MCQOPTIONS
Saved Bookmarks
MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 1251. |
If \[f(x)={{\log }_{x}}(In\,\,x)\], then at \[x=e,f'(x)\]equals- |
| A. | 0 |
| B. | 1 |
| C. | e |
| D. | 1/e |
| Answer» E. | |
| 1252. |
If \[y={{\tan }^{-1}}\left( \frac{{{2}^{x}}}{1+{{2}^{2x+1}}} \right),\]then \[\frac{dy}{dx}at\,x=0\] is |
| A. | \[\frac{3}{5}\log \,2\] |
| B. | \[\frac{2}{5}\log \,2\] |
| C. | \[-\frac{3}{2}\log \,2\] |
| D. | \[\log \,2\left( \frac{-1}{10} \right)\] |
| Answer» E. | |
| 1253. |
The number of points at which the function \[f(x)=\frac{1}{{{\log }_{e}}\left| x \right|}\] is discontinuous, is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» D. 4 | |
| 1254. |
Which of the following is correct for \[f(x)=\left\{ \begin{matrix} (x-e){{2}^{-{{2}^{\left( \frac{1}{(e-x)} \right)}},}} & x\ne e\,\,at\,\,x=e \\ 0, & x=e \\ \end{matrix} \right.\] |
| A. | f(x) is discontinuous at x = e |
| B. | f(x) is differentiable at x = e |
| C. | f(x) is non-differentiable at x = e |
| D. | None of these |
| Answer» D. None of these | |
| 1255. |
If the mean value theorem is\[f(b)-f(a)=(b-a)f'(c)\]. Then, for the function \[{{x}^{2}}-2x+3\] in \[\left[ 1,\frac{3}{2} \right]\] the value of c is |
| A. | 44322 |
| B. | 44291 |
| C. | 44259 |
| D. | 44354 |
| Answer» C. 44259 | |
| 1256. |
If \[f(x)=\cos \left[ \frac{\pi }{x} \right]\cos \left( \frac{\pi }{2}(x-1) \right);\] where [x] is the greatest integer function of x, then f(x) is continuous at |
| A. | x = 0 |
| B. | x = 1, 2 |
| C. | x = 0, 2, 4 |
| D. | None of these |
| Answer» C. x = 0, 2, 4 | |
| 1257. |
Let \[f:R\to R\] be a function defined by\[f(x)=min\{x+1,\left| x \right|+1\}\], Then which of the following is true? |
| A. | \[f(x)\] is differentiable everywhere |
| B. | \[f(x)\] is not differentiable at x = 0 |
| C. | \[f(x)\ge 1\] for all \[x\in R\] |
| D. | \[f(x)\] is not differentiable at \[x=1\] |
| Answer» B. \[f(x)\] is not differentiable at x = 0 | |
| 1258. |
If \[f''(x)=-f(x)\] and \[g(x)=f'(x)\] and \[F(x)={{\left( f\left( \frac{x}{2} \right) \right)}^{2}}+{{\left( g\left( \frac{x}{2} \right) \right)}^{2}}\] and given that \[F(5)=5\], then F (10) is equal to- |
| A. | 5 |
| B. | 10 |
| C. | 0 |
| D. | 15 |
| Answer» B. 10 | |
| 1259. |
Which of the following functions have finite number of points of discontinuity? (where \[[\cdot ]\]represents greatest integer functions) |
| A. | \[tan\,x\] |
| B. | \[x\text{ }\!\![\!\!\text{ }x]\] |
| C. | \[\frac{\left| x \right|}{x}\] |
| D. | \[\sin \,[n\pi x]\] |
| Answer» D. \[\sin \,[n\pi x]\] | |
| 1260. |
The number of points of non-differentiability for\[f(x)=max\,\{\left| \left| x \right|-1 \right|,1/2\}\] is |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 5 |
| Answer» E. | |
| 1261. |
