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This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 201. |
Find the slope of the tangent to the curve `y=x^3-3x+2` at the point whose x-coordinate is 3. |
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Answer» `y = x^(3)- 3x + 2` ` rArr (dy)/(dx) = 3x^(2)-3` at x = 3 Slope of tangent `m = 3 (3)^(2) - 3 = 24` |
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| 202. |
`f(x) = |x|` hasA. minimum at x= 0B. maximum at x = 0C. neither a maximum nor a minimum at x= 0D. none of these |
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Answer» Correct Answer - A `f(x) = |x| ge 0 " for all " x in R` The least value of `|x| " is " 0 " at " x = 0` `:. f(x) = |x|` has minima at x = 0 |
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| 203. |
Let `f(x) = {((x^(2) -2x -3)/(x + 1)","," when " x != -1),(" k,"," when " x = -1):}` If f(x) is continuous at `x = -1` then k = ?A. 4B. `-4`C. `-3`D. 2 |
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Answer» Correct Answer - B `underset(xrarr -1)(lim) f(x) = underset(x rarr -1)(lim) ((x -3) (x +1))/((x +1)) = underset(x rarr -1)(lim) (x -3) = -4` For continuity, we must have, `f(-1) = -4` |
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| 204. |
If the function `f(x) = {((sin^(2)ax)/(x^(2))","," when "x != 0),(" k,"," when " x = 0):}`is continuous at x = 0 then k = ?A. 3B. `-3`C. `-5`D. 6 |
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Answer» Correct Answer - D `f((pi)/(2) - 0) = underset(h rarr 0)(lim) (k cos ((pi)/(2) - h))/(pi -2((pi)/(2) - h)) = underset(h rarr 0)(lim) (k sin h)/(2h) = (k)/(2) underset(h rarr 0)(lim) (sin h)/(h) = ((k)/(2) xx 1) = (k)/(2)` `f((pi)/(2) + 0) = underset(h rarr 0)(lim) (k cos ((pi)/(2) + h))/(pi -2((pi)/(2) + h)) = underset(h rarr 0)(lim) (-k sin h)/(-2h) = (k)/(2) underset(h rarr 0)(lim) (sin h)/(h) = (k)/(2)` `:. (k)/(2) = 3 rArr k = 6` |
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| 205. |
If the function `f(x) = {((sin^(2)ax)/(x^(2))","," when "x != 0),(" k,"," when " x = 0):}`is continuous at x = 0 then k = ?A. aB. `a^(2)`C. `-2`D. `-4` |
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Answer» Correct Answer - B `underset(x rarr 0)(lim) f(x) = underset(x rarr 0)(lim) (sin^(2) ax)/(a^(2) x^(2)) xx a^(2) = a^(2). underset(ax rarr 0)(lim) ((sin ax)/(ax))^(2) = a^(2) xx 1^(2) = a^(2)` For continuity, we must have `f(0) = a^(2)` |
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| 206. |
If `y = e^(x) + e^(x + ...oo) " then " (dy)/(dx)=` ?A. `(1)/((1-y))`B. `(y)/((1 -y))`C. `(y)/((y -1))`D. none of these |
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Answer» Correct Answer - B `y = e^(x + y) rArr x + y = log y rArr 1 + (dy)/(dx) = (1)/(y) .(dy)/(dx)` |
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| 207. |
The value of k for which `f(x) = {((sin 5x)/(3x)","," if " x !=0),(" k,"," if " x = 0):}` is contnuous at x = 0 isA. `(1)/(3)`B. 0C. `(3)/(5)`D. `(5)/(3)` |
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Answer» Correct Answer - D For continuity at `x = 0`, we must have `underset(x rarr 0)(lim) f(x) = f(0)` `underset(x rarr 0)(lim) f(x) = underset(x rarr 0)(lim) (sin 5x)/(5x) xx (5)/(3) = (5)/(3) underset(5x rarr 0)(lim) (sin 5x)/(5x) = ((5)/(3) xx 1) = (5)/(3)` `:.` we must have, `f(0) = (5)/(3) hArr k = (5)/(3)` |
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| 208. |
