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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. |
Prove that .^(n)C_(r )+.^(n-1)C_(r )+..+.^(r )C_(r )=.^(n+1)C_(r+1) |
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Answer» Solution :`.^(r )C_(r ) + .^(r+1)C_(r ) + .^(r+2)C_(r) +.. + .^(n-1)C_(r )+ .^(n)C_(r )` `.^(r+1 )C_(r+1 )+ .^(r+1)C_(r )+ .^(r+2)C_(r ) +.. + .^(n-1)C_(r )+ .^(n)C_(r )` `= .^(r+1)C_(r+1) + .^(r+1)C_(r ) +.. + .^(n+1)C_(r )+ .^(n)C_(r )` `= .^(r+3)C_(r+1) +..+ .^(n-1)C_(r ) + .^(n)C_(r )` On adding similar WAY, we get L.H.S. `= .^(n-1)C_(r+1)+ .^(n-1)C_(r ) + .^(n)C_(r )` `= .^(n)C_(r+1) + .^(n)C_(r )` `= .^(n+1)C_(r+1)=R.H.S.` |
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| 2. |
A circle S cuts three circlesx^(2)+ y^(2) - 4x - 2y + 4 = 0 , x^(2) + y^(2) - 2x - 4y + 1 = 0 " and " x^(2) + y^(2) + 4x + 2y + 1 = 0orthogonally . Then the radius of S is |
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Answer» `(sqrt(29))/(8)` |
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| 3. |
Find points on the curve (x^(2))/(9)+(y^(2))/(16)=1 at which the tangents are(i) parallel to X - axis(ii) parallel to Y-axis. |
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| 4. |
Let f(x)={{:(|x-3|,"," "if " x lt -1),(3x+4,"," "if " x ge -1):}, g(x)=x^2-bx-2, (x in R) and b is a real constant. If gof is continuous at x=-1, then b is equal to _____ |
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| 5. |
P is a 2xx2 matrix . P'=P^(-1) then P=…….. |
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Answer» `[{:(COSTHETA,-SINTHETA),(-sintheta,costheta):}]` |
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| 6. |
A beam of ultraviolet light of all wavelengths passes through hydrogen gas at room temperature, in the x. dimction. Assume thatall photons emitted due to electron transitions inside the gas emerge in the y-diiection. LetA and B denote the lights emerging from the gas in the x- and y-directions respectively. Then, |
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Answer» some of the incident WAVELENGTHS WILLBE absent in A |
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| 7. |
Prove that the function defined by f(x)= cosx^2 is a continuous function |
| Answer» Solution :SINE `G(x)=x^2` and H (x)=cosx are CONTINUOUS functions,`(hog)(x)=h(g(x))=h(x^2)=cosx^2=f(x)` is a continuous FUNCTION composition of continuous functions is continuous | |
| 8. |
Find the range of the function "" f(x)=(sin^(2)x+sinx-1)/(sin^(2)x-sinx+2). |
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| 9. |
A bagcontains nwhiteand nblackballs. Pairsof ballsaredrawnat randomwithoutreplacementsuccessively, untilthe bagis empty. Ifthe numberof waysin whicheachpairconsistsof onewhiteand oneblackballis 14,400then n= |
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| 10. |
int((x+3)e^(x))/((x+4)^(2))dx is equal to |
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Answer» 1.`(1)/((x+4)^(2))+C` |
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| 11. |
Write down the power set of {a,b,c} |
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Answer» Solution :LET A ={a,B,C} `:.P(A)={{a},{b},{c},{a,b},{a,c},{b,c},A,pih}` |
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| 12. |
Integrate the following functions : int(x^(e-1)+e^(x-1))/(x^(e)+e^(x))dx |
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| 13. |
Let A=[{:(2,-1),(4,2):}]B=[{:(4,3),(-2,1):}]andC=[{:(-2,-3),(-1,2):]] Find the following (1) 2B+3C,(2)A+(B+C),(3)(2A-3B)-Cand4)(B+C)-2A. |
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Answer» (2) `[{:(4,-1),(1,5):}]` (3) `[{:(-6,-8),(15,-1):}]` (4)`[{:(-2,2),(-11,-1):}]` |
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| 14. |
