Derivatives: part 7: IITJEE tutorial problems practice

Problem 1: Differential coefficient of \sec{\arctan{x}} is

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

Problem 2: If \sin{(x+y)} = \log{(x+y)}, then \frac{dy}{dx} is equal to :

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

Problem 3: If y = \arcsin{\sqrt{x-ax}-\sqrt{a-ax}}, then \frac{dy}{dx} is equal to:

(a) \frac{1}{2\sqrt{x}\sqrt{1-x}} (b) \sin{(\sqrt{x})} \times \sin{(\sqrt{a})}

(c) \frac{1}{\sin{\sqrt{a-ax}}} (d) zero

Problem 4: For the differentiable function f, the value of : \lim_{h \rightarrow 0} \frac{(f(x+h))^{2}-(f(x))^{2}}{2h} is equal to:

(a) (f^{'}(x))^{2} (b) \frac{1}{2}(f(x))^{2} (c) f(x)f^{'}(x) (d) zero

Problem 5: The derivative of \arctan{\frac{\sqrt{1+x^{2}}-1}{x}} w.r.t. \arctan{(\frac{2x\sqrt{1-x^{2}}}{1-2x^{2}})} at x=0 is :

(a) \frac{1}{8} (b) \frac{1}{4} (c) \frac{1}{2} (d) 1

Problem 6: If x = e^{y+e^{y+e^{y+e^{y+ \ldots}}}} then \frac{dy}{dx} is

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

Problem 7: Consider the following statements:

(1) (\frac{f}{g})^{'} = \frac{f^{'}}{g^{'}} (2) \frac{(fg)^{'}}{fg} = \frac{f^{'}}{f} + \frac{g^{'}}{g}

(3) \frac{(f+g)^{'}}{f+g} = \frac{f^{'}}{f} + \frac{g^{'}}{g} (4) \frac{(f/g)^{'}}{f/g} = \frac{f^{'}}{f} + \frac{g^{'}}{g}

Which of the following statements are true?

(a) 1 and 2 (b) 2 and 3 (c) 2 and 4 (d) 3 and 4

Problem 8: If y=e^{x+3\log{x}} then \frac{dy}{dx} =

(a) e^{x+3\log{x}} (b) e^{x}.x^{2}(x+3) (c) e^{x}. e^{3\log{x}} (d) 3x^{2}e^{x}

Problem 9: If y=\sin^{2}(x \deg), then find the value of \frac{dy}{dx} is:

(a) \frac{\pi}{360}\sin{(2 x \deg)} (b) \frac{\pi}{2}\sin{(2x\deg)} (c) 180 \sin {(2x\deg)} (d) \frac{\pi}{180}\sin{(2x\deg)}

Problem 10: If y=\log_{a}{x} + \log_{x}{a} + \log_{x}{x}+ \log_{a}{a} then the value of \frac{dy}{dx} is:

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

Cheers,

Nalin Pithwa.

Derivatives: part 6: IITJEE tutorial practice problems

Problem 1:

If \sec {(\frac{x+y}{x-y})}=a, then \frac{dy}{dx} is (i) \frac{x}{y} (ii) \frac{y}{x} (iii) y (iv) x

Problem 2:

If f(x) = x+ 2, when -1<x<1;

f(x)=5, when x=3;

f(x) = 8-x, when x>3; then, at x=3, the value of f^{'}(x) is

(a) 1 (b) -1 (c) 0 (d) does not exist.

Problem 3:

If y = x \tan{y}, then \frac{dy}{dx} is equal to

(i) \frac{\tan{y}}{x-x^{2}-y^{2}} (ii) \frac{\tan{y}}{y-x}

(iii) \frac{y}{x-x^{2}-y^{2}} (iv) \frac{\tan{x}}{x-y^{2}}

Problem 4:

If g is the inverse function of f and f^{'}(x) = \frac{1}{1+x^{n}}, then g^{'}(x) is equal to

(i) 1 + (g(x))^{n} (ii) 1+g(x) (iii) 1-g(x) (iv) 1-(g(x))^{n}

Problem 5:

If f(x) = \log_{x^{2}}(\log{x}) then f(x) at x=c is :

(i) 0 (ii) 1 (iii) \frac{1}{e} (iv) \frac{1}{2e}

Problem 6:

If y = (\sin{x})^{\tan{x}} then \frac{dy}{dx} is equal to :

(i) (\sin{x})^{\tan{x}}(1+ \sec^{2}{x} \log{\sin{x}})

(ii) \tan{x}. (\sin{x})^{\tan{x}-1} \times \cos{x}

(iii) (\sin{x})^{\tan{x}}\sec^{2}{x} \times \log{\sin{x}}

(iv) \tan{x} (\sin{x})^{\tan{x}-1}

Problem 7:

If y = \sqrt{\sin{x}+y}, then \frac{dy}{dx} equals:

(i) \frac{\sin{x}}{2y-1} (ii) \frac{\sin{x}}{1-2y} (iii) \frac{\cos{x}}{1-2y}

(iv) \frac{\cos{x}}{2y-1}

Problem 8:

