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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. |
Consider the function f(x) = {{:( x^(2) - 1"," , -1 le x le 1) , ( lnx "," , 1 lt x le e):} Let f_(1) (x) = f (|x|) f_(2) (x) = |f(|x|)| f_(3) (x) = f (-x) Now answer the question Number of positive solution of the equation 2f_(2) (x) - 1= 0 is (are) |
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Answer» `{{:( 4 + x , : , x GE 0 ),( 4 - x , : , x LT 0):}` |
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| 3. |
Area of a rectangle having vertices A,B C and D with position vectors -hati+(1)/(2)hatj+4hatk,hati+(1)/(2)hatj+4hatkand-hati-(1)/(2)hatj+4hatk,respectively is |
| Answer» Answer :C | |
| 4. |
Differentiate the functions given in w.r.t. x. (logx)^(x)+ x^(log x). |
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| 5. |
The range of y = 2x^(2)+x+2/(2x^(2)+x+1) is |
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Answer» `(11-sqrt(2)/7 ,11+sqrt(2)/7 )` |
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| 6. |
{{:(-13=ay+24x),(9+6bx=5y):} If the system of equations above has no solutions, and a and b are constants, then what is the value of |a+b| ? |
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Answer» 0 `{{:(24 x + ay =-13),(6bx-5y=-9):}` In a system of equations that has no solution, the x-coefficients must equal each other and the y-coefficients must equal each other. Thus, for the x - coefficients, `24=6b,` which means that `b =4.` for the y-coefficients, `a =-5.` The question asks for the value of `|a+b|, ` which is `|-5+4|=|-1|=1,` CHOICE (B). (Note that if you USED the EQUATION `-bx+5y=9,` you would get `a =5and b =-4,` which still result in the correct answer.) |
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| 7. |
f:R rarr R , f(x) = {(12x+5,x gt1),(x-4,xle1):} then find f(0),f(-1/2), f(3),f(-5) . |
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| 8. |
Ify=cot ^(-1) sqrt(( x^(2) +a^(2)+x)/( sqrt( x^(2)+a^(2) -x)) ) ,then (dy)/(dx)= |
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Answer» ` (-a)/( 2(a^(2)+X^(2) ) ) ` |
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| 9. |
Diffusion of O_(2) and CO_(2)between lungs and blood present in :- |
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Answer» CONDUCTING ZONE |
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| 10. |
Given that the events A and B are such that P(A) =(1)/(2) , P(AuuB) = (3)/(5) and P(B) = p. Find p if they are (i) mutually exclusive, (ii) independent. |
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| 11. |
Find the second order derivatives of the functions given in Exercises 1 to 10. e^(6x) cos 3x. |
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| 12. |
Find that the function is continuous or discontinuous at the indicated point f(x) = {{:(|x-a|sin\ (1)/(x-a),if x ne a),(0, if x =a):} at x = a |
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Answer» At `LHL=UNDERSET(xrarra^(-))(lim)|x-a|sin'(1)/(x-a)` `=underset(hrarr0)(lim) |a-h-a| sin((h)/(a-h-a))` `= underset(hrarr0)(lim)-hsin(1/h) , [:' sin(-THETA)=-SINTHETA]` `=0 xx` [an oscillatingnumber between `-1` and 1] = 0 `RHL = underset (xrarra^(+))(lim)|x-a|sin((1)/(x-a))` `=underset(hrarr0)lim|a+h-a|sin((1)/(a+h-a))=underset(hrarr0)limh sin '1/h` `= 0xx` [ an oscillatingnumber between` -1` and 1] = 0 and `f(a) = 0` `:. LHL = RHL = f(a)` So, `f(x)` is continousat `x = a`. |
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| 13. |
Let a, b be the roots of the equationx^(2) - 4 x k_(1) = 0 and c , d the roots of the equationx^(2) - 36 x + k_(2) = 0Ifa lt b lt c lt dand a, b,c,dare in G.P. then the productk_(1) k_(2) equals |
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Answer» 81 |
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| 14. |
Let ABCDEFGHIJKL be a regular dodecagon. Then the value of (AB)/(AF) + (AF)/(AB) is equal to ____ |
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Answer» Similarly, `AN = R sin.(5pi)/(12) rArr AF = 2 R sin.(5pi)/(12)`  `:. (AB)/(AF) + (AF)/(AB) = (sin.(pi)/(12))/(sin.(5pi)/(12)) + (sin.(5pi)/(12))/(sin.(pi)/(12))` `= TAN.(pi)/(12) + cot.(pi)/(12)` `= (2 - sqrt3) + (2 + sqrt3) = 4` |
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| 15. |
Examine the continuity of the function f(x) = {((e^((1)/(x)))/(1 + e^((1)/(x)))",","if " x ne 0),(0",","if" x = 0):} at x= 0 |
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| 16. |
