28278 + Interview Questions in Maths in Class 12

1.	Find the second order derivatives of the function log (log x).
Answer» ANSWER :`-((1+LOG X))/((x log x)^2)`

Discussion

2.	L.P.P. has constraints of
Answer» ONE variables TWO variables one or two variables two or more variables Answer :D

Discussion

3.	Number of ways of selecting 4 shoes from 10 pairs such that exactly onee correct pair is selected is
Answer» ` .^(10)C_(1) .^(9)C_(2).2^(3)` ` .^(10)C_(1)xx9xx16` `(( .^(10)C_(1).^(4)P_(2).18xx16)/(4!))` `10xx18xx16` Solution :NUMBER of WAYS of selecting 1 pair from 10 pair is ` .^(10)C_(1)` number of ways of selecting 2 shoes from 18 shoes such that both are DIFFERENT is `(18xx16)/(2)`

Discussion

4.	Resolve (3x^(2)+1)/((x^(2)-3x+2)(2x+1)) into partial fractions.
Answer» ANSWER :`(3X^(2)+1)/((X^(2)-3x+2)(2x+1)) = (-4)/(3(x-1)) + (13)/(5(x-2)) + (7)/(15(2x+1))`

Discussion

5.	Identify the quantifiers of the following statements For every real number x, x^2 != x
Answer» SOLUTION :For EVERY

Discussion

6.	Let f(x) =g(x) \|(x-1)(x-2)(x-3)^(2)(x-4)^(3)\|, whereg(x)= x^(3)+bx^(2)+cx+d. if f(x)is differentiablefor all x in R and f' (3) + f'''(4)=0 thenthe value ofg(5) is ______.
Answer» ANSWER :12

Discussion

7.	sin 10^(@) sin50^(@)sin 60^(@) sin 70^(@)=m " and tan "20^(@) tan 40^(@) tan60^(@) tan80^(@)= n, then (n)/(m)=
Answer» `(3sqrt(3))/(16)` `16sqrt(3)` `(16)/SQRT(3)` `8sqrt(3)` ANSWER :B

Discussion

8.	The solution of ydx - xdy = xydx is
Answer» `(x)/(y) = C e^(x)` `xy = c e^(x)` `(x^(2))/(y^(2)) = c LOG x` `(x)/(y) = c e^(y)` Answer :A

Discussion

9.	Consider the function f(x)={{:(xsin(pi)/(x),"for"xgt0),(0,"for"x=0):} Then, the number of points in (0,1) where the derivative f'(x) vanishes is
Answer» 0 1 2 infinite ANSWER :D

Discussion

10.	If thecorrect formof 2i^(4) +4i^(3) - i^(2) + 3i is a + ib , wheretheimaginarynumberI issuchthat i^(2) = - 1 . Whatis the value of ( a+b) ?
Answer» ANSWER :2

Discussion

11.	If aN = {ax : x in N} " then " 3N cap 7N =
Answer» `{3,6,9,12,..}` `{7,14,21,28,…}` `{21,42,63,84,…}` `{5,10,15,……}` ANSWER :C

Discussion

12.	Differentiate: (2^(x))/(x)
Answer» SOLUTION :N/A

Discussion

13.	The tangent at a point P(theta) to the ellipse x^2/a^2+y^2/b^2=1 cuts the auxilliary circle at Q and R. If QR subtend a right angle atC (centre) then show that e=1/sqrt(1+sin^2 theta)
Answer» ANSWER :`(SQRT((a^2-R^2)(r^2-b^2)))/r`

Discussion

14.	The system of vector hat(i), hat(j), hat(k) is
Answer» orthogonal coplanar collinear parallel Answer :A

Discussion

15.	{:(" "Lt),(n rarr oo):} ((1)/(sqrt(4n^(2)-1))+(1)/(sqrt(4n^(2)-2^(2)))+....+(1)/(sqrt(3n^(2))))=
Answer» 0 1 `pi/3` `pi/6` ANSWER :D

Discussion

16.	If the vectors a=hati-hatj+2hatk, b=2hati+4hatj+hatk and c=lambdahati+hatj+mu hatk are mutually orthogonal, then (lambda, mu) is equal to
Answer» `(-3, 2)` `(2,-3)` `(-2, 3)` `(3, -2)` SOLUTION :SINCE,the givenvectors are mutually orthogonal, THEREFORE `a.B = 2 - 4 + 2 =0` `a .c = lambda - 1 + 2 mu = 0 "…."(i)` and `b .c = 2lambda + 4 +mu = 0 "….."(ii)` On solvingEqs. (i) and (ii), we get `mu = 2` and `lambda = - 3` `:. (lambda, mu) = (-3,2)`

