Let A= 1&0 2&1 , and U1, U2 are e first and second columns respectively of a 2 × 2 matrix U. Also, let the column matrices U1 and U2 satisfying AU_1= 1 0 and AU_2= 2 3 . Based on the above information, answer the following questions. 1. The matrix U1 + U2 is equal to: 1. 1 -1 2. 2 -2 3. 3 -3 4. 4 -4 2. The value of |U| is: 1. 2 2. -2 3. 3 4. -3 3. If X= 3&2 U 3 2 , then the value of |X| = 1. 3 2. -3 3. -5 4. 5 4. The minor of element at the position a22 in U is: 1. 1 2. 2 3. -2 4. -1 5. If U=[a_ij]_ 2 2 , then the value of a11A11 + a12A12, where Aij denotes the cofactor of aij, is: 1. 1 2. 2 3. -3 4. 3

1. (c) 3 -3 Solution: We have, A= 1&0 2&1 Let U_1= a and AU_1= 1 0 1&0 2&1 a = 1 0 a 2a+b = 1 0 ⇒ a = 1 and 2a + b = 0 ⇒ a = 1 and b = -2. Let U_2= c then AU_2= 2 3 1&0 2&1 c = 2 3 c 2c+d = 2 3 ⇒ c = 2 and 2c + d = 3 ⇒ c = 2 and d = 3 - 4= -1 Thus, U_1+U_2= 1 -2 + 2 -1 = 3 -3 2. (c) 3 Solution: Clearly, U= 1&2 -2&-1 |U|= 1&2 -2&-1 =-1+4=3 3. (d) 5 Solution: We have, X= 3&2 1&2 -2&-1 3 2 = 3&2 7 -8 =[21-16]=[5] |X|=5 4. (a) 1 Solution: a22 in U is -1 and its minor is 1. 5. (d) 3 Solution: Since, the sum of products of elements of any row (or column) with their corresponding cofactors is equal to the value of determinant. a11A11 + a12A12 = |U| = 3

Let A= 1&0 2&1 , and U1, U2 are e first and second columns respec…

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Let $\text{A}=\begin{bmatrix}1&0\\2&1\end{bmatrix},$ and U₁, U₂ are e first and second columns respectively of a 2 × 2 matrix U. Also, let the column matrices U₁ and U₂ satisfying $\text{AU}_1=\begin{bmatrix}1\\0\end{bmatrix}$ and $\text{AU}_2=\begin{bmatrix}2\\3\end{bmatrix}.$
Based on the above information, answer the following questions.

The matrix U₁ + U₂ is equal to:

$\begin{bmatrix}1\\-1\end{bmatrix}$
$\begin{bmatrix}2\\-2\end{bmatrix}$
$\begin{bmatrix}3\\-3\end{bmatrix}$
$\begin{bmatrix}4\\-4\end{bmatrix}$

The value of |U| is:

2
-2
3
-3

If $\text{X}=\begin{bmatrix}3&2\end{bmatrix}\text{U}\begin{bmatrix}3\\2\end{bmatrix},$ then the value of |X| =

3
-3
-5
5

The minor of element at the position a₂₂ in U is:

1
2
-2
-1

If $\text{U}=[\text{a}_\text{ij}]_{2\times2},$ then the value of a₁₁A₁₁+ a₁₂A₁₂, where A_ij denotes the cofactor of a_ij, is:

1
2
-3
3

✓

Answer

Solution:

We have, $\text{A}=\begin{bmatrix}1&0\\2&1\end{bmatrix}$

Let $\text{U}_1=\begin{bmatrix}\text{a}\\\text{b}\end{bmatrix}$ and $\text{AU}_1=\begin{bmatrix}1\\0\end{bmatrix}$

$\Rightarrow\begin{bmatrix}1&0\\2&1\end{bmatrix}\begin{bmatrix}\text{a}\\\text{b}\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}\Rightarrow\begin{bmatrix}\text{a}\\2\text{a}+\text{b}\end{bmatrix}=\begin{bmatrix}1\\0\end{bmatrix}$

⇒ a = 1 and 2a + b = 0 ⇒ a = 1 and b = -2.

