Mathematics Hothouse

Derivatives: part 4: IITJEE maths tutorial problems for practice

November 13, 2020 – 11:16 pm

Problem 1:

Given $x=x(t)$ , $y=y(t)$ , then $\frac{d^{2}y}{dx^{2}}$ is equal to

(a) $\frac{\frac{d^{2}y}{dt^{2}}}{\frac{d^{2}x}{dt^{2}}}$

(b) $\frac{\frac{d^{2}y}{dt^{2}} \times \frac{dx}{dt} - \frac{dy}{dt} \times \frac{d^{2}x}{dt^{2}}}{(\frac{dx}{dt})^{3}}$

(c) $\frac{\frac{dx}{dt} \times \frac{d^{2}y}{dt^{2}} - \frac{d^{2}x}{dt^{2}} \times \frac{dy}{dt}}{(\frac{dx}{dt})^{2}}$

(d) $\frac{1}{\frac{d^{2}x}{dy^{2}}}$

Problem 2:

$\frac{d}{dx}(\arctan{\sec{x}+ \tan{x}})$ is equal to

(a) 0 (b) $\sec{x}-\tan{x}$ (c) $\frac{1}{2}$ (d) 2

Problem 3:

If $y= \sqrt{x + \sqrt{x + \sqrt{x} + \ldots}}$ , then $\frac{dy}{dx}$ is equal to :

(a) 1 (b) \ $\frac{1}{xy}$ (c) $\frac{1}{2y-x}$ (d) $\frac{1}{2y-1}$

Problem 4:

If $f(x) = \left| \begin{array}{ccc} x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x \end{array} \right|$ , then $f^{'}(x) =$

(a) 12 (b) $6x^{2}$ (c) $6x$ (d) $12x^{2}$

Problem 5:

If $y = (\frac{x^{a}}{x^{b}}) ^{a+b} \times (\frac{x^{b}}{x^{c}})^{b+c} \times (\frac{x^{c}}{x^{a}})^{c+a}$ , then $\frac{dy}{dx}=$

(a) 0 (b) 1 (c) $a+b+c$ (d) abc

Problem 6:

If $y = \arctan{\frac{x-\sqrt{1-x^{2}}}{x+\sqrt{1-x^{2}}}}$ , then $\frac{dy}{dx}$ is equal to

(a) $\frac{1}{1-x^{2}}$ (b) $\frac{1}{\sqrt{1-x^{2}}}$ (c) $\frac{1}{1+x^{2}}$ (d) $\frac{1}{\sqrt{1+x^{2}}}$

Problem 7:

If $x=at^{2}$ , $y=2at$ , then $\frac{d^{2}y}{dx^{2}}=$

(a) $\frac{1}{t^{2}}$ (b) $\frac{1}{2at^{3}}$ (c) $\frac{1}{t^{3}}$ (d) $\frac{-1}{2at^{3}}$

Problem 8:

If $y=ax^{n+1} +bx^{-n}$ , then $x^{2}\frac{d^{2}y}{dx^{2}}=$

(a) $n(n-1)y$ (b) $ny$ (c) $n(n+1)y$ (d) $n^{2}y$

Problem 9:

If $x=t^{2}$ , $y=t^{3}$ , then $\frac{d^{2}y}{dx^{2}}=$

(a) $\frac{3}{2}$ (b) $\frac{3}{4t}$ (c) $\frac{3}{2t}$ (d) 0

Problem 10:

If $y=a+bx^{2}$ , a, b arbitrary constants, then

(a) $\frac{d^{2}}{dx^{2}} = 2xy$ (b) $x \frac{d^{2}y}{dx^{2}} - \frac{dy}{dx} + y=0$ (c) $x \frac{d^{2}y}{dx^{2}} = \frac{dy}{dx}$ (d) $x \frac{d^{2}y}{dx^{2}} = 2xy$

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

How to find square root of a binomial quadratic surd

November 12, 2020 – 10:47 pm

Assume $\sqrt{a+ \sqrt{b} + \sqrt{c} + \sqrt{d}}=\sqrt{x} + \sqrt{y} + \sqrt{z}$ ;

Hence, $a+\sqrt{b} + \sqrt{c} + \sqrt{d} = x+y+z+ 2\sqrt{xy} + 2\sqrt{yz}+ 2\sqrt{zx}$

If then, $2\sqrt{xy}=\sqrt{b}$ , $2\sqrt{yz}=\sqrt{c}$ , $2\sqrt{zx}=\sqrt{d}$ ,

And, if simultaneously, the values of x, y, z thus found satisfy $x+y+z=a$ , we shall have obtained the required root.

