#### 12th Standard English Medium Maths Reduced Syllabus Three Mark Important Questions With Answer Key- 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. Find the rank of the matrix $\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right]$ by reducing it to an echelon form.

2. Find the adjoint of the following:
$\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right]$

3. If zi =2− i and z2=-4+3i , find the inverse of z1z2 and $\cfrac { { z }_{ 1 } }{ { z }_{ 2 } }$

4. If $cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0$ then show that
(i) $cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )$
(ii) $sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right)$

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

6. If the equations x2+px+q= 0 and x2+p'x+q'= 0 have a common root, show that it must  be equal to $\frac { pq'-p'q }{ q-q' }$ or $\frac { q-q' }{ p'-p }$.

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

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

9. Find the equation of the tangent and normal to the circle x2+y2−6x+6y−8=0 at (2,2) .

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

11. With usual notations, in any triangle ABC, prove the following by vector method.
(i) a2=b2+c2−2bc cos A
(ii) b2=c2+a2−2ca cos B
(iii) c2= a2+b2−2ab cos C

12. If $\vec { a } =\hat { i } -\hat { k } ,\vec { b } =x\hat { i } +\hat { j } +(1-x)\hat { k } ,\vec { c } =y\hat { i } +x\hat { j } +(1+x+y)\hat { k }$show that $[\vec { a } ,\vec { b } ,\vec { c } ]$ depends on neither x nor y.

13. If $\vec { a } =\hat { i } -2\hat { j } +3\hat { k } ,\vec { b } =2\hat { i } +\hat { j } +\hat { k } ,\vec { c } =3\hat { i } +2\hat { j } +\hat { k }$ find
(i) $(\vec { a } \times \vec { b } )\times \vec { c }$
(ii) $\vec { a } \times (\vec { b } \times \vec { c } )$

14. Find the direction cosines of the straight line passing through the points (5,6,7) and (7,9,13) . Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points.

15. Find the equations of tangent and normal to the curve y = x2 + 3x − 2 at the point (1, 2)

16. Find the values in the interval $(\frac{1}{2},2)$ satisfied by the Rolle's theorem for the function $f(x)=x+\frac{1}{x}, x\in[\frac{1}{2},2]$

17. Write down the Taylor series expansion, of the function log x about x =1 upto three nonzero terms for x > 0.

18. Use linear approximation to find an approximate value of $\sqrt { 9.2 }$ without using a calculator.

19. A coat of paint of thickness 0.2 cm is applied to the faces of a cube whose edge is 10 cm. Use the differentials to find approximately how many cubic centimeters of paint is used to paint this cube. Also calculate the exact amount of paint used to paint this cube.

20. Let f (x, y) = 0 if xy ≠ 0 and f (x, y) =1 if xy = 0.
(i) Calculate: $\frac { \partial f }{ \partial x } (0,0),\frac { \partial f }{ \partial y } (0,0).$
(ii) Show that f is not continuous at (0,0)

21. For each of the following functions find the gxy, gxx, gyy and gyx
g(x, y) = x2 + 3xy − 7y + cos(5x)

22. A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x2 − 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively
Find $\frac { { \partial P } }{ \partial { x } }$ (1200, 1800) and $\frac { \partial v }{ \partial y}$ (1200, 1800)

23. If f (x) = f (a + x) , then $\int _{ 0 }^{ 2a }{ f(x)dx=2\int _{ 0 }^{ a }{ f(x)dx } }$

24. Evaluate the following definite integrals:
$\int _{ 0 }^{ 1 }{ \sqrt { \frac { 1-x }{ 1+x } } } dx$

25. If $\int _{ 0 }^{ \infty }{ { e }^{ -a{ x }^{ 2 } }{ x }^{ 3 }dx=32,\alpha >0,\quad find\alpha }$