List of practice Questions

Find \((a + b)^4 - (a - b)^4\). Hence, evaluate \((\sqrt3 + \sqrt2)^4 - (\sqrt3 - \sqrt2)^4\).

The number lock of a suitcase has 4 wheels, each labelled with ten digits i.e., from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase?

Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3).

Find the derivative of x at x = 1.

If a and b are distinct integers, prove that a - b is a factor of \(a^n - b^n\) , whenever n is a positive integer.
[Hint: write\( a ^n = (a - b + b)^n\) and expand]

Find the derivative of 99x at x = 100.

Find the derivative of the following functions from first principle.
(i) x³ − 27 (ii) ( x −1)( x- 2 )
(iii) \(\frac{1}{x^2}\) (iv) \(\frac{x+1}{x-1}\)

Find \((x + 1)^6 - (x - 1)^6\). Hence or otherwise evaluate \((\sqrt2 + 1)^6 + (\sqrt2 - 1)^6.\)

For the function f(x) = \(\frac{x^{100}}{100}\) + \(\frac{x^{99}}{99}\) + ......+ \(\frac{x^2}{2}\)+ x+1. Prove that f ′(1)= 100 f'(0 )

Find the derivative of xⁿ + ax ^n-1 + a²x^n-2+……...+a ^n-1x + aⁿ for some fixed real number a.

Expand using Binomial Theorem \(( 1+ \frac{X}{2} - \frac{2}{X})^4, X≠0\)

Find an approximation of (0.99)5 using the first three terms of its expansion

Evaluate\( ( √3 +√2) ^6 - ( √3 -√2) ^6.\)

Find the value of \(( a^2 + \sqrt{a^2-1})^4 + ( a^2 - \sqrt{a^2-1})^4\)

Show that \(9^{ n+1} -8n -9\) is divisible by \(64\), whenever n is a positive integer.

For some constants a and b, find the derivative of
(i) (x-a)(x-b) (ii) (ax²+b)² (iii)\(\frac{x-a}{x-b}\)

Find the derivative of \(\frac{x^n-a^n}{x-a}\) for some constant a.

Prove that \(\underset{r=0} {\overset{n}∑} { 3^r }\,{ ^nC_r} = 4^n\)

Find the derivative of
(i) 2x - \(\frac{3}{4}\)
(ii)(5x³+3x-1)(x-1)
(iii) x-3(5+3x) (iv)x⁵(3-6x-9)
(v) x-4(3-4x-5) (vi)\(\frac{2}{x+1}-\frac{x^2}{3x-1}\)

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): sinⁿ x

Find the derivative of cos x from first principle.

Find the derivative of the following functions (it is to be understood that \(a, \,b,\, c,\, d,\, p,\, q,\, r\) and \(s\) are fixed non-zero constants and \(m\) and \(n\) are integers), \((x+a)\).

Find the derivative of the following functions:
(i) sin x cos x
(ii) sec x
(iii) 5sec x+4cos x
(iv) cosec x
(v) 3cot x+5cosec x
(vi) 5sin x-6cos x+7
(vii) 2tan x-7sec x

Find the derivative of the following functions from first principle:
(i) −x (ii) (-x)-1 (iii) sin (x + 1) (iv) cos (x – π/ 8 )

Find the derivative of the following functions from first principle:
(i) −x (ii) (-x)-1 (iii) sin (x + 1) (iv) cos (x – \(\frac{\pi}{8}\))