#### 12th Standard English Medium Maths Reduced Syllabus Three Mark Important Questions - 2021(Public Exam )

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 75

3 Marks

25 x 3 = 75
1. Given A = $\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right]$, B = $\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right]$ and C = $\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right]$, find a matrix X such that AXB = C.

2. Find the rank of the following matrices by row reduction method:
$\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ 5 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} -1 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 4 \\ 11 \end{matrix} \end{matrix} \right]$

3. A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution ? (Use Cramer’s rule to solve the problem).

4. Find the adjoint of the following:
$\frac { 1 }{ 3 } \left[ \begin{matrix} 2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2 \end{matrix} \right]$

5. The complex numbers u,v, and w are related by $\cfrac { 1 }{ u } =\cfrac { 1 }{ v } +\cfrac { 1 }{ w }$ If v=3−4i and w=4+3i, find u in rectangular form.

6. If $\left| z-\cfrac { 2 }{ z } \right| =2$ show that the greatest and least value of |z| are $\sqrt { 3 } +1$ and $\sqrt { 3 } -1$ respectively.

7. Show that the equation ${ z }^{ 3 }+2\bar { z } =0$ has five solutions

8. Show that $\left( \cfrac { 19-7i }{ 9+i } \right) ^{ 12 }+\left( \cfrac { 20-5i }{ 7-6i } \right) ^{ 12 }$ is real

9. Obtain the Cartesian form of the locus of z=x+iy in
Im[(1−i)z+1]= 0

10. Obtain the Cartesian equation for the locus of z=x+iy in
|z-4|2-|z-1|2=16

11. Solve the equation 2x3+11x2−9x−18=0.

12. Solve the cubic equation : 2x3−x2−18x+9=0 if sum of two of its roots vanishes.

13. Solve the cubic equations:
8x3-2x2-7x+3=0

14. For what value of x , the inequality$\cfrac { \pi }{ 2 } <{ cos }^{ -1 }(3x-1)<\pi$ holds?

15. Find
i)  tan−1($-\sqrt3$)
ii)  tan−1$(tan\frac{3\pi}{5})$
iii) tan(tan-1(2019))

16. Prove that ${ tan }^{ -1 }x+{ tan }^{ -1 }\frac { 2x }{ 1-{ x }^{ 2 } } ={ tan }^{ -1 }\frac { 3x-{ x }^{ 2 } }{ 1-{ 3x }^{ 2 } } ,|x|<\frac { 1 }{ \sqrt { 3 } }$

17. Find the value of the expression in terms of x, with the help of a reference triangle.
tan$\left( { sin }^{ -1 }\left( x+\frac { 1 }{ 2 } \right) \right)$

18. Find the equations of the tangent and normal to the circle x2+y2=25 at P(-3,4).

19. A circle of area 9π square units has two of its diameters along the lines x+y=5 and x−y=1.
Find the equation of the circle.

20. Find the equation of the hyperbola with vertices (0,±4) and foci(0,±6).

21. Find the equation of the hyperbola in each of the cases given below:
(i) foci(±2,0), eccentricity =$\frac { 3 }{ 2 }$
(ii) Centre (2,1) , one of the foci (8,1) and corresponding directrix x = 4.
(iii) passing through(5,−2)and length of the transverse axis along x axis and of length 8 units.

22. Find the equation of the tangent to the parabola y2 =16xperpendicular to 2x+2y+3=0.

23. Find the equations of the tangent and normal to hyperbola 12x2−9y2=108 at $\theta =\frac { \pi }{ 3 }$ (Hint: use parametric form)

24. If D is the midpoint of the side BC of a triangle ABC, then show by vector method that ${ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })$

25. The vector equation in parametric form of a line is $\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )$. Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.