Two cards are drawn from pack of 52 cards one after another without replacement. Find the mean of the ranom variable X where X is number of Aces. |
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| 15. |
Integrate the following functions. int((1+sin2x)/(x+sin^(2)x))dx |
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| 16. |
If the origin is the limiting point of a system of coaxal of whichx^(2) + y^(2) + 2gx + 2gy+ c = 0is a member, then the equation of the circle of the orthogonal system is |
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Answer» `(x^(2) + y^(2)) (g + lambda f) + 4C (3x + LAMBDAY) = 0 ` |
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| 17. |
Find the area of bounded by the curve y=x^(3) and thr liney=x. |
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| 20. |
If alpha, beta are roots of the equationx^(2)-p(x+1) -q=0, the value of(alpha^(2)+2alpha+1)/(alpha^(2)+2alpha+q)+(beta^(2)+2beta+1)/(beta^(2)+2beta+q) is |
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Answer» 0 |
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| 21. |
A fair die is rolled. Consider the events E={1,3,5} and F={2,3}, find P(E|F). |
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| 22. |
Consider functions f and g such that composite gof is defined and is oneone. Are f and g both necessarily one-one. |
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| 23. |
A random variable X has the following probability distribution: find P(0 lt X lt 3) |
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Answer» Solution :`P(0 LT X lt 3)` = P(X=1 or 2) =P(X=1)+P(X=2) |
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| 24. |
Prove that : Find the 5^("th") term in the expansion of (3x-4y)^(7). |
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| 25. |
If[(a+b,2),(5,ab)]=[(6,2),(5,8)], then find the values of a and b respectively. |
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Answer» 2,4 |
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| 27. |
Find the area of the region bounded by the parabolas y^(2)=4x and x^(2)=4y |
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| 28. |
If(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + c_(3) x^(3) + …+ C_(n) x^(n), show that sum_(r=0)^(n) (C_(r) 3^(r+4))/((r+1)(r+2)(r+3)(r+4)) (1)/((n+1)(n+2)(n+3)(n+4))(4^(n+4) -sum_(t=0)^(3) ""^(n+4)C_(t^(3^(t)))). |
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Answer» SOLUTION :`LHS = sum_(r=0)^(n) (C_(r) 3^(r+4))/((r+1)(r+2)(r+3)(r+4))` ` =sum_(r=0)^(n)((C_(r)*3^(r+4))/((n+1)(n+2)(n+3)(n+4)))/(4!) 4!` `= sum_(r=0)^(n) (C_(r) 3^(r+4))/(""^(r+4)C_(4) *4!)= sum_(r=0)^(n)(n!)/(!(n-r)!) *(3^(r+4))/(((r+4)!)/(4!r!)*4!)` `=sum_(r=0)^(n) (n!*3^(r+4))/((n-r)!*(n+4)!)` `=sum_(r=0)^(n) (n!*3^(r+4))/((n-r)!*(n+4)!)*((n+1)(n+2)(n+3)(n+4))/((n+1)(n+2)(n+3)(n+4))` `=sum_(r=0)^(n) (n!*3^(r+4))/((n-r)!*(n+4)!(n+1)(n+2)(n+3)(n+4))` `=(1)/((n+1)(n+2)(n+3)(n+4))[sum_(r=0)^(n) ((n+4)!*3^(*r+4))/((n-r)!*(r+4)!)]` `=(1)/((n+1)(n+2)(n+3)(n+4))[sum_(r=0)^(n)""^(n+4)C_(r+4) 3^(r+4)]` `=(1)/((n+1)(n+2)(n+3)(n+4)){sum_(r=0)^(n)""^(n+4)C_(t)3^(t)}`[put r + 4 = t] `=(1)/((n+1)(n+2)(n+3)(n+4)){sum_(r=0)^(n)""^(n+4)C_(t)3^(t)- sum_(t=0)^(3) ""^(3^(t)n+4)C_(t)3^(t)} ` `=(1)/((n+1)(n+2)(n+3)(n+4)){(1+3)^(n+4) sum_(t=0)^(3) ""^(3^(t)n+4)C_(t)3^(t)}` `=(1)/((n+1)(n+2)(n+3)(n+4)){4^(n+4)- sum_(t=0)^(3) ""^(n+4)C_(t)3^(t)}` RHS . |
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| 29. |
If the lines y=4-3x,ay=x+10, 2y+bx+9=0 form three sides of the rectanglein order and the fourth side passes through (1, -2) then its equation is |
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Answer» `x-3y-7=0` |
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| 30. |
the valueof cos 20^@+ cos100^@+ cos140 ^@ isequalto |
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Answer» `1/2` |
