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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.
| 11902. |
Find the eccentricity and length of the latus rectum of the hyperbola (i) 4x^(2) - 9y^(2) = 27 (ii) x^(2) - 9y^(2) = 27 |
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Answer» (II) `(2)/(sqrt(3))` |
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| 11903. |
Simplify the following((sinalpha+icos alpha)^4)/((sin2theta-icos2theta)^3) |
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| 11904. |
x^("sin"x) + ("sin" x)^("cos"x) |
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Answer» SOLUTION :`"Let" y = x^("sin"x) + ("sin" x)^("COS"x)` `"Let" u = x^("sin"x) "and " v=("sin" x)^("cos"x)` `THEREFORE y=u+v` ` rArr (dy)/(dx) = (du)/(dx) + (DV)/(dx) "".....(1)` `"Now", u=x^("sin"x)` `rArr "LOG"u = "log"(x^("sin"x)) = "sin"x * "log"x` `rArr (1)/(u) (du)/(dx) = "sin"x * (d)/(dx)"log"x + "log"x * (d)/(dx)"sin"x` `rArr (du)/(dx) = u[("sin"x)/(x) + "log"x * "cos"x]` `rArr (du)/(dx) = x^("sin"x)[("sin"x)/(x) + "log"x * "cos"x]` `"and"v=("sin"x)^("cos"x)` ` rArr "log"v = "log" ("sin"x)^("cos"x)` `= "cos" x * "log"("sin"x)` `rArr (1)/(v) (dv)/(dx) = "cos"x (d)/(dx)"log"("sin"x) + "log"("sin"x)(d)/(dx)("cos"x)` `rArr (dv)/(dx) = v["cos"x * ("cos"x)/("sin"x) + "log" ("sin"x) * (-"sin"x)]` ` rArr (dv)/(dx) = ("sin"x)^("cos"x)["cos"* "cot"x + "sin"x "log"("sin"x)]` From equation (1) `(dy)/(dx) = x^("sin"x)[("sin"x)/(x) + "log"x * "cos"x] + ("sin"x)^("cos"x)["cos"x * "cot"x - "sin"x * "log"("sin"x)]` |
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| 11905. |
The projection of a line on the axes are 9,12, and 8. The length of the line is ....... |
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Answer» 7 |
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| 11906. |
Evaluate the following integrals using the definition of a definite integral as the limit of a sum. int_(0)^(1) e^(x) dx |
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| 11907. |
Evaluate the following integrals using the definition of a definite integral as the limit of a sum. int_(0)^((pi)/(2)) cos x dx |
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| 11908. |
The statement p rarr (q rarr p) is equivalent is |
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Answer» `p RARR (p vv Q)` |
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| 11909. |
A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are, drawn and are found to be both diamonds. Find the probability of the lost card being a diamond. |
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| 11910. |
Find the equation of plane perpendicular to the plane 2x + 3y - 5z - 6 = 0and passing through the point P(2,-1,-1) and Q(1,2,3) |
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| 11911. |
The longest geometric progression that can be obtained from the set (100,101,…,1000) has the number of terms equal to ___________ |
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| 11912. |
LetC_(1) -= x^(2) + y^(2) - 2x - 4y =0 . C_(2) = x^(2) + y^(2) + 2x + 10y + 10 = 0 " and " L -= 2x + 7y + 7 = 0Then L is the |
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Answer» COMMON CHORD of ` C_(1) " and " C_(2)` |
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| 11913. |
The solution of (dx)/(dy) + (x)/(y) = x^(2) is |
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Answer» `(1)/(y) = cx - X log x` |
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| 11914. |
What is geometrical significance of the relation |veca+vecb| = |veca-vecb| |
Answer» Solution : Let ABCD be a PARALLELOGRAM where `vec(AB) = veca, vec(BC) = vecb` Then `vec(AC) = veca+vecb` Again `vec(AD) = vecb` So `vec(DB) = veca-vecb` Now AC = `|vec(AC)| = |veca+vecb|` DB = `|vec(DB)| = |veca-vecb|` If `|veca+vecb| = |veca-vecb|` then AC = DB. Thus ABCD is a parallelogram where TWO diagonals are equal. Hence ABCD must be a TECTANGLE with adjacent vectors `veca` and `vecb`. |
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| 11915. |
If the tangent drawn at a point P on the curve y=3x^(2)-5x+7 is parallel to its chord joining the points (1,y_(1))and(2,y_(2)) on it, then the x-coordinates of the point P is |
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Answer» `SQRT2` |
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| 11916. |