Discussion

17.	Evaluate the definite integrals int_(0)^(1)(dx)/(1+x^(2))
Answer» ANSWER :`pi/4`

Discussion

18.	Find the equation of the common tangent of the following circles at their point of contact. x^2+y^2+10x-2y+22=0 x^2+y^2+2x-8y+8=0
Answer» ANSWER :4x+3y+7=0

Discussion

19.	The percentage error in measuring the side of a cube is 0.5, Then the percentage error in its volume is
Answer» `1//2` 1 `3//2` 2 Answer :C

Discussion

20.	Let C : x^(2) -3, D : y = kx^(2) be two parabolas and L_(1) : x a , L_(2) : x = 1 (a ne 0) be two straight lines. IF C and D intersect at apoint A on the lines L_(1) then the equation of the tangent line L at A to the parabola D is
Answer» `2(a^(3) -3) x - AY + (a^(3) -3A) =0` `2(a^(2) -3) x- ay -a^(3) + 3a = 0` `(a^(3) -3) x - 2ay -2A^(3) + 6a =0` NONE of these Answer :B

Discussion

21.	For points P-=(x_1,y_1) and Q-=(x_2,y_2) of the coordinates plane, a new distance d(P,Q) is defined by d(P,Q) =\|x_1-x_2\|+\|y_1-y_2\|. Let O-=(0,0)and A-=(3,2). Consider the set of points P in the first quadrant which are equidistant (with respect to the new distance) from O and A. The locus of point P is
Answer» one -one and onto function MANY one and onto function one-one and into function RELATION but not function Solution : OBVIOUSLY, the LOCUS of P is a relation but not a function.

Discussion

22.	f(x) = cot^(-1)x: R^(+) rarr (0,pi) and g(x) = 2x-x^(2): R rarrR then the range of f(g(x)) is ...........
Answer» `(0,(PI)/2)` `(0,pi/4)` `[(pi)/4 ,pi/2)` `{pi/4}` SOLUTION :N/A

Discussion

23.	Find the area of a parallelogram whose adjacent sides are given by the vectors veca=3hati+hatj+4hatkandvecb=hati-hatj+hatk
Answer» ANSWER :The REQUIRED AREA is `SQRT(42)`

Discussion

24.	Find the area of the region enclosed by the given curves . y=sinx ,y=cosx , x=0 , x=(pi)/(2)
Answer» ANSWER :`2(SQRT(2)-1)`

Discussion

25.	Solve abs(x/(x-1))+abs(x)=x^(2)/abs(x-1).
Answer» Answer :`X in {0} CUP (1,INFTY)`

Discussion

26.	Consider the function f(x) = \|{:(a^(2)+x,,ab,,ac),(ab,,b^(2)+x,,bc),(ac,,bc,,c^(2)+x):}\| which of thefollowingis true ?
Answer» f(x) has one+vepoint of MAXIMA. f(x) hasone -ve point of minima f(x)=0 has threedistinct ROOTS Localminimum value of f(x)is zero Solution :`Delta =(1)/(a) \|{:(a^(3)+ax,,ab,,ac),(a^(2)b,,b^(2)+x,,bc),(a^(2)c,,bc,,c^(2)+x):}\|` Applying `C_(1) to C_(1)+bC_(2) +cC_(2)` and taking `a^(2) +b^(2)+c^(2)+x` COMMON we GET `Delta =(1)/(a)(a^(2)+b^(2)+c^(2)+x) \|{:(a,,ab,,ac),(b,,b^(2)+x,,bc),(c,,bc,,c^(2)+x):}\|` Applying `C_(2) to C_(2)-bC_(1) " and " C_(3) to C_(3)-cC_(1)` we get `Delta =(1)/(a)(a^(2)+b^(2)+c^(2)+x) \|{:(a,,0,,0),(b,,x,,0),(c,,0,,x):}\|` `=(1)/(a) (a^(2) +b^(2)+c^(2)+x) (ax^(2))` `=x^(2) (a^(2)+b^(2)+c^(2)+x)` Thus `Delta ` is divisib le by x and `x^(2)`. alsograph of f(x) is

