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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.
| 6551. |
From a point on the circle x^(2) + y^(2) + 2gx + 2fy + c = 0 two tangents are drawn to the circle x^(2) + y^(2) 2 gx + 2fy + c sin ^(2) alpha + (g^(2) + f^(2)) cos^(2) alpha= 0 ( 0 lt alpha lt pi//2). |
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Answer» Hence PROVED. |
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| 6552. |
For any two positive integers x and y f(x,y)=1/((x+1)!)+1/((x+2)!)+1/((x+3)!)+……..+1/((x+y)!), then which of the following options is/are correct |
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Answer» `f(x,y) le 1/x (1/(x!)-1/((x+y)!))` `xsum_(i=1)^(y)1/((x+i)!) le 1/(x!)-1/((x+y)!)` …(1) `lim_(y to oo) (f(x,y)) le 1/(XX!) le 1/(x!)`…….(2) `f(x,x)=1/((x+1)!)+1/((x+2)!)+……….+1/((x+x)!) le x/(x!)=1/((x-1)!)` |
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| 6553. |
An urn contains w white balls and b black balls. Two players Q and R alternately draw a with replacement from the urn. The player that draws a white ball first wins the game. If Q begins the game, find the probability that Q wins the game. |
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Answer» |
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| 6554. |
The ratio in which the line y = x divides the segment joining (2, 3) and (8, 6) is |
| Answer» ANSWER :1 | |
| 6555. |
A factory manufactures two types of screws, A and B. Each type of screw requires the use of two machines, an automatic and a hand operated. It takes 4 minutes on the automatic and 6 minutes on hand operated machines to manufacture a package of screws A, while it takes 6 minutes on automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs. 7 and screws B at a profit of Rs. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximise his profit ? Determine the maximum profit. |
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| 6556. |
Consider the sequence in the form of group (1),(2,2)(3,3,3),(4,4,4,4),(5,5,5,5,5…..) The sum of first 2000 termsis |
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Answer» 84336 Let US write the TERMS in the groups as follows: 1,(2,2),(3,3,3),(4,4,4,4),… CONSISTING of 1,2,3,4,.. Terms. Let 2000th term fall in nth group. Then, `((n-1)n)/2lt2000le(n(n+1))/2` or n(n-1)`lt4000len(n+1)` Let us consider, `n(n-1)lt4000` or `n^(2)-n-4000lt0` or `nlt(1+sqrt(16001))/2` r `nlt64` We have `n(n+1)ge4000` or `n^(2)+n-4000ge0` or `nge63` That means 2000th term falls in 63RD group, which means that the 2000th term is 63. Now, the total number of terms up to 62nd group is `(62xx63)//2=1953`. Hence, the sum of first 2000 terms is `1^(2)+2^(2)+..+62^(2)+63(2000-1953)` `=(62(63)125)/6+63xx47=84336` Sum of the remaining terms=`63xx16=1008`. |
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| 6557. |
If theta is the angle between the lines AB and AC where A, B and C are the three points with coordinates (1, 2, -1), (2, 0, 3), (3, -1, 2) respectively, then sqrt(462)costheta is equal to |
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Answer» 20 |
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| 6558. |
Statement I In any triangle a cos A +b cos B+c cos C le s Statement II In any triangle sin ((A)/(2))sin ((B)/(2)) sin ((C )/(2)) le 1/8 |
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Answer» Both Statement I and Statement II are CORRECT and Statement II is the correct explanation of Statement I |
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| 6560. |
If z =x + iy is complex number satisfying |(z-2i)/(z + 2i)|=2 and the locus of z is a circle, then its radius is |
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Answer» A.`5/3` |
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| 6561. |
Find the value of sin^(-1)(cos(sin^(-1)x))+cos^(-1)(sin(cos^(-1)x)) |
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Answer» 0 |
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| 6562. |
The plane (x)/(1)+(y)/(2)+(z)/(3)=1 intersect x - axis, y - axis at A, B and C respectively. If the distance between the origin and the controid of DeltaABC is k_(1) units and the volume of the tetrahedron OABC is k_(2) cubic units, then the value of (k_(1)^(2))/(k_(2)) is equal to (where O is the origin) |
| Answer» Answer :B | |
| 6563. |
Delta=|{:(p,2-i,i+1),(2+i,q,3+i),(1-i,3-i,r):}| is always ……. |
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Answer» REAL |
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| 6564. |
If u=a-b and v=a+b and |a|=|b|=2, then |uxx v| is equal to |
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Answer» `2sqrt(16-(a.b)^(2))` `[because axxa =bxxb=0]` and `|axxb|^(2)+(a*b)^(2)=(ab SIN theta)^(2)+(ab cos theta)^(2)=a^(2)b^(2)` `|axxb|=sqrt(a^(2)b^(2)-(a*b)^(2))` So`|uxxv|=2|axxb|=2sqrt(a^(2)b^(2)-(a*b)^(2))` `=2sqrt(2^(2)2^(2)-(a*b)^(2))` `=2sqrt(16-(a*b)^(2))[because |a|=|b|=2]` |
