Write the negation of the following statements:
Question: Write the negation of the following statements: (i) Chennai is the capital of Tamil Nadu. (ii) $\sqrt{2}$ is not a complex number. (iii) All triangles are not equilateral triangle. (iv) The number 2 is greater than 7. (v) Every natural number is an integer. Solution: (i) Chennai is not the capital of Tamil Nadu. (ii) $\sqrt{2}$ is a complex number. (iii) All triangles are equilateral triangles. (iv) The number 2 is not greater than $7 .$ (v) Every natural number is not an integer....
Read More →Write the negation of the following statements:
Question: Write the negation of the following statements: (i) Chennai is the capital of Tamil Nadu. (ii) $\sqrt{2}$ is not a complex number. (iii) All triangles are not equilateral triangle. (iv) The number 2 is greater than 7. (v) Every natural number is an integer. Solution: (i) Chennai is not the capital of Tamil Nadu. (ii) $\sqrt{2}$ is a complex number. (iii) All triangles are equilateral triangles. (iv) The number 2 is not greater than $7 .$ (v) Every natural number is not an integer....
Read More →Give three examples of sentences which are not statements. Give reasons for the answers.
Question: Give three examples of sentences which are not statements. Give reasons for the answers. Solution: The three examples of sentences, which are not statements, are as follows. (i) He is a doctor. It is not evident from the sentence as to whom he is referred to. Therefore, it is not a statement. (ii) Geometry is difficult. This is not a statement because for some people, geometry can be easy and for some others, it can be difficult. (iii) Where is she going? This is a question, which also...
Read More →Which of the following sentences are statements? Give reasons for your answer.
Question: Which of the following sentences are statements? Give reasons for your answer. (i) There are 35 days in a month. (ii) Mathematics is difficult. (iii) The sum of 5 and 7 is greater than 10. (iv) The square of a number is an even number. (v) The sides of a quadrilateral have equal length. (vi) Answer this question. (vii) The product of (1) and 8 is 8. (viii) The sum of all interior angles of a triangle is 180. (ix) Today is a windy day. (x) All real numbers are complex numbers. Solution:...
Read More →Find the derivative of the following functions
Question: 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): $\frac{x}{\sin ^{n} x}$ Solution: Let $f(x)=\frac{x}{\sin ^{n} x}$ By quotient rule, $f^{\prime}(x)=\frac{\sin ^{n} x \frac{d}{d x} x-x \frac{d}{d x} \sin ^{n} x}{\sin ^{2 n} x}$ It can be easily shown that $\frac{d}{d x} \sin ^{n} x=n \sin ^{n-1} x \cos x$ Therefore, $f^{\prime}(x)=\frac{\sin ^{n} x \frac{d}{d x} x-x \fra...
Read More →Prove that:
Question: Prove that: (i) $\sqrt{3 \times 5^{-3}} \div \sqrt[3]{3^{-1}} \sqrt{5} \times \sqrt[6]{3 \times 5^{6}}=\frac{3}{5}$ (ii) $9^{3 / 2}-3 \times 5^{0}-\left(\frac{1}{81}\right)^{-1 / 2}=15$ (iii) $\left(\frac{1}{4}\right)^{-2}-3 \times 8^{2 / 3} \times 4^{0}+\left(\frac{9}{16}\right)^{-1 / 2}=\frac{16}{3}$ (iv) $\frac{2^{1 / 2} \times 3^{1 / 3} \times 4^{1 / 4}}{10^{-1 / 5} \times 5^{3 / 5}} \div \frac{3^{4 / 3} \times 5^{-7 / 5}}{4^{-3 / 5} \times 6}=10$ (v) $\sqrt{\frac{1}{4}+(0.01)^{-1 ...
