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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.
| 2003. |
Resolve (1)/((1-2x)^(2)(1-3x)) into partial fractions. |
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| 2004. |
The set of values of p for which the roots of the equation 3x^(2) + 2x + p(p-1) = 0 are of opposite signs is |
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Answer» `(-OO, 0)` |
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| 2005. |
If f : R to R^(2) and R^(+)to R are such that g{f(x)}=|sin x| and f{(gx)}=(sin sqrt(x))^(2), then a possible choice for f and g is |
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Answer» `f(x)=x^(2), G(x)=sin sqrt(x)` |
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| 2006. |
The number of ways of arranging n (gt 2) distinct objects in a line so that two particular objects are never together is |
| Answer» Answer :A::B | |
| 2007. |
Evaluate the following determinants: [[224,777,32],[735,888,105],[812,999,116]] |
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Answer» SOLUTION :`[[224,777,32],[735,888,105],[812,999,116]]` =`7`[[32,777,32],[105,888,105],[116,999,116]]=0` `(because C_1=C_2)` |
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| 2008. |
int _(0)^(pi//2) dx/(4cosx+9sinx)= |
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Answer» `1/sqrt(97) LOG ((13-sqrt(97))/(13+sqrt(97)))` |
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| 2009. |
Matrix A and B will be inverse of each other only if |
| Answer» Answer :D | |
| 2010. |
Show that the points(9, 1), (7, 9), (-2, 12), (6, 10) are concyclic and find the equation of the circle on which they lie. |
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| 2011. |
The number of ways of choosing triplet( x,y,z) such that zge"max"{x,y} and x,y,zepsilon"1,2,……..n,n+1} is |
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Answer» `.^(N+1)C_(3)+.^(n+2)C_(3)` |
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| 2012. |
Evaluation of definite integrals by subsitiution and properties of its : int_(0)^(100pi)sqrt(1-cos2x)dx=........... |
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Answer» `100SQRT2` |
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| 2013. |
Let X ={1,2,3,4}Determine whether f:X rarr Xdefined as given below have inverses. Find f^(-1) if it exist f={(1,1),(2,2),(2,3),(4,4)} |
| Answer» SOLUTION :F is not a fumction as `f(2)=2 "and"f(2)=3` | |
| 2014. |
For any vector vecr. I xx (vecr xx i)+j xx (vecr xx j) + xx (vecr+xx k) is equal to |
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Answer» 0 |
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| 2015. |
|{:(x,1,y+z),(y,1,z+x),(z,1,x+y):}|=....... |
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Answer» x+y+z |
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| 2016. |
If l(overline(b)timesoverline(c))+m(overline(c)timesoverline(a))+n(overline(a)timesoverline(b))=0 and at least one of thel, m, n is not zero , then the vectors overline(a), overline(b), overline(c) are |
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Answer» parallel |
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| 2017. |
Letn=10lambda+r", where " lambda,rinN, 0lerle9. A number a is chosen at random from the set {1, 2, 3,…, n} and let p_n denote the probability that (a^2-1) is divisible by 10. If r=0, then np_n equals |
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Answer» `2lambda` |
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| 2018. |
int (a cos^(3)x+b sin^(3)x)/(cos^(2)x sin^(2)x)dx= |
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Answer» a cosecx + b secx + C |
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| 2019. |
Match the following . {:("Limiting point "," Radical axis"),(I.(1","2)","(4,3), "a)x + y + 4 = 0 "),(II. (2","1)"," (-5","-6),"b)(3x + y - 10 =0 )"),(III."(3,2),(1,1)","c)(4x + 2y - 11=0)"):} |
| Answer» Answer :B::C | |
| 2020. |
Letn=10lambda+r, where lambda,rinN, 0lerle9. A number a is chosen at random from the set {1, 2, 3,…, n} and let p_n denote the probability that (a^2-1) is divisible by 10. If 1lerle8, then np_n equals |
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Answer» `2lambda-1` |
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| 2021. |
Letn=10lambda+r, where lambda,rinN, 0lerle9. A number a is chosen at random from the set {1, 2, 3,…, n} and let p_n denote the probability that (a^2-1) is divisible by 10. If r=9, then np_n equals |
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Answer» `2lambda` |
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| 2022. |
A hyperbola has foci (4, 2), (2, 2) and it passess through P(2, 4). The eccentricity of the hyperbola is |