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| 6565. |
Points z_1 & z_2 areadjacent vertices of a regular octagon. The vertex z_3 adjacent to z_2 (z_3 in z_1) is represented by |
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Answer» `z_2+1/SQRT(2)(1pmi)(z_1+z_2)` |
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| 6566. |
If alpha, beta are roots of the equation ax^(2)+bx+c=0 thenthe value of (1)/(a alpha +b)+(1)/(a beta +b)= |
| Answer» Answer :D | |
| 6567. |
If f(x)={{:(,x//2,(x lt 2)),(,x^(2)//3,(x ge 2)):}"then " underset(x to 2+)"Lt" f(x)= |
| Answer» Answer :B | |
| 6568. |
Draw a coordinate plane and plot the following points : |
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Answer» (3, 1) |
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| 6569. |
Feasible region (shaded) for a LPP is shown in Figure Maximise z = 5x + 7y. |
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Answer» |
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| 6570. |
The radius of a circular plate is increases at the rate of 0.01 cm/s when the radius is 12 cm. Then the rate at which the area increases, is |
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Answer» `0.24pi` SQ cm/s |
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| 6571. |
The valueofkso thatx^4 -4x^3 +5x^2 -2x +kis divisiblebyx^2-2x +2is |
| Answer» Answer :B | |
| 6572. |
Find the value of a if 2x^(2)+ay^(2)-2x+2y-1=0 represents a circle and also find its radius. (ii) Find the values of a,b if ax^(2)+bxy+3y^(2)-5x+2y-3=0 represents a circle. Also find the radius and centre of the circle. (iii) If x^(2)+y^(2)+2gx+2fy=0 represents a circle with centre (-4,-3) then find g,f and the radius of the circle. (iv)If the circle x^(2)+y^(2)+ax+by-12=0 has the centre at (2,3) then find a ,b and the raidus of the circle. (v) If the circle x^(2)+y^(2)-5x+6y-a=0 has radius 4 , find a |
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Answer» (ii) a=3, b=0 radius `=(sqrt(65))/6`, CENTRE `=(5/6,(-1)/3)` (iii) g=4,f=3 , radius =5 (iv) `a=-4, b=-6, 4=6` (v) a=-3 |
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| 6574. |
If f:R - {0} rarrR defined by 4(x)+5f((1)/(x))=(1-x)/(x) then f(2)= |
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Answer» `1//2` |
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| 6575. |
Find the slope of the normal to the curve x = a cos^(3)theta, y=a sin^(3)theta at theta = (pi)/(4). |
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| 6576. |
Iff:R to A , where A =[-1,1] , is defined as f(x)=cos x,thn find f is |
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Answer» into |
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| 6577. |
Show that if two of the lines ax^3+bx^2y+cxy^2+dy^2=0 (a ne 0) make complementary angles with X -axis in anti -clockwise sense, then a(a-c)+b(b-d)=0 . |
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| 6579. |
Coefficient of n^(-r) in the expansion of log_(10)((n)/(n-1)) is |
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Answer» `(1)/("rlog"_(e )10)` |
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| 6580. |
Let w be non-real fifth root of 3 and x=w^(3)+w^(4). If x^(5)=f(x), where f(x) is real quadratic polynominal, with roots alpha " and " beta, (alpha, beta in C), then determine f(x) and answer the following questions. Which of the following is not true? |
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Answer» `ALPHA+BETA=-3` |
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| 6581. |
Let w be non-real fifth root of 3 and x=w^(3)+w^(4). If x^(5)=f(x), where f(x) is real quadratic polynominal, with roots alpha " and " beta, (alpha, beta in C), then determine f(x) and answer the following questions. Every term of the sequence {f(x)}, n in N is divisible by |
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Answer» 12 |
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| 6582. |
Let w be non-real fifth root of 3 and x=w^(3)+w^(4). If x^(5)=f(x), where f(x) is real quadratic polynominal, with roots alpha " and " beta, (alpha, beta in C), then determine f(x) and answer the following questions. If alpha and beta are represented by points A and B in argand plane, then circumradius of /_\ OAB, where O is origin, is |
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Answer» `4//5` |
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| 6583. |
Two circles touching both the axes intersect at (3,-2) then the coordinates of their other point of intersection is |
| Answer» ANSWER :B | |
| 6584. |
Find the equation of the circle passing through (3,4)and having the centre at (-3,4) |
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| 6585. |