The triangle ABC, right angled at C, has median AD, BE and CF.AD lies along the line (1 – sqrt3)x + (sqrt( 3)+ 1) y – 6 = 0 , BE lies along the line (1 – 2sqrt 3)x + ( sqrt(3) + 2)y – 8 = 0 . If the length of the hypotenuse is 60, find the area of the triangle ABC. |
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| 17. |
therootsof theequationx^4 -6x^3 + 18x^2- 30 x+25=0are of theforma +-I band b+-iathen(a ,b)= |
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Answer» `(3,2)` |
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| 18. |
If the probability mass function of a discrete random variable X is P(x)=(C)/(x^(3)),x=1,2,3=0, otherwise. Then, E(X) is equal to |
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Answer» `(343)/(297)` |
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| 19. |
The total cost function C(x) = 2x^(3) - 5x^(2) + 7x. Check whether the MAC increases or decreases with increasein outputs . |
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| 20. |
If 16 x^(4) -32x ^(3) + ax ^(2) +bx + 1=0, a,b,in R has positive real roots only, then |b-a| is equal to : |
| Answer» ANSWER :B | |
| 21. |
Evaluate the following integrals. int(1)/(xsqrt(x^(2)+x+1))dx |
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| 22. |
A man takes a step forward with probability 0.4 and one step backward with probability 0.6, then the probability that at the end of eleven steps he is one step away from the starting point is |
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Answer» `""^(11)C_(6)xx (0.72)^(6)` |
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| 23. |
Givensin alpha+ cos alpha+tan alpha+ cot alpha+ sec alpha=7, find the value of sin 2 alpha. |
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| 24. |
How many on-to functions can be defined from a set A={a_(1),a_(2),......,a_(n)} to another set B={x,y,z} such that a_(1) is always mapped to x. |
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| 25. |
A weak acid HA (50.0ml) was titrated against 0.1 M NaOH. The pH values when 20 ml & 40 ml base have been added are found to be 4.898 & 5.324 respectively. Calculate the pH of the solution at equivalence point. (Given : log8//3=0.426,log2=0.301,log15=1.176) Mark your answer in single digit to the nearest integral digit, say your answer is 7.213 then mark 0007 or if your answer is 12.567 then mark as 0013. |
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Answer» `5.324=pk_(a)+"log"(4)/(a-4)` `0.426=log[((4)/(a-4))xx((a-2)/(2))]` `2.66686=(2(a-2))/((a-4))` `(8)/(3)=(2(a-2))/(a-4)` `4(a-4)=3(a-2)` `a=10` `4.898=pk_(a)+"log"(2)/(8)` `pk_(a)=505` `pH=7+(1)/(2)(5.5+"log"(10)/(150))` `=9.162` ANS. |
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| 26. |
If abs(z+4) le 3, the maximum value of abs(z+1) is |
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Answer» 4 |
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| 27. |
If ABC is nota right andgled triangle and sin((pi)/(4)-A)sin(pi)/(4-B)=-(1)(2sqrt(2))cosec ((pi)/(4)-C) then tan A tan B + tan B tan C + tan C tan A= |
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Answer» COT A +cot B + cot C |
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| 28. |
The plane 2x + 3y + kz - 7 = 0 is parallel to the line whose d.r's are (2,-3,1) then k= |
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Answer» 5 |
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| 29. |
Find the distance between the lines l_1 and l_2 given by vecr=hati+2hatj-4hatk+lambda(2hati+3hatj+6hatk) and vecr=3hati+3hatj-5hatk+mu(2hati+3hatj+6hatk). |
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| 30. |
Let rbe a real number andn in N be such that the polynomial 2x^(2)+ 2x + 1dividesthe polynomial (x + 1)^(n) - r. Then (n,r)can be-(A)(4000, 4^(1000)) (B)(4000, 1/4^(1000)) (C)(4^(1000),1/4^1000) (D)(4000,1/4000) |
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Answer» `(4000, 4^(1000))` `X = (-1+i)/(2), (-1-i)/(2)` x satisfies `(x + 1)^(N) - r = 0` `((-1+-i)/(2) + 1)^(n) - r = 0` `((1+-i)/(2))^(n) -r = 0` `((1)/(sqrt(2)))^(n) ((1+i)/(sqrt(2)))^(n) = r` `((1)/(sqrt(2)))^(n)(e^(+-( IPI)/(4)))^(n)= r` RHS= real LHS = realonly when n = multiply of 4 `n = 4000` `r = ((1)/(sqrt(2)))^(4000) = 1/(4^(1000))` |
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| 31. |
For the LPP, maximise z=x+4y subject to the constraints x+2yle2, x+2yge8,x,yge0. |
| Answer» Answer :D | |