Discussion

27.	Evaluate the following integrals. int(1)/(3+2sinx+cosx)dx
Answer» ANSWER :`TAN^(-1)(tan(x//2)+1)+C`

Discussion

28.	Evalaute intx^(_1//2)(2+3x^(1//3))^(-2)dx.
Answer» Solution :`I=intx^(-1//2)(2+3x^(1//3))^(-2)dx` Let ` x=t^(6) " or " dx=6T^(5)dt` ` :. I=int t^(-3)(2+3t^(2))^(-2) 6t^(5) dt` `=6 int (t^(2))/((2+3t^(2))^(2))dt` `=(6)/(9)int (t ^(2)dt)/(((2)/(3)+t^(2))^(2))` Now, Let ` t=sqrt(((2)/(3)))TAN theta` ` :. dt =sqrt(((2)/(3)))SEC^(2) theta d theta` ` :. I=(6)/(9)int((2)/(3)tan^(2) theta sqrt(((2)/(3)))sec^(2) theta d theta)/((4)/(9)sec^(4) theta)` `=sqrt((2)/(3))intsin^(2) theta d theta` `=(1)/(sqrt(6))int(1-cos 2 theta)d theta` `=(1)/(sqrt(6)){theta - (sin 2 theta)/(2)}+c` `=(1)/(sqrt(6)){theta - (tan theta)/(1+tan^(2)theta)}+c` `=(1)/(sqrt(6)){"tan"^(-1){sqrt((3)/(2))t}-(sqrt((3)/(2))*t)/(1+(3)/(2)t^(2))}+c` `=(1)/(sqrt(6)){"tan"^(-1){sqrt((3)/(2))x^(1//6)}-(sqrt(6)x^(1//6))/(2+3x^(1//3))}+c`

Discussion

29.	Find the second order derivatives of the following functions: sin^(2) x
Answer» ANSWER :`2 COS 2X`

Discussion

30.	Two lines are drawn at right angles one being a tangent to y^(2) = 12x and the other to x^(2) = 111116y. If the locus of their point of intersection is (x^(2)+y^(2))(lx+my)+(nx-3y)^(2)=0 then l+n-m is equal to
Answer» Solution :Tangets on both the parabolas `y = mx +(3)/(m),y = mx-4m^(2)` `m^(2)x- my +3=0,4m^(2)-mx+y=0` eliminating m we GET the LOCUS

Discussion

31.	The shortest distance between the skew lines l_(1): barr=bara_(1)+lambdabarb_(1) and l_(2): barr= bara_(2)+mubarb_(2) is
Answer» `((bara_(2)-bara_(1)).barb_(1)xxbarb_(2))/(\|barb_(1)xxbarb_(2)\|)` `((bara_(2)-bara_(1)).bara_(2)xxbarb_(2))/(\|barb_(1)xxbarb_(2)\|)` `((bara_(2)-barb_(2)).bara_(1)xxbarb_(1))/(\|barb_(1)xxbarb_(2)\|)` `((bara_(1)-barb_(2)).barb_(1)xxbara_(2))/(\|barb_(1)xxbara_(2)\|)` ANSWER :A

Discussion

32.	Find (dy)/(dx)y=cos^(-1)(e^2)
Answer» SOLUTION :`d/dxcos^-1(e^x)=(-1)/(SQRT(1-(e^x)^2))xxe^x`

Discussion

33.	Professor Malingowski, a chemist and teacher at to communite college, is organizing his graduated cylinders in the hopes of keeping his office tidy and setting a good example for his students. He has beakers with diameters, in inches, of 1/2,3/4,4/5,1 and 5/4. With his original five cylinders, Professor Malingowski realizes that he is missing a cylinder necessary for his upcominglab demonstration for Thurday's class. He remembers that the cylinder the needs, when added to the original five, will create a median diameter value of 9/10 for the set of six total cylinders. He also knows that the measure of the sixth cylinder will exceed the value of the range of the current five cylinders by a width of anywhere from 1/4 inches to 1/2 inches, inclusive. Based on the above data, which is one possible value of y, the diameter of this missing sixth cylinder?
Answer» ANSWER :`1 LE y le 1.25`

Discussion

34.	State with reason whether the following functions have inverse h : {2, 3, 4, 5} rarr {7, 9, 11, 13} with h = {(2,7), (3,9), (4, 11), (5, 13)}
Answer» SOLUTION :`h = {(2,7), (3,9), (4, 11), (5,13)}` YES, SINCE h is bijective `h^(-1)= {(7,2), (3,9), (11,4), (13,5)}`

