Author Archives: Nalin Pithwa

I am a DSP Engineer and a mathematician working in closely related areas of DSP, Digital Control, Digital Comm and Error Control Coding. I have a passion for both Pure and Applied Mathematics.

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

Wisdom of George Polya

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

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

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}

(c) \frac{\pi}{4} and 0 (d) \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

(c) no 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

(c) no discontinuity

(d) none

Regards,

Nalin Pithwa

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

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

(c) x^{2}-3x =2=0 (d) none

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

(c) is continuous 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

(c) f(x) is continuous in both A and B

(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

(c) discontinuous at x=1 and x=5

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

Regards,

Nalin Pithwa