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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.
| 6001. |
Integrate the following functions. int(1)/(x^(3)+1)dx |
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| 6002. |
In the group {1,2,3,4,5,6} under multiplication mod 7, 2^(-1)xx4= |
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Answer» 1 |
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| 6005. |
alphaandbeta are the roots of the equationThen (alpha-(1)/(beta))and(beta-(1)/(alpha)) are the roots of the equation - |
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Answer» `AX^(2)+a(B-1)^(x)+(a-1)^(2)=0` |
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| 6006. |
In the prime factorization of 37!""=2^(alpha_(2)).3^(alpha_(3)).5^(alpha_(5)).........37^(alpha_(37)) the ratio alpha_(3):alpha_(5)= |
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Answer» `3:5` |
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| 6007. |
Find the regression coefficient of x on y from the following data: sumx=15, sumy=15, sumy^(2)=49, sumxy=44, n=5, Also find the value of x when y = 7. |
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| 6008. |
Integrate the following functions : x^(3)sinx^(2) |
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| 6009. |
Evaluate the following integrals. int(1)/(1+tanx)dx |
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| 6010. |
Evaluate : int_(0)^(1)(log(1+x))/(1+x^2)dx |
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| 6012. |
Discuss the continuity of the function f given by {{:(x," if "x ge 0),(x^2," if "x lt 0):}. |
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| 6013. |
Find the equation of the circle with centre C and radius r where C = ((1)/(2) ,-9) ,r=5 |
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| 6014. |
Giventhat thesystem of equationsx=cy+bz ,y=az+cx , z=bx +ay has nonzerosolutions andand atleastone of the a,b,c is a properfraction. Systemhas solution such that |
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Answer» `X,y,z -= (1-2a^(2)):(1-2b^(2)):(1-2c^(2))` `-x+cy+bz=0` `cx-y+az=0` `bx+ay-z=0` has a nonzerosolution if `Delta = |{:(-1,,c,,b),(c,,-1,,a),(b,,a,,-1):}|=0` then CLEARLY the SYSTEMHAS infinitely manysolutions .From(1) and (2) we have `(x)/(ac+b) =(y)/(bc+a)=(z)/(1-c^(2)) ` `" or" (x^(2))/((1-a^(2))(1-c^(2)))=(y^(2))/((1-b^(2))(1-c^(2)a)) =(z^(2))/((1-c^(2))^(2))`[From (4)] `"or" (x^(2))/(1-a^(2))=(y^(2))/(1-b^(2)) =(z^(2))/(1-c^(2))` from (5)we see that `1-a^(2),1-b^(2),1-c^(2)` are allpositiveor allnegative .Giventhat oneof a,b,cis properfraction so `1-a^(2) gt ,1-b^(2) gt 0,1-c^(2) gt 0` WHICHGIVES `a^(2) +b^(2)+c^(2) lt 3` using(4) and (6) we get `1lt 3+2 abc` `"or" abc gt -1` |
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| 6015. |
Giventhat thesystem of equationsx=cy+bz ,y=az+cx , z=bx +ay has nonzerosolutions andand atleastone of the a,b,c is a properfraction. abc is |
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Answer» `gt-1` `-x+cy+bz=0` `cx-y+az=0` `bx+ay-z=0` has a nonzerosolution if `Delta = |{:(-1,,c,,b),(c,,-1,,a),(b,,a,,-1):}|=0` then Clearly the systemhas infinitely manysolutions .From(1) and (2) we have `(x)/(ac+b) =(y)/(bc+a)=(z)/(1-c^(2)) ` `" or" (x^(2))/((1-a^(2))(1-c^(2)))=(y^(2))/((1-b^(2))(1-c^(2)a)) =(z^(2))/((1-c^(2))^(2))`[From (4)] `"or" (x^(2))/(1-a^(2))=(y^(2))/(1-b^(2)) =(z^(2))/(1-c^(2))` from (5)we see that `1-a^(2),1-b^(2),1-c^(2)` are allpositiveor allnegative .Giventhat oneof a,b,cis properfraction so `1-a^(2) gt ,1-b^(2) gt 0,1-c^(2) gt 0` WHICHGIVES `a^(2) +b^(2)+c^(2) lt 3` using(4) and (6) we get `1lt 3+2 abc` `"or" abc gt -1` |
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| 6016. |
State which of the following statement is false ? |
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Answer» If `A ={0,1,2,3},B={-3,-2,-1,0,1} andf:A RARR B ` is the mapping defined by `f(x)=x-3` for all `x in A`, then f is a one-one mapping |
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| 6017. |
int(sin^2x)/(1+cosx)dx |
