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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 number of values of k for whilch the system of equations (k+1)x+8y=4k kx+(k+3)y=3k-1 has infinitely many solutions is |
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Answer» 0 |
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| 2. |
int_(0)^(infty)e^(-x)sin^(6) x dx = |
| Answer» Answer :D | |
| 3. |
One hundred indentical coins are thrown as each coin has the probability of head as p. Let x = number of coins showing heads then match the following conditions with p value. (A) P(x=49)=P(x=50)""1)""P=(1)/(2) (B) P(x=48)=P(x=52)""2)""(51)/(101) ( C ) P(x=r)=P(x=n-r)""3)""(50)/(101) ""4)" "(57)/(100) The correct matching is |
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Answer» `{:(A,B,C),(2,3,1):}` |
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| 4. |
Let A=[{:(3,7),(2,5):}] and B=[{:(6,8),(7,9):}] Verify that (AB)^(-1)=B^(-1)A^(-1) |
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| 5. |
If f(x)=(e^(x))/(1+e^(x)),I_(1)=int_(f(-a))^(f(a))xg{x(1-x)}dxandI_(2)=int_(f(-a))^(f(a))g{x(1-x)}dx, then the value of (I_(2))/(I_(1)) is |
| Answer» ANSWER :A | |
| 6. |
If A,B,C are point (1,0,4) , (0,-1,5) and (2,-3,1) respectively , then the coordinates of foot of the perpendicular drawn from A to the line BC are |
| Answer» Answer :D | |
| 7. |
Integrate the functions 1/x (logx)^(2) |
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| 8. |
End-product of which of following reaction give positive Iodoform test. |
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Answer» `H-overset(O)overset(||)C-Clunderset((ii)H^(o+))overset((i)CH_(3)MGBR("excess"))to` `H-overset(O)overset(||)C-Etunderset((ii)H^(o+))overset((i)MeMgBr("excess"))toH-underset(Me)underset(|)overset(OH)overset(|)C-Meto+ve" Iodoform test"` `CH_(3)-overset(O)overset(||)C-Hunderset((ii)H^(o+))overset((i)CH_(3)MgBr("excess"))toCH_(3)-overset(OH)overset(|)CH-CH_(3)to+ve" Iodoform test"` |
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| 9. |
If A(x)=[{:(cosx,-sinx),(sinx,cosx):}]then A((pi)/(2)).A(pi)=……… |
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Answer» A |
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| 11. |
Show that the locus of point of intersection of perpendicular tangents to the parabola y(2)=4ax is thedirectrix x+a=0. |
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Answer» <P> |
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| 12. |
Integration by partial fraction : int(dx)/(e^(x)+1-2e^(-x))=.... |
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Answer» `LOG(e^(x)-1)-log(e^(x)+2)+C` |
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| 13. |
Vector equation of the line 6x-2=3y+1=1-2z is |
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Answer» `BARR=(-(1)/(3)HATI+(1)/(3)hatj-(1)/(2)hatk)+LAMBDA(hati+2hatj-3hatk)` |
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| 14. |
If f: [0,2) rarr R is defined by f(x) = {(1 + (2k)/(k) " for " ,0 le x lt 1),(kx " for ",1 le x lt 2):} where k gt 0, and f is such that underset(x rarr 1)(lim) f(x) = underset(x rarr 1)(lim) f(x), then the value of k^(2) is |
| Answer» ANSWER :C | |
| 15. |
Find the angle between the following linesvecr=3veci+2vecj-veck+t(veci+2vecj+2veck)andvecr=5vecj+2veck+s(3veci+2vecj+6veck) |
