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This section includes 7 InterviewSolutions, each offering curated multiple-choice questions to sharpen your Current Affairs knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of |
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Answer» 1 m/h |
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| 2. |
Find the position vector of a point R which divides the line joing two points Pand Q whose position vectors are hati+2hatj-hatkand-hati+hatj+hatk repectively , in the ratio 2:1 (i) nternally (ii) externally |
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Answer» (ii) `-3hati+3hatk` |
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| 3. |
A={x_(1),x_(2),x_(3),x_(4),x_(5)}, B={y_(1),y_(2),y_(3),y_(4),y_(5)}. A one one mapping is selected at random from the set of mappings from A to B, the probability that it satisfies the condition f(x_(i))ne y_(i), i=1,2,3,4,5 is |
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Answer» `(1)/(3)` |
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| 4. |
Let f(x) =cosx sin2x. Then , min (f(x):-pilexlepi) is false |
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Answer» `-9//7` |
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| 5. |
Evaluate int sqrt(1 + sin 2 x ) " dx " (x in" R"). |
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| 6. |
Using the properties of determinats, prove that |{:(((a+b)^(2))/(c),c,c),(a,((b+c)^(2))/(a),a),(b,b,((c+a)^(2))/(b)):}|=2(a+b+c)^(3). Or, If p ne0,qne0and|{:(p,q,palpha+q),(q,r,qalpha+r),(palpha+q,q alpha+r,0):}|=0 then using thepropetries of eterminants, prove that at least one of the following statements is true: (a) p,q,r are in GP(b) alpha is a root of the equation px^(2)+2qx+r=0. |
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| 7. |
(x^(2))/(1+x^(6)) |
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Answer» SOLUTION :` INT(x^(2))/(1-x^(6))dx` `=int (x^(2))/(1-(x^(2))^(2))dx ""underset(rArr x^(2) dx=(DT)/(3))underset(rArr 3x^(2) dx=dt)(" Let " x^(3) =t)` `= int (1)/(1-t^(2)).(dt)/(3) =(1)/(3) int (1)/(1^(2)-t^(2))dt` ` =(1)/(3),(1)/(2) log |(1+t)/(1-t)| +c =(1)/(6) log |(1+x^(3))/(1-x^(3))|+c` |
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| 8. |
The smallest value of the constant m gt 0 for which f(x)=9mx-1+(1)/(x) ge 0 for all x gt 0, is |
| Answer» Answer :C | |
| 9. |
Find the shortest distance between the lines whose vector equations are vecr=hati(1+2lambda)+hatj(1-lambda)+lambda hatk and vecr=hati(2+3mu)+hatj(1-5 mu)+hatk(2mu-1) |
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Answer» |
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| 10. |
If int (1)/(5 + 4 sinx )dx = A tan^(-1) [B(5tanx /2+4] + then (A,B)= |
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Answer» `((1)/(3),(1)/(3))` |
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| 11. |
Evaluate the following integrals. intsin(tan^(-1)x)(1)/(1+x^(2))dx |
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| 12. |
Complex numbers z_(1), z_(2), z_(3) and z_(4) correspond to the points A, B, C and D respectively, on a circle abs(z) = 1. If z_(1) + z_(2) + z_(3) + z_(4) = 0. ThenABCD is necessarily |
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Answer» a rectangle |
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| 13. |
If thetais an angle between the two asymptotes of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 then cos (theta)/(2) is equal to |
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Answer» `(b)/SQRT(a^(2)+b^(2))` |
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| 14. |
Find local maximum and minimum value of f(x)=sin^(4)x + cos^(4)x, x in[0, (pi)/(2)]. |