If the difference between the mean and variance of a Binomial variate is 5/9 then find the probability for the event of 2 success when the experiment is conducted 5 times. |
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| 11917. |
If alpha,beta are the roots of equation x^(2)-10x+2=0 and a_(n)=alpha^(n)-beta^(n) then sum_(n=1)^(50)n.((a_(n+1)+2a_(n-1))/(a_(n))) is more than |
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Answer» 12500 `a_(n+1)-10a_(n)+2a_(n-1)=0` `(a_(n+1)+2a_(n-1))/(a_(n))=10` `sum_(n=1)^(50)n.(10)=10.(50xx51)/(2)` `=250xx51=12750` |
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| 11918. |
(cos75^0+isin75^0)/(cos30^0+isin30^0)= |
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Answer» `(1)/(SQRT2)(1+i)` |
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| 11919. |
The solution of the equation (dy)/(dx) +1/x =(1)/(x^(2))tan y sin y is: |
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Answer» `2Y =sin y (1-2CX ^(2))` |
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| 11920. |
Evaluate :int _1^(3)(x^(2) +5x) dx, expressing as a limit of sum. |
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| 11921. |
Ifveca andvecb are twonon- zerocollinearvectorsthen …… is correct . |
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Answer» `vecb nelamda VECA , AA lamdain R ` |
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| 11922. |
Statement I If I is incentre of DeltaABC and I_(1) excentre opposite to A and P is intersection of II_(1) and BC, then IP. I_(1)P=Bp. PC Statement II In a Delta ABC, I is incentre and I_(2) is excentre opposite to A, then IBI, I_(1), C must be square. |
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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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| 11923. |
Integrate the following functions. intx^(13/2)(1+x^(5/2))^(1/2)dx |
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| 11924. |
On Q, the set of rational numbers, define a relation R as follows: aRb If a cos 15^(@) + b sin 15^(@) is an irrational number, then |
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Answer» DOMAIN of R is Q |
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| 11925. |
Given that a.b=0 and axxb=0. What can you conclude about the vectors a and b? |
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| 11926. |
Find (dy)/(dx) of each of the functions expressed in parametric form:sin x= (2t)/(1 + t^(2)), tan y= (2t)/(1-t^(2)), t in R |
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| 11927. |
Sum of the roots of the equaiton4 (x - (1)/(x))^(2) - 4 (x - (1)/(x)) + 1 = 0 is |
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Answer» 5 |
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| 11928. |
lim_(xto0^(+)) (sum_(r=1)^(2n+1)[x^(r)]+(n+1))/(1+[x]+|x|+2x), where ninN and [.] denotes the greatest integer function, equals |
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Answer» 0 `=underset(xto0^(-))lim([x]+[x^(2)]+[x^(3)]+...+[x^(2n+1)]+(n+1))/(x)` `=underset(xto0^(-))lim(ubrace((-1)+(-1)+...+(-1))_((n+1)"times")+(n+1))/(x)=underset(xto0^(-))lim(0)/(x)=0` |
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| 11929. |
If P(not A) = 0.7, P(B) = 0.7 and P((B)/(A)) = 0. 5 thenfind P ((A)/(B)) and P (A cup B) |
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| 11930. |
Find the angle between the line (x - 6)/( 3) = (y - 7)/( 2) = (7 - z)/( 2) and the plane vec(r) * (hat(i) + hat(j) + 2 hat(k)) = 0 |
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| 11931. |
A metal belt buckle is being designed so that it has the shape of a regular hexagon inn the center and squares at opposite ends as shown in the figure above where ABCDEF is a regular hexagon and figures I and II are squares. The hexagon will be gold plated and the two squares silver plated. The length of a side of each square is 6 centimeters. Which of the following is closest to the percent of the total surface area of the buckle that will be silver plated? |
| Answer» Answer :B | |
| 11932. |
Rolle's Theorem for f (x)=sin^4x+cos^4x in [0,(pi)/2] is verified , then find c. |
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| 11933. |
Players A and B throws three and two dice, respectively, the trial go on simultaneously and successively until a 6 appears on at least one of the dice. Find the probabilities of the following eventsE_(1) = "Player A and not B obtain a 6".E_(2)= Player B and not A, obtain a 6".and E_(3)= "both players get a 6 simultaneously". |
