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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. |
If a matrix A=((3,2,0),(1,4,0),(0,0,5))show that A^2-7A+10I_(3) = 0 and hence find A^(-1) |
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| 2. |
Integrationof rationalfunctions I=int(x^(3)+1)/(x(x-1)^(3))dx. |
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| 3. |
x tan^(-1) x |
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Answer» Solution :` INT X. tan^(-1) x dx` ` =tan^(-1) x int x dx - int {(d)/(dx) tan^(-1) x int x dx }dx` ` =(x^(2))/(2) tan^(-1) x- int (1)/(1+x^(2)) .(x^(2))/(2) dx` ` =(x^(2))/(2) tan^(-1) x-(1)/(2) int (1-(1)/(1+x^(2)))dx` ` =(x^(2))/(2) tan^(-1) x-(1)/(2) (x-tan^(-1)x)+C` `=((x^(2)+1)/(2)) tan^(-1) x-(1)/(2) x+c` |
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| 4. |
Determine the differentials in each of the following cases. xy^2 + yx^2 = 1 |
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Answer» SOLUTION :`xy^2 +yx^2 = 1` `rArr dx CDOT y^2 + x cdot 2y dy +dy cdot x^2 + y cdot 2x dx = 0` `rArr (2 xy + x^2) dy = - (y^2 + 2xy) dx` `rArr dy=-(y^2+2xy)/(x^2+2xy) dx` |
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| 5. |
int(x^2+2)/((x-1)(x+2)(x+3))dx= |
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Answer» `(1)/(4)log|x-1|+2log|x+2|+(11)/(4)log|x+3|+C` |
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| 6. |
Let ** be a binary operation on the set Q of rational numbers as follows : a "*" b = ab^2 Find which of the binary operations are commutative and which are associative. |
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| 7. |
The sumof allpossiblenumbersgreaters greaterthan 2000formedby usingthedigits 2,3,4,5is |
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Answer» 93324 |
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| 8. |
If 4 different biscuts are distributed among 3 children at random, the probability that the first child receives exactly one biscut is |
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Answer» `(4)/(15)` |
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| 9. |
Let f(x) be continous on [0,6] and differentable on (0,6) Let f(0)=12 and f(6)=-4. If g(x) =f(x)/(x+1) then for some Lagrange's constantcin (0,6),g'( c)= |
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Answer» `-44/3` |
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| 10. |
vec(a)=2hati-2hatj+hatk,vec(b)=hati+2hatj-2hatk and vec( c )=2hati-hatj+4hatk then find the projection of vec(b)+vec( c ) on vec(a). |
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| 11. |
int_(0)^(oo) (ln x)/(x^(2)+a^(2))dx= |
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Answer» `((LN a)/(a))pi/4` |
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| 12. |
Find the vector from origin to the mid-point of the vector vec(P_1P_2) joining the points P_1 (4,3) and P_2(8, -5). |
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Answer» SOLUTION :`P_1` = (4,3) and `P_2` = (8, -5) If P is the mid-point of `P_1P_2` then P = (6, -1). Position vector of P = `vec(OP)` = `6hati-hatj` |
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| 13. |
Examine the continuity of f, where f is defined by f(x)={{:(sin x-cos x," if "x ne 0),(-1," if "x= 0):}. |
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| 14. |
A positive integer is chosen at random. The probability that the sum of the digits of its square is 33 is |
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Answer» `1/33` |
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| 15. |
Let A=[{:(1,sintheta,1),(-sintheta,1,sintheta),(-1,-sintheta,1):}] where 0lethetalepi then |
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Answer» DET(A) |
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| 16. |
Write down the set of letters forming that word Mathematics? |
| Answer» SOLUTION :{a,C,E,H,i,m,s,t} | |
| 17. |
If two switches S_(1) and S_(2) have respectively 90% and 80% chances of working. Find probabilities that each of the following circuits will work. |
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| 18. |
Verify the Rolle's theorem for each of the function in following questions: f(x)= log (x^(2)+2)- log 3, " in " x in [-1, 1] |
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| 19. |
ax + by - a^(2) = 0, where a, b non-zero, is the equation to the straight line perpendicular to a line l and passing through the point where I crosses the x-axis. Then equation to the line / is |
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Answer» `(X)/(B)-(y)/(a)=1` |
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| 20. |
The volume of the parallelopiped with edges 2i - 4j + 5k, I - j + k, 3i - 5j + 2k is -8 |
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| 21. |
If the polar of a point P w.r.t. the circle x^2+y^2=a^2touches the circle (x- a)^2 + y^2 = a^2, then P lies on: |
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Answer» `x^2+2ay=a^2` |
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| 22. |
Let two planes p _(1): 2x -y + z =2 and p _(2) : x + 2y - z=3 are given : Equation of the plane which passes through the point (-1,3,2) and is perpendicular to each of the plane is: |
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Answer» `x + 3Y - 5Z +2 =0` |
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| 24. |
If f(x)= {(x",",x in {0, 1}),(1",",x ge 1):} then, …… |
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Answer» F is continuous at x=1 only |
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| 25. |
If z_1 , z_2, z_3 are complex numbers such that |z_1|= |z_2| = |z_3|= |(1)/(z_1) + (1)/(z_2) + (1)/(z_3)|=1, then [ (1+ i)^(n_1) + (1-i)^(n_1)] + [ (1+ i)^(n_2) + (1-i)^(n_2)] is |
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Answer» EQUAL to 1 |
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| 26. |
(A xx B). {(B xx C) xx (C xx A)} = |
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Answer» `(B + C). {(C + A) xx (A + B)}` |