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Answer» SOLUTION :`INT(sin^2x)/(1+cosx)DX=int(sin^2x(1-cosx))/(1-cos^2x)dx` =`int(sin^2x(1-cosx))/(sin^2x)dx` =`int(1-cosx)dx` |
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| 6018. |
Giventhat thesystem of equationsx=cy+bz ,y=az+cx , z=bx +ay has nonzerosolutions andand atleastone of the a,b,c is a properfraction. a^(2)+b^(2)+c^(2) is |
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Answer» ` gt 2` `-X+cy+bz=0` `cx-y+az=0` `bx+ay-Z=0` has a nonzerosolution if `Delta = |{:(-1,,c,,b),(c,,-1,,a),(b,,a,,-1):}|=0` then Clearly the systemhas infinitely manysolutions .From(1) and (2) we have `(x)/(ac+b) =(y)/(bc+a)=(z)/(1-c^(2)) ` `" or" (x^(2))/((1-a^(2))(1-c^(2)))=(y^(2))/((1-b^(2))(1-c^(2)a)) =(z^(2))/((1-c^(2))^(2))`[From (4)] `"or" (x^(2))/(1-a^(2))=(y^(2))/(1-b^(2)) =(z^(2))/(1-c^(2))` from (5)we see that `1-a^(2),1-b^(2),1-c^(2)` are allpositiveor allnegative .Giventhat oneof a,b,cis properfraction so `1-a^(2) gt ,1-b^(2) gt 0,1-c^(2) gt 0` whichgives `a^(2) +b^(2)+c^(2) lt 3` using(4) and (6) we get `1lt 3+2 ABC` `"or" abc gt -1` |
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| 6019. |
One vertex of the triangle of maximum area that can be inscribed in the curve |z-(1+i)|=2sqrt2 is -2 + 2i, then the remaining vertices are |
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Answer» `((5- SQRT3)/(2))+i((1+3 sqrt3)/(2))` |
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| 6021. |
int (x^(3) - 2x^(2)+5 )e^(3x) dx. |
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| 6023. |
IFtan ( A-B )= 1,sec (A +B )= (2)/(sqrt(3)) , thesmallest positivevalueof B is |
| Answer» Answer :D | |
| 6024. |
If M is a 2xx2 matrix such that [(1),(-1)]=[(-1),(2)] and M^(2)[(1),(-1)]=[(1),(0)] then sum of elements of M is |
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Answer» SOLUTION :Let`M=[[a,b],[c,d]]` Now, `M [[1],[-1]] = [[-1],[2]] & M^(2) [[1],[-1]]=[[1],[0]]` `impliesa=-1,b=0, c=4,d=2` `THEREFORE (a+b+c+d) = 5` |
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| 6025. |
Foot of perpendicular of point (2,2,2) in the plane x+y+z=9 is |
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Answer» (1,1,1) |
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| 6026. |
A die is thrown (2n + 1) times. The probability of getting 1 or 3 or 4 atmost n times is |
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Answer» `(1)/(N)` |
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| 6027. |
int (5(x^(6)+1))/(x^(2)+1)dx=.... |
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Answer» `5(x^(7)+x)TAN^(-1)x+c` |
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| 6028. |
Let f(x)=int_0^x (cost)/tdt Then at x=(2n+1)pi/2 f(x) has |
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Answer» maxima when n=-2,-4,-6,.. and MINIMA when n=-1,3,-5,.. |
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| 6029. |
(sin 5 theta)/(sin theta)is equal to |
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Answer» `16 cos ^(4) theta - 12 cos ^(2) theta +1` |
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| 6030. |
The solution set of |x+1| + |x-1| lt 2is |
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Answer» `(-1,1)` |
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| 6031. |
In triangle ABCwhich of the following is not true : |
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Answer» `VEC(AB)+vec(BC)+vec(CA)=vec(0)` |
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| 6032. |
A(x_(1),y_(1)) and B(x_(2),y_(2)) are any two distinct points on the parabola y=ax^(2) +bx+c.if P(x_(3),y_(3)) be the point on the are AB where the tangent is parallel to the chord AB, then |
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Answer» <P>`x_(3)` is the A.M. between `x_(1)` and `x_(2)` `(x_(3),y_(3))=2ax_(3)+b=(x_(2)-y_(1))/(x_(2)-x_(1))[given]...(1)` `{As the tangent is (y+y_(3))/(2)=ax x_(3) +b((x+x_(3))/(2))+c}` `because` A and B LIE on the parabola, ` THEREFORE y_(1) = ax_(1)^(2)+bx_(1)+c...(i)` and `y_(2)=ax_(2)^(2)+bx_(2)+c....(ii)` `therefore y_(1)-y_(2) = [a(x_(1)+x_(2))(x_(1)-x_(2))+b(x_(1)+x_(2))]` `therefore (y_(2)-y_(1))/(x_(2)-x_(1))=a(x_(1)+x_(2))+b` `therefore "from" (1), a(x_(1)+x_(2))+b=2ax_(3)+b` `(x_(1)+x_(2))/(2)=x_(3)` |