Example:

Find the square root of $21-4\sqrt{5}+5\sqrt{3}-4\sqrt{15}$ .

Solution:

Clearly, we can’t have anything like

$21--4\sqrt{5}+8\sqrt{3}-4\sqrt{15}=\sqrt{x} + \sqrt{y} +\sqrt{z}$

We will have to try the following options:

$21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}=\sqrt{x} - \sqrt{y} - \sqrt{z}$

$21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}=\sqrt{x}-\sqrt{y}+\sqrt{z}$

$21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}=\sqrt{x}+\sqrt{y}-\sqrt{z}$ .

Only the last option will work as we now show:

So, once again, assume that $\sqrt{21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}}=\sqrt{x}+\sqrt{y}-\sqrt{z}$

Hence, $21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}=z+y+z+2\sqrt{xy}-2\sqrt{yz}+2\sqrt{zx}$

Put $2\sqrt{xy}=8\sqrt{3}$ , $2\sqrt{xz}=4\sqrt{15}$ , $2\sqrt{yz}=4\sqrt{5}$ ;

by multiplication, $xyz = 240$ ; that is $\sqrt{xyz}=4\sqrt{15}$ ; so it follows that : $\sqrt{x}=2\sqrt{3}$ , $\sqrt{y}=2$ , $\sqrt{z}=\sqrt{5}$ .

And, since, these values satisfy the equation $x+y+z=21$ , the required root is $2\sqrt{3}+2-\sqrt{5}$ .

That is all, for now,

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in algebra | Comments (0)

IITJEE Mains or Advanced Maths Doubt Solving Tutorials

November 9, 2020 – 1:17 pm

Any maths questions from any where or any other branded class problems sets.

Contact Nalin Pithwa

By Nalin Pithwa | Posted in IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha) | Comments (0)

Derivatives: part 3: IITJEE maths tutorial problems for practice

October 28, 2020 – 2:34 pm

Problem 1:

Differential coefficient of $\log[10]{x}$ w.r.t. $\log[x]{10}$ is

(a) $\frac{(\log{x})^{2}}{(\log{10})^{2}}$ (b) $\frac{(\log[x]{10})^{2}}{(\log{10})^{2}}$ (c) $\frac{(\log[10]{x})^{2}}{(\log{10})^{2}}$ (d) $\frac{(\log{10})^{2}}{(\log{x})^{2}}$

Problem 2:

The derivative of an even function is always:

(a) an odd function (b) does not exist (c) an even function (d) can be either even or odd.

Problem 3:

The derivative of $\arcsin{x}$ w.r.t. $\arccos{\sqrt{1-x^{2}}}$ is

(a) $\frac{1}{\sqrt{1-x^{2}}}$ (b) $\arccos{x}$ (c) $1$ (d) $\arctan{(\frac{1}{\sqrt{1-x^{2}}})}$

Problem 4:

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$ , then $\frac{dy}{dx}$ is

(a) $\frac{\sqrt{1-y^{2}}}{\sqrt{1-x^{2}}}$ (b) $\sqrt{1-x^{2}}$ (c) $\frac{\sqrt{1-x^{2}}}{\sqrt{1-y^{2}}}$ (d) $\sqrt{1-y^{2}}$

Problem 5:

$\frac{d}{dx} \arcsin{2x\sqrt{1-x^{2}}}$ is equal to

(a) $\frac{2}{\sqrt{1-x^{2}}}$ (b) $\cos{2x}$ (c) $\frac{1}{2\sqrt{1-x^{2}}}$ (d) $\frac{1}{\sqrt{1-x^{2}}}$

Problem 6:

