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1. Let \(z_1\) and \(z_2\) be two roots of the equation \(z^2+az+b=0\), \(z\) being complex. Further assume that the origin \(z_1\)and \(z_2\) form an equilateral triangle. Then

2. Let p and q be two statements. Amongst the following the statement that is equivalent to \(p \to q\) is

3. (Integer Answer ) Number of integral values of \(\alpha\) for which the equatin \(\frac{x^3}{3}-x=\alpha\) has 3 distinct solutions is

4. If \(y =2^{1/log_x8}\) then x equal to

5. The number of ways in which 6 men and 5 women can dine at a round table , if no two women are to sit together, is given by

6. (Integer Answer ) The number of integral terms in the expansion of \((\sqrt{3} +\sqrt[8]{5})^{256}\) is

7. If y =\(aln x +bx^2+x\) has its extremum values at \(x=-1\) and \(x=2\) then

8. If \(a > 2b >0\) then the positive value of m for which \(y =mx -b\sqrt{1+m^2}\) is a common tangent to \(x^2+y^2 =b^2/[latex] and [latex](x-a)^2 +y^2= b^2\) is

9. The sum of the series \(\frac{1}{1.2} -\frac{1}{2.3} +\frac{1}{3.4} -.....\) upto \(\infty\) is equal to

10. If \(x =\frac{2sin\alpha}{1+cos\alpha + sin\alpha}, \) then \(\frac{1-cos\alpha +sin\alpha}{1+sin\alpha}\) is equal to

11. The sum of the radii of inscribed and circumscribed circles for an n sided regular polygon of side a , is

12. If in \(\triangle \)ABC \(acos^2(\frac{C}{2}) +c cos^2(\frac{A}{2}) =\frac{3b}{2}\) then the sides a,b and c are in

13. (Integer Answer) If \(f(x) =x^n\) then the value of \(f(1) -\frac{f'(1)}{1!}+\frac{f''(1)}{2!}-\frac{f'''(1)}{3!}+......+ \frac{(-1)^n f^n (1)}{n!}\) is

14. Domain of definition of function \(f(x) =\frac{3}{4-x^2}+log_{10}(x^3-x)\) is

15. The slope of one of the common tangents to the hyperbolas \((x^2/a^2)-(y^2/b^2)=1\) and \((y^2/a^2)-(x^2/b^2)=1\) is

16. If the angle between the lines whose direction cosines are connected by the relations \(l+m+n=0\) and \(2lm+2nl-mn=0\) is \(\alpha\) , then the value of \(4sin^2\alpha\) is

17. (Integer Answer) Let f(a) =g(a) =k and their nth derivatives \(f^n(a), g^n (a) \) exist and are not equal for some n. Further , if \(\lim_{x \to a} \frac{f(a) g(x) -f(a) -g(a) f(x) +g(a)}{g(x) -f(x) }=4\) then value of k is equal to

18. If \(\alpha\) is the angle between the pair of tangents from point \((1,2)\) to the ellipse \(3x^2+2y^2=5\) then the value of \(tan\alpha[latex] is

19. If \(\int x log(1+1/x)dx = f(x) log(x+1) +g(x) x^2 +Ax +C\) then A =

20. If \(f: R \to R\) satisfies \(f(x+y) =f(x) +f(y), \) for all \(x,y \in R\) and f(1) =7 then \(\sum^n_{r=1} f(r) \) is

21. (Integer Answer) The value of the \(\lim_{x \to 0} \frac{\int^{x^2}_0 sec^2t dt}{x sinx}\) is

22. The value of the integral I = \(\int^1_0 x(1-x)^n dx\) is

23. \(\lim_{n \to \infty} \frac{1+2^4 +3^4 +....+n^4}{n^5} -\lim_{n \to \infty} \frac{1+2^3+3^3+...+n^3}{n^5}\)

24. (Integer Answer) Let \(\frac{d}{dx}F(x) =(\frac{e^{sinx}}{x}) ,x >0\) If \(\int^4_1 \frac{3}{x}e^{sinx^3}dx =F(k)-F(1)\) then one of the possible values of k is

25. The solution of the differential equation \((1+y^2) +(x-e^{tan^{-1}y})\frac{dy}{dx}=0\) is


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