MCQOPTIONS
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MCQOPTIONS
This section includes 657 Mcqs, each offering curated multiple-choice questions to sharpen your Testing Subject knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Let \[f(x)=\left\{ \begin{align} & {{x}^{p}}\sin \frac{1}{x},x\ne 0 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\] then \[f(x)\]is continuous but not differential at \[x=0\] if [DCE 2005] |
| A. | \[0<p\le 1\] |
| B. | \[1\le p<\infty \] |
| C. | \[-\infty <p<0\] |
| D. | p = 0 |
| Answer» B. \[1\le p<\infty \] | |
| 2. |
Let \[g(x)=x.\,f(x),\]where \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\] at \[x=0\] [IIT Screening 1994; UPSEAT 2004] |
| A. | g is differentiable but g' is not continuous |
| B. | g is differentiable while f is not |
| C. | Both f and g are differentiable |
| D. | g is differentiable and g' is continuous |
| Answer» B. g is differentiable while f is not | |
| 3. |
Let \[f(x)=\frac{1-\tan x}{4x-\pi },\ x\ne \frac{\pi }{4},\ \ x\in \left[ 0,\frac{\pi }{2} \right]\], If \[f(x)\]is continuous in \[\left[ 0,\frac{\pi }{2} \right]\], then \[f\left( \frac{\pi }{4} \right)\]is [AIEEE 2004] |
| A. | -1 |
| B. | \[\frac{1}{2}\] |
| C. | \[-\frac{1}{2}\] |
| D. | 1 |
| Answer» D. 1 | |
| 4. |
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{{{\sin }^{-1}}x}=\] [AI CBSE 1989, 90; DSSE 1989] |
| A. | 2 |
| B. | 1 |
| C. | ?1 |
| D. | None of these |
| Answer» C. ?1 | |
| 5. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{a}^{\cot x}}-{{a}^{\cos x}}}{\cot x-\cos x}=\] [Kerala (Engg.) 2001; J & K 2005] |
| A. | \[\log a\] |
| B. | \[\log 2\] |
| C. | a |
| D. | log x |
| Answer» B. \[\log 2\] | |
| 6. |
\[\underset{x\to \infty }{\mathop{\lim }}\,(\sqrt{{{x}^{2}}+1}-x)\]is equal to [RPET 1995] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 7. |
\[\underset{x\to -1}{\mathop{\lim }}\,\frac{\sqrt{\pi }-\sqrt{{{\cos }^{-1}}x}}{\sqrt{x+1}}\]is given by |
| A. | \[\frac{1}{\sqrt{\pi }}\] |
| B. | \[\frac{1}{\sqrt{2\pi }}\] |
| C. | 1 |
| D. | 0 |
| Answer» C. 1 | |
| 8. |
The value of \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{3x-4}{3x+2} \right)}^{\frac{x+1}{3}}}\] is equal to [Pb. CET 2004] |
| A. | \[{{e}^{-1/3}}\] |
| B. | \[{{e}^{-2/3}}\] |
| C. | \[{{e}^{-1}}\] |
| D. | \[{{e}^{-2}}\] |
| Answer» C. \[{{e}^{-1}}\] | |
| 9. |
\[\underset{x\to 0}{\mathop{\text{lim}}}\,\frac{{{x}^{3}}}{\sin {{x}^{2}}}=\] [AISSE 1984; AI CBSE 1984] |
| A. | 0 |
| B. | \[\frac{1}{3}\] |
| C. | 3 |
| D. | \[\frac{1}{2}\] |
| Answer» B. \[\frac{1}{3}\] | |
| 10. |
Let \[f(x)=\left\{ \begin{align} & {{x}^{2}}+k,\ \ \ \ \text{when}\ \ x\ge 0 \\ & -{{x}^{2}}-k,\ \ \text{when }x |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | ?2 |
| Answer» B. 1 | |
| 11. |
If \[f(x)=\left\{ \begin{align} & \frac{1}{x}\sin {{x}^{2}},\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\], then |
| A. | \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)\ne 0\] |
| B. | \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)\ne 0\] |
| C. | f(x) is continuous at\[x=0\] |
| D. | None of these |
| Answer» D. None of these | |
| 12. |