If $y=\arctan{\frac{x}{2}}-\arccos{\frac{x}{2}}$ , then $\frac{dy}{dx}$ is

(a) $\frac{2}{1+x^{2}}$ (b) $\frac{2}{4+x^{2}}$ (c) $\frac{4}{4+x^{2}}$ (d) $0$

Problem 7:

If $y=\arccos{(\frac{\sqrt{1+\sin{x}}+\sqrt{1-\sin{x}}}{\sqrt{1+\sin{x}}-\sqrt{1-\sin{x}}})}$ , then $\frac{dy}{dx}$ is equal to:

(a) $\frac{1}{2}$ (b) $\frac{2}{3}$ (c) $3$ (d) $\frac{3}{2}$

Problem 8:

If $y = \arctan{\frac{4x}{1+5x^{2}}} + \arctan{\frac{2+3x}{3-2x}}$ , then $\frac{dy}{dx}$ is

(a) $\frac{1}{1+x^{2}}$ (b) $\frac{5}{1+25x^{2}}$ (c) $1$ (d) $\frac{3}{1+9x^{2}}$

Problem 9:

If $2^{x}+2^{y}=2^{x+y}$ , then $\frac{dy}{dx}$ is equal to

(a) $\frac{2^{x}+2^{y}}{2^{x}-2^{y}}$ (b) $2^{x-y} \times \frac{2^{y}-1}{1-2^{x}}$ (c) $\frac{2^{x}+2^{y}}{1+2^{x+y}}$ (d) $\frac{2^{x+y}-2^{x}}{2^{y}}$

Problem 10:

If $y^{2}=p(x)$ , a polynomial of degree 3, then $2\frac{d}{dx}(y^{3}\frac{d^{2}y}{dx^{2}})$ is equal to

(a) $p^{'''}(x)+p^{'}(x)$ (b) $p^{''}(x).p^{'''}(x)$ (c) $p^{'''}(x).p(x)$ (d) a constant.

Regards,

Nalin Pithwa.

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Derivatives : part 2: IITJEE Maths : Tutorial problems for practice

October 26, 2020 – 4:22 am

Problem 1:

If $f(a)=2$ , $f^{'}(a)=1$ , $g(a)=-1$ , $g^{'}(a)=2$ , then the value of $\lim_{x \rightarrow a}\frac{g(x)f(a)-g(a)f(x)}{x-a}$ is

(a) -5 (b) $\frac{1}{5}$ (c) 5 (d) 0

Problem 2:

Let $y = \arcsin{(\frac{2x}{1+x^{2}})}$ , $0 < x <1$ and $0 < y < \frac{\pi}{2}$ , then $\frac{dy}{dx}$ is equal to :

(a) $\frac{2}{1+x^{2}}$ (b) $\frac{2x}{1+x^{2}}$ (c) $\frac{-2}{1+x^{2}}$ (d) none

Problem 3:

Let $f(x) = ax^{2}+1$ for $x \leq 1$

and $f(x)= x+a$ for $x \leq 1$ then f is derivable at $x=1$ , if

(a) $a=0$ (b) $a = \frac{1}{2}$ (c) $a=1$ (d) $a=2$

Problem 4:

If $f(x) = ax^{2}+b$ for $x \leq 1$

if $f(x)=b x^{2}+ax+c$ for $x>1$ , where $b \neq 0$ , then $f(x)$ is continuous and differentiable at $x=1$ , if

(a) $c=0, a=2b$ (b) $a=2b, c \in \Re$ (c) $a=b, c=0$ (d) $a=2b, c \neq 0$

Problem 5:

$\lim_{h \rightarrow 0} \frac{\cos^{2}(x+h)- \cos^{2}(x)}{h}$ is equal to

(a) $\cos^{2}(x)$ (b) $-\sin{2x}$ (c) $\sin{x} \cos{x}$ (d) $2\sin{x}$

Problem 6:

$\lim_{h \rightarrow 0} \frac{\sin{\sqrt{x+h}-\sin{\sqrt{x}}}}{h}$ is equal to

(a) $\cos {\sqrt{x}}$ (b) $\frac{1}{2\sin{\sqrt{x}}}$ (c) $\frac{\cos{\sqrt{x}}}{2\sqrt{x}}$ (d) $\sin{\sqrt{x}}$

