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InterviewSolution
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.
| 3551. |
Ifthecoefficient ofxinthe expansionof(x ^ 2+(k )/(x) ) ^5is 270 , then k isequalto |
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Answer» 1 Theco-efficient of` x ^m`intheexpansionof` (ax ^p +(B)/(x^Q))^N `is` ""^nC_r a ^( n - r) b ^ r `,where ` r = (np - m ) /( p + q )`, for all ` n in N `. Here, ` p = 2 , q = 1 ,n = 5, m = 1, a= 1 , b =k ` `thereforer = (np- m ) /(p +q ) ` `= ((5)/(2) - 1 )/( 2 + 1 ) ` `thereforer = 3` `therefore`co - efficientofxis ` ""^5C_3 ( 1 ) ^(5-3) (k) ^3 ` `therefore""5C_3 k^ 3=270` ` k^ 3=(270 )/(""5C _ 3 )` `k ^ 3=(270 xx 1xx 2 ) /(5 xx 4 ) ` `k^ 3=27` ` rArrk =3 ` |
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| 3552. |
Obatin the Maclaurin's series expansion for the following function . (i) e^(2x)(ii)sin^(2)x(iii) (1)/(1+x)(iv) tan x, -lt x lt (pi)/(2) |
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| 3553. |
(2x)/(x^(2)+1)+(1)/(3)((2x)/(x^(2)+1))^(3) +(1)/(5)((2x)/(x^(2)+1))^(5)+…..oo= |
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Answer» `log_(E )((x+1)/(x-1))` |
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| 3554. |
If int (sec^(2) x)/(3 + 4 tan x)dx = K log | 3 + 4 tan x | + C then 1/K = |
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Answer» `1//4` |
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| 3555. |
Let f:R to R defined by f(x)=x^(2)+1, forall x in R. Choose the correct answer |
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Answer» F is ONE-one onto |
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| 3557. |
For all a > 0, which of the following expression is equal to a^(-2)? |
| Answer» ANSWER :D | |
| 3558. |
The value of lambda for which the lines 3x + 4y=5, 2x+3y=4 and lambda x+4y=6 meet at a point is |
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Answer» 2 |
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| 3559. |
Lives of two models or refrigerators A and B obtained in a survey are given below: (##VIK_MAT_IIA_QB_C08_SLV_012_Q01.png" width="80%"> Which refrigerator model you suggest to purchase? |
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| 3560. |
Evaluate the following integrals int_1^4(dx/sqrtx)dx |
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Answer» SOLUTION :`int_1^4dx/(sqrtx)=int_1^4x^(-1/2)DX=[X^(1/2)/(1/2)]_1^4` `2{4^(1/2)-1^(1/2)}=2(2-1)=2` |
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| 3561. |
A manufactures makes two products, A and B. Product A sell at Rs. 200 each and takes 1/2 hours to make. Product B shell at Rs. 300 each and takes 1 hour to make. There is a permanent order for 14 of product A and 16 of product B.A working week consists of 40 hours of production and the weekly turnover must not be less than Rs. 10000. If the profit on each of the product A is Rs.20 and on product B, it is Rs.30 then how many of each should be produced so that the profit is maximum? Asso, find the maximum profit. |
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Answer» MAXIMIZE `Z=20x+30y.` subject to the constraints `x+2yle80,2x+3yge1000,xge14,yge16.` |
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| 3562. |
If matrix A = [a_(i j)]_(2 xx 2), where a_(i j) = {("1 if",i ne j),("0 if",i = j):} then A^(2) is equal to |
| Answer» Answer :A | |
| 3563. |