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| 32. |
If x_(n)=cos((pi)/(2^(n)))+isin((pi)/(2^(n))),ninN, then x_(1),x_(2),x_(3),…..x_(oo) is equal to |
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Answer» 1 |
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| 33. |
If the line of the vector bar (r ) = lambda I + 2j - kmakes angles alpha,beta,gamma with co - ordinate axes ,then : |
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Answer» 2 |
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| 34. |
If a normal to the parabola y^(2)=8x at (2, 4) is drawn then the point at which this normal meets the parabola again is |
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Answer» (18,-12) |
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| 36. |
A window is in the form of a rectangle surmounted bya semicircular opening. The total perimeter of the window is 10 m. Find thedimensions of the window to admit maximum light through the whole opening. |
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| 37. |
If a,b,c are positive integers such that agtbgtc and |(1,1,1),(a,b,c),(a^(2),b^(2),c^(2))|=-2 then 3a+7b-10c equals |
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Answer» 10 |
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| 39. |
If sinx+cosy=(1)/(3) and cos x+siny=(3)/(4), then the value of tan((x-y)/(2)) is equal to |
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Answer» `(5)/(13)` |
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| 40. |
If a and b are unit vectors, then what is the angle between a and b for sqrt3a-b to be a unit vector? |
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Answer» `30^(@)` `implies a.b =sqrt(3)/2 implies cos THETA=sqrt(3)/2 implies theta = 30^(@)` |
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| 42. |
Solve the following differential equations (i) (1+x^(2)) (dy)/(dx) + y = e^(tan^(-1))x (ii) (1+x^(2))(dy)/(dx) + y = tan^(-1)x |
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Answer» (ii) `y = Tan^(-1)x - 1 + c e^(-Tan^(-1)x)` |
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| 43. |
Evaluate the following integral int (cos 5x + cos 4 x)/(1- 2 cos 3x)dx |
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| 44. |
Assertion (A) : The coefficient of of x^(5) in the expansion log_(e ) ((1+x)/(1-x)) is (2)/(5) Reason (R ) : The equality log((1+x)/(1-x))=2[x+(x^(2))/(2)+(x^(3))/(3)+(x^(4))/(4)+….oo] is valid for |x| lt 1 |
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Answer» A is TRUE, R is true and R is CORRECT explanation of A |
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| 45. |
Find the values of the following integrals int_(0)^(1) x^(7//2) (1-x)^(5//2) dx |
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| 46. |
There are 20 points in a plane of which 5 are collinear and no three of the points are collinear unless all the three are from these 5 points. Find the number of different straight lines passing through these points |
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| 47. |
There are 20 points in a plane of which 5 are collinear and no three of the points are collinear unless all the three are from these 5 points. Find the number of different triangles fromed by joining these points |
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| 48. |
If x=2+3costheta and y=1-3sintheta represent a circle then the centre and radius is |
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Answer» 1)`(2,1),9` |
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| 49. |
Equation of the line passing through the point (1, -2, 5) and having d.r's 1, 2, 3 is |
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Answer» `(x-1)/(1) =(y+2)/(2) =(z-5)/(3)` |
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