Discussion

35.	Integrate the following function : int(2x)/(sqrt(1-x^(2)-x^(4)))dx
Answer» ANSWER :`SIN^(-1)((2x^(2)+1)/(SQRT2))+C`

Discussion

36.	Find area of the triangle withh vertices at the point given in each of the following : (11,8),(3,2),(8,12)
Answer» ANSWER :25

Discussion

37.	If adj [(1,0,2),(-1,1,-2),(0,2,1)]=[(5,a,-2),(1,1,0),(-2,-2,b)], then find a ,b
Answer» ANSWER :a = 4 , B = 1

Discussion

38.	a matrix H= [h_(ij)is used to keep track of three football players (numbered 1,2,and 3) in three matches 1^(st) ,2^(nd) and 3^(rd)Re(H_k) =number of matches in which both ))and k^(th)players or both did not play (if) jk it is 3) Img (h_(k) =( number of matches played byj^(th)player ) det (H) is
Answer» complex with imaginary PART a multiple of 4 non-positive multiple of 3 non-negative multiple of 4 none of these ANSWER :C

Discussion

39.	If f: [(2,oo) rarr R be the function defined by f(x) =x^(2) - 4x +5, then the range of f is ...........
Answer» R `[1,oo)` `[4,oo)` `[5,oo)` SOLUTION :N/A

Discussion

40.	a matrix H= h_(ij)is used to keep track of three football players (numbered 1,2,and 3) in three matches 1^(st) ,2^(nd) and 3^(rd)Re(H_k) =number of matches in which both )and k^(th)players or both did not play (if) jk it is 3) Img (h_(k) =( number of matches played byj^(th)player ) Let P= [P_(ik) ]. P_(ik) =1if player played in k^(th) match and p_k)=-i otherwise(i=sqrt(-1) )then , \|det(H) is equal to
Answer» <P>\|det (P) `\|_3` \|det (P)`\|^(2) ` \|det(P)`\|^(3) ` none of these Answer :B

Discussion

41.	In a bakery four types of biscuits are available. In how many ways a person can buy 10 biscuits if he decide to take atleast one biscuit of each variety?
Answer» ANSWER :84

Discussion

42.	In 1919, H. S. Reed and R. H. Holland published a paper on the growth of sunflowers. Included in the paper were the table and graph above, which show the height h, in centimeters, of a sunflower t days after the sunflower begins to grow. Over which of the following time periods is the average growth rate of the sunflower least?
Answer» Day 0 to Day 21 Day 21 to Day 42 Day 42 to Day 63 Day 63 to Day 84 Solution :The average growth rate of the sunflower over a certain time period is the increase in height of the sunflower over the period divided by the time. Symbolically, this rate is `(h(B)-h(a))/(b-a)` ,where a and b are the first and the last day of the time period, respectively. Sincethe time period for each option is the same (21 DAYS), the total growth over the period can be used to evaluate in which time period the sunflower grew the least. According to the graph, the sunflower grew the least over the period from day 63 to day 84. Therefore, the sunflower’s average growth rate was the least from day 63 to day 84. Alternate approach: The average growth rate of the sunflower over a certain time period is the slope of the line segment that joins the point on the graph at the beginning of the time period with the point on the graph at the end of the time period. Based on the graph, of the four time periods, the slope of the line segment is least between the sunflower’s height on day 63 and its height on day 84. Choices A, B, and C are incorrect. On the graph, the line segment from day 63 to 84 is less steep than each of the THREE other line segments representing other periods. Therefore, the average growth rate of the sunflower is the least from day 63 to 84.

Discussion

43.	For a matrix , A = [{:(1, 0, 0),(2, 1, 0),(3,2,1):}] , if U_(1), U_(2) and U_(3) are 3xx 1column matrices satisfying AU_(1) = [{:(1),(0),(0):}], AU_(2) = [{:(2),(3),(0):}], AU_(3) = [{:(2),(3),(1):}] and U is a 3xx 3 matrix whose columns are U_(1) , U_(2) and U_(3) . then sum of the element of U^(-1) is
Answer» 6 0 1 2`//`3 Answer :B