Let $\text{U}_2=\begin{bmatrix}\text{c}\\\text{d}\end{bmatrix}$ then $\text{AU}_2=\begin{bmatrix}2\\3\end{bmatrix}$

$\Rightarrow\begin{bmatrix}1&0\\2&1\end{bmatrix}\begin{bmatrix}\text{c}\\\text{d}\end{bmatrix}=\begin{bmatrix}2\\3\end{bmatrix}\Rightarrow\begin{bmatrix}\text{c}\\2\text{c}+\text{d}\end{bmatrix}=\begin{bmatrix}2\\3\end{bmatrix}$

⇒ c = 2 and 2c + d = 3

⇒ c = 2 and d = 3 - 4= -1

Thus, $\text{U}_1+\text{U}_2=\begin{bmatrix}1\\-2\end{bmatrix}+\begin{bmatrix}2\\-1\end{bmatrix}=\begin{bmatrix}3\\-3\end{bmatrix}$

Solution:

Clearly, $\text{U}=\begin{bmatrix}1&2\\-2&-1\end{bmatrix}$

$\therefore|\text{U}|=\begin{vmatrix}1&2\\-2&-1\end{vmatrix}=-1+4=3$

(d) 5

Solution:

We have, $\text{X}=\begin{bmatrix}3&2\end{bmatrix}\begin{bmatrix}1&2\\-2&-1\end{bmatrix}\begin{bmatrix}3\\2\end{bmatrix}$

$=\begin{bmatrix}3&2\end{bmatrix}\begin{bmatrix}7\\-8\end{bmatrix}=[21-16]=[5]$

$\therefore|\text{X}|=5$

(a) 1

Solution:

a₂₂ in U is -1 and its minor is 1.

(d) 3

Solution:

Since, the sum of products of elements of any row (or column) with their corresponding cofactors is equal to the value of determinant.

$\therefore$ a₁₁A₁₁ + a₁₂A₁₂ = |U| = 3

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A football match is organised between students of class XII of two schools, say school A and school B. For which a team from each school is chosen. Remaining students of class XII of school A and Bare respectively sitting
on the plane represented by the equation$ \vec{\text{r}}.(\hat{\text{i}}+\hat{\text{j}}+\hat{2\text{k}})=5$ and $ \vec{\text{r}}.(\hat{2\text{i}}-\hat{\text{j}}+\hat{\text{k}})=6$ to cheer up the team of their respective schools.

Based on the above information, answer the following questions.

The cartesian equation of the plane on which students of school A are seated is:

2x - y + z = 8
2x + y + z = 8
x + y + 2z = 5
x + y + z = 5

The magnitude of the normal to the plane on which students of school Bare seated, is:

$\sqrt{5}$
$\sqrt{6}$
$\sqrt{3}$
$\sqrt{2}$

The intercept form of the equation of the plane on which students of school Bare seated is:

$\frac{\text{x}}{6}+\frac{\text{y}}{6}+\frac{\text{z}}{6}=1$
$\frac{\text{x}}{3}+\frac{\text{y}}{(-6)}+\frac{\text{z}}{6}=1$
$\frac{\text{x}}{3}+\frac{\text{y}}{6}+\frac{\text{z}}{6}=1$
$\frac{\text{x}}{3}+\frac{\text{y}}{6}+\frac{\text{z}}{3}=1$

Which of the following is a student of school B?

Mohit sitting at (1, 2, 1)
Ravi sitting at (0, 1, 2)
Khushi sitting at (3, 1, 1)
Shewta sitting at (2, -1, 2)

The distance of the plane, on which students of school Bare seated, from the origin is:

6 units
$\frac{1}{\sqrt{6}}\text{ units}$
$\frac{5}{\sqrt{6}}\text{ units}$
$\sqrt{6}\text{ units}$

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If two vectors are represented by the two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in opposite order and this is known as triangle law of vector addition. Based on the above information, answer the following questions.