A furniture dealer deals only two items, tables and chairs. He has Rs. 500 to invest and a storage capacity for 60 pieces. A table costs him Rs. 250 and chair Rs. 50. He can sell a table for a profit of Rs. 50 and a chair for a profit of Rs. 15. The number of tables and chairs respectively to get maximum profit are |
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Answer» 10,50 |
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| 3564. |
Classify 1000 cm^3measures as scalar and vector. |
| Answer» SOLUTION :Volume-scalar | |
| 3565. |
Find the magnitude of two vectors vec(a) and vec(b) having the same magnitude and such that the angle between them is 60^@ and their scalar product is 1//2. |
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| 3566. |
consider the systemof equations : ltbr. 3x-y +4z=3 x+2y-3z =-2 6x+5y+lambdaz =-3 Prove thatsystemof equation has atleast one solutionforall realvaluesof lambda.also provethatinfinitesolutionsof the systemof equationssatisfy (7x-4)/(-5)=(7y+9)/(13)=z |
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Answer» Solution :`Delta= |{:(3,,-1,,4),(1,,2,,-3),(6,,5,,lambda):}|=7lambda +35` `" If "7lambda +35ne 0 i.e., lambda ne -5`then system has a unique solution (As `Delta ne 0` givesuniquesolution). But if `lambda =-5` then we have `Delta =0` . Solution exists in thiscase if `Delta_(X) =Delta_(y) =Delta_(Z)=0` `" For " lambda =-5` `Delta_(x)=|{:(3,,-1,,4),(-2,,2,,-3),(-3,,5,,5):}|=0` `Delta_(y)= |{:(3,,-3,,4),(1,,-2,,-3),(6,,-3,,-5):}|=0` `" and"Delta_(z)=|{:(3,,-1,,3),(1,,2,,-2),(6,,5,,-3):}|=0` Thus `Delta =Delta_(x)=Delta_(y)=Delta_(z)=0` and HENCE there exists infinite number of solutions. Now. eliminating x from the equations .we get `7y-13z =-9` Letus put `z = k in R.:. y= (13k -9)//7" and" so x=(4-5k)//7` Where k isany realnumber . `" Thus " .(7x-4)/(-5) =(7y+9)/(13)=z` |
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| 3567. |
For the given reaction , A(g) rarr 2B(g) , Delta_(r)H= 30 kJ//"mole" Delta_(r)S = 150 J//mol at 300 K If C_(P,A) = 20 J//K mol and C_(P,B) = 20 J//Kmol. (ln 3//2 = 0.4) Which of the following statement is/are correct ? |
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Answer» `DELTAH` will increase on increasing TEMPERATURE |
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| 3568. |
Evaluate the definite integral in exercise overset(1)underset(0)int x e^(x^(2))dx |
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| 3569. |
If O(vec0) is the circumcentre and O' the orthocentre of a triangle ABC, then prove that i. vec(OA)+vec(OB)+vec(OC)=vec(OO') ii. vec(O'A)+vec(O'B)+vec(O'C)=2vec(O'O) iii. vec(AO')+vec(O'B)+vec(O'C)=2 vec(AO)=vec(AP) where AP is the diameter through A of the circumcircle. |
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Answer» Solution :O is the CIRCUMCENTRE, which is the intersection of the right bisectiors of the sides of the triangle, and O' is the ORTHOCENTER, which is the point of intersection of ALTITUDES DRAWN from the vertices. Also, from geometry, we know that `""2OD=AO'` `therefore""2vec(OD)=VEC(AO')""(i)` i. To prove : `vec(OA)+vec(OB)+vec(OC)=vec(OO')` Now `vec(OB)+vec(OC)=2vec(OD)=vec(AO')` `rArr" "vec(OA)+vec(OB)+vec(OC)=vec(OA)+vec(AO')=vec(OO')""`[by (i)] ii. To prove : `vec(O'A)+vec(O'B)+vec(O'C)=2vec(OO')` `"""L.H.S."=2vec(DO)+2vec(O'D)""`[by (i)] `""2(vec(O'D)+vec(DO))=2vec(O'O)` iii. To prove : `vec(AO')+vec(O'B)+vec(O'C)=2vec(AO)=vec(AP)` `"""L.H.S."=2vec(AO')-vec(AO')+vec(O'B)+vec(O'C)` `""=2vec(AO')+(vec(O'A)+vec(O'B)+vec(O'C))` `""=2vec(AO')+2vec(O'O)=2vec(AO)` `""=vec(AP)` (where `AP` is the diameter through `A` of the circumcircle). 