Problem 7:

$(\arccos{x})^{'}= \frac{-1}{\sqrt{1-x^{2}}}$ where

(a) $-1 < x <1$ (b) $-1 \leq x \leq 1$ (c) $-1 \leq x < 1$ (d) $-1 < x \leq 1$

Problem 8:

$\frac{d}{dx}(\arctan{(\frac{3x-x^{2}}{1-3x^{2}})})$ is equal to

(a) $\frac{3}{1+x^{2}}$ (b) $\frac{3}{1+9x^{2}}$ (c) $\sec^{2}{x}$ (d) $\frac{1}{9+x^{2}}$

Problem 9:

If $x=a\cos^{3}(t)$ and $y=a\sin^{3}(t)$ , then $\frac{dy}{dx}$ is equal to

(a) $\cos{t}$ (b) $\cot{t}$ (c) $cosec{(t)}$ (d) $-\tan{t}$

Problem 10:

If $y = arcsin{\cos{x}}$ , then $\frac{dy}{dx}$ is equal to

(a) -1 (b) $\cos{t}$ (c) $cosec{(t)}$ (d) $-\tan{t}$

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Derivatives: part 1: IITJEE Maths Tutorial Problems Practice

October 26, 2020 – 12:36 am

Problem 1:

If $y=x^{x}$ , $x>0$ , then find $\frac{dy}{dx}$ .

Problem 2:

If $y= x^{x^{x^{\ldots}}}$ , then find the value of $x\frac{dy}{dx}$ .

Problem 3:

Find the derivative of $e^{\ln{x}}$ w.r.t. x.

Problem 4:

Let $f(x) = \log{(x+\sqrt{x^{2}+1})}$ , then find the value of $f^{'}(x)$ .

Problem 5:

If $y= \arctan{\frac{\sqrt{1+x^{2}}-1}{x}}$ , then find the value of $y^{'}(0)$ .

Problem 6:

If $y=t^{2}+t-1$ , then find the value of $\frac{dy}{dx}$ .

Problem 7:

If $x=a(t-\sin{t})$ , $y=a(1+\cos{t})$ , then evaluate $\frac{dy}{dx}$ .

Problem 8:

If $x^{y}=e^{x-y}$ , then evaluate $\frac{dy}{dx}$ .

Problem 9:

If $y= \sec^{-1}{(\frac{x+1}{x-1})} + \arcsin{(\frac{x-1}{x+1})}$ , then evaluate $\frac{dy}{dx}$ .

Problem 10:

If $y = \arctan{(\frac{\sin{x}+\cos{x}}{\cos{x}-\sin{x}})}$ , then find $\frac{dy}{dx}$

Problem 11:

If $\sqrt{x}+\sqrt{y}=4$ , then evaluate $\frac{dy}{dx}$ at $y=1$ .

Problem 12:

If $f(x) = \frac{x-4}{2\sqrt{x}}$ , then evaluate $f^{'}(0)$ .

Problem 13:

If $f^{'}(x) = \sin{\log{x}}$ and $y=f(\frac{2x+3}{3-2x})$ , find $\frac{dy}{dx}$ . One of the given choices is correct:

(a) $\frac{12\cos{(\log{x})}}{x(3-2x)^{2}}$

(b) $\frac{12\sin{\log{(\frac{2x+3}{3-2x})}}}{(3-2x)^{2}}$

(d) none of these

Problem 14:

If $f(0)=0=g(0)$ and $f^{'}(0)=6=g^{'}(0)$ , then $\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}$ is given by:

(a) 1 (b) 0 (c) 12 (d) -1

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Limits and Continuity: part 11: IITJEE Maths tutorial problems for practice

October 18, 2020 – 8:00 am

Problem 1:

A function $f(x)$ is defined as follows:

$f(x) = \frac{(e^{2x}-1)(1-\cos{x})}{\tan^{2}{(x)}\log{(1+2x)}}$ when $x \neq 0$

$f(0) = \log{a}$ is continuous at $x=0$ .

