GATE PYQ

Calculus

Q21.
Let u and v be two vectors in R^{2} whose Euclidean norms satisfy ||u||=2|| v|| . What is the value of \alpha such that w=u+\alphav bisects the angle between u and v ?

Q22.
The value of \lim_{x\rightarrow 1}\frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}

Q23.
\sum_{x=1}^{99}\frac{1}{x(x+1)}=__________ .

Q24.
If f(x)=R sin(\frac{\pi x}{2})+S,f'(\frac{1}{2})=\sqrt{2} and \int_{0}^{1}f(x)dx=\frac{2R}{\pi }, then the constants R and S are, respectively

Q25.
\lim_{x\rightarrow 4}\frac{sin(x-4)}{x-4}=_________.

Q26.
Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x)+ f(-x)) is 10, then the degree of (g(x)-g(-x)) is ________.

Q27.
If for non-zero x,af(x)+bf\left ( \frac{1}{x} \right )=\frac{1}{x}-25 where a\neq b then \int_{1}^{2}f(x)dx is

Q28.
\lim_{x\rightarrow 0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x} is given by

Q29.
The value of \lim_{x\rightarrow \infty }(1+x^{2})^{e^{-x}} is

Q30.
Which one of the following well-formed formulae in predicate calculus is NOT valid?