What is the set of all points, where the function \[f(x)=\frac{x}{1+\left| x \right|}\] is differentiable? |
| A. | \[(-\infty ,\infty )\] only |
| B. | \[(0,\infty )\] only |
| C. | \[(-\infty ,0)\cup (0,\infty )\]only |
| D. | \[(-\infty ,0)\] only |
| Answer» B. \[(0,\infty )\] only | |
| 1262. |
Which one of the following statements is correct in respect of the function\[f(x)={{x}^{3}}\sin x\]? |
| A. | f'(x) changes sign from positive to negative at x = 0 |
| B. | f '(x) changes sign from negative to positive at x = 0 |
| C. | Does not change sign at x = 0 |
| D. | \[f''(0)\ne 0\] |
| Answer» D. \[f''(0)\ne 0\] | |
| 1263. |
Which of the following function is continuous at for all value of x? (i) \[f\left( x \right)\] =sgn\[({{x}^{3}}-x)\](ii) \[f\left( x \right)\] =sgn \[(2\cos x-1)\](iii) \[f\left( x \right)\] =sgn \[({{x}^{2}}-2x+3)\] |
| A. | Only (i) |
| B. | Only (iii) |
| C. | Both (ii) and (iii) |
| D. | None of these |
| Answer» C. Both (ii) and (iii) | |
| 1264. |
\[\frac{{{d}^{n}}}{d{{x}^{n}}}(log\,x)=\] |
| A. | \[\frac{(n-1)!}{{{x}^{n}}}\] |
| B. | \[\frac{n!}{{{x}^{n}}}\] |
| C. | \[\frac{(n-2)!}{{{x}^{n}}}\] |
| D. | \[{{(-1)}^{n-1}}\frac{(n-1)!}{{{x}^{n}}}\] |
| Answer» E. | |
| 1265. |
The function \[f(x)=\sin ({{\log }_{e}}\left| x \right|),x\ne 0\], and 1 if \[x=0\] |
| A. | Is continuous at \[x=0\] |
| B. | Has removable discontinuity at \[x=0\] |
| C. | Has jump discontinuity at \[x=0\] |
| D. | Has oscillating discontinuity at \[x=0\] |
| Answer» E. | |
| 1266. |
Let f be a function which is continuous and differentiable for all real x. If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\in [2,4],\] then |
| A. | \[f(4)<8\] |
| B. | \[f(4)\ge 8\] |
| C. | \[f(4)\ge 12\] |
| D. | None of these |
| Answer» C. \[f(4)\ge 12\] | |
| 1267. |
Let \[f:R\to R\] be defined as \[f(x)=sin(\left| x \right|)\] Which one of the following is correct? |
| A. | f is not differentiable only at 0 |
| B. | f is differentiable at 9 only |
| C. | f is differentiable everywhere |
| D. | f is non-differentiable at many points |
| Answer» B. f is differentiable at 9 only | |
| 1268. |
What is the derivative of \[\left| x-1 \right|\] at\[x=2\]? |
| A. | -1 |
| B. | 0 |
| C. | 1 |
| D. | Derivative does not exist |
| Answer» D. Derivative does not exist | |
| 1269. |
If \[f(x)=\frac{1}{1-x}\], then the points of discontinuity of the function \[f[f\{f(x)\}]\] are |
| A. | {0, -1} |
| B. | {0, 1} |
| C. | {1, -1} |
| D. | None of these |
| Answer» C. {1, -1} | |
| 1270. |
If \[\theta \] are the points of discontinuity of \[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,{{\cos }^{2n}}x\] then the value of sin \[\theta \] is |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | ½ |
| Answer» B. 1 | |
| 1271. |
Let \[f(x)=\left\{ \begin{matrix} 3x-4, & 0\le x\le 2 \\ 2x+\ell , & 2 |
| A. | 0 |
| B. | 2 |
