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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.
| 1601. |
Find the number of proper divisors of 38808 and find their sum. |
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| 1602. |
Write the value ofunderset(xrarr0)lim[d/dx(underset0overset(x)intsqrt(1+t^2dt))]. |
| Answer» SOLUTION :`UNDERSET(xrarr0)limd/dx(undersetaoverset(X)intsqrt(1+t^2dt)) | |
| 1603. |
{:(" "Lt),(n rarr oo):} 3/n [1 + sqrt((n)/(n+3))+sqrt((n)/(n+6))+sqrt((n)/(n+9))+....+sqrt((n)/(4n-3))]= |
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Answer» 1 |
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| 1604. |
The position vector of the point of intersection of three planesvecr, vecn_(1) =q_(1), vecr, vecn_(2) =q_(2) , vecr, vecn_(3) =q_(3) where vecn_(1), vecn _(2) and vecn _(3) are non-coplanar vectors, is : |
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Answer» `(1)/([VECN _(1) vecn_(2)vecn_(3)])[q_(3) (vecn _(1)xx vecn_(2)) + q_(1)(vecn _(2) xx vecn _(3)) +q_(2) (vecn_(3) xx ecn_(1))]` |
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| 1605. |
For all real values of k, the polar of the point (2k, k - 4) with respect to x^(2) + y^(2) - 4x - 6y + 1 = 0 passes through the point |
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Answer» (1,1) |
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| 1606. |
If the sum of odd terms and the sum of even terms in the expansion of (x+a)^n are p and q respectively then p^2+q^2= |
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Answer» `((X+a)^(2N) - (x -a)^(2n))/(2)` |
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| 1607. |
If the mean of a set of observations x_(1), x_(2),….,x_(10) is 20 then the mean of x_(1)+4, x_(2)+8, x_(3)+12,….,x_(10) + 40 is |
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Answer» 34 |
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| 1608. |
IF8 cos2 theta+ 8sec2 theta= 65, 0 lt(pi)/(2) , thenthe valueof4 cos4 thetaisequal to |
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Answer» `(-33)/(8)` |
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| 1609. |
IF the area of the circle x^2+y^2=2 is divided into two parts by the parabola y=x^2, then the area (in sq units) of the larger part is |
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Answer» `(3PI)/2-1/3` |
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| 1610. |
If the equation x^(2)-y^(2)-x-ky-2=0 represents a pair of line, then k= |
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Answer» `PM3` |
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| 1611. |
Match the following: |
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Answer» <P>`{:(P, Q, R, S),(3, 2, 1, 4):}` |
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| 1612. |
Let a,b,c,d inN and a lt b lt c lt dsuch that the equation |x-a|+|x-b|+|x-c|+|x-d|=20-2(a+b) has infinite solutions then find the number of possible quadruplets (a,b,c,d). |
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Answer» Solution :552 f(b)=f( C)=d+c-a-b=20-2(a+b) `:. a+b+c+d=20` LET a=x,b=a+y,c=b+z, d=c+w `:.` 4x+3y+2z+w=20 As `23xx4!=552`. 
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| 1613. |
On the set of integers Z, relation R is defined as ''mRn iff m is a multiple of n'' then R is |
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Answer» REFLEXIVE, SYMMETRIC |
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| 1614. |
If alpha, beta , gamma are the roots of the equation x^3+px^2+qx+r=0 . Find the value of the following in terms of coefficients. (i) sum1/(betar) (ii) sum1/alpha (iii) sumalpha^2beta |
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Answer» (II) `-q/r` (III) `3r-pq` |
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| 1615. |
Letthetain [ 0, pi/2] whichone of thefollowingis true ? |
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Answer» `SIN^2 THETA gtcos^2 theta ` |
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| 1616. |
If the tangents at the extremities of a chord PQ of a parabola intersectat T, then the distances of the focus of the parabola from the point P,T,Q are in : |
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Answer» <P>AP `("at"_(1)t_(2),a(t_(1)+t_(2)))` then `SP=a(1+t_(1)^(2)),SQ=a(1+t_(2)^(2))` and `ST^(2)=("at"_(1)t_(2)-a)^(2)+(a(t_(1)+t_(2))-0)^(2)` `=a^(2)(t_(1)^(2)+t_(2)^(2)+t_(1)^(2)+t_(2)^(2)+1)=a^(2)(1+t_(1)^(2))(1+t_(2)^(2))=SP.SQ`. |
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| 1617. |
