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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. |
In the graph of theparametric equation {:{(x=t^(2)+t),(y=t^(2)-t):} |
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Answer» `x ge 0`  apparently the x values are always greater than some value. Usethe TRACE function to MOVE the cursoras FAR left on the gaph as it will go this leads to (correct) guessof `x ge-1/4` this can be verified by completing the squre on the x equation `x=(t^(2)+t+1/4)-1/4=(t+1/2)^(2)-1/4` This represent a PARABOLA that opens up with vertex at `(-1/2,-1/4)` Therefore `x ge -1/4` |
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| 2. |
int e^(x) [ "In " x + (1)/(x^(2)) ]dx = |
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Answer» `e^(x) `Inx+ C |
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| 3. |
Integrate the following functions 1/(cos^2x (1-tanx)^2) |
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Answer» SOLUTION :`1/(cos^2x(1-tanx)^2) = sec^x/(1-tanx)^2` Let 1-tanx = t Then `DT = -sec^2xdx` `GT sec^2x DX = -dt` therefore` int 1/(cos^2x(1-tanx)^2 dx` =`int -(dt)/t^2 = 1/t+c = 1/(1-tanx)+c` |
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| 4. |
If the line jjoining the point A (b cos alpha,b sin alpha)and B (alpha cosbeta,asin beta) is extended to the point N(x,y) such that AN : NB =b: alpha,then |
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Answer» Option1`" xcos"(alpha-beta)/(2)+ " y SIN" (alpha+beta)/(2)=0` |
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| 5. |
If the line y = mx + a meets the parabola x^(2)=4ay in two points whose abscissa are x_(1) and x_(2) then x_(1)+x_(2) =0 If |
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Answer» m=-1 |
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| 6. |
Ifx = log [ cot (pi/4 +theta )]then the value of sinhxis |
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Answer» `TAN 2THETA ` |
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| 7. |
Find (dy)/(dx) in the following x^(3) + x^(2)y + xy^(2) + y^(3) = 81 |
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Answer» |
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| 8. |
Find the values of a in the domain of the definition of the function, f(a)=sqrt(2a^(2)-a) for which theroots of the equation x^(2)+(a+1)x+(a-1)=0 lie between -2 & 1. |
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Answer» |
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| 9. |
Which one of the following is the true regarding the triangle shown in figure? |
| Answer» Answer :A | |
| 10. |
In the set Z of all integers, which of the following relation R is not an equivalence relation |
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Answer» `xRy:if X LE y` |
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| 11. |
Write the image on the point (2,1,3) with respect to yz-plane. |
| Answer» SOLUTION :DISTANCE of (3.1,5) from y-axis `SQRT(9+25)=sqrt34` | |
| 12. |
Let S be the set of all column matrices [(b_(1)),(b_(2)),(b_(3))] such that b_(1), b_(2), b_(2) in R and the system of equations (in real variables) -x+2y+5z=b_(1) 2x-4y+3z=b_(2) x-2y+2z=b_(3) has at least one solution. The, which of the following system (s) (in real variables) has (have) at least one solution for each [(b_(1)),(b_(2)),(b_(3))] in S ? |
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Answer» `x+2y+3z=b_(1), 4y+5z=b_(2)` and `x+2y+6z=b_(3)` `Delta=|(-1,2,5),(2,-4,3),(1,-2,2)|=0` Since no pair of planes is parallel, there are infinite NUMBER of solutions. Let `alphaP_(1)+betaP_(2)=P_(3)` `:. P_(1)+7P_(2)=13 P_(3)` `:. b_(1)+7b_(2)=13 b_(3)` (1) `x+2y+3x=b_(1), 4y+5z=b_(2)` and `x+2y+6z=b_(3)`. Since `Delta ne 0`, system has at least one solution for any SET of values of `b_(1), b_(2)` and `b_(3)`. (2) `x+y+3z=b_(1), 5x+2y+6z=b_(2)` and `-2x-y-3z=b_(3)` Since `Delta=0` and `b_(1)+7b_(2) ne 13 b_(3)`, system of equations has no solution. (3) `-x+2y-5z=b_(1), 2x-4y+10z=b_(2)` and `x-2y-5z=b_(1), 2x-4y+10z=b_(2)` and `x-2y+5z=b_(3)`. Since planes are parallel, there is no solution for any set of values of `b_(1), b_(2)` and `b_(3)`. (4) `x+2y+5z=b_(1), 2x+3z=b_(2)` and `x+4y-5z=b_(3)` since `Delta ne 0`, system has at least one solution for any set of values of `b_(1), b_(2)` and `b_(3)`. |
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| 13. |
Integrate thefunction in Exercise. (1)/(x+x log x) |
| Answer» | |
| 14. |
The number of arbitrary constants in the particular solution of a differential equation of third order is : |
| Answer» ANSWER :A | |
| 15. |
If f (x) =e^(x)+ int_(0)^(1) (e^(x)+te^(-x))f (t) dt, "find" f(x). |
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| 16. |
Find the value of (-i)^(4n+3),when n is positive. |
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Answer» SOLUTION :`(-i)^(4N+3)` `=(-i^(4n)(-i)^3=1(-i^3)=-(-i)=i` |
