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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. |
The real value of theta for which the expression (1 -isin theta)/(1 + 2 isin theta) is a real number is |
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Answer» `theta=npi` |
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| 2. |
There are five students S_(1),S_(2),S_(3),S_(4) and R_(5) arranged in a row, where initially the seat R_(1)is allotted to the students are randomly allotted the five seats R_(1),R_(2),R_(3),R_(4) and R_(6) arranged in a row, where initially the seat R_(i) is allotted to the student S_(i)i,=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on Paragraph, the question given below is one of them) For i=1,2,3,4. let T_(i) denote the event that the students S_(1) and S_(i+1) do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T_(1) cap T_(2) cap T_(3) cap T_(4) is |
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Answer» `(1)/(15)` Total `=-n(bar(T_(1)) CUP bar(T_(2)) cup bar(T_(3)) cup bar(T_(4))) ` `implies n(T_(1) cap T_(2) cap T_(3) cap T_(4))` `=5! -[""^(4)C_(1)4! 2!-(""^(3)C_(1) 3!2!+""^(3)C_(1)3!2!2!)+(""^(2)C_(1)2!2!+""^(4)C_(1) 2*2!)-2]` ` implies n(T_(1) cap T_(2) cap T_(3) cap T_(4))` `=120-[192-(36+72)+(8+16)-2]` `=120-[192-108+24-2]=14` `:.` Required PROBABILITY `=(14)/(120)=(7)/(60)` |
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| 3. |
Find the area of the region {(x, y) : x^(2) + y^(2) le 4, x + y ge 2}, using the method of integration. |
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| 4. |
There are five students S_(1),S_(2),S_(3),S_(4) and R_(5) arranged in a row, where initially the seat R_(1)is allotted to the students are randomly allotted the five seats R_(1),R_(2),R_(3),R_(4) and R_(6) arranged in a row, where initially the seat R_(i) is allotted to the student S_(i)i,=1,2,3,4,5. But, on the examination day, the five students are randomly allotted the five seats. (There are two questions based on Paragraph, the question given below is one of them) The probability that, on the examination day, the student S_(1) gets the previously allotted seat R_(1), and NONE of the remaining students gets the seat previously allotted to him/her is |
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Answer» `(3)/(40)` `:.` TOTAL number of arrangement of sitting five students is `5!=120`. Here, `S_(1)` gets previouslyalloted SEAT `R_(1)` . `:. S_(2) ,S_(3),S_(4) and S_(5)` not get previously seats . Totalnumber of WAY `S_(2) , S_(3), S_(4) and S_(5)` not get previously seat is `4! (1- (1)/(1!) +(1)/(2!) -(1)/(3!) +(1)/(4!))=24(1-1+(1)/(2)-(1)/(6)+(1)/(24))=24((12-4+1)/(24))=9` `:.` Required probability `=(9)/(!20)=(3)/(40)` |
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| 5. |
If it is raining, then I will not come. Give its contrapositive. |
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Answer» If I will COME, then it is not raining. |
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| 6. |
Find the distance of the point (2, 3, -2) from the point of intersection of the straight line passing through the points A(3, 0, 1) and B(6, 4, 3) with the plane x+y-z=7. |
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| 7. |
On R - {-1}, a binary operation ** defined by a"*"b = a + b +ab then find a^(-1). |
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| 8. |
P=[{:((2)/(3),3k,a),(-(1)/(3),-4k,b),((2)/(3),-5k,c):}]If PP^(T)=I and k=(1)/(sqrt(50)) then the value of a,b,c are respectively ………… |
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Answer» `+-(16)/(5sqrt(2)),+-(13)/(5sqrt(2)),+-(1)/(3sqrt(2))` |
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| 9. |
lim_(x to oo )((x^(3))/(3x^(2)-4)-(x^(2))/(3x+2)) is equal to |
| Answer» Answer :D | |
| 10. |
If f(x)=x^(2)," for x rational" =-x^(2), "for x irrational, then " |