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| 3570. |
If f(p,q)=int_(0)^(pi//2)cos^(p)x cos qx dx, then : |
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Answer» <P>`F(p, Q)=(q)/(p+q)f(p-1,q-1)` |
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| 3571. |
If f(x)=([{x}]tan^(-1)((x^(2)-3x-1)/(x^(2)-3x+5))+3-x^(7))^((1)/(7)). Where [k] and {k} denotes greatest integer and fractional part functions of k respectively, then the value of f^(-1)(50)-f(50)+f(f(100)), is |
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Answer» 0 |
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| 3572. |
Fortwonon - zerovectorsvecaand vecb| veca + vecb| = | veca|thenvectors2 veca+ vecbandvecbare ….. |
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Answer» PARALLEL |
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| 3573. |
Lt_(ntooo)[(n^(1//2))/(n^(3//2))+(n^(1//2))/((n+3)^(3//2))+(n^(1//2))/((n+6)^(3//2))+.......+(n^(1//2))/({n+3(n-1)}^(3//2))]= |
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Answer» `1/3` |
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| 3574. |
On R, the set of real numbers, define a relation ~ as follows: a, b in R , a ~ b if {a} = {b} Where {a} = a-[a], and [a] = greatest integer le a. Then ~ is an equivalence relation on R. Which of the following is an equivalence class containing sqrt(2) |
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Answer» `{a + 1/(sqrt(2)+1), a in Q}` |
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| 3575. |
Show that the points (a,b,c,)(b,c,a) and (c,a,b) from an equilateral triangle. |
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Answer» Solution :LET A =(a,b,c),B=(b,c,a) and C =c,a,b) ` "Then" AB = SQRT((b-a)^2+(c-b)^2+(a-c)^2) ` `sqrt((a-b)^2+(b-c)^2+(c-a)^2)` `"Similarly"BC = sqrt((a-b)^2+(b-c)^2+(c-a)^2) "and" CA =sqrt((a-b)^2+(b-c)^2+(c-a)^2)` THUS AB=BC=CA Hence `Delta ABC` is an equilateral TRIANGLE.(PROVED) |
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| 3576. |
Letf'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2)) and g(x) =f(sinx)+f(cosx) If x=4 is the only point of maxima in its neighborhood but x=3 is neither a point of maxima nor a point of minima then which of the following can be true? |
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Answer» `a lt 0, b gt 0` or `-9+12+a=3a+b or 2a+b=3` Also `f(4^(+)) or 4a+b=-b+6 r 2a+b=3` Thus f(x) is contnous for INFINITE values of a and b also `f(x)={{:(-2x+4,xlt3),(a,3ltxlt4),((-b)/(4),xgt4):}` For f(x) to be diffentiable `f(3^(-))=f(3^(+))` or `a=-2 and -(bb)/(4) =a=-2 or b=8` But these values do not SATISFY EQUATION (1) HENCE f(x) cannot be differentiable 
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| 3577. |
Statement -1: If a and b are positive real numbers and [.] denotes the greatest integer function , then |
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Answer» Statement -1 is true, Statement-2 is true,, Statement-2 is a CORRECT explanation for Statement-1. ` 0le {x}LT 1` `rArr 0le ({x})/(x) lt (1)/(x) " for all " x gt 0 rArr lim_(XTO oo) ({x})/(x)=0` Hence, statement -2 is true. Now, `lim_(xto0^+)(x)/(a) [(b)/(a)]=lim_(xto0 ^+)(x)/(a)((b)/(x)-{(b)/(x)})=lim_(xto0^+) ((b)/(a)-(x)/(a){(b)/(x)})` ` rArr lim_(xto 0^+)(x)/(a) [(b)/(x)]=(b)/(a)-(b)/(a)lim_(xto0^+)(x)/(b) {(b)/(x)}=(b)/(a)-(b)/(a)lim_(xto0^+)({(b)/(x)})/((b)/(x))` ` rArr lim_(xto 0^+)(x)/(a) [(b)/(x)]=(b)/(a)-(b)/(a)lim_(XTOOO) ({y})/(y)," where "y=(b)/(x)` ` rArr lim_(xto 0^+)(x)/(a) [(b)/(x)]=(b)/(a)-(b)/(a)xx0=(b)/(a)""["USING statements -2"]` |
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| 3578. |
Match the correct for blood group and donar compatibility :- {:(,"Blood group",,"Donar compatibility"),(a.,A,(i),"B,O"),(b.,B,(ii),O),(c.,AB,(iii),"A,O"),(d.,O,(iv),"AB, A, B, O"):} |
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Answer» a-(iii), B-(i), c-(IV), d-(II) |
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| 3580. |
Integration by partial fraction : int sin^(5)x cos^(4)x dx=... |
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Answer» `-(1)/(5)cos^(5)x+(2)/(7)x-(1)/(9)cos^(9)x+c` |
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| 3581. |
A stone is dropped into a quiet lake and waves move in a circle at a speed of 3.5 cm/sec.At the instant when the radius of the circular wave is 7.5 cm, how fast is the enclosed area increasing |
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| 3582. |
Find all the points of discontinuity of the greatest integer function defined by f(x) = [x] , where [x]denotes the greatest integer less than or equal to x. |
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Answer» Solution :Graph of the function is given in FIGURE. From the graph, it seems like that f(x) is dicontinuousat every integral point of x. Case I : Let c be a real number which , is not equal to any integer. It is evident from the graph that for all real numbers close to c the value of the function is equal to [c] , i.e,` underset (x to c) lim f(x) = underset(x to c) lim [x] = [c] ` , Also f (c) = [c] and hence the function is continuous at all real number not equal to integers. Case II : Let c be an integer. Then we can find a SUFFICIENTLY small real number h GT 0 i.e.0 LT h lt 1. such that [c -h] = c-1 WHEREAS =[c + h] =c Thus, ` underset( x to c^(-)) lim f(x) = c -1 and underset(x to c^(+)) f (x) =c ` Since these cannotbe equal to each other for any c, the function is discontinuous at every integral point. 