Find the cartesian equation fo the following planes. vecr.[(s-2t)hati+(3-t)hatj+(2s-t)hatk] = 15 |
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Answer» `(s-2t)X+(3-t)y+(2s+t)z = 15`, which is the cartesian EQUATION of the plane. |
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| 1618. |
Integration using trigonometric identities : int sec^(4)x tanx dx=.... |
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Answer» `(1)/(4)sec^(4)X+c` |
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| 1619. |
If the lines 3x-4y+4=0 and 6x-8y-7=0 are tangents to a circle, then the radius of the circle is |
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Answer» `3/4` |
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| 1620. |
If u + iv = (3i)/(x + iy + 2) , then y in terms of u , v is |
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Answer» `(3u)/(U^(2) + V^(2))` |
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| 1621. |
Draw the rough sketch of y=|x+2| |
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| 1622. |
Draw the rough sketch of y=|x^2-2| |
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| 1623. |
Evaluate : (i) int(dx)/(asinx+bcosx) (ii) int(dx)/(sinx+cosx) |
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Answer» Solution :(i) Put `a=rcosthetaandb=rsintheta" so that "` `r^(2)=(a^(2)+b^(2))andtheta=TAN^(-1)(b//a)`. `:.int(dx)/(asinx+bcosx)=int(dx)/(rcosthetasinx+rsinthetacosx)` `=(1)/(r)int(dx)/(sin(x+theta))=(1)/(r)*int"cosec "(x+theta)dx` `=(1)/(r)log{tan((theta+x)/(2))}+C` `=(1)/(SQRT(a^(2)+b^(2)))log{:[tan{(1)/(2)tan^(-1)((b)/(a))+(x)/(2)}]:}+C` (ii) We have write, `int(dx)/(sinx+COSX)=(1)/(sqrt(2))int(dx)/((1)/(sqrt(2))sinx+(1)/(sqrt(2))cosx)` `=(1)/(sqrt(2))int(dx)/(("COS"(pi)/(4)sinx+"sin"(pi)/(4)cosx))` `=(1)/(sqrt(2))int(dx)/(sin((pi)/(4)+x))=(1)/(sqrt(2))int"cosec"((pi)/(4)+x)dx` `=(1)/(sqrt(2))logtan((pi)/(8)+(pi)/(2))+C`. |
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| 1624. |
Let f : {1,3,4}to {1,2,5} and g : {1,2,5} to {1,3} be given by f = {(1,2), (3,5), (4,1) and g = {(1,3), (2,3) , (5,1)}. Write down gof. |
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| 1626. |
(Allocation problem) A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, no more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to to maximise the total profit of the society ? |
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| 1627. |
Three boxes numbered, I, II, III contain balls as follows {:(,"White","Black","Red"),(I, 1,2,3),(II, 2,1,1),(III, 4 ,5, 3):} One box is randomly selected and a ball is drawn from it. If the ball is red, then the probability that it is from box II. |
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| 1628. |
A tank with a capacity of 1000 litres originally contains 100 gms of salt dissolved in 400 litres of water. Beginning at time t = 0 and ending at time t = 100 minutes, water containing 1 gm of salt per litre enters the tank at the rate of 4 litre/minute and the wheel mixed solution is drained from the tank at a rate of 2 litre/minute. Find the differential equation for the amount of salt y in the tank at time t. |
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| 1629. |
Equation of a line passing through point (1,2,3) and parallel to XZ-plane is |
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Answer» `(x-1)/(a)=(y-2)/(0)=(z-3)/(0)` |
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| 1631. |
y+x^(2)=(dy)/(dx) has the solution |
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Answer» `y + X^(2) + 2x + 2 = C E^(x)` |
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| 1632. |
From a point A(t) on the parabola y^(2)=4ax, a focal chord and a tangent are drawn. Two circles are drawn in which one circle is drawn taking focal chord AB as diameter and other is drawn by taking the intercept of tangent between point A and point of the circles is |
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Answer» <P>the line joining focus and p  Hence, the common chord of the given CIRCLES is line AP (which is the intercept of tangent at point A between point A and directrix). |
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| 1633. |
Find the values of the following :( sqrt3/2 + i/2)^(5) - ( sqrt3/2 - i/2)^(5) |
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Answer» Re `(Z) GT 0` , IM `(z) LT 0` |
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| 1634. |
If |z| ge 3 , then the least value of |z + (1)/(z)| is |