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| 17. |
If alpha, beta, gamma are respectively the acute angles made by any line with the coordinate axes then |
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Answer» `alpha+beta+gamma=90^(@)` |
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| 18. |
Find the angle between the vectors hat(i)-2hat(j)+3hat(k) and 3hat(i)-2hat(j)+hat(k). |
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Answer» Solution :Let `vec(a)=HAT(i)-2hat(j)+3hat(k) and vec(b)=3hat(i)-2hat(j)+hat(k)`. Magnitude of `vec(a),|vec(a)|=sqrt(1^2+(-2)^2+3^2)=sqrt(1+4+9)=sqrt(14)`. Magnitude of `vec(b),|vec(b)|=sqrt(3^2+(-2)^2+1^2)=sqrt(9+4+1)=sqrt(14)`. Now, `vec(a)vec(b)=(hat(i)-2hat(j)+3hat(k)).(3hat(i)-2hat(j)+hat(k))=1.3+(-2).(-2)+3.1=3+4+3=10` (Do product of TWO VECTORS is equal to the sum of the products of their corresponding components.) Let `theta` be the required angle between a and b then `cos theta =(vec(a).vec(b))/(|vec(a)||vec(b)|)=(10)/(sqrt(14)sqrt(14))=(10)/(14)=(5)/(7)rArr theta= cos^(-1)((5)/(7))`. |
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| 19. |
the sumof theradii of inscribedandcircumscribedcirculesfor ann sidesregularpolygonof sidea, is |
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Answer» `a/2cot ((pi)/(2 n))` |
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| 20. |
If A+B +C= pi /4 then 4 cos ""A/2 cos ""B/2 cos ""C/2 - cos ""pi/8= |
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Answer» `COS ((pi)/(4) -A)+ cos ((pi)/(4)-B)+ cos ((pi)/(4) -C)` |
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| 21. |
Find the number of positive integral solutions of x_1x_2x_3x_4x_5=840, such that x_1 must be even. |
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| 23. |
Find the volume of the solid generated by revolving about the line y= -2a the figure bounded by the parabola y^(2)= 4ax and the straight line x=a |
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| 24. |
Assertion (A): The locus of the centres of the circle through the points of intersection of the circles x^2+y^2-2x+y=0, x^2+y^2=1 is 2x-y+1=0 Reason (R) : The locus of the centres of the circles through the intersection of the two circles is its radical axis. |
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Answer» Assertion is true, Reason is FALSE |
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| 25. |
Angle between tangents drawn from originto parabola y^(2)= 4a(x-a)is |
| Answer» Answer :A | |
| 26. |
A: In a DeltaABC,(1-r_1/r_2)(1-r_1/r_2)=2 then the triangle is right angled. R: In a DeltaABC,r_1r_2+r_2r_3+r_3r_1=2r^2 |
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Answer» A is TRUE, R is true and R is correct EXPLANATION of A |
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| 27. |
If veca and vec(to)b are two non-zero perpendicular vectors, then a vector y satisfying equations veca. vecy=c (where c is scalar) and veca xx vec(to)y=b is |
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Answer» `|a|^(2) [CA-(a XX B)]` |
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| 28. |
A particle moves so that the space described in time 't' is square root of a quadratic function of 't', then |
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Answer» ACC. VARIES as `s^(3)` |
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| 29. |
(i) A shop-keeper buys a particular type of electric bulbs from three manfacturers M_(1), M_(2) and M_(3). He buys 25% of his requirement from M_(1), 45% from M_(2) and 30% from M_(3). Based on the past experience, he found that 2% of type M_(3) bulbs are defective, where as only 1% of type M_(1) and Type M_(2) are defective . If a bulb chosen by him at random is found defective find the probability that it was of type M_(3). (ii) In a certain college, 25% of the boys and 10% of the girls are studying mathematics. The girls constitute 60% of the student strength . If a student is selected at random is found studying mathematics, find the probability that the student is a girl. |
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| 30. |
Solve(x + 3y^(2))(dy)/(dx) = y ( y rt 0). |
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| 31. |
By using the properties of definite integrals, evaluate the integrals int_(0)^(pi/2)(sqrt(sinx))/(sqrt(sinx)+sqrt(cosx))dx |
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| 32. |
For the circles 3x^2+3y^2+x+2y-1=0 , 2x^2+2y^2+2x-y-1=0 radical axis is |
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Answer» 4x-7y=1 |
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| 33. |
What is the value of sum _(1+j = odd 1 le | lt | le 10) (i +j) - sum_(1 + j = even 1 le | lt | le 10)(i+j) ? |
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| 34. |