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| 17. |
A right triangle , shown below , has a longer leg measing 16sqrt3 centimeters . How long is the hypotenuse of the triangle , in centimeters ? |
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Answer» 8 |
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| 18. |
Integration of a binomialdifferential intx^(-(2)/(3))(1+x^((2)/(3)))^(-1)dx. |
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| 19. |
The term independent of x in the expansion of ((2sqrtx)/(5)-(1)/(2xsqrtx))^11 is |
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Answer» 5TH term |
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| 20. |
If the circle x^2+y^2+8x-4y+c=0 touches the circle x^2+y^2+2x+4y-11=0 externally and cuts the circle x^2+y^2-6x+8y+k=0 orthogonally then k = |
| Answer» ANSWER :B | |
| 21. |
Let N be the set of naturalnumbers and relation R on N be defined by R={(x,y):x,y in N, x+4y=10} check whether R is reflexive , symmetric and transitive. |
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| 22. |
Differentiate w.r.t.x the function in Exercises 1 to 11. (5x)^( 3 cos 2x) |
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| 23. |
Which of the above is many to many relation ? |
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Answer» f |
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| 24. |
Find |x|, if for a unit vector a, (vec(x)-vec(a)).(vec(x)+vec(a))=12. |
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Answer» SOLUTION :Given, `|a|=1`. (`because` It is a UNIT vector, magnitude of a unit vector is 1.) and `(VEC(x)-vec(a)).(vec(x)+vec(a))=12` `RARR vec(x).vec(x)+vec(x).vec(a)-vec(a).vec(x)-vec(a).vec(a)=12 ( because a.a =|a|^2 and a.b =b.a)` `|vec(x)|^2-|vec(x)|^2=12 rArr |vec(x)|-12=12 [because |a|=1 "as a is a unit vector"]` `rArr |vec(x)|^2=13 rArr |vec(x)|=sqrt(13)`. |
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| 25. |
Find the point on the curve y = x^(3) – 11x + 5 at which the tangent is y = x – 11. |
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| 26. |
Differentiate the functions with respect to x . cos ( sin x) |
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| 27. |
Evaluate the following integrals int (2x^2 - 3 sinx +5sqrtx) dx |
| Answer» Solution :int (2x^2-3sinx+5sqrtx) dx = 2 x^3/3 - 3(-COSX) + 5 x^(3/2)/(3/2)+C = 2x^3/3+3cosx+10/3 x^(3/2) +c` | |
| 28. |
Evaluate int_(0)^(n)(x-[x])dx, n in N, where [ ] denotes the GIF |
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| 30. |
Determine order and degree (if defined) of the following differential equations y'' + 2y' + sin y = 0 |
| Answer» Solution :The highest ORDER DERIVATIVE in the DIFFERENTIAL EQUATION is y.. and its degree is 1. `therefore` The order and the degree of the differential equation are 2 and 1 respectively. | |
| 31. |
Find the equation of the circle which intersects each of the following circles orthogonlly i)x^2 + y^2 + 2x + 4y + 1 = 0. x^2 + y^2 - 2x + 6y - 3 = 0.2(x^2 + y^2) + 6x + 8y - 3= 0. |
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| 32. |
Consider f(x)=int_(-1)^(x)(e^((x-t)/(x-2-t))dt)/(x-2-t)^(2) Q. The y-intercept of tangent drawn to graph of y=f(x) at x=-1 is |
