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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. |
Write the simplest form of tan^(-1) [( 3 cos x - 4 sin x )/( 4 cos x + 3 sin x) ], if (3)/(4) tan x gt -1 |
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| 2. |
If A(3,5), B(-5, - 4), C(7, 10) are the vertices of a parallelogram, taken in the order, then the co-ordinates of the fourth vertex are |
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Answer» (10, 19) |
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| 3. |
Let R be the region containing the point (x, y) on the X-Y plane, satisfying 2le|x+3y|+|x-y|le4. Then the area of this region is |
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Answer» 5 SQ. units The area `=8-2=6.` |
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| 4. |
The point (3,1) is a point on a circle C with centre (2,3) and C is orthogonal to x^(2) + y^(2) = 8. The conjugate point of (3,1) w.r. tox^(2) + y^(2) = 8which lies on C is |
| Answer» ANSWER :C | |
| 5. |
If the volume of parallelopiped with coterminous edges 4hati+5hatj+hatk, - hatj+hatk and 3hati+9hatj+phatk is 34 cu units, then p is equal to |
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Answer» Solution :SINCE, volume of parallelopiped `=34` `:. |(4,5,1),(0,-1,1),(3,9,p)|=34` `IMPLIES 4(-p-9)-5(-3)+1(3)=34` `implies -4p-36+15+3=34` `implies 4p=-52` `implies p=-13` |
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| 6. |
Find adjoint of each of the matrices in Exercises 1 and 2 [{:(1,-1,2),(2,3,5),(-2,0,1):}] |
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| 7. |
Find the distance of the point (2,1,-1) is equidistant from the plane x-2y+2z+4z=9. |
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| 8. |
Find the d.r. of the normal to the plane 4x + 3y - z + 6 = 0 and |
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Answer» `LT -4, -3, 1 GT` |
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| 9. |
If int(x+(cos^(-1)3x)^(2))/(sqrt(1-9x^(2)))dx=Psqrt(1-9x^(2))+Q(cos^(-1)3x)^(3)+c then |
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Answer» `P=-(1)/(9)` |
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| 10. |
Show that the foot of the perpendicular from focus to the tangent of the parabola y^(2)=4ax lies on the tangent at vertex. |
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| 11. |
If the area of the triangle formed by the pair of lines 8x^(2) - 6xy + y^(2) =0 and the line 2x + 3y = a is 7 , then a is equal to |
| Answer» ANSWER :D | |
| 12. |
int_(-pi//2)^(pi//2) cos theta (1+sin theta)^(2) d theta= |
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Answer» `(14)/(3)` |
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| 13. |
The position vector of a point A is (4, 2, -3). If the distance of the point A from XY- plane is p_(1) and from Y - axis is p_(2) then p_(1)+p_(2)= …………. |
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Answer» 2 |
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| 14. |
int (log x )^(3) dx = |
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Answer» `X [ (log x)^(3) - (log x)^(2) + (log x) - 6]` + c |
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| 15. |
Find the are length of the cardioid rho= a (1 + cos varphi) (a gt 0, 0 le varphi le 2pi) |
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| 16. |
Find the values of the following integrals (v) int_(0)^(pi/4) sec^(6)x dx |
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| 18. |
If the radius of a sphere is measured as 9 m with an error of 0.03 m, then find the approximate error in calculating its surface area. |
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| 19. |
Coefficient of x^(n) in log_(e )(1+(x)/(2)) is |
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Answer» `(1)/(N.2^(n))` |
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| 20. |
Find the number of 4-digit numbers that can be formed using the digits 0, 1,2, 3, 4, 5 which are divisible by 6 when repetition of the digits is allowed |