If $\vec{\text{p}},\vec{\text{q}},\vec{\text{r}}$ are the vectors represented by the sides of a triangle taken in order, then $\vec{\text{q}},+\vec{\text{r}}=$

$\vec{\text{p}}$
$2\vec{\text{p}}$
$-\vec{\text{p}}$
None of these

If ABCD is a parallelogram and AC and BD are its diagonals, then $\overline{\text{AC}}+\overline{\text{BD}}=$

$2\overline{\text{DA}}$
$2\overline{\text{AB}}$
$2\overline{\text{BC}}$
$2\overline{\text{BD}}$

If ABCD is a parallelogram, where $\overline{\text{AB}}=2\vec{\text{a}}$ and $\overline{\text{BC}}=2\vec{\text{b}},$ then $\overline{\text{AC}}-\overline{\text{BD}}=$

$3\vec{\text{a}}$
$4\vec{\text{a}}$
$2\vec{\text{b}}$
$4\vec{\text{b}}$

If ABCD is a quadrilateral whose diagonals are $\overline{\text{AC}}$ and $\overline{\text{BD}},$ then $\overline{\text{BA}}+\overline{\text{DC}}=$

$\overline{\text{AC}}+\overline{\text{DB}}$
$\overline{\text{AC}}+\overline{\text{BD}}$
$\overline{\text{BC}}+\overline{\text{AD}}$
$\overline{\text{BD}}+\overline{\text{CA}}$

If T is the mid point of side YZ of $\triangle\text{XYZ},$ then $\overline{\text{XY}}+\overline{\text{XZ}}=$

$2\overline{\text{YT}}$
$2\overline{\text{XT}}$
$2\overline{\text{TZ}}$
None of these

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Three slogans on chart papers are to be placed on a school bulletin board at the points A, Band C displaying A (Hub of Learning), B (Creating a better world for tomorrow) and C (Education comes first). The coordinates of these points are (1, 4, 2), (3, -3, -2) and (-2, 2, 6) respectively.

Based on the above information, answer the following questions.

Let $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ be the position vectors of points A, B and C respectively, then $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$ is equal to:

$2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}$
$2\hat{\text{i}}-3\hat{\text{j}}-6\hat{\text{k}}$
$2\hat{\text{i}}+8\hat{\text{j}}+3\hat{\text{k}}$
$2(7\hat{\text{i}}+8\hat{\text{j}}+3\hat{\text{k}})$

Which of the following is not true?

$\overline{\text{AB}}+\overline{\text{BC}}+\overline{\text{CA}}=\vec{0}$
$\overline{\text{AB}}+\overline{\text{BC}}-\overline{\text{AC}}=\vec{0}$
$\overline{\text{AB}}+\overline{\text{BC}}-\overline{\text{CA}}=\vec{0}$
$\overline{\text{AB}}-\overline{\text{CB}}+\overline{\text{CA}}=\vec{0}$

Area of $\triangle\text{ABC}$ is:

19 sq. units
$\sqrt{1937}\text{sq}.\text{units}$
$\frac{1}{2}\sqrt{1937}\text{sq}.\text{units}$
$\sqrt{1837}\text{sq}.\text{units}$

Suppose, if the given slogans are to be placed on a straight line, then the value of $|\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}|$ will be equal to:

-1
-2
2
0

If $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}},$ then unit vector in the direction of vector $\vec{\text{a}}$ is:

$\frac{2}{7}\hat{\text{i}}-\frac{3}{7}\hat{\text{j}}-\frac{6}{7}\hat{\text{k}}$
$\frac{2}{7}\hat{\text{i}}+\frac{3}{7}\hat{\text{j}}+\frac{6}{7}\hat{\text{k}}$
$\frac{3}{7}\hat{\text{i}}+\frac{2}{7}\hat{\text{j}}+\frac{6}{7}\hat{\text{k}}$
None of these

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Ginni purchased an air plant holder which is in the shape of a tetrahedron.
Let A, B, C, and Dare the coordinates of the air plant holder where $\text{A}\equiv(1,1,1),\text{B}\equiv(2,1,3),\text{C}\equiv(3,2,2)$ and $\text{D}\equiv(3,3,4).$

Based on the above information, answer the following questions.