Discussion

44.	In 1919, H. S. Reed and R. H. Holland published a paper on the growth of sunflowers. Included in the paper were the table and graph above, which show the height h, in centimeters, of a sunflower t days after the sunflower begins to grow. The function h, defined by h(t) = at + b , where a and b are constants, models the height, in centimeters, of the sunflower after t days of growth during a time period in which the growth is approximately linear. What does a represent?
Answer» The predicted NUMBER of centimeters the sunflower grows each day during the period The predicted height, in centimeters, of the sunflower at the beginning of the period The predicted height, in centimeters, of the sunflower at the end of the period The predicted total increase in the height of the sunflower, in centimeters, during the period Solution :Based on the DEFINITION and contextual interpretation of the function h, when the value of t increases by 1, the height of the sunflower increases by a centimeters. Therefore, a represents the predicted AMOUNT, in centimeters, by which the sunflower grows each day during the period the function MODELS. Choice B is incorrect. In the given model, the beginning of the period corresponds to t = 0, and since h(0) = b, the predicted height, in centimeters, of the sunflower at the beginning of the period is represented by b, not by a. Choice C is incorrect. If the period of time modeled by the function is c days long, then the predicted height, in centimeters, of the sunflower at the end of the period is represented by ac + b, not by a. Choice D is incorrect. If the period of time modeled by the function is c days long, the predicted total increase in the height of the sunflower, in centimeters, during that period is represented by the difference h(c) − h(0) = (ac + b) − (a · 0 + b), which is equivalent to ac, not a.

Discussion

45.	Let x+1/x=1 and a, b and c are distict positive integer such that (x^(a)+1/x^a)+(x^b+1/x^(b))+(x^(c) +1/x^(c))=0 Then the minimum value of (a+b+c) is
Answer» 7 8 9 10 Answer :C

Discussion

46.	Whichof the followingcould bethe equation of the graph shown above ?
Answer» ANSWER :B

Discussion

47.	Integrate the following rational functions : int(1)/(6e^(2x)+5e^(x)+1)dx
Answer» ANSWER :`X+2log\|2e^(x)+1\|-3log\\|3e^(x)+1\|+c`

Discussion

48.	Product of perpendiculars from the foci of x^(2)/4-y^(2)/9=1" to "y=mx+sqrt(4m^2- 9) where m gt 3/2 is
Answer» 4 `(36)/(13)` 3 9 Answer :D

Discussion

49.	If y=f(x) and x=g(y) are inverse of each other. Then g'(y) and g"(y) in terms of derivative of f(x) is
Answer» `-(F"(X))/([f'(x)]^(3))` `(f"(x))/([f'(x)]^(2))` `-(f"(x))/([f'(x)]^(2))` NONE of these Answer :A

Discussion

50.	If overset(-)(a)=2hati-3hatj+5hatk, overset(-)(b)=3hati-4hatj+5hatk and overset(-)(c)=5hati-3hatj-2hatk, then the volume of the parallelopiped with co-terminus edges overset(-)(a)+overset(-)(b)+ overset(-)(c), overset(-)(c)+overset(-)(a) is
Answer» 1 5 8 16 Answer :D

Discussion

Explore topic-wise InterviewSolutions in Current Affairs.

Find the second order derivatives of the function log (log x).

L.P.P. has constraints of

Number of ways of selecting 4 shoes from 10 pairs such that exactly onee correct pair is selected is

Resolve (3x^(2)+1)/((x^(2)-3x+2)(2x+1)) into partial fractions.

Identify the quantifiers of the following statements For every real number x, x^2 != x

Let f(x) =g(x) |(x-1)(x-2)(x-3)^(2)(x-4)^(3)|, whereg(x)= x^(3)+bx^(2)+cx+d. if f(x)is differentiablefor all x in R and f' (3) + f'''(4)=0 thenthe value ofg(5) is ______.

sin 10^(@) sin50^(@)sin 60^(@) sin 70^(@)=m " and tan "20^(@) tan 40^(@) tan60^(@) tan80^(@)= n, then (n)/(m)=

The solution of ydx - xdy = xydx is

Consider the function f(x)={{:(xsin(pi)/(x),"for"xgt0),(0,"for"x=0):} Then, the number of points in (0,1) where the derivative f'(x) vanishes is

If thecorrect formof 2i^(4) +4i^(3) - i^(2) + 3i is a + ib , wheretheimaginarynumberI issuchthat i^(2) = - 1 . Whatis the value of ( a+b) ?

