Equivalence Relations and Equivalence Classes

Exercise 6.3 Question 18: Prove that the function given by {\displaystyle f(x) = x^3 - 3{x^2} + 3x - 100 } is increasing in R.

NCERT Exercise 6.2 Question 3: Show that the function given by {\displaystyle f(x) = \sin x } (a) increasing in {\displaystyle \left ( 0, \frac{\pi}{2} \right ) } (b) decreasing in {\displaystyle \left ( \frac{\pi}{2}, \pi \right ) } (c) neither increasing nor decreasing in {\displaystyle (0, \pi) }

Question: Show that {\displaystyle \int{\frac{\sin x}{1 + \sin x}} = \sec x - \tan x + x + \text{C} }

NCERT Exercise 6.1 Question 10 A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

If {\displaystyle x = \sin t} and {\displaystyle y = \sin pt } , prove that {\displaystyle (1 - x^2)\frac{d^{2}y}{dx^2} - x\frac{dy}{dx} + p^2y = 0 } .

If {\displaystyle a x^2 + 2hxy + b y^2 + 2gx + 2fy + c = 0 } , then show that {\displaystyle \frac{dy}{dx} . \frac{dx}{dy} = 1 } .

NCERT Example – 47 Continuity and Differentiability Chapter 5: Find {\displaystyle \frac{dy}{dx} } in the following parametric function {\displaystyle x = a^{\left ( t+\frac{1}{t} \right )}, y = {\left ( t+\frac{1}{t} \right )}^a} .

Differentiate the following function w. r. t. x, Question 12: {\displaystyle y = (\log x)^x + (\sin^{-1}x)^{\sin x} }

Question: Find the derivative of the following function w.r.t x, {\displaystyle y = \cos^{-1}\left( \frac{a + b\cos x}{b + a \cos x} \right) }

Question 8. Find the value of k, if the following function is continuous {\displaystyle f(x)=\begin{cases} \frac{log (1+ax)-log (1-bx)}{x}, & \text{ if } x\ne 0 \\ k, & \text{ if } x=0 \end{cases} }

Question 15. If { \displaystyle (x - a)^2 + (y - b)^2 = c^2 } , for some c > 0, prove that {\displaystyle \frac{\left[ 1 + \left( \frac{dy}{dx}\right )^2 \right ]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} } is a constant independent of a and b.

Question 14. If { \displaystyle y = Ae^{mx} + Be^{nx}} , show that { \displaystyle \frac{d^2{y}}{dx^2} - (m + n)\frac{dy}{dx} + mny = 0 } .

Question 7 : { \displaystyle x = \frac{\sin^3{t}}{\sqrt{\cos 2t}},  y = \frac{\cos^3{t}}{\sqrt{\cos 2t}} }

Example 26 Find the derivative of f given by { \displaystyle f(x) = \sin^{–1} x } assuming it exists.

Find the value of a, if the following function is continuous
Question 17: {\displaystyle f(x) = \begin{cases} a \sin \frac{\pi }{2} (x+1), & \text{ if } x \le 0 \\ \frac{\tan x-\sin x}{x^{3} }, & \text{ if } x > 0 \end{cases}}

Question: Given a square matrix of order 3×3 such that | A | = 12 find the value of ∣A adj(A)∣

Question 4. Express the following matrices as the sum of symmetric and a skew symmetric matrix: {\displaystyle {\begin{bmatrix} 2&{ - 2}&{ - 4}\\ { - 1}&3&4\\ 1&{ - 2}&{ - 3} \end{bmatrix}} }

NCERT Miscellaneous Exercise Question 5 : Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

There are two families A and B. There are 4 men, 2 women and 1 child in family A and 2 men, 3 women and 2 children in family B. They recommended daily allowance for calories i.e. Men: 2000, Women: 1500, Children: 1200 and for proteins is Men: 50 gms., Women: 45 , Children: 30 gms. Represent the above information by matrices, using matrix multiplication calculate the total requirements of calories and proteins for each of the families.

Question 7. If {\displaystyle A = { \begin{bmatrix} {\cos^2\theta} & {\cos \theta \sin \theta} \\ {\cos \theta \sin \theta } & {\sin^2 \theta} \end{bmatrix} } } , {\displaystyle B = {\begin{bmatrix} {\cos^2 \phi} & { \cos \phi \sin \phi } \\ {\cos \phi \sin \phi } & {\sin^2 \phi } \end{bmatrix}} } then show that AB is a zero matrix, provided {\displaystyle (\theta  - \phi )} is an odd multiple of {\displaystyle \frac{\pi }{2}} .

Simplify: {\displaystyle \sin ^{-1} \left ( \frac{5}{13} \cos x+\frac{12}{13} \sin x \right ) }

NCERT Exercise 2.2 Question 9 Inverse Trigonometric Functions { \displaystyle {\tan }^{-1} \left ( \frac{x}{\sqrt{a^2-x^2}} \right ), | x | < a}

Question. Let A = R – {3} and B = R – {1}. Consider the function {\displaystyle f: A \rightarrow B } defined by {\displaystyle f(x) = \frac{x-2}{x-3} } . Is f one-one and onto? Justify your answer.