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Answer» `tan.(3PI)/(10)` |
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| 2023. |
Ab, AC and AD are three adjacent edges of a parallelpiped. The diagonal of the praallelepiped passing through A and direqcted away from it is vector veca. The vector of the faces containing vertices A, B , C and A, B, D are vecb and vecc, respectively , i.e. vec(AB) xx vec(AC) and vec(AD) xx vec(AB) = vecc the projection of each edge AB and AC on diagonal vector veca is |veca|/3 vector vec(AB) is |
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Answer» `1/3 veca+ (vecaxx(VECB-vecc))/|veca|^(2)` `vec(AB)xxvec(AC)=vecb` `vec(AD)xxvec(AB)=vecc` `vec(AB).veca/(|veca|)=|veca|/3 Rightarrowvec(AB).veca= (|veca|^(2))/3` `vec(AB).veca/(|veca|)=|veca|/3 Rightarrowvec(AC).veca= (|veca|^(2))/3` ` (vec(AB) XX vec(AC))xxveca = vecb xxveca` `vec(AC)-vec(AB)=3(vecbxxveca)/(|veca|^(2))` `|veca|^(2)=vec(AB).veca+vec(AC).veca+vec(AD).veca` `(|veca|^(2))/3=vec(AD).veca` `(vec(AD)xxvec(AB))xxveca=veccxxveca` `vec(AB)- vec(AD) = 3 (vecc xx veca)/(|veca|^(2))` Now from (II) and (III), we get `vec(AC) and vec(AD)`as `vec(AC)=1/3veca+ (vecaxx(vecb xx vecc))/(|veca|^(2))+(3(vecbxxveca))/(|veca|^(2))` ` vec(AD)= 1/3veca+ (vecaxx(vecb-vecc))/(|veca|^(2))- (3(vec cxxveca))/(|veca|^(2))` 
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| 2024. |
Statement-I: int sec^(n) " x dx " = (sec^(n-2) "x tanx")/(n -1) + (n -2)/(n -1) I_(n-2) Statement- II : int sec^(3) " xdx " = (sec x " " tanx )/(2) + (1)/(2)log | sec x - tan x | + k Which of the following is true ? |
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Answer» Only I |
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| 2025. |
Show that the following system is inconsistent. (a-b)x+(b-c)y+(c-a)z=0 (b-c)x+(c-a)y+(a-b)z=0 (c-a)x+(a-b)y+(b-c)z=1 |
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Answer» Solution :`DELTA=[[a-b,b-c,c-a],[b-c,c-a,a-b],[c-a,a-b,b-c]] [R_1rarrR_1+R_2+R_3]` =`[[0,b-c,c-a],[0,c-a,a-b],[0,a-b,b-c]]`=0 `DELTAX=[[0,b-c,c-a],[0,c-a,a-b],[1,a-b,b-c]]` =`(a-b)(b-c)-(c-a)^2ne0` `therefore` the SYSTEM is inconsistent. |
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| 2026. |
If a and b are arbitrary constants then the differential equation having (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 as its gengeral solution is |
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Answer» `((d^(2)y)/(DX^(2)))^(2)=[1+((dy)/(dx))^(2)]^(3)` |
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| 2027. |
Integrate the following rational functions : int(1)/(sinx(3+2cosx))dx |
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| 2028. |
Let L_(1): (x-5)/3 = (y-7)/-1 = (z+2)/1 L_(2) : (x-3)/3 = (y-3)/2 = (z-6)/4 be two lines in space. If L_(3) is a line with direction ratios lt 2, 7,-5 gt meets L_(1) and L_(2) at A and B respectively, then the value of (AB)^(2)is |
Answer» Solution : `A(5+ 3lambda, 7-lambda, -2 + lambda)` `B(3+ 3mu, 3 + 2mu, 6 + 4mu)` DIRECTION of `AB =LT 2 + 3lambda - 3mu, 4- lambda-2mu, -8 + lambda - 4mu gt` AB is PERPENDICULAR to `L_(2)` `11 lambda - 29 mu - 18=0`........(1) Also `(2+3lambda - 3mu)/2 = (4- lambda - 2mu)/7 = (-8 + lambda - 4mu)/-5` `rArr lambda + m +2=0`.........(2) from (1) and (2) `lambda=-1` `mu=-1` `rArr A(2,8,-3)`and `B(0,1,2)` `(AB)^(2) = 4+49 + 25 = 78` |
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| 2029. |
Choose the correct answer . Which of the following differential equation has y = c_1e^x + c_2e^(-x) as the general solution ? |
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Answer» `(d^2y)/(dx^2) + y = 0` |
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| 2030. |
int(dx)/(3+2sinx)= |
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Answer» `(2)/(sqrt(5))tan^(-1)((1)/(sqrt(5))(3TAN((x)/(2))+2))+c` |
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| 2031. |
Find the coefficient of x^(4) in the expansion of (3x)/((x-2)(x+1)) in powers of x specifying the interval in which the expansion is valid. |
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Answer» `-2 lt X lt OO` |
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| 2032. |
Two perpendicular to S intersect at Q, then |OQ| is equal to (O being origin) |
| Answer» Answer :A | |
| 2033. |
If A=[(cosalpha,-sinalpha),(sinalpha,cosalpha)] then A+A'=I, the value of alpha is : |
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Answer» `(PI)/(6)` |
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| 2034. |
a,b,c,d in R^+ such that a,b and c are in H.P and ap.bq, and cr are in G.P then p/r+t/p is equal to |
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Answer» ab=cd `rArr2bd=c(b+d)` `RARR(a+c)d=c(b+d)` [as 2b=a+c] `rArrad+cd=bc+cd` `rArr bc=ad` |
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| 2035. |