Read More →Find the equations of the tangent and normal to the given curves at the indicated points:
Question: Find the equations of the tangent and normal to the given curves at the indicated points: (i) $y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$ at $(0,5)$ (ii) $y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$ at $(1,3)$ (iii) $y=x^{3}$ at $(1,1)$ (iv) $y=x^{2}$ at $(0,0)$ (v) $x=\cos t, y=\sin t$ at $t=\frac{\pi}{4}$ Solution: (i) The equation of the curve is $y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$. On differentiating with respect tox, we get: $\frac{d y}{d x}=4 x^{3}-18 x^{2}+26 x-10$ $\left.\frac{d y}{d x}\right]_{(0,5)...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers): (x+ secx) (x tanx) Solution: Let $f(x)=(x+\sec x)(x-\tan x)$ By product rule, $f_{2}^{\prime}(x)=\lim _{h \rightarrow 0}\left(\frac{f_{2}(x+h)-f_{2}(x)}{h}\right)$ $=\lim _{h \rightarrow 0}\left(\frac{\sec (x+h)-\sec x}{h}\right)$ $=\lim _{h \rightarrow 0} \frac{1}{h}\left[\frac{1}{\cos (x+h)}-\frac{1}{\cos x}\right]$ $=\lim _{h \rightarrow 0}...
Read More →Find the derivative of the following functions
Question: 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): $\frac{x}{1+\tan x}$ Solution: Let $f(x)=\frac{x}{1+\tan x}$ $f^{\prime}(x)=\frac{(1+\tan x) \frac{d}{d x}(x)-x \frac{d}{d x}(1+\tan x)}{(1+\tan x)^{2}}$ $f^{\prime}(x)=\frac{(1+\tan x)-x \cdot \frac{d}{d x}(1+\tan x)}{(1+\tan x)^{2}}$ (i) Let $g(x)=1+\tan x$. Accordingly, $g(x+h)=1+\tan (x+h)$. By first principle, $g^{\pr...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers): $\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$ Solution: Let $f(x)=\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$ By quotient rule, $f^{\prime}(x)=\cos \frac{\pi}{4} \cdot\left[\frac{\sin x \frac{d}{d x}\left(x^{2}\right)-x^{2} \frac{d}{d x}(\sin x)}{\sin ^{2} x}\right]$ $=\cos \frac{\pi}{4} \cdot\left[\frac{\sin x \cdot 2 x-x^{...
Read More →Find points on the curve
Question: Find points on the curve $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$ at which the tangents are (i) parallel tox-axis (ii) parallel toy-axis Solution: The equation of the given curve is $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$. On differentiating both sides with respect tox, we have: $\frac{2 x}{9}+\frac{2 y}{16} \cdot \frac{d y}{d x}=0$ $\Rightarrow \frac{d y}{d x}=\frac{-16 x}{9 y}$ (i) The tangent is parallel to the $x$-axis if the slope of the tangent is i.e., $0 \frac{-16 x}{9 y}=0$, which is...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\frac{4 x+5 \sin x}{3 x+7 \cos x}$ Solution: Let $f(x)=\frac{4 x+5 \sin x}{3 x+7 \cos x}$ By quotient rule, $f^{\prime}(x)=\frac{(3 x+7 \cos x) \frac{d}{d x}(4 x+5 \sin x)-(4 x+5 \sin x) \frac{d}{d x}(3 x+7 \cos x)}{(3 x+7 \cos x)^{2}}$ $=\frac{(3 x+7 \cos x)\left[4 \frac{d}{d x}(x)+5 \frac{d}{d x}(\sin x)\right]-(4 x+5 \sin x)\left[3 \frac{d}...
Read More →Find the derivative of the following functions
Question: 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): $(x+\cos x)(x-\tan x)$ Solution: Let $f(x)=(x+\cos x)(x-\tan x)$ By product rule, $f^{\prime}(x)=(x+\cos x) \frac{d}{d x}(x-\tan x)+(x-\tan x) \frac{d}{d x}(x+\cos x)$ $=(x+\cos x)\left[\frac{d}{d x}(x)-\frac{d}{d x}(\tan x)\right]+(x-\tan x)(1-\sin x)$ $=(x+\cos x)\left[1-\frac{d}{d x} \tan x\right]+(x-\tan x)(1-\sin x)$ ...
Read More →Find the equations of all lines having slope 0 which are tangent to the curve .