Let `f(x) = {(x "sin "(1)/(x)","," if " x != 0),(" 0,"," where " x = 0):}` Then, which of the following is the true statement ?A. f(x) is not defined at x = 0B. `underset(x rarr 0)(lim) f(x)` does not existC. f(x) is continuous at x = 0D. `f(x)` is discontinuous at x = 0 |
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Answer» Correct Answer - C `f(0) = 0` `underset(x rarr 0)(lim) f(x) = underset(x rarr 0)(lim) x "sin " (1)/(x) = 0 xx ("a finite quantity") = 0` `:. f(x)` is continuous at `x = 0` |
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| 209. |
If the function `f(x) {((1 - cos 4x)/(8x^(2))",",x !=0),(" k,",x = 0):}`is continuous at x = 0 then k = ?A. 1B. 2C. `(1)/(2)`D. `(-1)/(2)` |
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Answer» Correct Answer - C `underset(x rarr 0)(lim) f(x) = underset(h rarr 0)(lim) f(0 +h) = underset(h rarr 0)(lim) (1 - cos 4h)/(8h^(2)) = underset(h rarr 0)(lim) (2 sin ^(2) 2h)/(8h^(2))` `= (1)/(2) underset(h rarr 0)(lim) ((sin 2h)/(2h))^(2) = ((1)/(2) xx 1^(2)) =(1)/(2)` `underset(x rarr 0)(lim) f(x) = underset(h rarr0)(lim) f(0 -h) = underset(h rarr 0)(lim) (1 - cos 4 (-h))/(8(-h)^(2)) = underset(h rarr0)(lim) ((1 - cos 4h))/(8h^(2)) = (1)/(2)` `:. underset(x rarr 0^(+))(lim) f(x) = (1)/(2)` For continuity, we must have `f(0) = (1)/(2)` |
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| 210. |
The value of k for which `f(x) = {((3x + 4 tan x)/(x)","," where " x != 0),(" k,"," where " x = 0):}` is continuous at x = 0, isA. 7B. 4C. 3D. none of these |
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Answer» Correct Answer - A `f(0) = k` `underset(x rarr 0)(lim) f(x) = underset(x rarr 0)(lim) (3x + 4 tan x)/(x) = underset(x rarr 0)(lim) {3 +(4 tan x)/(x)} = (3 + 4) = 7` `:. f(x)` is continuous at `x = 0 hArr f(0) = 7 hArr k = 7` |
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| 211. |
If the function `f(x) = {(kx + 5 ", when " x le 2),(x -1 ", when " x gt 2):}` is continuous at x = 2 then k = ?A. 2B. `-2`C. 3D. `-3` |
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Answer» Correct Answer - B `f(2) = underset(x rarr 2)(lim) f(x)` `underset(x rarr 2^(+))(lim) f(x) = underset(h rarr 0)(lim) f(2 + h) = underset(h rarr 0)(lim) (2 + h -1) = 1` `underset(x rarr 2)(lim) f(x) = underset(h rarr 0)(lim) f(2 -h) = underset(h rarr 0)(lim){k (2 -h) + 5} = 2k + 5` `:. 2k + 5 = 1 rArr k == -2` Also, `f(2) = 2k + 5 = 1`. Hence, `k = -2` |
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| 212. |
The function `f(x)-x^(3)-6x^(2)+12x-16, x in R` isA. increasing for all `x in R, x ne 2`B. decreasingC. neither increasing nor decreasingD. decreasing for all `x in R, x ne 2` |
| Answer» Correct Answer - A | |
| 213. |
The function `f(x)=cos x, 0 le x le pi` isA. decreasingB. increasingC. neither increasing nor decreasingD. increasing for `0 le x le pi` |
| Answer» Correct Answer - A | |
| 214. |
The function `f(x)=x^(3)-6x^(2)+12x-16, x in R` isA. [1,2]B. [1,2)C. (1,2]D. (1,2) |
| Answer» Correct Answer - D | |
| 215. |
The sum of the perimeter of a circle and square isk, where k is some constant. Prove that the sum of their areas is least whenthe side of square is double the radius of the circle. |