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Answer» Solution :Put `(1)/(X-2-t)=U` `implies(1)/((x-2-t)^(2))dt=du` `f(x)=int_((1)/(x-1))^(-(1)/(2))e.e^(2u)du` `f(x)=(e)/(2).(e^(-1)-e^((2)/(x-1)))` `f(x)=(1)/(2)-(e)/(2).e^((2)/(x-1))` (1) `f(x)lt(1)/(2)` for all `xepsilonR` `implies` Greatest INTEGER in the Range `=0` (2) `f^(')(x)=(e)/((x-1)^(2))e^((2)/(x-1))` `f^(')(-1)=(1)/(4)(x+1)` `impliesy` intercept`=(1)/(4)` |
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| 33. |
If theta inR and (1-costheta)/(1+2icostheta) is real number, then theta will be ( when T : set of integers) |
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Answer» `(2n+1)pi/2,NINI` |
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| 34. |
It has been found from an experiment that 40% of rats stimulated on administering a particular drug. If 5 rats are given drug. Find probability that exactly three rats stimulated. |
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| 35. |
Equation of the motion of the particle is S = t^(3)-6t^(2)+9t, where S is in mets and t is in seconds.(i) Find the instantaneous velocity when t = 2.(ii) When particle is at rest ?(iii) Find distance travelled by particle in first 5 second. |
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Answer» (II) t = 1 sec, t = 3 sec (iii) 20 met. |
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| 36. |
Find the area of the loop and the curve 3ay^(2)=x(x-a)^(2)1. |
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| 37. |
Let A and B be any two point on eachof the circles x^(2) +y^(2) -8x -8y +28 =0 and x^(2) +y^(2) -2x -3= 0respectively . If d is the distance between A and B then the set of all possible values of d is |
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Answer» ` 1 le d le 9` |
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| 38. |
Statement 1: The system of linear equations x|(sinalpha)y+(cos alpha)z=0 x+(cos alpha)y+(sin alpha)z=0 x-(sin alpha)y-(cos alpha)z=0 has a non trivial solution for only one value of alpha lying in the interval (0,pi//2) Statement 2: The equation in alpha Delta=|(cos alpha, sin alpha, cos alpha),(sin alpha, cos alpha, sin alpha),(cos alpha, -sin alpha, -cos alpha)|=0 has only one solution lyining in the interval (0,pi//2) |
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| 39. |
Let Delta=|{:(a,p,x),(b,q,y),(c,r,z):}|=16" then "Delta_1=|{:(p+x,a+x,a+p),(q+y,b+y,b+q),(r+z,c+z,c+r):}|=32 |
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| 40. |
If the vectors a=hati-hatj+2hatk, b=2hati+4hatj+hatk and c=lambdahati+hatj+muhatk are mutually orthogonal, then (lambda, mu) is equal to |
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Answer» `(- 3, 2)` ` ( hati- hatj +2HATK ).(LAMDA hati +hatj +MU hatk )=0` `implies lamda -1+2 mu =0` `implies lamda +2 mu =1` `b.c=0` ` implies 2 lamda +4+mu =0` `implies2 lamda+mu =-4` on solvingEqs,(i) and (ii)we GET `lamda =- 3 , mu= 2 ` |
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| 41. |
If f(x) is continuous at x=pi/2, wheref(x)=(1-sin x)/((pi-2x)^(2)), for x != pi/2, then f(pi/2)= |
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Answer» `(-1)/(4)` |
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| 42. |
A building has two lifts. L_(1) and L_(2) are events when lifts are working. Probability P(L_(1) ) = 0.01 = P(L_(2))* L_(1) and L_(2) are independent events. What is the probability of atleast one lift is not working ? |