Find the position vector of $\overline{\text{AB}}.$

$-\hat{\text{i}}-2\hat{\text{k}}$
$2\hat{\text{i}}+\hat{\text{k}}$
$\hat{\text{i}}+2\hat{\text{k}}$
$-2\hat{\text{i}}-\hat{\text{k}}$

Find the position vector of $\overline{\text{AC}}.$

$2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$
$2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$
$-2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$
$\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$

Find the position vector of $\overline{\text{AD}}.$

$2\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}$
$\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$
$3\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$
$2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$

Area of $\triangle\text{ABC}=$

$\frac{\sqrt{11}}{2}\text{sq}.\text{units}$
$\frac{\sqrt{14}}{2}\text{sq}.\text{units}$
$\frac{\sqrt{13}}{2}\text{sq}.\text{units}$
$\frac{\sqrt{17}}{2}\text{sq}.\text{units}$

Find the unit vector along $\overline{\text{AD}}.$

$\frac{1}{\sqrt{17}}(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
$\frac{1}{\sqrt{17}}(3\hat{\text{i}}+3\hat{\text{j}}+2\hat{\text{k}})$
$\frac{1}{\sqrt{11}}(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$
$(2\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})$

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If a real valued function f{x) is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
For example, every polynomial, constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
Based on the above information, answer the following questions.

If $\text{f}(\text{x})=\begin{cases}\text{x},\text{for x}\leq0\\0,\text{for x}>0\end{cases},$ then at x = 0

f(x) is differentiable and continuous.
f(x) is neither continuous nor differentiable.
f(x) is continuous but not differentiable.
None of these.

If $\text{f}(\text{x})=|\text{x}-1|,\text{x }\in\text{ R},$ then at x = 1

f(x) is not continuous.
f(x) is continuous but not differentiable.
f(x) is continuous and differentiable.
None of these.

f(x) = x³ is:

Continuous but not differentiable at x = 3
Continuous but not differentiable at x = 3
Neither continuous nor differentiable at x = 3
None of these.

If $\text{f}(\text{x})=[\sin\text{x}],$ then which of the following is true?

f(x) is continuous and differentiable at x = 0.
f(x) is discontinuous at x = 0.
f(x) is continuous at x = 0 but not differentiable.
f(x) is differentiable but not continuous at $\text{x}=\frac{\pi}{2}.$

If $\text{f}(\text{x})=\sin^{-1}\text{x},-1\leq\text{x}\leq1,$ then:

f(x) is both continuous and differentiable.
f(x) is neither continuous nor differentiable.
f(x) is continuous but not differentiable.
None of these.

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Consider the following equations of curves $\text{y}=\cos\text{x},\text{y}=\text{x}+1$ and y = 0.
On the basis of above information, answer the following questions.

The curves $\text{y}=\cos\text{x}$ and y = x + 1 meet at:

(1, 0)
(0, 1)
(1, 1)
(0, 0)

$\text{y}=\cos\text{x}$ meet the x-axis at:

$\Big(\frac{-\pi}{2},0\Big)$
$\Big(\frac{\pi}{2},0\Big)$
Both (a) and (b).
None of these.

Value of the integral $\int\limits_{-1}^{0}(\text{x}+1)\text{dx}$ is:

$\frac{1}{2}$
$\frac{2}{3}$
$\frac{3}{4}$
$\frac{1}{3}$

Value of the integral $\int\limits_{0}^{\frac{\pi}{2}}\cos\text{x dx}$ is:

0
-1
2
1

Area bounded by the given curves is:

$\frac{1}{2}\text{ sq}.\text{units}$
$\frac{3}{2}\text{ sq}.\text{units}$
$\frac{3}{4}\text{ sq}.\text{units}$
$\frac{1}{4}\text{ sq}.\text{units}$

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A building is to be constructed in the form of a triangular pyramid, ABCD as shown in the figure.

Let its angular points are A(0, 1, 2), B(3, 0, 1), C(4, 3, 6), and D(2, 3, 2), and G be the point of intersection of the medians of $\triangle\text{BCD}.$
Based on the above information, answer the following questions.

The coordinates of point Gare:

(2, 3, 3)
(3, 3, 2)
(3, 2, 3)
(0, 2, 3)

The length of vector $\overline{\text{AG}}$ is:

$\sqrt{17}\text{ units}$
$\sqrt{11}\text{ units}$
$\sqrt{13}\text{ units}$
$\sqrt{19}\text{ units}$

Area of $\triangle\text{ABC}$ (in sq. units) is:

$\sqrt{10}$
$2\sqrt{10}$
$3\sqrt{10}$
$5\sqrt{10}$

The sum of lengths of $\overline{\text{AB}}$ and $\overline{\text{AC}}$ is:

5 units
9.32 units
10 units
11 units

The length of the perpendicular from the vertex D on the opposite face is:

$\frac{6}{\sqrt{10}}\text{ units}$
$\frac{2}{\sqrt{10}}\text{ units}$
$\frac{3}{\sqrt{10}}\text{ units}$
$8\sqrt{10}\text{ units}$

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A thermometer reading 80°F is taken outside. Five minutes later the thermometer reads 60°F. After another 5 minutes the thermometer reads 50°F. At any time t the thermometer reading be T°F and the outside temperature be S°F.