If I_(n) = int_(0)^(pi//2) x^(n).cos x dx then the value of I_(9)+72I_(7)= |
| Answer» Answer :A | |
| 2036. |
Consider f(x) =x^(2)+ax+3 and g(x)=x+bandF(x) = lim_( n to oo)(f(x)+x^(2n)g(x))/(1+x^(2n)) IfF(x) is continuousatx=-1, then |
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Answer» a+b=-2 |
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| 2037. |
Find the probability distribution of the number of green balls drawn when 3 balls are drawn one by one withoutreplacementfrom a bag containing 3 green and 5 white balls. |
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| 2038. |
The vectors vec(a),vec(b),vec(c) are such that the projection of vec(c)" on "vec(a) is equal to the projection of vec(c)" on "vec(b). If |vec(a)|=2,vec(b)|=1,|vec(c)|=3andvec(a).vec(b)=1, then |vec(a)-2vec(b)-vec(c)| is equal to |
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Answer» 3 |
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| 2039. |
Let ,alpha ,beta,gammabe the roots of x^3+px^2+qx+r=0. Then find the (ii) sum 1/(alpha) |
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| 2041. |
Evaluate the following lim_(xto3)(x^2+2x-15)/(x^2-x-6) |
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Answer» SOLUTION :`lim_(xto3)(X^2+2x-15)/(x^2-x-6)` `=lim_(xto3)(x^2+5x-3x-15)/(x^2-3x+2x-6)` `=lim_(xto3)(x(x+5)-3(x-5))/(x(x-3)+2(x-3))` `=lim_(xto3)((x-3)(x+5))/((x-3)(x+2))` `=lim_(xto3)(x+5)/(x+2)=(3+5)/(3+2)=8/5` |
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| 2042. |
A manufacturing company makes two modeis A and B of a product. Each piece of model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Fach piece of model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours availableare 180 and 30 respectively. The company makes a profit of Rs. 8,000on each piece of modal A and Rs. 12,000 on week maximum profit ? What is the maximum profit per week ? |
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| 2043. |
The tangent drawn to the hyperbola (x^(2))/(16)-(y^(2))/(9)=1, at point P in the first quadrant whose abscissa is 5, meets the lines 3x-4y=0 and 3x+4y=0 at Q and R respectively. If O is the origin, then the area of triangle OQR is (in square units) |
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Answer» 6 |
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| 2044. |
The locus of the point (x, y) whose distance from the line y=2x+2 is equal to the distance from (2, 0), is a parabola with the length of latus rectum same as that of the parabola y=Kx^(2), then the value of K is equal to |
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Answer» `(SQRT5)/(12)` |
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| 2045. |
Match the following lists : |
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Answer» a. TANGENT to the parabola having slope m is `ty=X+t^(2)`. It passes through the point (2,3). Then, `3t=2+t^(2)`, i.e., t=1 or 2. The point of contact is (1,2) or (4,4) B. Let a point on the circle be `P(x_(1),y_(1))`. Then the chord of contact of the parabola w.r.t P is `yy_(1)=2(x+x_(1))`. Comparing with y=2(x-2), we have `y_(1)=1andx_(1)=-2`, which also satisfy the circle. c. Point Q on the parabola is at `(t^(2),2t)`. Now, the area of triangle OPQ is `|(1)/(2)|:(0,0),(4,-4),(t^(2),2t),(0,0):||=6or8t+4t^(2)=pm12` For`t^(2)+2t-3=0,(t-1)(t+3)=0`. Then t=1 or t=-3. Then point Q is (1,2) or (9,-6). d. Point (1,2) and (-2,1) satisfy both the curves. |
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| 2046. |
If the circleS -= x^(2) + y^(2) - 16 = 0intersects another circle S' = 0 f radius 5 in such a manner that the common chord is f maximum length and has a slope equal to 3/4 then the centre ofS'= 0 is |
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Answer» (9/5,-12/5) R (-9/5,125) |
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| 2047. |
Show, that the angle between the tangents to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 and the circle x^(2)+y^(2)=ab at a pointofintersection is "tan"^(-1)(a-b)/(sqrt(ab)) |
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| 2048. |
Integrate the following intdx/(x^2+2x+2) |
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Answer» SOLUTION :`INTDX/(x^2+2x+2)`[put(1+x=t then dx=dt] `INT(dt)/(1+t^2)=tan^(-1)t+C` `tan^(-1)(1+x)+C` |
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| 2049. |
By using the properties of definite integrals evaluate the integrals in exercise. overset(a)underset(0) int (sqrt(x))/(sqrt(x)+sqrt(a-x))dx |
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| 2050. |
If A+B+C+D=180^(@) " then " cos A cos B+ cos C cos D = |
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Answer» `SIN A sin B + sin C sin D ` |
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