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| 6033. |
Find values of k if area of triangle is 4 sq. units and vertices are (2,2), (6,6) and (5,k). |
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| 6034. |
An open box with square base is to be made of a given quantity of card board of area c^(2). Show that the maximum volume of the box is (c^(3))/(6sqrt(3)) cubic units. |
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| 6035. |
By using the properties of definite integrals, evaluate the integrals int_(0)^((pi)/(2))(2logsinx-logsin2x)dx |
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| 6036. |
Find the equation of the circle with its centre at (3, 2) and which touches to the line x+2y-4=0. |
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Answer» Solution :`ABS(bar(PC))=abs((ax_1+by_1+c)/SQRT(a^2+b^2))` = `(1 XX 3 + 2 xx 2 - 4)/sqrt(1^2+2^2) = 3/sqrt5` = radius  `therefore` Equation of the CIRCLE is `(x-h)^2+(y-k)^2=a^2` or, `(x-3)^2+(y-2)^2=9/5` |
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| 6037. |
sin((pi)/3-"sin"^(-1)(-1/2)) is equal to |
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Answer» `1/2` |
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| 6038. |
Find the values of the following correct to five decimals. (1)/(root(3)(128)) |
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| 6039. |
Find the middle term(s) in the expansionof n in N (x sqrt(x) -(2)/(x))^(15) |
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| 6040. |
int_(0)^((pi)/(2))(sin^(2)x)/(sin x + cos x) dx = |
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Answer» `(3)/(SQRT(2))LOG (sqrt(2) + 1)^((1)/(2))` |
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| 6041. |
Solve (x^(2)+(1)/(x^(2)))-5(x+(1)/(x))+6=0," when "x ne0 |
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Answer» `{2+-(SQRT(3) )/( 2)}` |
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| 6042. |
A letter is known to have come either from 'TATANAGAR' or 'CALCUTTA' . On the envelope. Just two consecutive letter TA are visible . The probability that the letters comes from 'TATA NAGAR' is |
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Answer» `(4)/(11)` |
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| 6043. |
Let f(x){:(=x^(2),","" when "xlt1),(=1,",""when"xle1):} Draw the graph of the function f (x) and evaluateunderset(xrarr1)"lim"f(x). |
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| 6044. |
If A = (1,2),B = ( 3 ,-2 ) and P moves in the plane such that AP + BP = 7 ,then the locus of P has two axes of symmetry. Their equations are : |
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Answer» x-2y+3=0, 2x+y=4 |
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| 6045. |
The points on the curve 9y^(2)=x^(3), where the normal to the curve makes equal intercepts with the axes are ……….. |
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Answer» `(-4, -(8)/(3))` |
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| 6046. |
If cos(theta - theta_(1)) =1/(2sqrt(3)) and sin(theta-theta_(2))=1/(3sqrt(2)), where 0 lt theta - theta_(1), theta-theta_(2) lt pi/2, then the value of 108/5{ cos^(2) (theta_(1)-theta_(2)) + sqrt(6)/18 sin(theta_(1)-theta_(2))} is ………….. |
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| 6047. |
The parabola y= (1)/(2)x^(2) divides the circle x^(2) + y^(2)=8 into two parts find the area of each part. |
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| 6048. |
If the ecentricity of a hyperbola is sqrt3 then the ecentricity of its conjugate hyperbola is |
| Answer» Answer :C | |
| 6049. |
A is the set of the first five positive odd integers . B is the set of the first five positions even integers. {:("Quantity A","Quantity B"),("The standard deviation of A","The standard deviation of B"):} |
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