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| 3583. |
If x^(2) +y^(2) -4x + 6y + c = 0represents a circle with radius 6 then find the value of c. |
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| 3584. |
One in 9 ships is likely to be wrecked when they are set on a sall. When 6 ships are on sail, find the probability for (i) atleast one will arive safety. (ii) exactly three will arrive safety |
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| 3585. |
Find the value of a such that the function f defined by f(x)={((sinax)/(sinx) if xne0),(1/a if x=0):} is continuous at x=0. |
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Answer» Solution :F(0)=1/a `if f(x) is CONTINUOUS at x=0 then `lim_(xto0)f(x)=f(0)` `implieslim_(xto0)(sinax)/(sinx)=1/a` `a=1/a[becauselim_(thetato0)(sintheta)/(THETA)=1` a^2=1impliesa=+-1` |
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| 3586. |
Evaluate the following integrals using properties of integration : int_(0)^(pi) x [ sin^(2) ( sin x ) + cos^(2) ( cos x ) ]dx |
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| 3587. |
The volume of a cube increases at a consant rate. Prove that the increase in its surface area varies inversely as the length of the side. |
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| 3588. |
Evaluate the following integrals intsin^(-1)((2x)/(1+x^(2)))dx |
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| 3589. |
If x=(5)/((2!).3)+(5.7)/((3!).3^(2))+(5.7.9)/((4!).3^(3))+…. then find the value of x^(2)+4x. |
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| 3590. |
Side AB of triangle ABC slides on coordinate axis if tan A = p ,tan B = 1 / p, find the locus of vertexC. |
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| 3591. |
If the function f(x) =[[(sin 3x/x), x!=0] ,[k , x=0]] is continuous at x=0, then k= |
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Answer» 1 |
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| 3592. |
Find the variance and standard deviation of the following data(##VIK_MAT_IIA_QB_C08_SLV_009_Q01.png" width="80%"> |
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| 3593. |
If x= e^((x)/(y)), then prove that (dy)/(dx)= (x-y)/(x.log x) |
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| 3594. |
Solve the following linear programming problems graphically : Minimise : Z = 18x +10y subject to constraints 4x+y ge 20, 2x+3y ge 30, x, y ge 0. |
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| 3595. |
A variable plane makes intercepts on x, y and z axes and it makes a tetrahedron of volume 64 cu. Units. The locus of foot of perpendicular from origin on the plane is |
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Answer» `(x^(2)+y^(2)+z^(2))^(2)=384 xyz` VOLUME of TETRAHEDRON OABC is `v=(abc)/6=64 rArr abc=384` Foot of perpendicular forrm (0,0,0) on this plane is `x/(1//a)=y/(1//b)=z/(1//c)=1/(1/a^(2) +1/b^(2)+1/c^(2))`=k `rArr x=k/a, y=k/b, z=k/c` and `1/k=1/a^(2)+1/b^(2)+1/c^(2)` `rArr 1/k=(x^(2)+y^(2)+z^(2))/k^(2) rArr x^(2)+y^(2)+z^(2)=`k `therefore (x^(2)+y^(2)+z^(2))^(2)=abc xyz=384 xyz` is the required locus. |
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| 3596. |
If therange, menadeviation variancestandarad deviation of 1,2,3,4,5 are respectivelydenotedby a,b,c then |
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Answer» ` a LT B lt C lt d` |
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| 3597. |
Statement I The system of linear equations x+(sin alpha )y+(cos alpha )z=0 x+(cos alpha ) y+(sin alpha )z=0 -x+(sin alpha )y-(cos alpha )z=0 has a not trivial solution for only one value ofalphalying between0 and pi. Statement II |{:(sin x, cos x , cos x),( cos x , sin x , cos x) , (cos x , cos x , sin x ):}|=0 has no solution in the interval -pi//4 lt x lt pi//4. |
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Answer» Statement I is TRUE , Statement II is true , Statement II is a CORRECT explanation for Statement I. |
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| 3599. |
If a=1/sqrt(10)(3i+k)" and "b=1/7(2i+3j-6k), then the value of (2a-b).[(a times b) times (a+2b)] is |
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Answer» 3 |
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| 3600. |
If int (cos x+x)/(1+sin x) dx=f(x)+int (3"cos"x/2-"sin"x/2)/("cos"x/2+"sin"x/2) dx+c, then f(x)= |
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Answer» `(-2 X)/(1+"TAN"x/2)` |
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