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Answer» F is CONTINUOUS at x=0 and x=1/2 |
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| 11. |
If abs{:(x-4, 2x-6, 3x-8),(x-8, 2x-18, 3x-32),(x-16, 2x-54, 3x-128):}=0, then x^(2)= |
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Answer» 16 |
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| 12. |
Delta ABCEF is a regular hexagon whose centre is O. Then overline(AB)+overline(AC)+overline(AD)+overline(AE)+overline(AF)= |
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Answer» `2overline(AO)` |
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| 13. |
int (sin^(-1) x -cos^(-1)x)/(sin^(-1) x + cos^(-1)x)dx = |
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Answer» `(4)/(PI) [ XSIN^(-1) x + sqrt(1 -x^(2)) ] - x + C` |
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| 14. |
A bag contains 2 white and 1 red ball. One ball is drawn át random and then put back in the box after noting its colour. The process is repeated again. If X denotes the number of red balls recorded in the two draws, describe X. |
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| 15. |
Integrate the following functions sec^2 (7-4x) |
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Answer» SOLUTION :`INT sec^2(7-4x)DX` = TAN(7-4x)/-4 +c` =`-1/4 tan(7-4x)+c` |
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| 16. |
The point M (x,y) of the graph of the function y= e^(-|x|) so that area bounded by the tangent at M and the coordinate axes is greatest is |
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Answer» `(1, E^(-1))` |
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| 17. |
D,E and F are the middle points of the sides of the triangle ABC, then |
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Answer» CENTROID of the TRIANGLE DEF is the same as that of ABC |
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| 18. |
Determine whether each of the following relations is reflexive, symmetric and transitive.Relation in the set N of natural numbers defined as R= {(x,y): y = x+5 and x < 4} |
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Answer» Solution :R = {(1,6) (2,7) (3,8)} R is not reflexive because`(1,1)!in R` R is not symmetric because `(1,6) in` R but `(6,1)!in R` R is clearly TRANSITIVE because there can.t exist two ordered PAIRS in the FORMS (a,B)and (b,c) in R. 
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| 19. |
The sum of all three digit numbers each of which is equal to 11 times the sum of the squares of its digits is lamda . Find the sum of digits of lamda |
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| 20. |
To find the point of contact P-=(x_(1), y_(1)) of a tangent to the graph of y=f(x) passing through origin O, we equate the slope of tangent to y=f(x) at P to the slope of OP. Hence we solve the equation f'(x_(1))=(f(x_(1)))/(x_(1)) to get x_(1) and y_(1). The equation |ln mx|=px where m is a positive constant has a single root for |
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Answer» <P>`0 LT p lt m//e` |
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| 21. |
Find the 2xx2 mtrix X if X+[[0,1], [1,0]]=[[2,0],[0,2]] |
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Answer» SOLUTION :`X+[[0,1], [1,0]]=[[2,0],[0,2]]` `:. X=[[2,0],[0,2]]-[[0,1],[1,0]]=[[2,-1],[-1,2]]` |
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| 22. |
Ifvec(a) = hat(i) - 2 hat(j) + 3 hat(k) and vec(b) = 2 hat(i) - 3 hat(j) + 5 hat(k) , then angle between vec(a) and vec(b) is |
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Answer» `(PI)/(2)` |
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| 23. |
Solve the differential equation (1+e^(x//y))sx+e^(x//y)(1-(x)/(y))dy=0. Or Find the general solution of the differential equation (1+tany)(dx-dy)+2xdy=0. |