| C. | -2 |
| D. | -1 |
| Answer» D. -1 | |
| 1272. |
Which of the following function(s) has/have removable discontinuity at \[x=1\]? |
| A. | \[f(x)=\frac{1}{In\left| x \right|}\] |
| B. | \[f(x)=\frac{1}{{{x}^{3}}-1}\] |
| C. | \[f(x)={{2}^{{{2}^{\frac{1}{1-x}}}}}\] |
| D. | \[f(x)=\frac{\sqrt{x+1}-\sqrt{2x}}{{{x}^{2}}-x}\] |
| Answer» E. | |
| 1273. |
If \[{{x}^{a}}{{y}^{b}}={{(x-y)}^{a+b}},\] then the value of \[\frac{dy}{dx}-\frac{y}{x}\] is equal to |
| A. | \[\frac{a}{b}\] |
| B. | \[\frac{b}{a}\] |
| C. | 1 |
| D. | 0 |
| Answer» E. | |
| 1274. |
Consider the following in respect of the function \[f(x)=\left\{ \begin{matrix} 2+x, & x\ge 0 \\ 2-x, & x |
| A. | 1 only |
| B. | 3 only |
| C. | 2 and 3 only |
| D. | 1 and 3 only |
| Answer» E. | |
| 1275. |
If function \[\text{f(x)=}\left\{ \begin{align} & \text{x,if}\,\,\text{x}\,\,\text{is}\,\,\text{rational} \\ & \text{1-x,if}\,\,\text{x}\,\,\text{is}\,\,\text{irrational} \\ \end{align} \right.\text{,then}\] the number of points at which f(x) is continuous, is- |
| A. | \[\infty \] |
| B. | 1 |
| C. | 0 |
| D. | None of these |
| Answer» C. 0 | |
| 1276. |
Let f'(x) be continuous at x = 0 and f"(0) = 4. Then value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}\] is |
| A. | 12 |
| B. | 10 |
| C. | 6 |
| D. | 4 |
| Answer» B. 10 | |
| 1277. |
Suppose that \[f(0)=-3\] and \[f'(x)=\,\le 5\] for all values of x. Then, the largest value which f(2) can attain is???? |
| A. | 7 |
| B. | 10 |
| C. | 2 |
| D. | 9 |
| Answer» B. 10 | |
| 1278. |
Let f be a continuous function on R such that\[f(1/4n)=(\sin {{e}^{n}}){{e}^{-{{n}^{2}}}}+\frac{{{n}^{2}}}{{{n}^{2}}+1}\]. Then the value of f (0) is |
| A. | 1 |
| B. | 44228 |
| C. | 0 |
| D. | None of these |
| Answer» B. 44228 | |
| 1279. |
If \[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\], where \[0 |
| A. | \[\frac{1}{2}\] |
| B. | 2 |
| C. | \[\sin x+\cos x\] |
| D. | \[\sin x-\cos x\] |
| Answer» B. 2 | |
| 1280. |
If \[y=\frac{1}{{{t}^{2}}+t-2}\] where \[t=\frac{1}{x-1}\], then find the number of points of discontinuities of \[y=f(x),\], |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» D. 4 | |
| 1281. |
If \[f(x)=\left| 1-x \right|,\] then the points where \[{{\sin }^{-1}}(f\left| x \right|)\] is non-differentiable are |
| A. | \[\left\{ 0,1 \right\}\] |
| B. | \[\left\{ 0,-1 \right\}\] |
| C. | \[\left\{ 0,1,-1 \right\}\] |
| D. | None of these |
| Answer» D. None of these | |
| 1282. |
If \[y=lo{{g}_{10}}x+lo{{g}_{x}}10+lo{{g}_{x}}x+lo{{g}_{10}}10\] then what is \[{{\left( \frac{dy}{dx} \right)}_{x=10}}\] equal to? |
| A. | 10 |
| B. | 2 |
| C. | 1 |
| D. | 0 |
| Answer» E. | |
| 1283. |
Given \[f(x)=b({{[x]}^{2}}+[x])+1\] for \[x\ge -1\]\[=\sin (\pi (x+a))\] for \[x |
| A. | \[a=2n+(3/2);b\in R;n\in I\] |
| B. | \[a=4n+2;b\in R;n\in I\] |