If f(x), g(x) and h(x) are polynomials of degree 4 and |{:(f(x),g(x),h(x)),(a,b,c),(p,q,r):}|=mx^4+nx^3+rx^2+t is an identity in x,then |{:(f"''(0)-f''(0),g''(0)-g''(0),h''(0)-h''(0)),(a,b,c),(p,q,r):}| is |
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Answer» an even NUMBER `AAN,rinZ` |
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| 35. |
Find the number of numbers less than 2000 that can be formed using the digits, 1,2,3,4 if repetition is allowed. |
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| 36. |
The normal at an end of a latus rectum of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 passes through an end of the minor axis if: |
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Answer» `e^4+e^2=1` |
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| 37. |
Find the number of selections of 10 balls from unlimited number of red, black, white and green balls so that each selection must contain atleast one ball of each colour. |
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| 38. |
(a,0) and (b,0) are centres of two circles belonging to a coaxial system of which y-axis is the radical axis. If radius of one of the circle is r then the radius of the other circles is |
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Answer» `(r^(2) + B^(2) + a^(2))^(1//2)` LET the radius of other circle be `r_(2)` Equation of circles are `(x - a)^(2) + y^(2) = r^(2)` and `(x - b)^(2) + y^(2) = r_(2)^(2)` `implies S -= x^(2) + y^(2) 2ax - r^(2) + a^(2) = 0` and `S^(1) -= x^(2) + y^(2) - 2bx - r_(2)^(2) + b^(2) = 0` Radical AXIS is, `S - S^(1) = 0` `implies -2ax - r^(2) + 2bx + r_(2)^(2) + a^(2) - b^(2) = 0` `implies` since radical axis is y-axis `implies x = 0` `implies r_(2)^(2) = r^(2) + a^(2) - b^(2) = 0` `implies r_(2)^(2) = r^(2) + b^(2) - a^(2)` `:. r_(2) = (r^(2) + b^(2) - a^(2)y^(2))` |
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| 39. |
If x^(2) + y^(2) - 4x - 2y + 5 = 0 and x^(2 + y^(2) - 6x - 4y = 0 are membes of a coaxal system of circles then centre of a point circle in the systems is |
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Answer» (-5, -6) and `S^(1) -= x^(2) + y^(2) - 6x - 4y - 3 = 0` Equation of coaxial system of circles is `S + lambda L - 0` `L = S - S^(1) = 0` `implies 2X + 2y + 8 = 0` `implies x + y + 4 = 0` `implies x^(2) + y^(2) - 4x - 2y + 5 + lambda x + lambda y +4 lambda = 0` `implies x^(2) + y^(2) + x (lambda - 4) + y (lambda - 2) + (5 + 4 lambda) = 0` centre, `C = [(4 - lambda)/(2), (2 - lambda)/(2)]` and radius, r = 0 `implies (16 - 8lambda + lambda^(2) + 4 - 4 lambda + lambda^(2) - 20 - 16 lambda)/(4) = 0` `implies 2 lambda^(2) - 28 lambda = 0` `implies lambda (lambda - 14) = 0` `implies lambda 0, 14` `implies` centre `C = ((4)/(2), (2)/(2))` (or) `((-10)/(2), (-12)/(2))` `:. C = (2,1)` (or) (-5, -6) |
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| 40. |
Find the equation of tangent to the parabola y(2)=16x inclined at an angle 60^(@) with its axis and also find the point of contact. |
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| 41. |
From origin chords are drawn to the cirlce x^(2)-y^(2)-2px=0 then locus of midpoints of all such chords is |
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Answer» `X^(2)+y^(2)-px=0` |
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| 43. |
Let the line L having equation (x-1)/(2)=(y-3)/(5)=(z-1)/(3) intersects the plane P, having equation x-y+z=5 at the point A. Statement-I Equation of the line L' thorugh the point A, lying in the plane P and having minimum inclination with line L is 8x+y-72-4=0=x-y+z-5 Statement-II Line L' must be projection of the line L in the plane P. |
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Answer» Statement I is TRUE, Statement II is also true, Statement-II is the CORRECT EXPLANATION of Statement-I. |
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| 44. |
(1)/(1!) + (1 + 3)/(2!) xx + (1 + 3 + 5)/(3!) x^(3) … |
| Answer» Answer :1 | |
| 45. |
Ifalpha, beta are the roots of x^(2)-2x+4 = 0and for any n in N, a^(n) +B^(n) = k cos""(npi)/(3) then k= |
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Answer» `2^(N)` |
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| 47. |
Which of the following facts is true about the human blood ? |
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Answer» The ABSOLUTE number of lymphocyte is greater than that of neutrophill |
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| 48. |
a_1,a_2,a_3,…., a_nfrom an A.P.Then the sum sum_(i=1)^10(a_i a_(i+1)a_(i+2))/(a_i + a_(i+2)) where a_1=1 and a_2=2 is : |
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Answer» `495/3` |
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| 49. |
If A and B are square matrices of order 3 such that absA = -1, absB = 3 then the value of determinant of 3AB is |
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Answer» -9 |
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