The value of a should be

(i) $\frac{e}{2}$ (b) $\frac{1}{2e}$ (c) 2 (d) none

Problem 2:

If $f(x) = \frac{e^{(2x)} + e^{(-2x)} -2}{1-\cos{(4x)}}$ when $x \neq 0$ is continuous at $x=0$ , then what is the value of $f(0)$

Problem 3:

Given $f(x) = x + a$ , when $-1 \leq x \leq 0$

and $f(x) = x + b$ when $0 < x \leq 1$

and $f(x) = c -x$ when $1 < x \leq 2$

if f is continuous at $x=0$ and $x=1$ and $f(2)=1$ , then the value of $3a+b-2c=$

(i) 0 (ii) 1 (iii) 2 (iv) 3

Problem 4:

If the function $f(x)$ is continuous on its domain where

$f(x) = x^{2} + ax + b$ for $0 \leq x < 2$

$f(x)=4x-1$ for $2 \leq x < 4$

$f(x)=ax^{2+17b}$ for $4 \leq x \leq 6$

then the quadratic equation whose roots are 2a and 2b is:

(i) $x^{2}+2x-8$ (b) $x^{2}-2x-8=0$ (c) $x^{2}+2x+8$ (d) $x^{2}-2x+8=0$

Problem 5:

The value of c for which the function

$f(x) = \frac{\sin{(x)} + \sin{((a+1)x)}}{x}$ when $x<0$

$f(x) = c$ when $x=0$

$f(x) = \frac{(x+bx^{2})^{\frac{1}{2}}-x^{\frac{1}{2}}}{bx^{\frac{3}{2}}}$

is continuous at $x=0$ is

(i) 1/2 (ii) -1/2 (iii) 2 (iv) -2

Problem 6:

If $f(x) = \frac{\sin{x\pi}}{x-1}+a$ when $x<1$

$f(x) = 2x$ , when $x=1$

$f(x)= \frac{1+\cos{x\pi}}{\pi (1-x)^{2}} + b$ when $x>1$

is continuous at $x=1$ , then a and b have the values:

(i) $3\pi, 3\frac{\pi}{2}$ (ii) $3\pi, \frac{\pi}{2}$ (iii) $\pi, \frac{\pi}{2}$ (iv) $\pi, 3\frac{\pi}{2}$

Problem 7:

If $f(x) = \frac{(\sin{x} - \cos{x})^{2}}{\sqrt{2}-\sin{x}-\cos{x}}$ , when $x \neq \frac{\pi}{4}$ is continuous at $x=\frac{\pi}{4}$ then $f(\frac{\pi}{4})=$

(a) 1/2 (b) -1/2 (c) 2 (d) none of these

Problem 8:

If $f(x)= \frac{x+1}{x+2}$ and $g(x)=\frac{1}{x}$ , then $\lim_{x \rightarrow 2} (g+f)(x)=$

(i) 4/3 (b) 5/3 (c) 2 (d) 7/3

Problem 9:

Evaluate the following: $\lim_{x \rightarrow 4} \frac{(x^{2}-x-12)^{18}}{(x^{3}-8x^{2}+16x)^{9}}$

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Wisdom of George Polya

October 17, 2020 – 6:50 am

A great discovery solves a great problem. But there is a grain of discovery in the solution of any problem.

— George Polya, How to Solve it

By Nalin Pithwa | Posted in mathematicians, memory power concentration retention, miscellaneous, motivational stuff | Comments (0)

Limits and Continuity: Part 10: Tutorial Problems for IITJEE Maths

October 16, 2020 – 3:24 pm

Problem 1:

The point of discontinuity of the function:

$f(x) = \frac{1}{\sin{x} - \cos{x}}$ in the closed interval $[0, \frac{\pi}{2}]$ are:

(a) 0 and $\frac{\pi}{2}$ (b) $\frac{\pi}{2}$ and $\frac{\pi}{4}$

Problem 2:

Given $f(x) = \frac{x^{2}-9}{x-3}$ for $0 \leq x <3$ and $f(x) = 4x-5$ for $3 \leq x \leq 6$

Consider:

(i) f(x) is discontinuous in $(0,3)$

(ii) f(x) is discontinuous in $(3,6)$

(iii) f(x) is continuous in $[0,6]$

(iv) $\lim_{x \rightarrow 3} f(x)$ exists.