| C. | \[a=4n+(3/2);b\in {{R}^{+1}};n\in I\] |
| D. | \[a=4n+1;b\in {{R}^{+}};n\in I\] |
| Answer» B. \[a=4n+2;b\in R;n\in I\] | |
| 1284. |
If the function \[f(x)=\left[ \frac{{{(x-2)}^{3}}}{a} \right]\sin (x-2)+a\,\cos (x-2),\] [.] denotes the greatest integer function is continuous and differentiable in [4, 6], then |
| A. | \[a\in [8,\,64]\] |
| B. | \[a\in (0,\,8]\] |
| C. | \[a\in [64,\,\infty )\] |
| D. | None of these |
| Answer» D. None of these | |
| 1285. |
Consider the following: 1. \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\] exists. 2. \[\underset{x\to 0}{\mathop{\lim }}\,\,\,{{e}^{\frac{1}{x}}}\] does not exist. Which of the above is/are correct? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» C. Both 1 and 2 | |
| 1286. |
If the derivative of the function\[f(x)=\left\{ \begin{matrix} a{{x}^{2}}+b & x |
| A. | a=2, b=3 |
| B. | a=3, b=2 |
| C. | a=-2, b=-3 |
| D. | a=-3, b=-2 |
| Answer» B. a=3, b=2 | |
| 1287. |
A value of c for which conclusion of Mean Value Theorem holds for the function \[f(x)={{\log }_{e}}x\] on the interval [1, 3] is |
| A. | \[lo{{g}_{3}}e\] |
| B. | \[lo{{g}_{e}}3\] |
| C. | \[2lo{{g}_{3}}e\] |
| D. | \[\frac{1}{2}{{\log }_{3}}e\] |
| Answer» D. \[\frac{1}{2}{{\log }_{3}}e\] | |
| 1288. |
If \[f(x)=\left\{ \begin{align} & \left( {{x}^{2}}/a \right)-a,\,\,when\,\,\,xa \\ \end{align} \right.\] |
| A. | \[\underset{x\to a}{\mathop{\lim }}\,f(x)=a\] |
| B. | \[f(x)\] is continuous at x = a |
| C. | \[f(x)\] is discontinuous at x = a |
| D. | None of these |
| Answer» C. \[f(x)\] is discontinuous at x = a | |
| 1289. |
Let \[f(x)=\frac{{{({{e}^{x}}-1)}^{2}}}{\sin \left( \frac{x}{a} \right)\log \left( 1+\frac{x}{4} \right)}\] for \[x\ne 0,\] and \[f(0)=12\]. If f is continuous at \[x=0\], then the value of a is equal to |
| A. | 1 |
| B. | -1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 1290. |
If \[f(x)=\sqrt[3]{\frac{{{x}^{4}}}{\left| x \right|}},x\ne 0\] and f(0) = 0 is: |
| A. | Continuous for all x but not differentiable for any x |
| B. | Continuous and differentiable for all x |
| C. | Continuous for all x and differentiable for all \[x\ne 0\] |
| D. | Continuous and differentiable for all \[x\ne 0\] |
| Answer» D. Continuous and differentiable for all \[x\ne 0\] | |
| 1291. |
If the polynomial equation \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+.....+{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}=0\],n positive integer, has two different real roots \[\alpha \]and \[\beta \], then between \[\alpha \text{ }and\text{ }\beta \], the equation \[n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{x}^{n-2}}+....+{{a}_{1}}=0\] has |
| A. | Exactly one root |
| B. | At most one root |
| C. | At least one root |
| D. | No root |
| Answer» D. No root | |
| 1292. |