Consider the following two binary relations on the set A={a,b,c} R_(1) ={(c,a), (b,b), (a,c),(c,c), (b,c), (a,a)} R_(2) = {(a,a),(b,a),(c,c),(b,a),(b,b),(a,c)}, Then |
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Answer» `R_2` is SYMMETRIC but it is not TRANSITIVE |
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| 6586. |
A particle moving in a straight line covers half the distance with speed of 3 m/s. The other half of the distance covered in two equal time intervals with speed of 4.5 m/s and 7.5 m/s respectively. The average speed of the particle during this motion is :- |
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Answer» 4.0 m/s |
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| 6587. |
Solve the following system of equations by Cramer's Rule : {:((b+c)(y+z)-zx=b-c),((c+a)(z+x)-by=c-a),((a+b)(x+y)-cz=a-b):} |
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| 6588. |
If the equation x^(2) + bx + ca = 0 and x^(2) + cx + ab = 0 have a common root and b ne c , then their other roots will satisfy the equation |
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Answer» `x^(2) + (B + c) x + BC = 0` |
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| 6589. |
A square is inscribed inside the ellipse (x^(2))/( a^(2)) +(y^(2))/( b^(2)) =1 then the length of the side of the square is |
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Answer» ` (ab)/( SQRT( a^(2) +B^(2)))` |
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| 6590. |
Evaluate the following integrals. int(1)/((x^(2)+1)sqrt(x^(2)+2))dx |
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Answer» |
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| 6591. |
Three numbers are in G.P. if we double the middle term, we get an A.P. Then the common ratio of G.P equals |
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Answer» `2 +- SQRT(3)` |
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| 6592. |
3 small squares 1 xx 1 size are selected from a chess board. Find the probability that the selected 3 squares are not in the colour. |
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Answer» |
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| 6593. |
(i) Find the equations of the tangent and normal at the positive end of the latusrectum of the ellipse 9x^(2) + 1 6 y^(2) = 144 (ii) Find the equations of the tangent and normal to the ellipse 2x^(2) + 3y^(2) = 11 at the point whose ordinate is one. |
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Answer» (II) 0 |
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| 6594. |
A particle is ..................... |
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Answer» `v=sqrt(((DR)/(dt))^(2)+(R omega)^(2))` `=sqrt(beta^(2)+(R_(0)+beta t)^(2))` |
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| 6595. |
Select the correct alternative for the following questions: The coordinates of the point case (i) are: |
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Answer» `(-1,0)` |
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| 6596. |
Integrate the following functions sinx/(1+cosx). |
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Answer» Solution :LET t = 1+cosx. Then DT = -SINX DX therefore` int sinx/(1+cosx) dx = int 1/t xx -dt` =`-log|t|+c` =`-log|1+cosx|+c` |
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| 6597. |
If vec(d)=lamda(vec(a)xxvec(b))+mu(vec(b)xxvec(c))+omega(vec(c)xxvec(a))and|vec(c)xxvec(a)|=(1)/(8)" then "lamda+mu+omega is ………….. |
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Answer» 0 |
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| 6598. |
Right triangle ABC with right angle A has an area equal to its perimeter. If the incentre of the triangle ABC is I and IB is of length sqrt(13), find its area (in sq. units). |
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Answer» Solution :`rcosec(B)/(2)=sqrt(13)` ALSO, `Delta=rs` but `Delta=2s("given")` `:.r=2` Hence `"sin"(B)/(2)=(2)/(sqrt(13))` `:.cosB=1-2"sin"^(2)(B)/(2)=1-(8)/(13)=(5)/(13)`  now using `2s=Delta` `30K=(5K.12K)/(2)` `:.""k=1` hence area of `Delta=30` SQ. units. 
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| 6599. |
Let A= { - 4, - 2, -1, 0, 3, 5} andf : A to IRbe defined by{:f(x)={(3x - 1,"for",x gt3),(x^(2) +1,"for",-3le xle 3),(2x -3,"for",x lt - 3):}Thenthe range of f is |
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Answer» { - 11, 5, 2, 1, 10, 14} |
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| 6600. |
Let A_(1) (r in N) be the area of the bounded region whose boundary is defined by (6y^(2) r - x) (6 pi^(2) y - x) = 0 then the value of underset(n to oo)(lim) (sqrt(A_(1) A_(2) A_(3)) + sqrt(A_(2) A_(3) A_(4)) + ….. Terms ) is |
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Answer» `pi^(9)` |
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