Mathematics

Q.

find the smallest Number which leaves remainder 8 and 12 when divided by 28 and 32 respectively

Q.

Prove that$\sqrt{2}$ is an irrational number.

Q.

What is the smallest number that, when divided by 35, 56 and 91 leaves remainder of 7 in each case

Q.

Find the greatest number of 6 digits exactly divisible by 24, 15, 36.

Q.

Find the least number that is divisible by all the numbers between $1$ and $10$ (both inclusive).

Q. Find the smallest number which when increased by 17 is exactly divisible by 520 and 468?
Please Explain it briefly.

Q.

Find the zeros of the following quadratic polynomials $6x^2-3-7x$ and verify the relationship between the zeros and the coefficients.

Q. If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, then evaluate :

(i) α − β
(ii) $\frac{1}{\mathrm{\alpha }}-\frac{1}{\mathrm{\beta }}$
(iii) $\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}-2\mathrm{\alpha \beta }$
(iv) α²β − αβ²
(v) α⁴ + β⁴
(vi) $\frac{1}{a\mathrm{\alpha }+b}+\frac{1}{a\mathrm{\beta }+b}$
(vii) $\frac{\mathrm{\beta }}{a\mathrm{\alpha }+b}+\frac{\mathrm{\alpha }}{a\mathrm{\beta }+b}$
(viii) $a\left(\frac{{\mathrm{\alpha }}^2}{\mathrm{\beta }}+\frac{{\mathrm{\beta }}^2}{\mathrm{\alpha }}\right)+b\left(\frac{\mathrm{\alpha }}{\mathrm{\beta }}+\frac{\mathrm{\beta }}{\mathrm{\alpha }}\right)$

Q.

Check whether $6^n$ can end with the digit 0 for any natural number n.

Q.

A rectangular Courtyard is $18m72cm$ long and $13m20cm$ broad it is to be paved with square tiles of the same size and the least possible number of such tiles?

Q.

Find the largest number which divides 615 and 963 leaving the remainder 6 in each case.

Q.

Prove that the following is irrational : $\frac{1}{\sqrt{2}}$

Q.

Find the zeros of the quadratic polynomial √3x² - 8x + 4√3.

Q.

$5$ pencils and $7$ pens together cost $Rs.50,$ whereas $7$ pencils and $5$ pens together cost $Rs.46.$ Find the costs of one pencil and one pen.

Q.

If $\alpha$, $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=2x^2-5x+7$. find a polynomial whose zeros are $2\alpha +3\beta$ and $3\alpha +2\beta$

Q.

Find the zeros of the following quadratic polynomials $4s^2-4s+1$ and verify the relationship between the zeros and the coefficients.

Q.

Prove that $\sqrt{2}+\sqrt{3}$ is irrational.

Q.

Find a quadratic polynomial, the sum and product of whose zeros are $-3$ &$2$

Q.

Find the greatest number that will divide 445, 572 and 699 leaving remainder 4, 5, 6 effectively

Q.

Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.

Q.

If $\alpha$ and $\beta$ are the zeros of the polynomial $f\left(x\right)=x^2+x-2$, find the value of $(\frac{1}{\alpha }-\frac{1}{\beta })$.

Q.

Find the largest four-digits number which when divided by 4, 7 and 13 leaves a remainder of 3 in each case.

Q.

In a seminar, the number of participants in hindi, English and mathematics are 60, 84 and 108 respectively.find the maximum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same sunject

Q.

The number of polynomials having zeroes as $-2$and $5$ is

1. $1$

2. $2$

3. $3$

4. more than $3$

Q.

Factorize the equation: $x^3-3x^2-9x-5$.

Q.

Prove that $\frac{2}{\sqrt{7}}$ is irrational.

Q.

The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs. 155. What are the fixed charges and the charge per kilometer? How much does a person have to pay for travelling a distance of 25 km?

Q.

Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.

Q.

Prove that for any positive integer $n,n^3-n$ is divisible by $6$.

Q.

If $\alpha$ and $\beta$ are the zeros of the polynomial $f\left(x\right)=6x^2+x-2$, find the value of $(\frac{\alpha }{\beta }+\frac{\beta }{\alpha }).$