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Answer» Or x(cosy+siny)e^(y)=e^(y)siny+C` |
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| 24. |
For nge 2, " let "a_(n)=Sigma_(r=0)^(n) (1)/(C_(r)^(2)),then value of b_(n)=Sigma_(r=1)^(n) (1)/(r^(2)C_(r)^(2)) equals……(where C_(r) denotes ""^(n)C_(r)). |
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Answer» `(1)/(n^(2))a_(n)` `F(alpha,BETA)=|{:(COSALPHA,-sinalpha,,""1,),(sinalpha,cosalpha,,""1,),(0," "0,,1-cosbeta+sinbeta,):}|=1-cosbeta+sinbeta` |
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| 25. |
Evalute the following integrals int (log x)/(x^(2))dx |
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Answer» |
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| 26. |
If the function f:[-1,1]to R defined by |
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Answer» a MAXIMUM |
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| 27. |
If one root of the equation x^(2) + px + 12 = 0 is 4, while the equation x^(2)+ px + q = 0 has equal roots, then the value of 'q' is |
| Answer» Answer :C | |
| 28. |
If A and B are invertible matrices of order 3 |A|=2and |(AB)^(-1)|=-(1)/(6) find |B|. |
| Answer» Answer :C | |
| 30. |
The values that m can take, so that the straightline y=4x+m touches the curve x^(2)+4y^(2)=4 is |
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Answer» `+-sqrt(45)` |
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| 31. |
Centre of circle passing through A(0,1), B(2,3), C(-2,5) is |
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Answer» (-1,10) |
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| 32. |
If (cos alpha)/(cos A)+(sin alpha)/(sin A)+(sin beta)/(sin A)=1, where alpha and beta do not differ by an even multiple of pi, prove that (cos alpha cos beta)/(cos^(2)A)+(sin alpha sin beta)/(sin^(2)A)= |
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Answer» `-2` `rArr ((cos theta)/(cos A))^(2)=(1-(sin theta)/(sin A))^(2)` `rArr (1-sin^(2)theta)/(cos^(2)A)=1-2(sin theta)/(sin A)+((sin theta)/(sin A))^(2)` `rArr (1)/(cos^(2)A)=1-2(sin theta)/(sin A)+(sin^(2)theta)/(sin^(2)A)+(sin^(2))/(cos^(2)A)` `rArr -(1)/(cos^(2)A)-2(sin theta)/(sin A)+(sin^(2)theta)/(sin^(2)A cos^(2)A)=0` `rArr -(sin^(2)A)/(cos^(2)A)-2(sin theta)/(sin A)+(sin^(2)theta)/(sin^(2)A cos^(2)A)=0` `rArr -(sin^(2)A)/(cos^(2)A)-2(sin theta)/(sin A)+(sin^(2)theta)(sin^(2)A cos(2)A)=0` `rArr ((sin theta)/(sin A))^(2)-2((sin theta)/(sin A))cos^(2) A - sin^(2) A = 0` Clearly, `(sin alpha)/(sin A)` and `(sin beta)` are the roots of this equation `therefore (sin alpha)/(sin A)(sin B)/(sin A)=-sin^(2)A` `rArr (sin alpha sin beta)/(sin^(2)A)=-sin^(2)A` Similarly, by MAKING a quadratic in ......(1) `(cos theta)/(cos A)`, we get `(cos alpha cos beta)/(cos^(2)A)=-cos^(2)A`.....(2) Adding (1) and (2), we get `(cos alpha cos beta)/(cos^(2)A)+(sin alpha sin beta)/(sin^(2)A)=-(cos^(2)A+sin^(2)A)=-1` |
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| 33. |
Find the distance between the planes 2x - 3y + 6z+8= 0 and vecr.(2hati-3hatj+6hatk)=-4 |
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| 34. |
The mean and s.d. of a sample of size 10 were found to be 9.5 and 2.5 respectively . Later on , an additional observation 15 was added to the original data. The s.d. of 11 observation is |
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Answer» `2.6` |
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| 35. |
Evaluate:int ( (6x+ 5))/( sqrt(6+ x-2x^(2)))dx |
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| 36. |
Measure of the angle between the vector vec(a)=hati-hatj+hatk and vec(b)=hati+hatj+hatk is ………….. |
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Answer» ` SIN^(-1)"" ( 2 sqrt2)/( 3)` |
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| 37. |
Ifa circle is inscribed in a square of side 10, so that the circle touches the four sides of thesquare internally then radius of the circle is |