Consider the function\[f(x)=\left\{ \begin{matrix} {{x}^{2}}, & x>2 \\ 3x-2, & x\le 2 \\ \end{matrix} \right.\]. Which one of the following statements is correct in respect of the above function? |
| A. | f(x) is derivable but not continuous at x = 2. |
| B. | f(x) is continuous but not derivable at x = 2. |
| C. | f(x) is neither continuous nor derivable at x = 2. |
| D. | f(x) is continuous as well as derivable at x = 2. |
| Answer» C. f(x) is neither continuous nor derivable at x = 2. | |
| 1293. |
If\[f(x)=\left\{ \begin{matrix} mx+1x\le \frac{\pi }{2} \\ \sin x+nx>\frac{\pi }{2} \\ \end{matrix}\,\,\,\text{is}\,\,\text{continuous}\,\,\text{at} \right.\]\[x=\frac{\pi }{2}\], then which one of the following is correct? |
| A. | m = 1, n = 0 |
| B. | \[m=\frac{n\pi }{2}+1\] |
| C. | \[n=m\left( \frac{\pi }{2} \right)\] |
| D. | \[m=n=\frac{\pi }{2}\] |
| Answer» D. \[m=n=\frac{\pi }{2}\] | |
| 1294. |
The value of p for which the function\[f(x)=\left\{ \begin{matrix} \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},x\ne 0 \\ 12{{(log\,4)}^{3}},x=0 \\ \end{matrix} \right.\]may be continuous at \[x=0\], is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | None of these |
| Answer» E. | |
| 1295. |
Let \[f'(x)=-1+\left| x-2 \right|,\] and \[g(x)=1-\left| x \right|;\] then the set of all points where fog is discontinuous is: |
| A. | {0, 2} |
| B. | {0, 1, 2} |
| C. | {0} |
| D. | An empty set |
| Answer» E. | |
| 1296. |
Which one of the following is correct in respect of the function \[f(x)=\frac{{{x}^{2}}}{\left| x \right|}\] for \[x\ne 0\] and f(0) = 0? |
| A. | f (x) is discontinuous every where |
| B. | f (x) is continuous every where |
| C. | f(x) is continuous at x = 0 only |
| D. | f(x) is discontinuous at x = 0 only |
| Answer» C. f(x) is continuous at x = 0 only | |
| 1297. |
Consider the function \[f(x)=\left\{ \begin{matrix} ax-2 & for & -2 |
| A. | -1 |
| B. | 1 |
| C. | 0 |
| D. | 2 |
| Answer» B. 1 | |
| 1298. |
Let \[f(x+y)=f(x)+f(y)\] and \[f(x)={{x}^{2}}g(x)\] for all\[x,\text{ }y\in R\], where g(x) is continuous function. Then f?(x) is equal to |
| A. | g'(x) |
| B. | g(0) |
| C. | g(0)+g'(x) |
| D. | 0 |
| Answer» E. | |
| 1299. |
Let \[f(x)=\left\{ \begin{align} & {{5}^{1/x}},x |
| A. | f is discontinuous |
| B. | f is continuous only, if \[\lambda =0\] |
| C. | f is continuous only, whatever \[\lambda \] may be |
| D. | None of these |
| Answer» B. f is continuous only, if \[\lambda =0\] | |
| 1300. |
If \[y=\frac{(a-x)\sqrt{a-x}-(b-x)\sqrt{x-b}}{\sqrt{a-x}+\sqrt{x-b}}\], then \[\frac{dy}{dx}\] wherever it is defined is |
| A. | \[\frac{x+(a+b)}{\sqrt{(a-x)(x-b)}}\] |
| B. | \[\frac{2x-a-b}{2\sqrt{a-x}\sqrt{x-b}}\] |
| C. | \[-\frac{(a+b)}{2\sqrt{(a-x)(x-b)}}\] |
| D. | \[\frac{2x+(a+b)}{2\sqrt{(a-x)(x-b)}}\] |
| Answer» C. \[-\frac{(a+b)}{2\sqrt{(a-x)(x-b)}}\] | |