Question: Find the equations of all lines having slope 0 which are tangent to the curve $y=\frac{1}{x^{2}-2 x+3}$. Solution: The equation of the given curve is $y=\frac{1}{x^{2}-2 x+3}$. The slope of the tangent to the given curve at any point (x,y) is given by, $\frac{d y}{d x}=\frac{-(2 x-2)}{\left(x^{2}-2 x+3\right)^{2}}=\frac{-2(x-1)}{\left(x^{2}-2 x+3\right)^{2}}$ If the slope of the tangent is 0, then we have: $\frac{-2(x-1)}{\left(x^{2}-2 x+3\right)^{2}}=0$ $\Rightarrow-2(x-1)=0$ $\Righta...
Read More →Find the equation of all lines having slope 2 which are tangents to the curve
Question: Find the equation of all lines having slope 2 which are tangents to the curve $y=\frac{1}{x-3}, x \neq 3$. Solution: The equation of the given curve is $y=\frac{1}{x-3}, x \neq 3$. The slope of the tangent to the given curve at any point (x,y) is given by, $\frac{d y}{d x}=\frac{-1}{(x-3)^{2}}$ If the slope of the tangent is 2, then we have: $\frac{-1}{(x-3)^{2}}=2$ $\Rightarrow 2(x-3)^{2}=-1$ $\Rightarrow(x-3)^{2}=\frac{-1}{2}$ This is not possible since the L.H.S. is positive while t...
Read More →Simplify:
Question: Simplify: (i) $\left(16^{-1 / 5}\right)^{5 / 2}$ (ii) $\sqrt[5]{(32)^{-3}}$ (iii) $\sqrt[3]{(343)^{-2}}$ (iv) $(0.001)^{1 / 3}$ (v) $\frac{(25)^{3 / 2} \times(243)^{3 / 5}}{(16)^{5 / 4} \times(8)^{4 / 3}}$ (vi) $\left(\frac{\sqrt{2}}{5}\right)^{8} \div\left(\frac{\sqrt{2}}{5}\right)^{13}$ (vii) $\left(\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right)^{7 / 2} \times\left(\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right)^{-5 / 2}$ Solution: (ii) $\sqrt[5]{(32)^{-3}}$ $=\sqrt[5]...
Read More →Find the equation of all lines having slope
Question: Find the equation of all lines having slope $-1$ that are tangents to the curve $y=\frac{1}{x-1}, x \neq 1$. Solution: The equation of the given curve is $y=\frac{1}{x-1}, x \neq 1$. The slope of the tangents to the given curve at any point (x,y) is given by, $\frac{d y}{d x}=\frac{-1}{(x-1)^{2}}$ If the slope of the tangent is 1, then we have: $\frac{-1}{(x-1)^{2}}=-1$ $\Rightarrow(x-1)^{2}=1$ $\Rightarrow x-1=\pm 1$ $\Rightarrow x=2,0$ Whenx= 0,y= 1 and whenx= 2,y= 1. Thus, there are...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\left(a x^{2}+\sin x\right)(p+q \cos x)$ Solution: Let $f(x)=\left(a x^{2}+\sin x\right)(p+q \cos x)$ By product rule, $f^{\prime}(x)=\left(a x^{2}+\sin x\right) \frac{d}{d x}(p+q \cos x)+(p+q \cos x) \frac{d}{d x}\left(a x^{2}+\sin x\right)$ $=\left(a x^{2}+\sin x\right)(-q \sin x)+(p+q \cos x)(2 a x+\cos x)$ $=-q \sin x\left(a x^{2}+\sin x\r...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\left(x^{2}+1\right) \cos x$ Solution: Let $f(x)=\left(x^{2}+1\right) \cos x$ By product rule, $f^{\prime}(x)=\left(x^{2}+1\right) \frac{d}{d x}(\cos x)+\cos x \frac{d}{d x}\left(x^{2}+1\right)$ $=\left(x^{2}+1\right)(-\sin x)+\cos x(2 x)$ $=-x^{2} \sin x-\sin x+2 x \cos x$...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$x^{4}(5 \sin x-3 \cos x)$ Solution: Let $f(x)=x^{4}(5 \sin x-3 \cos x)$ By product rule, $f^{\prime}(x)=x^{4} \frac{d}{d x}(5 \sin x-3 \cos x)+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^{4}\right)$ $=x^{4}\left[5 \frac{d}{d x}(\sin x)-3 \frac{d}{d x}(\cos x)\right]+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^{+}\right)$ $=x^{4}[5 \cos x-3(-\sin x)]...