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| 27. |
If 4x + y+ p= 0( p gt 0) is tangent to the ellipse x ^(2) + 3y ^(2) =3 and 16x + qy + 14 =0 (1 gt 0) is a normal to the ellipse x ^(2) + 8y ^(2) =33, then p + q= |
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Answer» 8 |
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| 28. |
e^([(x-1)-(1)/(2)(x-1)^(2)+(1)/(3)(x-1)^(3)-(1)/(4)(x-1)^(4)+………oo])= |
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Answer» X |
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| 29. |
Let S denote the set of all 6-tuples (a, b, c, d, e, f) of positive integers such that a^2 + b^2 + c^2 + d^2+ e^2=Consider the set |
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| 31. |
Find the angle between the pair of planes 7x+5y+6z+30=0 and 3x-y-10z+4=0 |
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| 32. |
Triangle ABD is inscribed in a circle, such that AD is a diameter of the circle. (Refer to the same figure as for problem #11.) If the area of triangle ABD is 84.5 square units, what is the length of arc BCD? |
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| 33. |
Number of triangle ABC /_B=90^(@) such that point B is vertex and A & C are point of zeros of a Quadratic expressioni y=ax^(2)+bx+c where b is an odd integer & a,cepsilonz, is |
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Answer» Slope of `BA` Slope of `BC=-1`  `(D/(4a))/(x_(1)+b/(2a)).(D/(4a))/(x_(2)+b/(2a))=-1` `implies(D^(2))/(16A^(2))+(b^(2))/(4a^(2))+(x_(1)+x_(2)) b/(2a)+x_(1)x_(2)=0` `implies(D^(2))/(16a^(2))+(b^(2))/(4a^(2))-b/a(b/(2a))+c/a=0` `impliesD=0` or `4` |
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| 34. |
There are nine different books on a shelf, four red and five are green. Number of ways in which it is possible to arrange these books if Column – I"" Column – II (P) the red books must""(1)(4) 6! be together and green books together (Q) the red books must be""(2)(8) 6! together whereas the green books may or may not be together (R) no two books of the same""(3)(20) 6! colour are adjacent (S) if the books are arranged on a round table and no two of the three specified books are together""(4)(24) 6! |
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Answer» <P>`{:(P,Q,R,S),(2,4,3,1):}` |
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| 36. |
Out of 1000 boys in a school, 300 played cricket, 380 played hockey and 420 played basketball. Of the total, 120 played both basketball and hockey,100 played cricket and basketball, 70 played cricket and hockey and 56 played all the three games. If the probability of the number of boys who did not play any game is k, then 200kis equal to |
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| 38. |
Maximize the function z = 11x + 7y , subject to the constraints : x le 3, y le 2, x ge 0, y ge 0 |
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| 40. |
Statement -1 the numberof waysof distributing10identicalballsin 4distinctboxessuch thatno boxisemptyis""^(9)C_(3) . Statement-2thenumberof waysof choosingany3placesfrom9differentplacesis""^(9) C_(3). |
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Answer» Statement-1 is TRUE, Statement-2 is FALSE |
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| 41. |
Mr. Lastorka's science class is running experiments with an energy efficient model electric car, As the initial rate of energy delivered to the car, measured in watt, increases, the number of millimeters moved by the car from its starting positions increases exponentially . The results the several trial runs are shown on the scatterplot graph below. Based on the data, the students in Mr . Lastroka's class determine the exact equation involving Watts, x and total distance from start,y. They call the function y=f(x) over the x-axis. He challenges each student to determine the new function and what it would mean from a physics perspective. Four students pairs gave their answers below. Who is correct, and for what reasons? |
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Answer» CHARLES and Shannon, who identify the new equation as `y=-2^x` and EXPLAIN that the new graph indicates that the car is still moving forward at the same RATE as before |
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| 42. |
If P is a point on the line passing through A(3i + j - k) and parallel to 2i - j + 2k such that AP = 15 then position vector of P is |
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Answer» 13I - 4J + 9K |
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| 43. |
The solution of 2(dy)/(dx_(1)) = (y)/(x)+(y^(2))/(x^(2)) is |
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Answer» `(1)/(y) = (1)/(X) + csqrt(x)` |
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| 44. |
Find the coordinates of the foot of the perpendicular from the point (5, 2, -8) to the straight line vecr=2hati-hatj+2hatk+t(hati+3hatj-hatk). Find also the shortest distance from the point to this straight line. |
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| 45. |
If y= int_(1)^(t^3) root(3)(z) log z dz and x = int_(sqrtt)^(3)z^(2) log z dz then (dy)/( dx) is |
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Answer» `-4T^(5//2)` |
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| 46. |
If both roots are less than 3 then 'a' belongs to |
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Answer» `((3)/(2),3)` |
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| 47. |
Examine the continuity of the function f(x) = {((|x-4|)/(2(x-4))",","if" x ne 4),(0",","if" x = 4):} at x=4 |
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| 48. |
The volume of solid obtained by revolving (x^2)/(9)+(y^2)/(16)=1 about the minor axis : |
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Answer» `48pi` |
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| 49. |
Find the values of the following integrals (where [.] is gif and {.} is fractional part) Evaluate the following int_(0)^(1.5)[x^(2)]dx |
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