#### 12th Standard English Medium Maths Reduced Syllabus Five Mark Important Questions With Answer Key - 2021(Public Exam )

12th Standard

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Maths

Time : 02:45:00 Hrs
Total Marks : 125

5 Marks

25 x 5 = 125
1. Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

2. If a1, a2, a3, ... an is an arithmetic progression with common difference d, prove that tan $x = {-b \pm \sqrt{b^2-4ac} \over 2a}\quad \left[ tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +....tan^{ -1 }\left( \frac { d }{ 1+{ a }_{ n }{ a }_{ n-1 } } \right) \right] =\frac { { a }_{ n }-{ a }_{ 1 } }{ 1+{ a }_{ 1 }{ a }_{ n } }$

3. Provethat ${ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\$

4. Find the equation of the ellipse whose eccentricity is $\frac { 1 }{ 2 }$, one of the foci is(2,3) and a directrix is x = 7 . Also find the length of the major and minor axes of the ellipse.

5. Certain telescopes contain both parabolic mirror and a hyperbolic mirror. In the telescope shown in figure the parabola and hyperbola share focus F1 which is 14mabove the vertex of the parabola. The hyperbola’s second focus F2 is 2m above the parabola’s vertex. The vertex of the hyperbolic mirror is 1m below F1. Position a coordinate system with the origin at the centre of the hyperbola and with the foci on the y-axis. Then find the equation of the hyperbola.

6. On lighting a rocket cracker it gets projected in a parabolic path and reaches a maximum height of 4m when it is 6m away from the point of projection. Finally it reaches the ground 12m away from the starting point. Find the angle of projection.

7. A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t2 in t seconds.
How long does the camera fall before it hits the ground?

8. Find the equation of the tangent and normal to the Lissajous curve given by x = 2cos3t and y = 3sin 2t, t ∈ R

9. Find the asymptotes of the following curves $f(x)=\cfrac { 3x }{ \sqrt { { x }^{ 2 }+2 } }$

10. Find the intervals of monotonicity and local extrema of the function f (x) = x log x + 3x.

11. If the curves 4x=y2 and 4xy=k cut at right angles show that k2=512.

12. Evaluate: $\int _{ 0 }^{ \frac { \pi }{ 2 } }{ (\sqrt { tan\quad x } +\sqrt { cot\quad x } )dx }$

13. Find the area of the region bounded between the curves y = sin x and y = cos x and the lines x = 0 and x =$\pi$

14. Find the area of the curve y2=(x-5)2(x-6) between
(i) x=5 and x=6
(ii) x=6 and x=7

15. Find the area of the region bounded by the curves x2+2y2=0 and x+3y2=1.

16. AOB is the positive quadrant of the ellipse $\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1$ where OA=a and OB=b.Find the area between the arc AB and chord AB of the elipse.

17. Solve the differential equation: (x2+y2)dy = xy dx. It is given that y(1) = 1 and y(x0) = e. Find the value of x0.

18. Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years?

19. Solve : $\cfrac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx$

20. Solve : $\cfrac { dy }{ dx } =-\cfrac { x+ycos }{ 1+sinx }$ .Also find the domain of the function.

21. The mean and standard deviation of a binomial variate X are respectively 6 and 2.
Find
(i) the probability mass function
(ii) P(X = 3)
(iii) P(X$\ge$2).

22. The difference between the mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution.

23. Verify (p ∧ -p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.

24. Show that (Z7-[0],X7) satisfies closure, identity, inverse and commutative properties.

25. Prove that (2019)10+(2020)10≡1025(mod 2018)