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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. |
Find the absolute maximum and minimum values of the function f given by f(x) = cos^(2)x+sinx, x in [ 0,pi] |
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| 2. |
If alpha, beta are the roots of x^(2)-px+q=0 then (alpha+beta)x-(alpha^(2)+beta^(2))(x^(2))/(2)+(alpha^(3)+beta^(3))(x^(3))/(3)-….oo= |
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Answer» `LOG(1+px+qx^(2))` |
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| 3. |
For k=2,3,…, let S_(k) denote the sum of the infinite G.P. whose first term is k^(2)+k-2 and common ratio is (1)/(k)," then "overset(oo)underset(k=1)Sigma (S_(k))/(2^(k))= |
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| 4. |
Find the derivative of f given by f(x)= sin^(-1)x assuming it exists. |
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| 6. |
Show that points (0,1,2),(2,5,8),(5,6,6) and (3,2,0) from a parallelogram. |
Answer» SOLUTION :LetA=`(0,1,2),B=(2,5,8)C=(5,6,6), D= (3,2,0)` `"Then "AB=sqrt(4+16+36)=sqrt(56)` `DC =sqrt(4+16+36)=sqrt(56) ` ` "Thus" AB=DC ` `"Again" BC=sqrt(9+1+4) = sqrt(14)` `AD=sqrt(9+1+4) =sqrt(14)` `:. BC=AD ` Thus the OPPOSITE sides of the QUADRILATERAL ABCD are equal, hence it is a PARALLELOGRAM. |
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| 7. |
a and b are positive integers that a^(2) + 2b = b^(2) + 2a +5. The value of b is……………. |
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| 8. |
Let E_(1) and E_(2) two ellipse whose centres are at the orgin. Then major axes of E_(1) and E_(2) lie along the x-axis and the y-axis, respectively. LetS be the circle x^(2)+(y-1)^(2)=2 the straight line x + y = 3 touches the curves S, E_(1) and E_(2) at P, Q and R, respectively. Suppose that PQ = PR = (2sqrt2)/(3), if e_(1) and e_(2) are the eccentricities of E_(1) and E_(2), respectively, then the correct expression(s) is (are) |
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Answer» `e_(1)^(2)+e_(2)^(2)=(43)/(40)` `e_(1)^(2)=1(b^(2))/(a^(2))and e_(2)^(2)=1(A^(2))/(B^(2))` The slope of the x+y = 3 is -1. It TOUCHES ellipses `E_(1) and E_(2)`. Therefore, `a^(2)+b^(2)=3 and A^(2)+B^(2)=3`. The COORDINATES of Q and R are `((A^(2))/(3),b^(2)/(3))and((A^(2))/(3),(B^(2))/(3))` respectively. The equation of the line passing through the centre (0, 1) of the circle and perpendicular to `x+y=3 " is " y-1=1(x-0) or x-y+1=0`. This line intersects `x+y=3` at (1, 2). Thus, the line `x+y=3` touches the circle S at (1,2). The equation of the line`x+y=3` in distance form is `(x-1)/(cos3pi/4)=(y-2)/(sin3pi//4)or(x-1)/(-1//sqrt2)=(y-2)/(1//sqrt2)` The coordinates of Q and R are given by `(x-1)/(-1//sqrt2)=(y-2)/(1//sqrt2)=(2sqrt2)/(3)and, (x-1)/(-1//sqrt2)=(y-2)/(1//sqrt2)=(2sqrt2)/(3)` respectively. The coordinates of Q and R are `(5//3,4//3)and(1//3, 8//3)` respectively. but, the coordinates of Q and R are `(a^(2)//3, b^(2)//3) and (A^(2)//3,B^(2)//3)`respectively. `therefore (a^(2))/(3)=(5)/(3),(b^(2))/(3)=(4)/(3),(A^(2))/(3)=(1)/(3) and (B^(2))/(3)=(8)/(3)` `a^(2)=5,b^(2)=4, A^(2)=1 and B^(2)=8` `therefore e_(1)^(2)=1-(4)/(5)and e_(1)^(2)=1-(1)/(8)=(7)/(8)` `rArr e_(1)^(2)+e_(2)^(2)=(43)/(40)and e_(1)e_(2)=(sqrt7)/(2sqrt10)` |
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| 9. |
Find the equation of the common normal to the parobolas y^(2) = 4ax and x^(2) = 4ay. |
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| 10. |
By the definition of the definite integral, the value of underset(n-oo)(lim) ((1^4)/(1^5 +n^5)+(2^4)/(2^5 + n^5)+(3^4)/(3^5 + n^5) + …+ (n^4)/(n^5 + n^5)) is |
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Answer» `log 2` |
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| 11. |