Which of the above statements are false?

(a) only 1 and 2 (b) only 2 and 4 (c) only 1 and 3 (d) none of a, b, c

Problem 3:

For the following function:

$f(x) = \frac{x^{2}-3x+2}{x-3}$ for $0 \leq x \leq 4$ , and

$f(x) = \frac{x^{2}+1}{x-2}$ for $4 < x \leq 6$

Consider

(i) f(x) is discontinuous in $(0,4)$

(ii) f(x) is discontinuous in $(4,6)$

(iii) f(x) is discontinuous in $[0,6]$

(iv) $\lim_{x \rightarrow 3}f(x)$ exists

Which of the above statements are true?

(a) only 1 and 2 (b) only 2 and 4 (c) only 1 and 3 (d) none of a, b and c

Problem 4:

If the function $f(x)$ where

$f(x) = \frac{(3^{x}-1)^{2}}{\tan{x} \log{(1+x)}}$ for $x \neq 0$

$f(x) = \log{k} . \log{\sqrt{3}}$ for $x=0$

is continuous at $x=0$ , then $k=$

(a) 6 (b) $\sqrt{3}$ (c) 9 (d) $\frac{3}{2}$

Problem 5:

At $x = \frac{3 \pi}{4}$ , the function $f(x)$ where

$\frac{\cos{x} + \sin{x}}{3\pi -4x}$ , where $x \neq \frac{3\pi}{4}$

$f(x) = \frac{1}{\sqrt{2}}$ where $x = \frac{3\pi}{4}$

has

(a) removable discontiuity

(b) irremovable discontinuity

(d) none

Problem 6:

If $f(x)$ is given to be continuous at $x=0$ , where

$f(x) = \frac{(e^{kx}-1) \sin{(kx)}}{x^{2}}$ for $x \neq 0$ and $f(0)=4$ , then the value of k is:

(a) 2 (b) -2 (c) $\pm {2}$ (d) $\pm {\sqrt{2}}$

Problem 7:

If given function $f(x)$ is continuous at zero and if

$f(x) = \frac{4^{x}-2^{x+1}+1}{1-\cos{x}}$ when $x \neq 0$ and $f(0)=k$ , then the value of k is :

(a) $\frac{1}{2}(\log{2})^{2}$ (b) $2(\log{2})^{2}$ (c) $4 \log{2}$ (d) $\frac{1}{4} \log{2}$

Problem 8:

If $f(x)$ is continuous at $x=3$ , where

$f(x) = \frac{(2^{x}-8) \log{(x-2)}}{1- \cos{(x-3)}}$ when $x \neq 3$ and $f(3)=k$ then the value of k is:

(a) $16 \log{2}$ (b) $4 \log{2}$ (c) $8\log{2}$ (d) $2 \log{2}$

Problem 9:

A function $f(x)$ is defined as follows:

$f(x) = \frac{ab^{x}-ba^{x}}{x^{2}-1}$ where $x \neq 1$ and $f(1)=k$ is continuous at $x=1$ , then find the value of k.

Problem 10:

At the point $x=0$ the function $f(x)$ where

$f(x) = \frac{\log{\sec^{2}{(x)}}}{x \sin{x}}$ , when $x \neq 0$

$f(x) =e$ when $x=0$ possesses

(a) removable discontinuity

(b) irremovable discontinuity

(d) none

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Limits and Continuity: Part 9: Tutorial Practice for IITJEE Maths

October 16, 2020 – 1:28 pm

Problem 1:

If $f(x) = \frac{(5^{\sin{x}}-1)^{2}}{x \log{(1+2x)}}$ , where $x \neq 0$ is continuous at zero, then find the value of $f(0)$ .

Problem 2:

If $f(x) = 2x + a$ for $0 \leq x <1$ and $f(x) = 3x+b$ for $1 \leq x \leq 2$ is continuous at $x=1$ and $a+b=1$ , then the find the value of $3a-4b$ .

Problem 3:

If $f(x) = \frac{2^{3x}-3^{x}}{x}$ for $x<0$ and $f(x) = \frac{1}{x} \log{(1+ \frac{8x}{3})}$ for $x>0$ .