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Answer» only I |
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| 38. |
Find A^(-1)" if "A=[{:(0,1,1),(1,0,1),(1,1,0):}] and show that A^(-1)=(A^2-3I)/2 |
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| 39. |
If A={0,1},B={1,2},C={2,3} then (A xx B)nn(A xx C)= |
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Answer» `{(0,1),(1,2)}` |
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| 40. |
Find the mean deviation about the median for the following data 13,17,16,11,13,10,16,11,18,12,17 |
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| 41. |
AA a,b, in A (set of all real numbers ) a R b harr sec^(2)a - tan^(2) b=1. Prove that R is an equivalence relation. |
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| 42. |
A quality control engineer inspects a random sample of 3 calculators from a lot of 20 calculators. If such a lot contains 4 slightly defective calculators, what is the probability that the inspectors sample will contain (i) no slightly defective calculators (ii) one slightly defective calculators |
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| 43. |
Let (a_(1), a_(2), a_(3)…, a_(2011)) be a permutation (that is a rearrangement) of the numbers 1,2,3…, 2011.Show that there exist two numbers j,k such that 1 le j lt k le 2011 and |a_(i)-j|= |a_(k)-k| |
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| 44. |
Integrate the function is Exercise. (e^(5 log)-e^(4logx))/(e^(3logx)-e^(2logx)) |
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| 45. |
At what points on the curve x^(2)+y^(2)-2x-4y+1=0, the tangents are parallel to the Y-axis ? |
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| 46. |
OABC is a unit square where O is origin and B = (1,1) . The curves y^2=x and x^2=y divide the are of the square into three three parts OABO,OBO and OBCO, if a_1,a_2,a_3 are the areas (in sq units) of these parts respectively, then a_1 +2a_2+3a_3= |
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Answer» 1 |
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| 47. |
Let f(x) =x^(3)-3(7-a)X^(2)-3(9-a^(2))x+2 The values of parameter a if f(x) has a negative point of local minimum are |
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Answer» `pi` f(X)=`3x^(2)-6(7-a)x-3(9-a^(2))` for real root `Dge0` or `49+a^(2)-14a+9-a^(2)ge0 or ale58/14` when point of minma is negativepoint of maxima is also NEGATIVE HENCE EQUATION f(x) =`3xA^(2)-6(7-a)x-3(9-a^(2))` =0 has both roots negative sum or roots =`2(7-a)lta or a gt 7 ` which is not possible as FORM (1) `ale58/14` When pointof maxima is positive point of minima is also positive ltrbgt Hence equation `f(x) =3x^(2)-6(7-a)x-3(9-a^(2))=0` has both roots positive sum roots =`2(7-a)gt0 or alt7` Also product of roots is positive or `-(9-a^(2))gt0 or a^(32)gt9 or a in (-oo,-3)cup(3,oo)` From (1),(2) and (3)in `(-oo,-3)cup(3,58//14)` For points of extrema of opposite sign equation (1) has roots of opposite sign Thus a in (-3,3). |
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| 48. |
Let two independent eventsA and Bsuch that P(A)=0.3,P(B)=0.6 FindP(A and not B) |
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Answer» <P> SOLUTION :GIVEN that P (A) = 0.3 and P(B) = 0.6therefore `P(A^c)`= 1-P(A) = 1-0.3=0.7`P(B^c)` = 1-P(B) = 1-0.6=0.4 =`P(A)P(B^c)` `0.3xx0.4`=0.12 |
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| 49. |
If (1+x+x^2+x^3)^5=sum_(k=0)^(15) a_k x^k then sum_(k=0)^(7) a_24= |
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Answer» 128 |
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| 50. |
Prove that the curve y = x^(2) and, x=y^(2) divide the square bounded by x = 0, y = 0, x = 1, y = 1 into three equal parts. |
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