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Answer» Local minimum value `= (1)/(2)` Globle maximum value = 1, Globle minimum value `= (1)/(2)` |
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| 15. |
Solve 8x^4 -2x^3 - 27 x^2 +6x +9=0giventhattworootshavethe sameabsolutevalue, butareoppositein sign. |
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Answer» |
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| 17. |
Which of the following statement is/are correct for sodium phosphite ? |
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Answer» Ionic bond is present |
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| 18. |
There are 5 letters and 5 addressed envelopes. If the letters are placed at random in the envelopes. Find the chance that atleast one letter goes into wrong envelope. |
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| 19. |
int_(0)^(3)[x]dx=…, where [x] is greatest integer function. |
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Answer» 3 |
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| 20. |
int_(0)^(3) (3x+1)/(x^2 + 9) dx is equal to |
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Answer» `LOG (2 sqrt2)+ (pi)/(12)` |
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| 21. |
show that the given differential equation is homogeneous and solve it. cos^(2)x(dy)/(dx) + y = tan x (o le x le (pi)/(2)) |
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Answer» |
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| 22. |
C_0+(C_1 x)/(2)+(C_2 x^2)/(3)+…...+(C_n x^n)/(n+1)= |
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Answer» `(1)/((N+1)X)` |
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| 23. |
Let P and Q be two points on the circle |w|=r represented by w_1 and w_2 respectively, then the complex number representing the point of intersection of the tangents of P and Q is : |
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Answer» `(w_1w_2)/(2(w_1+w_2))` |
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| 24. |
Let F and g be two polynomials then int(f(x)g''(x)-f''(x)g(x)dx is equal to (ignoring the constant of integration) |
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Answer» `=(F(X))/(G'(x))` |
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| 25. |
A path of length n is a sequence of points (x_(1),y_(1)), (x_(2),y_(2)),….,(x_(n),y_(n)) with integer coordinates such that for all i between 1 and n-1 both inclusive, either x_(i+1)=x_(i)+1and y_(i+1)=y_(i) (in which case we say the i^(th) step is rightward) or x_(i+1)=x_(i) and y_(i+1)=y_(i)+1 ( in which case we say that the i^(th) step is upward ). This path is said to start at (x_(1),y_(1)) and end at (x_(n),y_(n)). Let P(a,b), for a and b non-negative integers, denotes the number of paths that start at (0,0) and end at (a,b). The value of sum_(i=0)^(10)P(i,10-i) is |
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Answer» `1024` `=^(10)C_(0)+^(10)C_(1)+….+^(10)C_(10)=2^(10)=1024` |
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| 26. |
A path of length n is a sequence of points (x_(1),y_(1)), (x_(2),y_(2)),….,(x_(n),y_(n)) with integer coordinates such that for all i between 1 and n-1 both inclusive, either x_(i+1)=x_(i)+1and y_(i+1)=y_(i) (in which case we say the i^(th) step is rightward) or x_(i+1)=x_(i) and y_(i+1)=y_(i)+1 ( in which case we say that the i^(th) step is upward ). This path is said to start at (x_(1),y_(1)) and end at (x_(n),y_(n)). Let P(a,b), for a and b non-negative integers, denotes the number of paths that start at (0,0) and end at (a,b). Number of ordered pairs (i,j) where i ne j for which P(i,100-i)=P(i,100-j) is |
