6 cm. If a chord parallel to these chords and midway between them is length √a, find the value of a.

(b) The line joining two points A(2, 0) and B(3, 1) is rotated about point A in anticlockwise direction through an angle of $$15^{\circ}$$. Find the equation of the line in the new position. IF B goes to C in new position, find the coordinates of C.

9Q) Find the number of numbers $$\leq 10^{8}$$ which are neither perfect squares, nor perfect cubes, nor perfect fifth powers .

10Q) Let PQRS be a rectangle such that PQ= a and QR =b. Suppose $$r_{1}$$ is the radius of the circle passing through P and Q and touching RS and $$r_{2}$$ is the radius of the circle passing through Q and R and touching PS. Show that :

$$5(a+b) \leq 8(r_{1}+r_{2} )$$.