A metal box with a square base and vertical sides is to contain 1024 cm³. The material for the top and bottom costs Rs. 5 per cm² and the material for the sides cost Rs. 2.50/cm². Find the least cost of the box.

6

(a)

AB is a diameter of a circle and C is any point on the circle. Show that the area of ΔABC is maximum when it is isosceles.

6

(a)

If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of triangle is maximum when the angle between them is \(60^\circ\)
\((i.e., \frac{\pi}{3})\)

6

(a)

Using properties of determinants, show that triangle ABC is isosceles if: 
\(\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0\)

6

(a)

Show that triangle ABC is an isosceles triangle,  if the determinant
\( \triangle =\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ C } +cosC \end{matrix} \right| \)

6

(a)

A company manufactures two types of sweaters, type A and B. It costs Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make at most 300 sweaters and can spend Rs. 72,000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs 200 on each unit of type A. The company charging a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve for max. profit.

6

(a)

Using integration, find the area of the region bounded by the two parabolas y² = 4x and x² = 4y

6

(a)

If A = \(\left[ \begin{matrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{matrix} \right] \) are square matrices, find A, B and hence solve the system of equation:
x - y = 3, 2x + 3y + 4z = 17 and y + 2z = 7

6

(a)

If a + b + c \(\neq \) 0 and \(\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|=0\) then prove that  a = b = c.

6

(a)

Using the properties of determinants, prove that 
\(\left|\begin{array}{ccc} (b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2} \end{array}\right|=2 a b c(a+b+c)^{3}\)

6

(a)

If \(A=\left[\begin{array}{rrr} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{array}\right]\) then findA^-1.using A^-1,equations x - 2y = 10,,2x - y- z = 8 and - 2y +z = 7.

6

(a)

Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, then find the rate at which they are being separated.

6

(a)

Evaluate the following integral
\(\int \frac{(\cos 5 x+\cos 4 x)}{1-2 \cos 3 x} d x\)

6

(a)

Evaluate the following integral
\(\int e^{-3 x} \cos ^{3} x d x\)

6

(a)

Evaluate the following integral.
\(\int_{0}^{\pi} x \log |\sin x| d x\)

6

(a)

Draw a rough sketch of the region \(\left\{(x, y): y^{2} \leq 6 \text { ax and } x^{2}+y^{2} \leq 16 a^{2}\right\}\). Also, find the area of the region sketched using method of integration.

6

(a)

Find the area bounded by the lines y = 4x + 5, y = 5 - x and 4y = x + 5.

6

(a)

Find the equation of the plane through the intersection of the planes \(\vec{r} \cdot(\hat{i}+3 \hat{j})-6=0\) and \(a\vec{r} \cdot(3 \hat{i}-\hat{j}-4 \hat{k})=0\) whose perpendicular distance from origin is unity.

6

(a)

A shopkeeper sells three types of flower seeds A₁, A₂ and A₃. They are sold as a mixture, where the proportions are 4 : 4 : 2, respectively. The germination rates of the three types of seedsare 45%, 60% and 35%. Calculate the probability.
(i) of a randomly chosen seed to germinate
(ii) that it willhot germinate given that the seed is of type A_3.
(iii) that it is of the type A₂ given that a randomly chosen seed does not germinate.

6

(a)