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Answer» `50` and `P(j,100-j)=^(100)C_(j)` Given `'^(100)C_(i)=^(100)C_(j)` `impliesi+j=100` NUMBER of non negative integral solutions for this EQUATION `=101` (including `i=j=50`) HENCE required number of ORDERED pairs `=100` |
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| 27. |
A path of length n is a sequence of points (x_(1),y_(1)), (x_(2),y_(2)),….,(x_(n),y_(n)) with integer coordinates such that for all i between 1 and n-1 both inclusive, either x_(i+1)=x_(i)+1and y_(i+1)=y_(i) (in which case we say the i^(th) step is rightward) or x_(i+1)=x_(i) and y_(i+1)=y_(i)+1 ( in which case we say that the i^(th) step is upward ). This path is said to start at (x_(1),y_(1)) and end at (x_(n),y_(n)). Let P(a,b), for a and b non-negative integers, denotes the number of paths that start at (0,0) and end at (a,b). The sum P(43,4)+sum_(j=1)^(5)P(49-j,3) is equal to |
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Answer» `P(4,48)` Now `P(43,4)+sum_(j=1)^(5)P(49-j,3)` `=('^(47)C_(4)+^(47)C_(3))+^(48)C_(3)+^(49)C_(3)+^(50)C_(3)+^(51)C_(3)` `=('^(48)C_(4)+^(48)C_(3))+^(49)C_(3)+^(50)C_(3)+^(51)C_(3)` `=('^(48)C_(4)+^(49)C_(3))+^(50)C_(3)+^(51)C_(3)=('^(50)C_(4)+^(50)C_(3))` `=^(51)C_(4)+^(51)C_(3)=^(52)C_(4)=P(48,4)=P(4,48)` |
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| 28. |
Numbers of balls in three boxes are as follows. One box is selected of random and two balls are drawn which are of red colour and white colour each. What is the probability that selected ball is of from box 1 ? |
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| 29. |
Show that the tangent at (-1, 2) of the circle x^(2) + y^(2) - 4x-8y+ 7 = 0 touches the circle x^(2) + y^(2) + 4x + 6y= 0 and also find itspoint of contact. |
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| 30. |
Prove that from underset(n to oo) x_n=aa it follows that underset(n to oo)lim |x_n|=|a|. Is the converse true ? |
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| 31. |
If x=1/3, then the value of cos(2cos^(-1)x+sin^(-1)x) = ____ |
| Answer» Answer :A | |
| 32. |
The vectors vec(a) and vec(b) are not perpendicular. The vectors vec( c ) and vec(d) are such that vec(b)xx vec( c )=vec(b)xx vec(d) and vec(a).vec(d)=0 then vec(d) = …………… |
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Answer» `VEC( C )+((vec(a)*vec( c ))/(vec(a)*vec(B)))vec(b)` |
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| 34. |
if 0 le x le 1 prove that y=x en x -(x^(2))/(2)+(1)/(2) is a function such that (d^(2) y)/(dx^(2)) le 0. Deduce that x en x gt (x^(2))/(2)-(1)/(2). |
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| 35. |
Classify 10g//m^3measures as scalar and vector. |
| Answer» SOLUTION :Density-scalar | |
| 36. |
Statement 1 : The function f(x)=|x-a_(1)|+|x-a_(2)|+……+|x-a_(2n-1)| where, a_(1), a_(2), …… , a_(2n-1) are distinct numbers has no local minima . because Statement 2 : There does not not exist an interval in domain f for which f'(x)=0 throughout in the interval. |
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Answer» STATEMENT - 1 is TRUE, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 4 |
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| 37. |
If A and B are two events such that P(A) = (1)/(4), P(B) = (1)/(2) and P(A cap B) = (1)/(8), find P (not A and not B). |
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| 38. |
If alpha, beta are the roots of x^(2)-px +1=0 and gamma is a root of x^(2)+px+1=0, then (alpha+gamma)(beta+gamma) is |
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Answer» 0 |
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| 39. |
If a gt 0, x inR then 1+ x log_(e) a +x^2/(2!) (log_(e) a)^2+x^3/(3!)(log_ea)^3+....oo= |