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| 21. |
If f(x)=||x-1|+|x-3|-|2x-1||+||x-1|+|x-3|+|2x-1||, then |
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Answer» `F(X)` is non-differentiable at `x=1` `:.x=1, 3/2` point of non-differentiability and minimum value is 2 |
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| 22. |
The symmetric equation of lines 3x+2z-5=0 and x+y-2z-3=0 is |
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Answer» `(x-1)/(5)=(y-4)/(7)=(z-0)/(1)` `therefore 3a+2b+c=0` and a+b-2c=0 `Rightarrow (a)/(-4-1)=(b)/(1+6)=(c)/(3-2)` `Rightarrow (a)/(-5)=(b)/(7)=(c)/(1)` In order to find on the required line, we put z=0 in the two given equations to obtain 3x+2y=5 and x+y=3. On SOLVING these two equations, we get x=-1, y=4 `therefore` Coordintaes of POINT on required line are (-1,4,0) Hence, required line is `(x+1)/(-5)=(y-4)/(7)=(z-0)/(1)` |
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| 23. |
Probability that A speaks truth is 4/5. A coin tossed. A reports that a head appears. The probability that actually there was head is a)4/5 b)1/2 c)1/5 d)2/5 |
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| 24. |
((cos3theta-isin3theta)^4)/((sin2theta+icos2theta)^5)= |
| Answer» SOLUTION :-(sin2theta+icos2theta) | |
| 25. |
Statement-1 :Delta_(f)H_(400K)^(@) is zero for O_(2)(g) Statement-2 : Delta_(f)H_(298K)^(@) is zero for O(g). |
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Answer» STATEMENT-1 is TRUE, statement-2 is true and statement-2 is correct explanation for statement -2 |
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| 26. |
Integrate the functions (cossqrtx)/(sqrtx) |
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| 29. |
The unit vector in the direction of bar(x)=(-2,1,-2) is ………….. |
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Answer» `((2)/(3),-(1)/(3),(2)/(3))` |
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| 30. |
Differentiate the following w.r.t.x sin^-1(x^3) |
| Answer» SOLUTION :`d/dx[sin^-1(x^3)]=1/(SQRT(1-(x^3)^2)d)d/dx(x^3)=(3x^2)/sqrt(1-x^6)` | |
| 31. |
Let f(x) =(x -2) (x^(4) -4x^(3) + 6x^(2) - 4x +1) then value of local minimum of f is |
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Answer» `-2//3` |
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| 32. |
The set of all solutions of the inequation x^(2) - 2x +5 le 0 in R is |
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Answer» `R-(-infty, -5)` |
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| 33. |
Using the fact that 0 le f(x) le g(x) , c lt x lt d rArr int_(c)^(d) f(x) dx le int_(c)^(d) g(x) dx, we can conclude that int_(1)^(3) sqrt(3+x^3) lies in the interval |
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Answer» `((1)/(2),3)` |
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| 34. |
If int_(1)^(2) (dx)/( (x^(2) -2x + 4)^(3/2))=(k)/( k+5) then k is equal to |
| Answer» ANSWER :A | |
| 35. |
Find (dy)/(dx) in the following: y= tan^(-1) ((a cos x- b sin x)/(b cos x + a sin x)) |
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| 36. |
Differentiate tan^2x+sec^2x |
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Answer» SOLUTION :`y=tan^2x+sec^2x` `dy/dx=2tanxcdot d/dx(TANX)+2secxd/dx(SECX)` `=2tanxcdot sec^2x+2sec^2xcdot tanx=4sec^2x tanx` |
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| 37. |
If the normal at any point P on the ellipse x^2/a^2+y^2/b^2=1 meets the axes in G and g respectively. Find the ratio PG:Pg |
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| 38. |
Which of the following matrice is invertible? [[2,1,-2],[1,2,1],[3,6,4]] |
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Answer» Solution :LET A=`[[2,1,-2],[1,2,1],[3,6,4]]` `therefore ABSA =[[2,1,-2],[1,2,1],[3,6,4]]` =`2[[2,1],[6,4]]-1[[1,1],[3,4]]-2[[1,2],[3,6]]` =2(8-6)-(4-3)-2(6-6) =4-1-0=`3 ne 0` |
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| 39. |
