## IITJEE Foundation Maths : Tutorial Problems III

1. When $a=-3, b=5, c=-1, d=0$, find the value of $26c\sqrt[3]{a^{3}-c^{2}d+5bc-4ac+d^{2}}$
2. Solve the equations: (a) $\frac{1}{3}x - \frac{2}{7}y = 8 -2x$ and $\frac{1}{2}y - 3x =3-y$ (b) $1 = y+z =2(z+x)=3(x+y)$
3. Simplify: (a) $\frac{a-x}{a+x} - \frac{4x^{2}}{a^{2}-x^{2}} + \frac{a-3x}{x-a}$ (b) $\frac{b^{2}-3b}{b^{2}-2b+4} \times \frac{b^{2}+b-30}{b^{2}+3b-18} \div \frac{b^{3}-3b^{2}-10b}{b^{2}+8}$
4. Find the square root of : $9-36x+60x^{2}-\frac{160}{3}x^{3}+\frac{80}{3}x^{4}-\frac{64}{3}x^{5}+\frac{64}{81}x^{6}$
5. In a cricket match, the extras in the first innings are one-sixteenth of the score, and in the second innings the extras are one-twelfth of the score. The grand total is 296, of which 21 are extras; find the score in each innings.
6. Find the value of $\frac{a^{2}-x^{2}}{\frac{1}{a^{2}} - \frac{2}{ax} + \frac{1}{x^{2}}} \times \frac{\frac{1}{a^{2}x^{2}}}{a+x}$
7. Find the value of : $\frac{1}{3}(a+2) -3(1-\frac{1}{6}b) - \frac{2}{3}(2a-3b+\frac{3}{2}) + \frac{3}{2}b - 4(\frac{1}{2}a-\frac{1}{3})$.
8. Resolve into factors: (i) $3a^{2} -20a-7$ (ii) $a^{4}b^{2}-b^{4}a^{2}$
9. Reduce to lowest terms: $\frac{4x^{3}+7x^{2}-x+2}{4x^{3}+5x^{2}-7x-2}$
10. Solve the following equations: (a) $x-6 -\frac{x-12}{3}= \frac{x-4}{2} + \frac{x-8}{4}$; (b) $x+y-z=0$, $x-y+z=4$, $5x+y+z=20$; (c) $\frac{ax+b}{c} + \frac{dx+e}{f} =1$
11. Simplify: $\frac{x+3}{x^{2}-5x+6} - \frac{x+2}{x^{2}-9x+14} + \frac{4}{x^{2}-10x+21}$
12. A purse of rupees is divided amongst three persons, the first receiving half of them and one more, the second half of the remainder and one more, and the third six. Find the number of rupees the purse contained.
13. If $h=-2, k=1, l=0, m=1, n=-3$, find the value of $\frac{h^{2}(m-l)-\sqrt{3hn}+hk}{m(l-h)-2hm^{2}+\sqrt[3]{4hk}}$
14. Find the LCM of $15(p^{3}+q^{3}), 5(p^{2}-pq+q^{2}), 4(p^{2}+pq+q^{2}), 6(p^{2}-q^{2})$
15. Find the square root of (i) $\frac{4x^{2}}{9} + \frac{9}{4x^{2}} -2$; (ii) $1-6a+5a^{2}+12a^{3}+4a^{4}$
16. Simplify $\frac{20x^{2}+27x+9}{15x^{2}+19x+6} + \frac{20x^{2}+27x+9}{12x^{2}+17x+6}$
17. Solve the equations: (i) $\frac{a(x-b)}{a-b} + \frac{b(x-a)}{b-a} =1$ (ii) $\frac{9}{x-4} + \frac{3}{x-8} = \frac{4}{x-9} + \frac{8}{x-3}$
18. A sum of money is to be divided among a number of persons; if Rs. 8 is given to each there will be Rs. 3 short, and if Rs. 7.50 is given to each there will be Rs. 2 over; find the number of persons.

Regards,

Nalin Pithwa

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

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