The coordinates of a point which trisect the line segment joining the points P(4,2,-6) and Q(10,-16 ,6) |
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| 12. |
Equation of a line passing through (1,-2, 3) and parallel to the plane 2x+3y+z+5 = 0 is |
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Answer» `(x-1)/(-1)=(y+2)/(1)=(z-3)/(-1)` |
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| 13. |
Find the values of a and b such that the function f defined by f(x) = {((x-4)/(|x-4|) +a ",","if "x lt 4 ),(a+b,"if " x=4),((x-4)/(|x-4|)+b,"if " x gt 4):} is a continuous function at x= 4 |
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| 14. |
The median of the following data is{:("Marks obtained ","No. of students "),("less than 20",0),("less than 30",4),("less than 40",16),("less than 50",30),("less than 60",46),("less than 70",66),("less than 80",82),("less than 90",92),("less than 100",100):} |
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Answer» 62 |
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| 15. |
int_(pi/2)^(pi) e^(x) ((1-sinx)/(1-cosx)) dx |
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Answer» Solution :Let `I= int_(pi//2)^(pi) E^(x) ((1-sin x)/(1-cos x))DX` `=int_(pi//2)^(pi)e^(x)[(1-2SIN((x)/(2))cos ((x)/(2)))/(2sin^(2)((x)/(2)))]dx` `=int_(pi//2)^(pi) e^(x) ((1)/(2)"cosec"^(2) (x)/(2) -cot .(x)/(2))dx` `int_(pi//2)^(pi) e^(x) (-cot .(x)/(2) +(1)/(2) " cosex"^(2) .(x)/(2))dx` `[underset(inte^(x) {f(x) +f(x)}dx =e^(x)f(x))("Here " (d)/(dx) (-cot.(x)/(2))=(1)/(2)"cosec"^(2).(x)/(2))]` `:. I=[e^(x) (-cot .(x)/(2))]_(pi//2)^(pi)` `=-e^(x) cot ((pi)/(2))-[-e^(pi//2)cot ((pi)/(4))]` `=-e^(pi).0+e^(pi//2).1=e^(pi//2)` |
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| 16. |
If int_(0)^(pi//2) sin^(6) x dx= (pi)/(32) then int_(-pi)^(pi) (sin^(6)x + cos^(6) x)dx= |
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Answer» `(5pi)/(8)` |
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| 17. |
The remainder obtained when the polynomial x^(3)-3x^(2)+2x-3 is divided by x-2 is |
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Answer» 3 |
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| 19. |
Integrate the following functions e^(tan^-1 x)/(1+x^2) |
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Answer» SOLUTION :Let t = `tan^-1x`. Then dt = `1/(1+x^2) dx` THEREFORE` int e^(tan^-1x)/(1+x^2) dx = int e^x dt` =`e^t+C = e^(tan^-1x) +c` |
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| 20. |
The value of int sin 3sqrt(x) dx is |
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Answer» `{(2-x^(2//3))COSX^(1//3)+2x^(1//3)SINX^(1//3)}+C` |
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| 21. |
A binomial random variable X, 5P(X = 3) = 2P(X = 2), when n = 5. Find the value ofparameter p. |
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| 22. |
Is the function f defined by {{:(x," if "x le 1),(5," if "x gt 1):} continuous at x= 0? At x=1? At x=2?. |
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| 23. |
For a certainfrequency table which has been partly reproducedhere, the arithmetic meanwas found to be Rs. 28.07 {:("Income (in Rs.)",15,20,25,30,35,40),("No. of workers",8,12,?,16,?,10 ):} If the total number of workers is 75, then the missing frequencies are |
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Answer» 14,15 |
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| 24. |
If the points (0,0),(2,0),(0,-2) and (k,-2) are concylic then k= |
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Answer» 2 |
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| 25. |
Integrate the following rational functions : int(2x+3)/(x^(2)-2x-3)dx |
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| 26. |