Consider the following statements:

i) $\lim_{x \rightarrow 0} f(x)$ does not exist.

ii) $\lim_{x \rightarrow 0^{+}} f(x)$ exists but $f(0)$ is not defined.

iii) $f(x)$ is discontinuous at zero

iv) $\lim_{x \rightarrow 0^{-}} f(x)$ exists, but $f(0)$ is not defined.

Which of the above statements are false?

(a) all four (b) (ii) and (iv) (c) (i) and (iii) (d) none

Problem 4:

If $f(x) = \frac{\log{x} - \log{2}}{x-2}$ for $x >2$ and $f(x) = \frac{1-\cos{(x-2)}}{(x-2)^{2}}$ for $x <2$

Consider the following statements:

(i) $\lim_{x \rightarrow 2^{-}} f(x)$ does not exist.

(ii) $\lim_{x \rightarrow 2^{+}}$ does not exist.

(iii) $f(x)$ is continuous at $x=2$

(iv) $f(x)$ is discontinuous at $x=2$ .

Which of the above statements are true?

(a) none (b) iv (c) iii (d) ii

Problem 5:

If the function f is continuous at $x=0$ and is defined by

$f(x) = \frac{\sin{4x}}{5x}+a$ for $x>0$

$f(x) = x+4-b$ for $x <0$

$f(x) = 1$ for $x =0$

The quadratic equation whose roots are values of 5a and 2b is

(a) $x^{2}-2x+3=0$ (b) $x^{2} + 3x +2=0$

Problem 6:

The function $f(x) = \frac{(e^{3x}-1)^{2}}{x \log{(1+3x)}}$ for $x \neq 0$ and $f(0)=\frac{1}{3}$

(a) has a removable discontinuity at $x=0$

(b) has irremovable discontinuity at $x=0$

(d) none of the above.

Problem 7:

If $f(x)$ is continuous in $[0,8]$ and

$f(x) = x^{2} + ax + b$ when $0 \leq x <2$

$f(x) = 3x+2$ when $2 \leq x \leq 4$

$f(x) = 2ax + 5b$ when $4 < x \leq 8$

Then, values of a and b are:

(a) 3,2 (b) 1, -2 (c) -3, 2 (d) 3,-2

Problem 8:

The value of $a^{2} - b^{2}$ if f is continuous on $[-\pi, \pi]$ where

$f(x) = -2\sin{x}$ for $-\pi \leq x \leq -\frac{\pi}{2}$

$f(x) = a \sin{x} + b$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$

$f(x) = \cos{x}$ for $\frac{\pi}{2} \leq x \leq \pi$ is

(a) 0 (b) 2 (c) $\infty$ (d) indeterminate

Problem 9:

Given $f(x) = \frac{x^{2}+3x+5}{x^{2}-7x+10}$ . Let $A \equiv [-2,3]$ and $B \equiv [6,10]$ then

(a) f(x) is continuouis in B but discontinuous in A

(b) f(x) is discontinuous in B but continuous in A

(d) f(x) is discontinuous in both A and B

Problem 10:

The function $f(x) = \frac{2x^{2}-x+7}{x^{2}-4x+5}$ is

(a) continuous for all real values of x

(b) discontinuous for all real values of x

(d) discontinuous at $x=2$ and $x=3$

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Mathematics Hothouse

Derivatives: part 4: IITJEE maths tutorial problems for practice

How to find square root of a binomial quadratic surd

IITJEE Mains or Advanced Maths Doubt Solving Tutorials

Derivatives: part 3: IITJEE maths tutorial problems for practice

Derivatives : part 2: IITJEE Maths : Tutorial problems for practice

Derivatives: part 1: IITJEE Maths Tutorial Problems Practice

Limits and Continuity: part 11: IITJEE Maths tutorial problems for practice

Wisdom of George Polya

Limits and Continuity: Part 10: Tutorial Problems for IITJEE Maths

Limits and Continuity: Part 9: Tutorial Practice for IITJEE Maths

Pages

Categories

Archives

Search

Blogroll

RSS Feeds

Meta