If aN = {ax : x in N} " then " 3N cap 7N =

Differentiate: (2^(x))/(x)

The tangent at a point P(theta) to the ellipse x^2/a^2+y^2/b^2=1 cuts the auxilliary circle at Q and R. If QR subtend a right angle atC (centre) then show that e=1/sqrt(1+sin^2 theta)

The system of vector hat(i), hat(j), hat(k) is

{:(" "Lt),(n rarr oo):} ((1)/(sqrt(4n^(2)-1))+(1)/(sqrt(4n^(2)-2^(2)))+....+(1)/(sqrt(3n^(2))))=

If the vectors a=hati-hatj+2hatk, b=2hati+4hatj+hatk and c=lambdahati+hatj+mu hatk are mutually orthogonal, then (lambda, mu) is equal to

Evaluate the definite integrals int_(0)^(1)(dx)/(1+x^(2))

Find the equation of the common tangent of the following circles at their point of contact. x^2+y^2+10x-2y+22=0 x^2+y^2+2x-8y+8=0

The percentage error in measuring the side of a cube is 0.5, Then the percentage error in its volume is

Let C : x^(2) -3, D : y = kx^(2) be two parabolas and L_(1) : x a , L_(2) : x = 1 (a ne 0) be two straight lines. IF C and D intersect at apoint A on the lines L_(1) then the equation of the tangent line L at A to the parabola D is

f(x) = cot^(-1)x: R^(+) rarr (0,pi) and g(x) = 2x-x^(2): R rarrR then the range of f(g(x)) is ...........

Find the area of a parallelogram whose adjacent sides are given by the vectors veca=3hati+hatj+4hatkandvecb=hati-hatj+hatk

Find the area of the region enclosed by the given curves . y=sinx ,y=cosx , x=0 , x=(pi)/(2)

Solve abs(x/(x-1))+abs(x)=x^(2)/abs(x-1).

Consider the function f(x) = |{:(a^(2)+x,,ab,,ac),(ab,,b^(2)+x,,bc),(ac,,bc,,c^(2)+x):}| which of thefollowingis true ?

Evaluate the following integrals. int(1)/(3+2sinx+cosx)dx

Evalaute intx^(_1//2)(2+3x^(1//3))^(-2)dx.

Find the second order derivatives of the following functions: sin^(2) x

Two lines are drawn at right angles one being a tangent to y^(2) = 12x and the other to x^(2) = 111116y. If the locus of their point of intersection is (x^(2)+y^(2))(lx+my)+(nx-3y)^(2)=0 then l+n-m is equal to

The shortest distance between the skew lines l_(1): barr=bara_(1)+lambdabarb_(1) and l_(2): barr= bara_(2)+mubarb_(2) is

Find (dy)/(dx)y=cos^(-1)(e^2)

State with reason whether the following functions have inverse h : {2, 3, 4, 5} rarr {7, 9, 11, 13} with h = {(2,7), (3,9), (4, 11), (5, 13)}

Integrate the following function : int(2x)/(sqrt(1-x^(2)-x^(4)))dx

Find area of the triangle withh vertices at the point given in each of the following : (11,8),(3,2),(8,12)

If adj [(1,0,2),(-1,1,-2),(0,2,1)]=[(5,a,-2),(1,1,0),(-2,-2,b)], then find a ,b

a matrix H= [h_(ij)is used to keep track of three football players (numbered 1,2,and 3) in three matches 1^(st) ,2^(nd) and 3^(rd)Re(H_k) =number of matches in which both ))and k^(th)players or both did not play (if) jk it is 3) Img (h_(k) =( number of matches played byj^(th)player ) det (H) is

If f: [(2,oo) rarr R be the function defined by f(x) =x^(2) - 4x +5, then the range of f is ...........

In a bakery four types of biscuits are available. In how many ways a person can buy 10 biscuits if he decide to take atleast one biscuit of each variety?

Let x+1/x=1 and a, b and c are distict positive integer such that (x^(a)+1/x^a)+(x^b+1/x^(b))+(x^(c) +1/x^(c))=0 Then the minimum value of (a+b+c) is

Whichof the followingcould bethe equation of the graph shown above ?

Integrate the following rational functions : int(1)/(6e^(2x)+5e^(x)+1)dx

Product of perpendiculars from the foci of x^(2)/4-y^(2)/9=1" to "y=mx+sqrt(4m^2- 9) where m gt 3/2 is

If y=f(x) and x=g(y) are inverse of each other. Then g'(y) and g"(y) in terms of derivative of f(x) is

If overset(-)(a)=2hati-3hatj+5hatk, overset(-)(b)=3hati-4hatj+5hatk and overset(-)(c)=5hati-3hatj-2hatk, then the volume of the parallelopiped with co-terminus edges overset(-)(a)+overset(-)(b)+ overset(-)(c), overset(-)(c)+overset(-)(a) is