Find the value of (dy)/(dx) if y= x^(tan x) + sqrt((x^(2) +1)/(2)) |
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| 27. |
int(x^(3)-1)/(4x^(3)-x) is equal to |
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Answer» `(1)/(4)xlog|x|-(3)/(16)log|2x-1|-(9)/(16)log|2x+1|+C` |
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| 28. |
Compute the volume of the solid generated by revolving about the y-axis the figure bounded by the parabolas y= x^(2) and 8x= y^(2) |
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| 29. |
If A=[{:(2x,9),(-3,-2):}]and|A|=3 then x = ....,x in R |
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Answer» 7.5 |
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| 30. |
The position vector of four points A, B, C and D are a,b,c and d respectively . If a- b = 2(d-c) the |
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Answer» <P>AB and CD bisects ` therefore2c + a = 2d + b` `implies(2c+a) /( 2+1 )=(2d+b)/(2+1)=p` Hence, point p(p) bisects AC and BD. |
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| 31. |
Find all triples (p, 9,r) of primes such that pq = r + 1 and 2(p^2+ q^2) =r^2 + 1 |
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| 32. |
Integrate the followingintcosaxdx |
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Answer» SOLUTION :`intcosaxdx=intcosthetacdot(1/a)d"THETA ` PUT `ax=theta` then ADX=`d"theta` or dx=1/a`(dtheta)` `(1/a)sintheta+C=(1/a)sinax+C` |
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| 33. |
The sum of the products of the non-conjugate root of i^(-1//4) taken two at a time is |
| Answer» ANSWER :B | |
| 34. |
Find the slope of the normal to the curve y=xe^-x at x=2. |
| Answer» SOLUTION :`y=xe^-xrArr(DY)/dx=x(-e^-x)+e^-x=e^-x(1-x)` SLOPE of the TANGENT at x=2 | |
| 35. |
Which of the following sentences are propositions and which are not ? Write with reason :May God grant you long life . |
| Answer» SOLUTION :"MAY God GRANT you LONG life" is not a statenet as it neither true nor false. | |
| 36. |
The sumof the series1/(1*2*3)+1/(3*4*5)+1/(5*6*7)+... is |
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Answer» ` log_(e)2-1/2` ` :. T_(n) = 1/((2n-1)(2n)(2n+1))` ` = 1/(2n-1)=1/(2n) +1/(2(2n+1))` ` = 1/2 [ 1/(2n-1)-1/(2n)] -1/2 [ 1/(2n)- 1/(2n+1)]` On PUTTING n =1,2,3,..., ` T_(1)= 1/2 [ 1/1 -1/2 ] -1/2 [ 1/2 -1/3]` ` T_(2) = 1/2 [ 1/3 - 1/4 ] - 1/2 [ 1/4 -1/5] ` `{:(".................."),(".................."):}` ` :. S = T_(1) +T_(2) +T_(3) +...+ T_(n) +...` ` =1/2 [ 1- 1/2 + 1/3 +1/4 +1/5 -1/6 +1/7-...]` ` -1/2 [ 1/2 - 1/3- 1/4 -1/5 +1/6 -1/7 +...]` ` -1/2 [1/2 -1/3 +1/4 -1/5 +1/6 -1/7 +...]` `= 1/2 log_(e)(1+1)+1/2 [ -1+{1-1/2 +1/3 -1/4 +...}]` ` = 1/2 log_(e) 2 - 1/2 +1/2 log_(e) (1+1) = log_(e) 2 - 1/2 ` |
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| 37. |
If veca, vecb ,vecc are vectors such that veca + vecb + vecc=0 and |veca| =7, |vecb| =5, |vecc| =3 then angle between vector vecb and vecc is |
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| 38. |
Find area of the region bounded by the curve y^(2)=4x, y - aixs and the line y=3. |
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| 39. |
Each of the following defines a relation of N : x+4y =10 , x , y in N Determine which of the above relations are reflexive , symmetric and transitive . |
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| 40. |
Write{x:x is a prime number and 1 le xle 100 } set in the form of lists? |
| Answer» SOLUTION :`{2,3,5,7,……….97}` | |
| 41. |
