## IITJEE Foundation Maths: Tutorial Problems IV

1. Resolve into factors: (a) $2x^{2}-3ab+(a-6b)x$ (b) $4x^{2}-4xy-15y^{2}$
2. In the expression, $x^{3}-2x^{2}+3x-4$, substitute $a-2$ for x, and arrange the result according to the descending powers of a.
3. Simplify: (i) $\frac{x}{1-\frac{1}{1+x}}$ (ii) $\frac{x^{2}}{a+\frac{x^{2}}{a+\frac{x^{2}}{a}}}$
4. Find the HCF of $3x^{3}-11x^{2}+x+15$ and $5x^{4}-7x^{3}-20x^{2}-11x-3$
5. Express in the simplest form: (i) $\frac{\frac{x}{y}-\frac{y}{x}}{\frac{x+y}{y}-\frac{y+x}{x}}$ (ii) $(\frac{x^{3}-1}{x-1} + \frac{x^{3}+1}{x+1})\div (\frac{1}{x-1} + \frac{1}{x+1})$
6. A person possesses Rs. 5000 stock, some at 3 per cent; four times as much at 3.5 %, and the rest at 4 %; find the amount of each kind of stock when his income is Rs. 178.
7. Simplify the expression: $-3[(a+b)-{(2a-3b) -( 5a+7b-16c) - (-13a +2b -3c -5d)}]$, and find its value when $a=1, b=2, c=3, d=4$.
8. Solve the following equations : (i) $11y-x=10$ and $11x-101y=110$ (ii) $x+y-z=3$, and $x+z-y=5$, and $y+z-x=7$.
9. Express the following fractions in their simplest form: (i) $\frac{32x^{3}-2x+12}{12x^{5}-x^{4}+4x^{2}}$ (ii) $\frac{1}{x + \frac{1}{1+ \frac{x+3}{2-x}}}$
10. What value of a will make the product of $3-8a$ and $3a+4$ equal to the product of $6a+11$ and $3-4a$?
11. Find the LCM of $x^{3}-x^{2}-3x-9$ and $x^{3}-2x^{2}-5x-12$
12. A certain number of two digits is equal to seven times the sum of its digits; if the digit in the units’ place be decreased by two and that in the tens place by one, and if the number thus formed be divided by the sum of its digits, the quotient is 10. Find the number.
13. Find the value of $\frac{6x^{2}-5xy-6y^{2}}{2x^{2}+xy-y^{2}} \times \frac{3x^{2}-xy-4y^{2}}{2x^{2}-5xy+3y^{2}} \div \frac{9x^{2}-6xy-8y^{2}}{2x^{2}-3xy+y^{2}}$
14. Resolve each of the following expressions into four factors: (i) $4a^{4}-17a^{2}b^{2}+4b^{4}$; (ii) $x^{8}-256y^{8}$
15. Find the expression of highest dimensions which will divide $24a^{4}b -2a^{2}b^{2}-9ab^{4}$ and $18a^{6}+a^{4}b^{2}-6a^{3}b^{3}$ without remainder.
16. Find the square root of : (i) $x(x+1)(x+2)(x+3)+1$ (ii) $(2a^{2}+13a+15)(a^{2}+4a-5)(2a^{2}+a-3)$
17. Simplify: $x - \frac{2x-6}{x^{2}-6x+9} - 3 + \frac{x^{2}+3x-4}{x^{2}=x-12}$
18. A quantity of land, partly pasture and partly arable, is sold at the rate of Rs. 60 per acre for the pasture and Rs. 40 per acre for the arable, and the whole sum obtained at Rs. 10000. If the average price per acre were Rs. 50, the sum obtained would be 10 per cent higher; find how much of the land is pasture and how much is arable.

Regards,

Nalin Pithwa

Purva building, 5A
Flat 06
Mumbai, Maharastra 400101
India

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