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Answer» 0.9999 |
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| 44. |
if a,b,c,d,e and f are six real numbers such that a+b+c=d+e+fa^2+b^2+c^2=d^2+e^2+f^2 and a^3+b^3+c^3=d^3+e^3+f^3 , prove by mathematical induction that a^n+b^n+c^n=d^n+e^n+f^n forall n in N. |
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Answer» Solution :Let `P(n):a^(n)+b^(2)+c^(n)=d^(n)+e^(n)+f^(n),AA n in N ""..(i)` where `a+b+c+d=e+""...(ii)` `a^(2)+b^(2)+c=d^(2)+e^(2)+f ""....(III)` and `a^(2)+b^(3)+c^(3)=d^(3)+e^(2)+^(3)""...(IV)` Step I from n from Eq. (i) we get `P(1): a+b+c=d+e+f"" ` [ given] Hence the result is true for n 1 Also, for n=2 from Eq(i), we get `P(2): a^(2)+b^(2)+c^(2)=d^(2)+e^(3)+f^(3) ""` [ given] Hennce the result trueor n=3 Therefore, P(1) , P(2) and P (3) are true. Step II Assume that `P(k-2),P(k-1)and P(k)` are true, then `P(k-2), a^(k-2)+b^(k-2)= d^(k-2)+e^(k-2)+f^)k-2) ""...(v)` `p(k-1):a^(-1)+b^(k-1)+c^(k-1)=d^(k-1)+e^(k-1)+f^(k-1) ""....(vi)` and `P(k): a^(k)+b^(k)+c^(k)=d^(k)+e^(k)+f^(k) "m"...(vii)` Step III for ` xn=k+1` we SHALL to prove that `P(k+1):a^(k+1)+b^(k+1)=d^(k+1)+e^(k+1)+f^(k+1)` LHS `=a^(k+1)+b^(k+1)+c^(k+1)` `=(a^(k)+b^(k)(a+b+c)-(a^(k-1)+b^(k-1)+c^(k-1))` `(ab+bc+ca)+abc(a^(k-2)+b^(k-2)+b^(k-2)+c^(k-2))` `=(d^(k)+e^(f)+f^(k))(d+e+f)-(d^(k-1)+e^(k-1)+c^(k-2))` `(de+ef+fd)+def(d^(k-2)+e^(k-2)+f^(k-2))` [ using Eqs. (ii), (iii), (iv), (v), (vi), (vii)] `:. (a+b+c)^(2)=(d+e+f)^(2)` `RARR a^(2)+b^(2)+c^(2)+2(ab+bc+ca)` `=d^(2)+e^(2)+f^(2)+2(de+ef+fd)` `rArr ab+bc+ca=de+ef+fd` `[ :. a^(2)+b^(2)+c^(2)=d^(2)+e^(2)+f^(2)]` and `a^(3)+b^(3)+c^(3) -3abc` `(=d+e+f)(d^(2)+e^(2)+f^(2)-de-ef-fd)` `=d^(3)+e^(3)+f^(3)-EDF` `rArr abc=def [ :. a^(3)+b^(3)+c^(3)=d^(3)+e^(3)+f^(3))` `=d^(k+1)+e^(k+1)+f^(k-1)=RHS` This shows that result is true for n=k+1. Hence by second perincipal of mathmatical inducition, the result is true for all `n in N`. |
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| 45. |
For all a,b in R we define a*b =|a-b|Show tha t* commutative but not associative |
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Answer» SOLUTION :(i) `FORALL` a,b in R we have `a*b=|a-b|=|-(a-b)|=|b-a|=(b*a)` `therefore`* Is commutative we have `(2*3)*4=|2-3|*4=|-1|*4=1*4=|1-4|=|-3|=3` `therefore` (2*3)*4 ne 2 * (3*4) Hence * is not associative |
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| 47. |
Which of the following functions is continuousat x = 0 ? |
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Answer» `F(x)={((sin2x)/x,",", x != 0),(1,",",x = 0):}` |
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| 48. |
If |{:(x,e^(x-1),(x-1)^(3)),(x-lx,cos(x-1),(x-1)^(2)),(tanx,sin^(2)x,cos^(2)x):}|=a_(0)+a_(1)(x-1)+a_(2)(x-1)^(2)cdots The eqution whose roots are a_(0) "and " a_(1)is |
| Answer» Answer :D | |
| 49. |
If thetain(0,2pi) then number of solution of the equationcos^(4)2theta+2sin^(2)2theta=17(sintheta+costheta)^(8) is greater than of equal to |
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Answer» 2 `1+t^(4)=17(1+t)^(4)` `8y^(2)+34y+35=0" where "y=t+(1)/(t)` Solving :`y=-(5)/(2)` `:.t=-(1)/(2)` `sin2theta=-(1)/(2)` `:.theta=(7pi)/(12),(11pi)/(12),(19pi)/(12),(23pi)/(12)` `:.4" sollutions ".` |
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