If \[{{S}_{n}}=\sum\limits_{k=1}^{n}{{{a}_{k}}}\]and\[\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a,\]then \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{S}_{n+1}}-{{S}_{n}}}{\sqrt{\sum\limits_{k=1}^{n}{k}}}\]is equal to |
| A. | 0 |
| B. | a |
| C. | \[\sqrt{2}a\] |
| D. | \[2a\] |
| Answer» B. a | |
| 13. |
If \[f(x)=\left\{ \begin{matrix} \frac{{{x}^{2}}-9}{x-3}\,, & \text{if }x\ne 3 \\ 2x+k\,, & \text{otherwise} \\ \end{matrix} \right.\], is continuous at \[x=3,\] then \[k=\] [Kerala (Engg.) 2002] |
| A. | 3 |
| B. | 0 |
| C. | ?6 |
| D. | 1/6 |
| Answer» C. ?6 | |
| 14. |
If \[f(x)=\left\{ \begin{align} & x\sin \frac{1}{x},\,\,x\ne 0 \\ & \,\,\,\,\,\,\,\,\,\,\,\,k,\,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\], then the value of k is [MP PET 1999; AMU 1999; RPET 2003] |
| A. | 1 |
| B. | ?1 |
| C. | 0 |
| D. | 2 |
| Answer» D. 2 | |
| 15. |
Which one of the following is a objective function on the set of real numbers [Kerala (Engg.) 2002] |
| A. | \[2x-5\] |
| B. | \[|x|\] |
| C. | \[{{x}^{2}}\] |
| D. | \[{{x}^{2}}+1\] |
| Answer» B. \[|x|\] | |
| 16. |
\[\underset{x\to 2}{\mathop{\lim }}\,\frac{|x-2|}{x-2}=\] [AI CBSE 1985] |
| A. | 1 |
| B. | ?1 |
| C. | Does not exist |
| D. | None of these |
| Answer» D. None of these | |
| 17. |
The equation \[{{\log }_{3}}({{3}^{x}}-8)=2-x\] has the solution |
| A. | \[x=1\] |
| B. | \[x=2\] |
| C. | \[x=3\] |
| D. | \[x=4\] |
| Answer» C. \[x=3\] | |
| 18. |
The solution set of the inequalities \[3x-7>2(x-6)\] and \[6-x>11-2x,\] is |
| A. | \[(-5,\infty )\] |
| B. | \[[5,\infty )\] |
| C. | \[(5,\infty )\] |
| D. | \[[-5,\infty )\] |
| Answer» D. \[[-5,\infty )\] | |
| 19. |
The marks obtained by a student of class \[XI\]in first and second terminal examinations are 62 and 48, respectively. The minimum marks he should get in the annual examination to have an average of at least 60 marks, are |
| A. | 70 |
| B. | 50 |
| C. | 74 |
| D. | 48 |
| Answer» B. 50 | |
| 20. |
A company manufactures cassettes. Its cost and revenue functions are \[C(x)=26000+30x\] and \[R(x)=43x,\] respectively, where x is the number of cassettes produced and sold in a week. The number of cassettes must be sold by the company to realise some profit, is |
| A. | More than 2000 |
| B. | Less than 2000 |
| C. | More than 1000 |
| D. | Less than 1000 |
| Answer» B. Less than 2000 | |
| 21. |
The equation \[\left| |x-1|+a \right|=4\] can have real solution for x if 'a' belongs to the interval |
| A. | \[(-\infty ,+\infty )\] |
| B. | \[(-\infty ,4]\] |
| C. | \[(4,+\infty )\] |
| D. | \[[-4,4]\] |
| Answer» C. \[(4,+\infty )\] | |
| 22. |
The set of real values of x satisfying \[\left| x-1 \right|\le 3\] And \[\left| x-1 \right|\ge 1\] is |
| A. | \[[2,4]\] |
| B. | \[(-\infty ,2]\cup [4,+\infty )\] |
| C. | \[[-2,0]\cup [2,4]\] |
| D. | None of these |
| Answer» D. None of these | |
| 23. |
The number of integral roots of the equation\[\left| x-1 \right|+\left| x+2 \right|-\left| x-3 \right|=4\] is |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» D. 4 | |
| 24. |
If \[\frac{3x-4}{2}\ge \frac{x+1}{4}-1,\] then \[x\in \] |
| A. | \[[1,\infty )\] |
| B. | \[(1,\infty )\] |
| C. | \[(-5,5)\] |
| D. | \[[-5,5]\] |