Read More →Find the point on the curve
Question: Find the point on the curve $y=x^{3}-11 x+5$ at which the tangent is $y=x-11$. Solution: The equation of the given curve is $y=x^{3}-11 x+5$. The equation of the tangent to the given curve is given as $y=x-11$ (which is of the form $y=m x+c$ ). $\therefore$ Slope of the tangent $=1$ Now, the slope of the tangent to the given curve at the point $(x, y)$ is given by, $\frac{d y}{d x}=3 x^{2}-11$ Then, we have: $3 x^{2}-11=1$ $\Rightarrow 3 x^{2}=12$ $\Rightarrow x^{2}=4$ $\Rightarrow x=\...
Read More →Find the derivative of the following functions
Question: Find the derivative of the following functions (it is to be understood thata,b,c,d,p, q,randsare fixed non-zero constants andmandnare integers):$\frac{\sin (x+a)}{\cos x}$ Solution: Let $f(x)=\frac{\sin (x+a)}{\cos x}$ By quotient rule, $f^{\prime}(x)=\frac{\cos x \frac{d}{d x}[\sin (x+a)]-\sin (x+a) \frac{d}{d x} \cos x}{\cos ^{2} x}$ $f^{\prime}(x)=\frac{\cos x \frac{d}{d x}[\sin (x+a)]-\sin (x+a)(-\sin x)}{\cos ^{2} x}$ $\ldots$ (i) Let $g(x)=\sin (x+a)$. Accordingly, $g(x+h)=\sin (...
Read More →Find a point on the curve
Question: Find a point on the curve $y=(x-2)^{2}$ at which the tangent is parallel to the chord joining the points $(2,0)$ and $(4,4)$. Solution: If a tangent is parallel to the chord joining the points (2, 0) and (4, 4), then the slope of the tangent = the slope of the chord. The slope of the chord is $\frac{4-0}{4-2}=\frac{4}{2}=2$. Now, the slope of the tangent to the given curve at a point (x,y) is given by, $\frac{d y}{d x}=2(x-2)$ Since the slope of the tangent = slope of the chord, we hav...
Read More →Find points at which the tangent to the curve
Question: Find points at which the tangent to the curve $y=x^{3}-3 x^{2}-9 x+7$ is parallel to the $x$-axis. Solution: The equation of the given curve is $y=x^{3}-3 x^{2}-9 x+7$. $\therefore \frac{d y}{d x}=3 x^{2}-6 x-9$ Now, the tangent is parallel to thex-axis if the slope of the tangent is zero. $\therefore 3 x^{2}-6 x-9=0 \Rightarrow x^{2}-2 x-3=0$ $\Rightarrow(x-3)(x+1)=0$ $\Rightarrow x=3$ or $x=-1$ When $x=3, y=(3)^{3}-3(3)^{2}-9(3)+7=27-27-27+7=-20$ When $x=-1, y=(-1)^{3}-3(-1)^{2}-9(-1...
Read More →Find the slope of the normal to the curve
Question: Find the slope of the normal to the curve $x=1-a \sin \theta, y=b \cos ^{2} \theta$ at $\theta=\frac{\pi}{2}$. Solution: It is given that $x=1-a \sin \theta$ and $y=b \cos ^{2} \theta$. $\therefore \frac{d x}{d \theta}=-a \cos \theta$ and $\frac{d y}{d \theta}=2 b \cos \theta(-\sin \theta)=-2 b \sin \theta \cos \theta$ $\therefore \frac{d y}{d x}=\frac{\left(\frac{d y}{d \theta}\right)}{\left(\frac{d x}{d \theta}\right)}=\frac{-2 b \sin \theta \cos \theta}{-a \cos \theta}=\frac{2 b}{a}...
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