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Answer» a |
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| 41. |
Two cards are drawn at random from a well shuffled pack of 52 cards without replacement, find the mean and variance of number of aces. |
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| 42. |
Mr. B has two fair 6-sided dice, one whose faces are numbered 1 to 6 and the second whose faces are numbered 3 to 8. Twice, he randomly picks one of dice (each dice equally likely) and rolls it. Given the sum of the resulting two rolls is 9, The probability he rolled same dice twice is (m)/(n) where m and n are relatively prime positive integers. Then the value of (m+n) is |
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| 43. |
Find the centre of gravity of the figure bounded by the ellipse 4x^(2) + 9y^(2)= 36 and the circle x^(2) + y^(2) = 9 and situated in the first quadrant |
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| 44. |
Evaluate : (i) int(1)/((1+tanx))dx (ii) int(1)/((1+cotx))dx (iii) int((1-tanx)/(1+tanx))dx (iv) int(tanx)/((secx+cosx))dx |
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Answer» Solution :(i) `INT(1)/((1+tanx))dx=int(1)/((1+(sinx)/(cosx)))dx` `=int(cosx)/((cosx+sinx))dx=int((cosx+sinx)+(cosx-sinx))/(2(cosx+sinx))dx` `=(1)/(2)intdx+(1)/(2)int((cosx-sinx))/((cosx+sinx))dx` `=(1)/(2)intdx+(1)/(2)int(1)/(t)dt," where"(cosx+sinx)=tand(cosx-sinx)dx=dt` `=(1)/(2)X+(1)/(2)log|t|+C=(1)/(2)x+(1)/(2)log|{:cosx+sinx:}|+C` (ii) `int(1)/((1+cotx))dx=int(1)/((1+(cosx)/(sinx)))dx=int(sinx)/((sinx+cosx))dx` `=int((sinx+cosx)-(cosx-sinx))/(2(sinx+cosx))dx` `=(1)/(2)intdx-(1)/(2)int((cosx-sinx))/((sinx+cosx))dx` `=(1)/(2)intdx-(1)/(2)int(1)/(t)dt`, where sin x + cos x = t and (cos x - sin x)dx=dt `=(1)/(2)x-(1)/(2)log|t|+C=(1)/(2)x-(1)/(2)log|{:sinx+cosx:}|+C`. (iii) `int((1-tanx)/(1+tanx))dx=int((1-(sinx)/(cosx)))/((1+(sinx)/(cosx)))dx=int((cosx-sinx))/((cosx+sinx))dx` `=int(1)/(t)dt," where"(cosx+sinx)=t and(cosx+sinx)dx=dt` `=log|t|+C=log|{:(cosx+sinx):}|+C`. (IV) `int(tanx)/((secx+cosx))dx=int(sinx)/(1+cos^(2)x)dx` `=-int(1)/((1+t^(2)))dt," where"cosx=t and sinx dx=-dt`, `=-TAN^(-1)t+C=-tan^(-1)(cosx)+C`. |
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| 45. |
Observe the following statements : (A) : 10 is the mean of a set of 7 obervations and 5 is the mean of a set of 3 observations. The mean of a combined set is 9. (R ) : If bar(x)_(i)(i = 1,2,…,k) are the means of k - series n_(i)(I = 1,2,3,…,k) respectively, then the combined or composite mean is bar(x) = (n_(1)bar(x)_(1) + n_(2) bar(x)_(2) + ...+ n_(k)bar(x_(k)))/(n_(1)+n_(2)+...+n_(k)) |
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Answer» both A, R are TRUE and `R RARR A` |
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| 46. |
Prove the sum of 99t^(th) powers of the roots of the equation x^7-1 = 0 is zero and hence deduce the roots of x^6+x^5+x^4+x^3+x^2+x+1=0 |
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| 47. |
int _(-1)^(1) (x+1)dx |
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Answer» SOLUTION :`OVERSET(1)UNDERSET(-1)(int)(x+1)DX =[(x^(2))/(2)+x]_(-1)^(1)` `=[(1^(2))/(2)+1]-[((-1)^(2))/(2)-1]` `=(3)/(2)+(1)/(2)=(4)/(2)=2` |
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| 49. |
f(x)= {(2x+3",","if" x le 2),(2x-3",","if" x gt 2):} |
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| 50. |
(x - 2)^(2) + (y + 3)^(2) = 16 touching the ellipse (x-2)^(2)/(p^(2))+(y+3)^(2)/(q^(2))=1 from inside if (2,-6) is one focus of the ellipse then (p,q)= |
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Answer» 4,5 |
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