| Answer» B. \[(1,\infty )\] | |
| 25. |
Solution set of the inequality \[{{\log }_{3}}(x+2)(x+4)+lo{{g}_{1/3}}(x+2) |
| A. | \[(-2,-1)\] |
| B. | \[(-2,3)\] |
| C. | \[(-1,3)\] |
| D. | \[(3,\infty )\] |
| Answer» C. \[(-1,3)\] | |
| 26. |
For positive real numbers a, b, c such that \[a+b+c=p,\] which one does not hold? |
| A. | \[(p-a)(p-b)(p-c)\le \frac{8}{27}{{p}^{3}}\] |
| B. | \[(p-a)(p-b)(p-c)\ge 8abc\] |
| C. | \[\frac{bc}{a}+\frac{ca}{b}+\frac{ab}{c}\le p\] |
| D. | None of these |
| Answer» D. None of these | |
| 27. |
The shaded region shown in the figure is given by the in equations |
| A. | \[14x+5y\ge 70,\,\,\,y\le 14\] and \[x-y\ge 5\] |
| B. | \[14x+5y\le 70,\,\,\,y\le 14\] and \[x-y\ge 5\] |
| C. | \[14x+5y\ge 70,y\ge 14\] and \[x-y\ge 5\] |
| D. | \[14x+5y\ge 70,y\le 14\] and \[x-y\le 5\] |
| Answer» E. | |
| 28. |
Evaluate \[\underset{x\to \infty }{\mathop{\lim }}\,{{2}^{x-1}}\tan \left( \frac{a}{{{2}^{x}}} \right).\] |
| A. | \[a\] |
| B. | \[2a\] |
| C. | \[\frac{a}{2}\] |
| D. | \[4a\] |
| Answer» D. \[4a\] | |
| 29. |
\[\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \frac{1+\operatorname{tanx}}{1+\sin x} \right\}}^{\cos ecx}}\] is equal to |
| A. | \[\frac{1}{e}\] |
| B. | 1 |
| C. | \[e\] |
| D. | \[{{e}^{2}}\] |
| Answer» C. \[e\] | |
| 30. |
The limit \[\underset{x\to 0}{\mathop{\lim }}\,{{(cosx)}^{1/\sin x\frac{1}{\sin x}}}\] is equal to |
| A. | \[e\] |
| B. | \[{{e}^{-1}}\] |
| C. | 1 |
| D. | Does not exist |
| Answer» D. Does not exist | |
| 31. |
\[\underset{x\to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,{{[1+{{(cos\,x)}^{\cos x}}]}^{2}}\] is equal to |
| A. | Does not exist |
| B. | 1 |
| C. | e |
| D. | 4 |
| Answer» E. | |
| 32. |
Let \[f(x)=4\] and \[f'(x)=4.\]Then \[\underset{x\to 2}{\mathop{\lim }}\,\frac{xf(2)-2f(x)}{x-2}\] is given by |
| A. | 2 |
| B. | -2 |
| C. | -4 |
| D. | 3 |
| Answer» D. 3 | |
| 33. |
What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{\sqrt{1-\cos x}}\] equal to? |
| A. | \[\sqrt{2}\] |
| B. | \[-\sqrt{2}\] |
| C. | \[\frac{1}{\sqrt{2}}\] |
| D. | Limit does not exist |
| Answer» E. | |
| 34. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,{{\log }_{e}}{{(sinx)}^{\tan x}}\] is |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | None of these |
| Answer» D. None of these | |
| 35. |
The value of \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,{{\left[ {{1}^{1/{{\cos }^{2}}x}}+{{2}^{1/{{\cos }^{2}}x}}+...+{{n}^{1/co{{s}^{2}}x}} \right]}^{{{\cos }^{2x}}}}\] is |
| A. | 0 |
| B. | n |
| C. | \[\infty \] |
| D. | \[\frac{n(n+1)}{2}\] |
| Answer» C. \[\infty \] | |
| 36. |
\[\underset{x\to 1}{\mathop{\lim }}\,\frac{(1-x)(1-{{x}^{2}})...(1-{{x}^{2n}})}{{{\{(1-x)(1-{{x}^{2}})...(1-{{x}^{n}})\}}^{2}}},n\in N,\] equals |
| A. | \[^{2n}{{P}_{n}}\] |
| B. | \[^{2n}{{C}_{n}}\] |
| C. | \[(2n)!\] |
| D. | None of these |
| Answer» C. \[(2n)!\] | |
| 37. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \cos e{{c}^{3}}x.\cot x-2{{\cot }^{3}}x.\cos ecx+\frac{{{\cot }^{4}}x}{\sec x} \right]\] is equal to |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | None of these |
| Answer» B. -1 | |
| 38. |