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Answer» Let x be the side of the square and r be the radius of the circle. According to the problem Let P = perimeter of square+ circumference of circle ` rArr P = 4x + 2 pi r` ….(1) Let `A = pi r^(2) + x^(2)` ` = pi r^(2) = ((P-2 pi r )/4)^(2)` [ From eqn. (1)] ` = pi r^(2) + 1/16 (P-2pi r)^(2)` ` rArr (dA)/(dr) = 2 pi r +2/16 (-2pi)(p-2 pi r)` ` = 2 pi r - pi/4 (P-2 pi r)` and ` (d^(2)A)/(dr^(2)) = 2 pi + pi^(2)/2` From maxima/minima ` (dA)/(dr) = 0 ` ` 2 pi r - pi/4 (P-2 pi r) =0` ...(2) ` rArr 2 pi r = pi/4 4 x` ` rArr 2 r = x ` at x = 2r, ` (d^(2)A)/(dr^(2)) gt 0 ` ` rArr` A is minimum. Therefore, total area will be minimum if the side of the square is equal to the diameter of the circle. |
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| 216. |
The function `f(x)=x^(4)-2x^(3)+1` is decreasing forA. `x ge (3)/(2)`B. `xgt(3)/(2)`C. `x le (3)/(2)`D. `x lt (3)/(2)` |
| Answer» Correct Answer - D | |
| 217. |
The value of a for which the sum of the square of the roots of the equation `x^(2)-(a-2)x-a+1=0` is least, is |
| Answer» Correct Answer - B | |
| 218. |
The least value of the sum of any positive real number and its reciprocal isA. 3B. 4C. 1D. 2 |
| Answer» Correct Answer - D | |
| 219. |
The function `f(x)=x log x`A. has maximum value `(1)/(e )`B. has maximum value `(-1)/(e )`C. has minimum value `(1)/(e )`D. has minimum value `(-1)/(e )` |
| Answer» Correct Answer - D | |
| 220. |
The function `f(x)=x^(x)` hasA. minima at x=eB. maxima at x=eC. maxima at `x=(1)/(e )`D. minima at `x=(1)/(e )` |
| Answer» Correct Answer - D | |
| 221. |
For what values of x, the function `f(x) = x^(5)-5x^(4)+5x(3)-1` is maximum or minimum? Prove that at x = 0, the function is neither maximum nor minimum. |
| Answer» Correct Answer - Min value = 3 and max at x = 1 | |
| 222. |
If x = -1 and x = 2 are extreme points of f(x) = `alpha log|x| + beta x^2 + x`, thenA. `alpha=-6, beta =(1)/(2)`B. `alpha=-6, beta=(-1)/(2)`C. `alpha=2, beta =(-1)/(2)`D. `alpha=2, beta=(1)/(2)` |
| Answer» Correct Answer - C | |
| 223. |
Maximum value of `sin theta + cos theta" in "(0,(pi)/(2))` is |
| Answer» Correct Answer - C | |
| 224. |
The real numbers which must exceeds its cube isA. `(-1)/(sqrt(3))`B. `(1)/(sqrt(3))`C. `(1)/(2)`D. `(1)/(sqrt(2))` |
| Answer» Correct Answer - B | |
| 225. |
On [1,e] the greatest value of `x^(2) log x` isA. `-e^(2)`B. `e^(2)`C. `(1)/(e )log""(1)/(sqrt(e ))`D. `e^(2) log sqrt(e )` |
| Answer» Correct Answer - B | |
| 226. |
The maximum value of `f(x)=a sin x +b cos x` isA. `-sqrt(a^(2)+b^(2))`B. `sqrt((a^(2)+b^(2))`C. `(-1)/(sqrt(a^(2)+b^(2)))`D. `(1)/(sqrt(a^(2)+b^(2)))` |
| Answer» Correct Answer - B | |
| 227. |
The function f(x)=log xA. has maxima at x=eB. has minima at x=eC. has neither maxima nor minimaD. has maximum value 1 |
| Answer» Correct Answer - C | |
| 228. |
The function `f(x)=x^(2)e^(x)` has maximum value |
| Answer» Correct Answer - D | |
| 229. |
Find the maximum and minimum values of the function `f(x) = sin x + cos 2x`. |
| Answer» Correct Answer - Max value ` = 9/8, ` min value = 0 | |
| 230. |
`f(x)=(e^(2x)-1)/(e^(2x)+1)` isA. trigonometricB. evenC. decreasingD. increasing |
| Answer» Correct Answer - D | |
| 231. |
Let f(x) be a polynomial of degree four having extreme values at x=1 and x=2. If `lim_(x to 0)(1+(f(x))/(x^(2)))=3,` then f(2) is equal to |
| Answer» Correct Answer - A | |
| 232. |
The maximum value of ` f(x) =(log x)/(x) (x ne 0, x ne 1)` isA. eB. `(1)/(e )`C. `e^(2)`D. `(1)/(e^(2))` |
| Answer» Correct Answer - B | |