A is a set containg n elements. A subset P of A is chosen at random and the set A is reconstructed by replacing the random. Fond the probability that Pcup Q contains exactoly r elements with1 le r le n. |
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| 40. |
If four positive integers are taken at random and multiplied together, then the probability that the last digit is 1, 3, 7 or 9 is |
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Answer» `1/8` |
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| 41. |
The set of point (x,y) satisfying x+y le 1 , -x-yle 1 lie in the region bounded by the two straight lines passing through the respective pair of points. |
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Answer» `{(1,0) ,(0,1)} and {(-1,0) ,(0,-1)}` |
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| 42. |
A factory has three machines, X,Y and Z, producing 1000,2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% defective bolts and Z produces 2% defective bolts. At the end of the day, a bolt is drawn at randomand it is found to be defective. What is the probability that this defective bolt has been produced by the machine X? |
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Answer» <P> Solution :Total number of bolts produced in DAY`=(1000+2000+3000)=6000`. Let `E_1,E_2and E_3` b the events of drawing a bolt produced by MACHINES X, Y and Z respectively. Then, `P(E)=1000/6000=1/6,P(E_2)=2000/6000=1/3andP(E_3)=3000/6000=1/2`. Let E be the event of drawing a defective bolt. Then, `P(E//E_1)`= probability of drawing a defective bolt, GIVEN that it is produced by the machine X `=1/100`. `P(E//E_2)`= probability of drawing a defective bolt, given that it is produced by the machine Y `1.5/100=15/1000=3/200`. `P(E//E_3)`=probability of drawing a defective bolt, given that it is produced by the machine Z `=2/100=1/50`. Required probability `=P(E_1//E)` = probability that the bolt drawn is produced by X, given that it is defective `(P(E_1).P(E//E_1))/(P(E_1).P(E//E_1)+P(E_2).P(E//E_2)+P(E_3).P(E//E_3))` `=((1/6xx1/100))/((1/6xx1/100)+(1/3xx3/200)+(1/2xx1/50))` `=(1/600xx600/10)=1/10=0.1`. Hence, the required probability is 0.1. |
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| 43. |
If S-= x^(2) + y^(2) + 2gx + 2fy + c= 0repre- sents a circle then show that the straight line lx + my + n = 0 (ii) meet the circle S = 0 in two points if g^(2)+f^(2)-c gt ((gl+mf-n)^(2))/((l^(2)+m^(2))) |
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| 44. |
For a DeltaABC the vertices are A(0, 3), B(0, 12) and C(x, 0). If the circumcircle of DeltaABC touches the x - axis, then the area (in sq. units) of the DeltaABC is |
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Answer» 36 |
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| 45. |
The maximum area of an isosceles triangle inscribed in the ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 with the vertex at one end of the major axis is |
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Answer» `SQRT(3)AB` |
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| 46. |
Find |vecaxxvecb|, if veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk |
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| 47. |
If P is any point common to the hyperbola (x^(2))/(16)-(y^(2))/(25)=1 and the circle having line segment joining its foci as diameter then sum of focal distances of point P is |
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Answer» `6sqrt(2)` `:. P` LIES on the circle `x^(2) + y^(2) = 41` Any point on hyperbola is `(4 sec theta, 5 tan theta)` `rArr 16 sec^(2) theta + 25 tan^(2) theta = 41` `rArr tan theta = (5)/(sqrt(41))` and `sec theta = sqrt((66)/(41))` `:. PF_(1) + PF_(2) = e (4 sec theta -(a)/(e)) +e (4 sec theta +(a)/(e))` `= 8 esec theta = 2 sqrt(66)` |
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| 48. |
Integrate the following functions. intsqrt(tanx)dx |
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| 49. |
Probability distribution of random variable X is as follows: Find its standard deviation. |
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