If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{(sinnx)[(a-n)nx-tanx]}{{{x}^{2}}}=0,\] then the value of a |
| A. | \[\frac{1}{n}\] |
| B. | \[n-\frac{1}{n}\] |
| C. | \[n+\frac{1}{n}\] |
| D. | None |
| Answer» D. None | |
| 39. |
\[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\sin [x-3]}{[x-3]} \right],\] where \[[.]\] denotes greatest integer function is |
| A. | 0 |
| B. | 1 |
| C. | Does not exist |
| D. | sin 1 |
| Answer» D. sin 1 | |
| 40. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{100}}}{{{e}^{x}}}+{{\left( \cos \frac{2}{x} \right)}^{{{x}^{2}}}} \right)=\] |
| A. | \[{{e}^{-1}}\] |
| B. | \[{{e}^{-4}}\] |
| C. | \[(1+{{e}^{-2}})\] |
| D. | \[{{e}^{-2}}\] |
| Answer» E. | |
| 41. |
\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{x}^{2}}+5x+3}{{{x}^{2}}+x+2} \right)}^{x}}\] |
| A. | \[{{e}^{4}}\] |
| B. | \[{{e}^{2}}\] |
| C. | \[{{e}^{3}}\] |
| D. | 1 |
| Answer» B. \[{{e}^{2}}\] | |
| 42. |
\[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{\left[ \frac{x}{2} \right]}{ln\,(sin\,x)}\] (where [.] denotes the greatest integer function) |
| A. | Does not exist |
| B. | Equals 1 |
| C. | Equals 0 |
| D. | Equals -1 |
| Answer» D. Equals -1 | |
| 43. |
If [.] denotes the greatest integer function, then \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{[x]+[2x]+...+[nx]}{{{n}^{2}}}\] is |
| A. | 0 |
| B. | \[x\] |
| C. | \[\frac{x}{2}\] |
| D. | \[\frac{{{x}^{2}}}{2}\] |
| Answer» D. \[\frac{{{x}^{2}}}{2}\] | |
| 44. |
If \[{{A}_{i}}=\frac{x-{{a}_{i}}}{\left| x-{{a}_{i}} \right|},i=1,2,3,....,n\] and \[{{a}_{1}} |
| A. | Is equal to \[{{(-1)}^{m}}\] |
| B. | Is equal to \[{{(-1)}^{m+1}}\] |
| C. | Is equal to \[{{(-1)}^{m-1}}\] |
| D. | Does not exist |
| Answer» E. | |
| 45. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{{{x}^{2}}}{4}\log (1+3x)},\] is |
| A. | \[\frac{4}{3}{{(in4)}^{2}}\] |
| B. | \[\frac{4}{3}{{(In4)}^{3}}\] |
| C. | \[\frac{3}{2}{{(In4)}^{2}}\] |
| D. | \[\frac{3}{2}{{(In4)}^{3}}\] |
| Answer» C. \[\frac{3}{2}{{(In4)}^{2}}\] | |
| 46. |
If \[f(x)=\sqrt{{{x}^{2}}-10x+25},\] then the derivative of f(x) on the interval \[[0,7]\] is |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | None of these |
| Answer» E. | |
| 47. |
The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{2}-\sqrt{1+\cos x}}\] is |
| A. | \[4\sqrt{2}{{(log3)}^{2}}\] |
| B. | \[8\sqrt{2}{{(log3)}^{2}}\] |
| C. | \[2\sqrt{2}{{(log3)}^{2}}\] |
| D. | None of these |
| Answer» C. \[2\sqrt{2}{{(log3)}^{2}}\] | |
| 48. |
If \[f(x)={{\left( \frac{{{\sin }^{m}}x}{{{\sin }^{n}}x} \right)}^{m+n}}.{{\left( \frac{{{\sin }^{n}}x}{{{\sin }^{p}}x} \right)}^{n+p}}.{{\left( \frac{{{\sin }^{p}}\,x}{{{\sin }^{m}}x} \right)}^{p+m}}\] Then \[f'(x)\]is equal to |
| A. | 0 |
| B. | 1 |
| C. | \[{{\cos }^{m+n+px}}\] |
| D. | None of these |
| Answer» B. 1 | |
| 49. |
If \[f(x)=\frac{\sin ({{e}^{x-2}}-1)}{in(x-1)},\] then \[\underset{x\to 2}{\mathop{\lim }}\,f(x)\] is equal to |
| A. | \[-2\] |
| B. | \[-1\] |
| C. | \[0\] |
| D. | 1 |
| Answer» E. | |
| 50. |
\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{n}^{p}}{{\sin }^{2}}(n!)}{n+1},0 |
| A. | 0 |
| B. | \[\infty \] |
| C. | 1 |
| D. | None of these |
| Answer» B. \[\infty \] | |