| 32. |
Given that the vectors a and b are non-collinear, the values of x and y for which the vector equality 2u - v = w holds true if u = xa + 2y b, v = - 2y a + 3xb, w = 4a - 2b are |
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Answer» `x=(4)/(7),y=(6)/(7)` `implies 2(xa+2yb)-(-2ya+3xb)=4a-2b` `implies (2x+2y-4)a+(-3x+4y+2)b=0` `implies 2x+2y-4=0 and -3x+4y+2=0`[`because` a, b are non-collinear] `x=(10)/(7), y=(4)/(7)` |
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| 33. |
If A=[{:(1,2,0),(-2,-1,-2),(0,-1,1):}] find A^(-1). Using A^(-1), solve the system of linear equations x-2y=10, 2x-y-z=8, -2y+z=7. |
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| 34. |
Differentiate (x + 3)^(2) (x + 4)^(3)(x+5)^(4) w.r.to x. |
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| 35. |
Consider the function g(x) defined as g(x)*(x^((2^(2008)-1))-1=(x+1)(x^(2)+1)(x^(4)+1)...(x^(2^(2007))+1)-1 the value of g(2) equals ……. . |
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| 36. |
If (3)/(2)+(7)/(2)i is a solution of the equation ax^(2)-6x+b=0, where a and b are real numbers, then the value of a+b is equal to |
| Answer» ANSWER :D | |
| 37. |
Let veca=hati+hatj+hatk, vecb=-hati+hatj+hatk, vecc=hati-hatj+hatk and vecd=hati+hatj-hatk. Then, the line of intersection ofplanes one determined by veca, vecb and other determined by vecc, vecd is perpendicular to |
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Answer» `x`-axis `(VECA xx vecb)xx (vecc xx vecd)` `[veca vec C vecd]vecb-[vecb vecc vecd]vecd` `=4vecb-veca=-8hati` Which is perpendicular to y-axis and z-axis. |
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| 38. |
If in the expansion of (1+x)^(m)(1-x)^(n) the coefficients of x and x^(2) are 3 and -6 respective then m is |
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Answer» 6 |
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| 39. |
Without actually solving show that the equation x^(4)+2x^(3)-2=0 has only one real root in the interval (0,1) . |
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| 40. |
Evaluation of definite integrals by subsitiution and properties of its : If int_(-1)^(4)f(x)dx=4 and int_(2)^(4)(3-f(x)dx=7 then int_(2)^(-1)f(x)dx=....... |
| Answer» Answer :C | |
| 41. |
If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (-4, 3, -6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD. |
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Answer» `therefore` Hence,the angle between the LINES AB and CD is `0^@`. |
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| 42. |
Let Z in C with Im (z) = 10 and it satisfies =(2z-n)/(2z+n)=2i-1 For some natural number n. then |
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Answer» n=20 and Re (z) =-10 Since z satisfies `(2z-n)/(2z +n)=2i -1 , n ne N` `therefore (2x+20 i-n)=(2i -1)(2x+20 I +n)` `RARR (2x-n)+20 i=(-2 -n- 40 )+(4x+2N- 20)i` On comparing real imaginary parts ,we get `2x-n= -2 x-n 40 and 20 = 4x+2n-20` `rArr4x = -40 and 20 = 4x 2n = 40 ` `rArr x=-10 and - 40 +2n = 40 rArr n = 40` So , n = 40 and x= Re (z) = -10 |
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| 43. |
What is the area of shaded region in the figure? |
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Answer» `10PI + 27sqrt(3)` |
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| 44. |
Three lines are drawn from the origin O with direction cosines proportional to (1, -1, 1), (2, -3, 0) and (1, 0,3). The three lines are |
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Answer» not COPLANAR |
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| 45. |
For a, b, c in Q and b + c ne a, the roots of ax^(2)-(a+b+c)x+(b+c)=0 |
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Answer» RATIONAL and unequal |
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| 46. |
Solve the following system of linear equaltions, using matrix in inversion method: 5x+2y=3,3x+2y=5. |
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| 47. |
integrate the following inte^(3x)dx |
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Answer» Solution :`inte^(3X)DX` `[put3x=t Then 3dx=DT or dx=(1/3)dt `inte^tcdot(1/3)dt=(1/3)E^t+C` `(1/3)e^(3x)+C` |
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| 48. |
If x= overset(oo)underset(n =0)Sigma a^(n),y= overset(oo)underset(n=0)Sigma b^(n), 0 lt a lt b lt 1, and z=overset(oo)underset(n=0)Sigma ((a)/(b))^(n), then |
| Answer» Answer :A | |