| 233. |
The function `f(x)=(3)/(x)+10, x ne 0` isA. neither increasing nro decreasingB. increasing for all `x in R, x ne 0`C. decreasing for all `x in R, x ne 0`D. decreasing for all `x lt 0` |
| Answer» Correct Answer - C | |
| 234. |
If `x ,y in R^+`satisfying `x+y=3,`then the maximum value of `x^2y`is.A. 2B. -2C. 4D. -4 |
| Answer» Correct Answer - C | |
| 235. |
If `f(x)=x e^(x(1-x))`, then f(x) isA. decreasing on `((-1)/(2),1)`B. increasing on `((-1)/(2),1)`C. decreasing on RD. increasing on R |
| Answer» Correct Answer - B | |
| 236. |
The function `f(x)=x^(2)e^(x)` has minimum value |
| Answer» Correct Answer - A | |
| 237. |
If `a^(2)gtb^(2),` then the minimum value of `f(x)=a^(2) cos^(2)x+b^(2) sin^(2) x` isA. `a^(2)-b^(2)`B. `a^(2)+b^(2)`C. `a^(2)`D. `b^(2)` |
| Answer» Correct Answer - D | |
| 238. |
The minimum value of `f(x)= a sin x+b cos x` isA. `-sqrt(a^(2)+b^(2))`B. `sqrt((a^(2)+b^(2))`C. `(-1)/(sqrt(a^(2)+b^(2)))`D. `(1)/(sqrt(a^(2)+b^(2)))` |
| Answer» Correct Answer - A | |
| 239. |
Function `f(x)=x^(2)-3x+4` has minimum value at x = ……………………. |
| Answer» Correct Answer - D | |
| 240. |
The function `f(x)=(7)/(x)-3, x in R, x ne 0` isA. increasing for `x gt 0`B. decreasing for `x lt 0`C. increasing for all `x in R, x ne 0`D. decreasing for all `x in R, x ne 0` |
| Answer» Correct Answer - D | |
| 241. |
A particle moves in a straight line so that its velocity at any point is given by `v^(2)=a+bx`, where `a,b ne 0` are constants. The acceleration isA. zeroB. uniformC. non-uniformD. indeterminate |
| Answer» Correct Answer - B | |
| 242. |
The function `f(x)=x-(1)/(x), x in R, x ne 0` isA. decreasing for all `x in R, x ne 0`B. increasing for all `x in R, x ne 0`C. neither increasing nor decreasingD. dereasing for all `x ne R` |
| Answer» Correct Answer - B | |
| 243. |
If `f(x)=2x^(3)-21x^(2)+36x-20`, thenA. f has maxima at x=6B. f has minima at x=1C. f has maximum value -3D. f has minimum value -3 |
| Answer» Correct Answer - C | |
| 244. |
If `f(x)=2x^(3)-21 x^(2)+36x-20`, thenA. f has maxima at x=1B. f has minima at x=1C. f has maximum value -128D. f has minimum value -3 |
| Answer» Correct Answer - A | |
| 245. |
If `f(x)=2x^(3)-21x^(2)+36x-20`, thenA. f has maxima at x=6B. f has minima at x=6C. f has maximum value -128D. f has minimum value -3 |
| Answer» Correct Answer - B | |
| 246. |
The function `f(x)=2-3x+3x^(2)-x^(3), x in R` isA. neither increasing nro decreasingB. increasingC. decreasing for all `x in R, x ne 1`D. increasing for all `x in R, x ne 1` |
| Answer» Correct Answer - C | |
| 247. |
The function `f(x)=(x-1)/(x+1), x ne -1` isA. decreasing for all `x in R, x ne -1`B. increasing for all `x in R, x ne -1`C. neither increasing nor decreasingD. decreasing for all `x in R` |
| Answer» Correct Answer - B | |
| 248. |
The function `f(x)=(x)/(x^(2)+1)` decreasing, ifA. `x lt -1 and x gt 1`B. `-1 lt x lt1`C. `x lt -1" or "x gt 1`D. `x le 1 and x ge 1` |
| Answer» Correct Answer - C | |
| 249. |
Find the values of x such that `f(x)=x^(3)+12x^(2)+36x+6` is an increasing function.A. `x in (-oo, -6)" or "x in (2,oo)`B. `x in (-oo, -6)" or "x in (-2,oo)`C. `x in (-oo,6)" or "x in (-2,oo)`D. `x in (-oo, 6)" or "x in (2,oo)` |
| Answer» Correct Answer - B | |
| 250. |
The function `f(x)=(x)/(x^(2)-1)` increasing, ifA. `-1 lt x`B. `x gt1`C. `-1 lt x" or "x gt1`D. `-1lt xlt 1` |
| Answer» Correct Answer - D | |