If function f(x) is continuous in the interval (a, b) and having same definition between a and b, then we can find int _(a) ^(b) f (x) dx if f (x) is discontiuous and not same definition between a and b, then we must break the interval such that f(x) becomes continuous and having same definition in the breaking intervals. Now, if f (x) is discontinuous at x =c (a lt c lt b), then int _(a)^(b) f (x) dx = int _(a)^(c ) f (x) dx + int _(c ) ^(b) f (x) dx andalso if f (x) is discontinous at x =a in (0, 2a), then we can write int _(0) ^(2a) f(x) dx = int _(0) ^(a ) {f (a-x) + f(a+x) } dx On the basis of above information, answere the following questions : int _(0) ^(10) [ (x ^(2) + 2)/(x ^(2) +1 ]dx (where [.] denotes greatest integer function ) is equal to |
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Answer» 0 |
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| 43. |
Compute the following: [[-1,4,-6],[8,5,16],[2,8,5]]+[[12,7,6],[8,0,5],[3,2,4]] |
| Answer» SOLUTION :`[[3,-1,3],[-1,0,2]],[[2,-3],[1,0],[3,1]]=[[6-1+9, -9+0+3],[-2+0+6, 3+0+2]]= [[14,16],[4,5]]` | |
| 44. |
If function f(x) is continuous in the interval (a, b) and having same definition between a and b, then we can find int _(a) ^(b) f (x) dx if f (x) is discontiuous and not same definition between a and b, then we must break the interval such that f(x) becomes continuous and having same definition in the breaking intervals. Now, if f (x) is discontinuous at x =c (a lt c lt b), then int _(a)^(b) f (x) dx = int _(a)^(c ) f (x) dx + int _(c ) ^(b) f (x) dx andalso if f (x) is discontinous at x =a in (0, 2a), then we can write int _(0) ^(2a) f(x) dx = int _(0) ^(a ) {f (a-x) + f(a+x) } dx On the basis of above information, answere the following questions : int _(-1)^(1) [|x|] d ((1)/(1+ e ^(-1//x))) (where [.] denotes the greatest integer functions ) is equal to |
| Answer» ANSWER :D | |
| 45. |
If function f(x) is continuous in the interval (a, b) and having same definition between a and b, then we can find int _(a) ^(b) f (x) dx if f (x) is discontiuous and not same definition between a and b, then we must break the interval such that f(x) becomes continuous and having same definition in the breaking intervals. Now, if f (x) is discontinuous at x =c (a lt c lt b), then int _(a)^(b) f (x) dx = int _(a)^(c ) f (x) dx + int _(c ) ^(b) f (x) dx andalso if f (x) is discontinous at x =a in (0, 2a), then we can write int _(0) ^(2a) f(x) dx = int _(0) ^(a ) {f (a-x) + f(a+x) } dx On the basis of above information, answere the following questions : int _(0) ^(1) sin ([x]+[2x])dx (where [.] denotes the greatest integer function) is equal to |
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Answer» `sin 1` |
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| 46. |
If z = sinthet a - icostheta then for any integer n |
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Answer» `Z^(N)+1/(z^(n))=2cos((npi)/2-ntheta)` |
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| 47. |
If two events A and B are such that P(A) gt 0 andPB()ne1,thenP(barA//braB)) is equal to: |
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Answer» <P>`1-P(A//B)` |
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| 48. |
On the set N of all natural numbers define the ooperation * on N by m*n = gcd (m,n) for all m,n in N Showat * is commutative as well as associative |
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Answer» Solution :(i) commutativity for all m,n in N we have gcd (m,n) =gcd(n.m) Therefore m*n=n*m,n in N (ii) Associativity LET m,n p in N Then (m*n)*p=[gcd(m,n,p}] =gcd[{gcd{(m,n),p}] [`therefore` gcd of three NUMBERS =gcd {(gcd of any t wo THIRD )}] =gcd (m,n*p)=m*(n*p) Hence * is associatie on N |
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| 49. |
The vectors 3a - 2b - 4c, -a + 2c, - 2a + b + 3c are |
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Answer» LINEARLY DEPENDENT |
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