If x = \sqrt{\frac{1-t^{2}}{1+t^{2}}} and y = \sqrt{\frac{\sqrt{1+t^{2}}-sqrt{1-t^{2}}}{\sqrt{1+t^{2}}+\sqrt{1-t^{2}}}}

then the value of \frac{d^{2}y}{dx^{2}} at t=0 is given by:

(a) 0 (b) 1/2 (c) 1 (d) -1

Problem 9:

If x = a \cos^{3}{\theta}, y = a \sin^{3}{\theta}, then \sqrt{1 + (\frac{dy}{dx})^{2}} is equal to:

(i) \sec^{2}{\theta} (ii) \tan^{2}{\theta} (iii) \sec{\theta} (iv) |\sec{\theta}|

Problem 10:

If y = \arcsin{\sqrt{1-x}} + \arccos{\sqrt{x}}, then \frac{dy}{dx} equals:

(a) \frac{1}{\sqrt{x(1-x)}} (b) \frac{1}{x(1+x)} (c) \frac{-1}{\sqrt{x(1-x)}} (d) none

Regards,

Nalin Pithwa

Derivatives: part 5: IITJEE maths tutorial problems for practice

Problem 1:

The derivative of arcsec (\frac{1}{1-2x^{2}}) w.r.t. \sqrt{1-x^{2}} at x=\frac{1}{2} is

(a) 2 (b) -4 (c) 1 (d) -2

Problem 2:

If y = \sin{\sin{x}} and \frac{d^{2}y}{dx^{2}} + \frac{dy}{dx} \tan{x} + f(x)=0, then f(x) =

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

Problem 3:

If f(x) = \log_{a}{\log_{a}{x}}, then f^{'}(x) is

(a) \frac{\log_{a}{e}}{x \log_{e}{x}} (b) \frac{\log_{e}{a}}{x} (c) \frac{\log_{e}{a}}{x\log_{a}{x}} (d) \frac{x}{\log_{e}{a}}

Problem 4:

If y=\log {\tan{\frac{x}{2}}} + \arcsin{\cos{x}}, then \frac{dy}{dx} is

(a) cosec (x) -1 (b) cosec (x) +1 (c) cosec (x) (d) x

Problem 5:

If y^{x}=x^{y}, then \frac{dy}{dx} is

(a) \frac{y}{x} (b) \frac{x}{y} (c) \frac{y(x\log{y}-y)}{x(y\log{x}-x)} (d) \frac{x \log{y}}{y \log{x}}

Problem 6:

Let f, g, h and k be differentiable in (a,b), if F is defined as F(x) = \left | \begin{array}{cc} f(x) & g(x) \\ h(x) & k(x) \end{array} \right | for all a, b, then F^{'} is given by:

(i) \left | \begin{array}{cc} f & g \\ h & k \end{array} \right| + \left | \begin{array}{cc}f & g \\ h^{'}  & k \end{array} \right |

(ii) \left | \begin{array}{cc}f & g^{'} \\ h & k^{'} \end{array}\right | + \left | \begin{array}{cc} f^{'} & g \\ h & k^{'} \end{array} \right |

(iii) \left | \begin{array}{cc}f^{'} & g^{'} \\ h & k \end{array} \right | + \left | \begin{array}{cc}f & g \\ h^{'} & h^{'} \end{array} \right |

(iv) \left | \begin{array}{cc}f & g \\ h^{'} & k^{'} \end{array} \right | + \left | \begin{array}{cc}f^{'} & g \\h & k \end{array} \right |

Problem 7:

If pv=81, then \frac{dp}{dv} at v=9 is equal to:

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

Problem 8:

If x^{2}+y^{2}=1, then

(i) yy^{''}-2(y^{'})^{2}+1=0 (ii) yy^{''} - (y^{'})^{2}-1=0 (iii) yy^{''} + (y^{'})^{2} + 1 = 0 (iv) yy^{''} - 2(y^{'})^{2}-1=0

Problem 9:

If y = \arctan{\frac{\sqrt{x}-1}{\sqrt{x}+1}} + \arctan{\frac{\sqrt{x}+1}{\sqrt{x}-1}}, then the value of \frac{dy}{dx} will be

(i) 0 (ii) 1 (iii) -1 (iv) - \frac{1}{2}

Problem 10:

Let f(x) = \left | \begin{array}{ccc} x^{3} & \sin{x} & \cos{x} \\ 0 & -1 & 0 \\ p & p^{2} & p^{3} \end{array} \right |, where p is a constant, then \frac{d^{3}}{dx^{3}}(f(x)) at x=0 is

(a) p (b) p+p^{2} (c) p+p^{3} (d) independent of p

Regards,

Nalin Pithwa

Derivatives: part 4: IITJEE maths tutorial problems for practice

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

How to find square root of a binomial quadratic surd

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

IITJEE Mains or Advanced Maths Doubt Solving Tutorials

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

Contact Nalin Pithwa

Derivatives: part 3: IITJEE maths tutorial problems for practice

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.

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

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

Derivatives: part 1: IITJEE Maths Tutorial Problems Practice

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}}

(c) \frac{12\cos{\log{(\frac{2x+3}{3-2x})}}}{x(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

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

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