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| 11934. |
ABCD is a parallelogram and P is the mid point of the side AD. The line BP meets the diagonal AC in Q. Then the ratio AQ:QC=. |
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Answer» `1:2` |
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| 11935. |
Integrate the function is Exercise. (I)/(x^(2)(x^(4)+1)^((3)/(4))) |
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| 11936. |
The ordinate of a point on the parabola y^(2) = 18x is one third of its length of the latusrectom. Theo the length of subtangent at the point is |
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Answer» 12 |
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| 11937. |
The shortest distance between the curves y = |x^(2) – 6x – 27| and (x – 3)^(2) + (y – 39)^(2) = 4 is |
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Answer» `=|(x - 9) (x -3)|` `y(3) = |(-6)6| = 36` `(x - 3)^(2) + (y - 39)^(2) = 4` CENTRE `x = 3, y = 39 & R = 2`. Shortest distance will be along x = 3, Shortest distance = `(39 - 2) - 36 = 1`. |
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| 11938. |
Describe geometrically the following subsets of C : {z inC |z-1 + i|=1} |
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| 11939. |
lim_(n to oo)((sum_(r=1)^(n)r^(2))(sum_(r=1)^(n)r^(3)))/((sum_(r=1)^(n)r^(6)))= |
| Answer» ANSWER :D | |
| 11940. |
Let the quadratic equation ax^2+bx+c=0 has two purely complex roots. IF ap=b and aq=c, then- |
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Answer» <P>p is purely IMAGINARY and q is purely real |
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| 11941. |
Using binomial theorem find (i) (101)^(5) (ii) 51^(6) |
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Answer» Solution :(i) We can WRITE `(101)^(5)=(100+1)^(5)` Now by using binomial theorem, we GET `(100+1)^(5)=(100)^(5)+.^(5)C_(1)(100)^(4)+.^(5)C_(2)100^(3)+.^(5)C_(3)100^(2)+.^(5)C_(4)(100)+.^(5)C_(5)` `=10000000000+5xx100000000+10xx1000000+10xx10000+5xx100+1` `=10000000000+500000000+10000000+100000+500+1` `=1051010001` (ii) We can write `(51)^(6)=(50+1)^(6)` By using binomial theorem, we get `(1+50)^(6)=1+.^(6)C_(1)50+.^(6)C_(2)(50)^(2)+.^(6)C_(3)(50)^(3)+.^(6)C_(4)(50)^(4)+.^(6)C_(5)(50)^(5)+.^(6)C_(6)(50)^(6)` `=1+6xx50+15xx2500+20xx125000+15xx6250000+6xx312500000+15625000000` `=1+300+37500+2500000+93750000+1875000000+15625000000` `=17596287801` |
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| 11942. |
The number of values of k for which the equation x^(2) - 2x + k = 0 has two distinct roots lying in the interval (0,1) is |
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Answer» 0 |
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| 11943. |
If a,b,c are non- coplaner unit vectors such that a xx (b xx c)= (b+c)/(sqrt(2)), then the angle between a and b is |
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Answer» A`pi/6` |
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| 11944. |
A die is thrown 2n+1 times. The probability of getting 1 or 4 atmost n times is |
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Answer» `1//2` |
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| 11945. |
State which of the following are positive ?cosec 126^@ |
| Answer» SOLUTION :`COSEC 126^@` is +ve as cosec is +ve in 2ND QUADRANT. | |
| 11946. |
Solve graphically x le y |
Answer» SOLUTION :
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| 11947. |
If the length of the latus rectum is half the length of the conjucate axes of a hyperbola then its eccentricity is: |
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Answer» `SQRT(5)` |
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| 11948. |
{:("Column A","x in an even integer", "Column B"),("The number of distinct prime factors of 4x",,"The number of distinct prime factors of x"):} |
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Answer» If column A is LARGER |
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| 11950. |
If ax^2 + bx + c = 0, a, b, c in R has no real zeros, and if a + b + c + lt 0, then |
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Answer» `c GT 0` |
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