Find the value of $1(1!)+2(2!)+3(3!)+\cdots+20(20!)$.
If $x+\dfrac{1}{x-1}=5$, then find the value of $\,(x-1)^2+\dfrac{1}{(x-1)^2}\,$?
Use suitable identities to find the following products:
(i) (x + 4) (x + 10)
(ii) (x + 8) (x – 10)
(iii) (3x + 4) (3x – 5)
(iv) \((y^ 2 + \frac{3 }{ 2}) (y^ 2 – \frac{3 }{ 2}) \)
(v) (3 – 2x) (3 + 2x)
Evaluate the following products without multiplying directly:
(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96
Factorise the following using appropriate identities:
(i) 9x 2 + 6xy + y 2
(ii) 4y 2 – 4y + 1
(iii) x 2 – \(\frac{y^2 }{ 100}\)
Expand each of the following, using suitable identities:
(i) (x + 2y + 4z) 2 (ii) (2x – y + z) 2 (iii) (–2x + 3y + 2z) 2
(iv) (3a – 7b – c) 2 (v) (–2x + 5y – 3z) 2 (vi) [ \(\frac{1 }{ 4}\) a - \(\frac{1 }{ 2}\) b + 1]2
Factorise:
(i) 4x 2 + 9y 2 + 16z 2 + 12xy – 24yz – 16xz
(ii) 2x 2 + y 2 + 8z 2 – 2√2 xy + 4√2 yz – 8xz
Write the following cubes in expanded form:
(i) (2x + 1)3 (ii) (2a – 3b) 3 (iii) [\(\frac{3}{2}\) x + 1]3 (iv) [x - \(\frac{2 }{ 3} \)y]3
Factorise each of the following:
(i) 8a 3 + b 3 + 12a 2b + 6ab2
(ii) 8a 3 – b 3 – 12a 2b + 6ab2
(iii) 27 – 125a 3 – 135a + 225a 2
(iv) 64a 3 – 27b 3 – 144a 2b + 108ab2
(v) 27p 3 – \(\frac{1}{ 216}\) – \(\frac{9 }{ 2}\) p2 + \(\frac{1 }{4}\) p
(i) 27y 3 + 125z 3
(ii) 64m3 – 343n 3
[ Hint : See Question 9. ]