Based on the above information, answer the following questions.

If $\lambda$ is posinve constant of propornonality, then $\frac{\text{dn}}{\text{dt}}$ is:

$\lambda(\text{T - S})$
$\lambda(\text{T + S})$
$\lambda\text{T S}$
$-\lambda(\text{T - S})$

The value of T(S) is:

30°F
40°F
5D°F
60°F

The value of T(10) is:

50°F
60°F
80°F
90°F

Find the general solution of differential equation fanned in given situation.

$\log\text{T}=\text{St + c}$
$\log(\text{T}-\text{S})=\lambda{\text{t + c}}$
$\log\text{T}=\text{tT + c}$
$\log(\text{T + S})=\lambda{\text{t + c}}$

Find the value of constant of integration c in the solution of differential equation formed in given situation.

$\log(60\ -\ \text{S})$
$\log(80\ +\ \text{S})$
$\log(80\ -\ \text{S})$
$\log(60\ +\ \text{S})$

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A company produces three products every day. Their production on certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product.

Using the concepts of matrices and determinants, answer the following questions.

If x, y and z respectively denotes the quantity (in tons) of first, second and third product produced, then which of the following is true?

x + y + z = 45
x + 8 = z
x - 2y + z = 0
All of these.

If $\begin{pmatrix}1&1&1\\1&0&-2\\1&-1&1\end{pmatrix}^{-1}=\frac{1}{6}\begin{pmatrix}2&2&2\\3&0&-3\\1&-2&1\end{pmatrix}$ then the inverse of $\begin{pmatrix}1&1&1\\1&0&-1\\1&-2&1\end{pmatrix}$ is:

$\begin{pmatrix}\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\\frac{1}{2}&0&\frac{-1}{2}\\\frac{1}{6}&\frac{-1}{3}&\frac{1}{6}\end{pmatrix}$
$\begin{pmatrix}\frac{1}{2}&0&-\frac{1}{2}\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\\frac{1}{6}&\frac{-1}{3}&\frac{1}{6}\end{pmatrix}$
$\begin{pmatrix}\frac{1}{3}&\frac{1}{2}&\frac{1}{6}\\\frac{1}{3}&0&\frac{-1}{3}\\\frac{1}{3}&\frac{-1}{2}&\frac{1}{6}\end{pmatrix}$
None of these.

x : y : z is equal to:

12 : 13 : 20
11 : 15 : 19
15 : 19 : 11
13 : 12 : 20

Which of the following is not true?

|A| = |A'|
(A')^-1 = (A^-1)'
A is skew synunetric matrix of odd order, then |A| = 0
|AB| = |A| + |B|

Which of the following is not true in the given determinant of A, where A $=[\text{a}_\text{ij}]_{3\times3}?$

Order of minor is less than order of the det (A).
Minor of an element can never be equal to cofactor of the same element.
Value of a determinant is obtained by multiplying elements of a row or column by corresponding cofactors.
Order of minors and cofactors of same elements of A is same.

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A trust fund has $₹ 35000$ that must be invested in two different types of bonds, say $\mathrm{X}$ and $\mathrm{Y}$. The first bond pays $10 \%$ interest p.a. which will be given to an old age home and second one pays $8 \%$ interest p.a. which will be given to WWA (Women Welfare Association). Let A be a $1 \times 2$ matrix and B be a $2 \times 1$ matrix, representing the investment and interest rate on each bond respectively.

(i) Represent the given information in matrix algebra.

(ii) If ₹ 15000 is invested in bond $\mathrm{X}$, then find total amount of interest received on both bonds?

(iii) If the trust fund obtains an annual total interest of ₹ 3200 , then find the investment in two bonds.

If the amount of interest given to old age home is ₹500, then find the amount of investment in bond Y.

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