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Answer» `(7)/(3)` |
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| 1635. |
Let vec(a)=2hat(i)-3hat(j)+6hat(k)andvec(b)=-2hat(i)+2hat(j)-hat(k), then ("Projection of "vec(a)" on "vec(b))/("Projection of "vec(b)" on "vec(a))= |
| Answer» Answer :B | |
| 1636. |
A tower PQ subtends an angle alpha at a point A on the same level as the foot Q of the tower. It also subtends the same angle alpha at a point B where AB subtends the angle alpha with AP then |
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Answer» AB = BQ |
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| 1637. |
If the sum of 1^(2) + (1^(2) + 2^(2) )…m terms = (m(m+1))/(6)k then k is equal to |
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Answer» `(m+2)` |
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| 1638. |
Find the angle between the curves, (x^(2))/(8) + (y^(2))/(2) = 1 and (x^(2))/(4) - (y^(2))/(2) = 1 |
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| 1639. |
Let S be the set of all real matrices A = [(a,b),(c,d)] such that a+d =2 and A' =A^(2) -2A. Then S: |
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Answer» is an empty SET |
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| 1640. |
Two point P, Q are taken at random on a straight line OA oflength a. The chance that PQgtb, where blta is |
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Answer» `(|a-B|)/a` |
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| 1641. |
Form the differential equation of circles touching the x-axis at origin: |
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Answer» |
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| 1642. |
The function f(x) =x^3+ax^2+bx+c,a^2lt=3b has |
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Answer» positive real numbers `in a^2le 3b` |
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| 1643. |
If x,y,z are all different from zero and |{:(1+x,1,1),(1,1+y,1),(1,1,1+z):}|=0 then value of x^(-1)+y^(-1)+z^(-1) is ".........." |
| Answer» Answer :D | |
| 1644. |
Let f : A to B and g : B to Cbe the bijective functions. Then (g of )^(-1) is |
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Answer» `F^(-1) o G^(-1)` |
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| 1645. |
The value of int(cosec^2x)/sec^2xdx is |
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Answer» `(- (COT x)/TAN x +c)` |
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| 1646. |
If |z+3|^(2)-|z-3|^(2)=8 then the locus of z is |
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Answer» straight line PARALLEL to the imaginary axis |
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| 1647. |
Which of the following is/are always false? |
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Answer» A QUADRATIC equation with RATIONAL coefficients haszero or two irrational roots. |
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| 1648. |
-1lexlt0,0leylt1,1lezlt2 and [" " ] is greatest interger function then, |{:([x]+1,[y],[z]),([x],[y]+1,[z]),([x],[y],[z]+1):}|=...... |
| Answer» ANSWER :C | |
| 1649. |
Column -1 : real valued function, Column -2: continuity of the function, Column - 3: differentiability of the function, Match the following Column(s) (Whre [.] denotes the greatest integer function and {.} fractional at function) Which of the following combination combination is correct? |
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Answer» (I)(i)`(R)` (II) `f(x)=(x^(2)-9)|x^(2)+11x+24|+sin|x-7|+cos|x-4|+(x-1)^(3//5)sin(x-1)` is continuous `AA x epsilonR` and not differentiable at `x=-8` & `7` (III) `f(x)={((x+1)^(3//5)-(3pi)/2, : xlt-1),((x-1/2)cos^(-1)(4x^(3)-3x), : -1le x le 1), ((x-1)^(5//3),:1ltxlt2):}` is discontinuous at `x=-1` & 1 not differentiable at `x=-1, -1/2` & 1 (IV) `f(x)=P{sinx}{cosx}+(sin^(3)pi{x})([x]), x epsilon [-1, 2pi]` Let `G(x)=underset("cont. at" x=I) ubrace((sinpi{x})([x]))(sin^(2)pi{x})` `g^(')(I^(+))=g^(')(I^(-))` so differentiable at `x=I` and for `{sinx}{cos}` Doubtful poins for non differentiabililty are `x=0, (pi)/2, pi, (3pi)/2` `:.{sinx}.{cosx}` is discontinuous at `x=0, (pi)/2, 2pi` So not differentiable at `x=2NPI, 2npi+(pi)/2` |
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| 1650. |
Solve the following differential equations. (dy)/(dx)+y tan x=cos^(3)x |
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Answer» (ii) `y sec x = TAN x + (1)/(3) tan^(3) x + c` |
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