CBSE Class 12 Mathematics Set 1- (65/1/1) Question Paper 2026 with Solution PDF

Step 1: Understanding the Concept:

The principal value range of the inverse cosine function \( \cos^{-1}x \) is \( [0, \pi] \).

Hence, for every permissible value of \( x \), the output of \( \cos^{-1}x \) lies within \( 0 \) and \( \pi \), including both endpoints.


Step 2: Key Formula or Approach:

The range of the base inverse function is:
\[ 0 \leq \cos^{-1}x \leq \pi \]


Step 3: Detailed Explanation:

Given the function:
\[ y = 2 \cos^{-1}x \]

We begin with the known range of \( \cos^{-1}x \):
\[ 0 \leq \cos^{-1}x \leq \pi \]

To determine the range of \( y \), multiply the entire inequality by the positive constant \( 2 \):
\[ 2 \times 0 \leq 2 \times \cos^{-1}x \leq 2 \times \pi \]
\[ 0 \leq 2 \cos^{-1}x \leq 2\pi \]

Replacing \( 2 \cos^{-1}x \) with \( y \), we obtain:
\[ 0 \leq y \leq 2\pi \]

This corresponds to option (C).


Step 4: Final Answer:

The range of \( y \) is \( 0 \leq y \leq 2\pi \).
Quick Tip: First determine the principal range of the inverse trigonometric function.
If the function is multiplied by a constant \( k \), the endpoints of the range are multiplied by \( k \).
If a constant is added to the function, the entire range shifts accordingly.

Step 1: Understanding the Concept:

A row-matrix is a type of matrix that consists of only one horizontal row of elements.

The order of a matrix is written as \( (number of rows) \times (number of columns) \).


Step 2: Key Formula or Approach:

For a matrix to be classified as a row-matrix, its order must be of the form \( 1 \times n \), where \( n \geq 1 \).


Step 3: Detailed Explanation:

Let us examine each of the given options:

(A) \( 2 \times 1 \): This matrix contains 2 rows and 1 column. Since it has more than one row, it represents a column-matrix rather than a row-matrix.

(B) \( 1 \times 2 \): This matrix has 1 row and 2 columns, which satisfies the definition of a row-matrix.

(C) \( 1 \times 1 \): This matrix has a single row and a single column. Because it has exactly one row, it can also be considered a row-matrix (and simultaneously a column-matrix).

(D) \( 1 \times n \): This represents the general form of a row-matrix containing \( n \) columns.

Therefore, option (A) is the only order that cannot represent a row-matrix.


Step 4: Final Answer:

The order \( 2 \times 1 \) cannot represent a row-matrix.
Quick Tip: Remember:
Row matrix: \( 1 \times n \) (exactly one row)
Column matrix: \( m \times 1 \) (exactly one column)

Step 1: Understanding the Concept:

The transpose of a matrix, represented by \( A' \), is obtained by switching its rows with columns.

Transpose operations satisfy several standard algebraic properties.


Step 2: Key Formula or Approach:

Important transpose properties include:

1. \( (A + B)' = A' + B' \)

2. \( (A - B)' = A' - B' \)

3. \( (AB)' = B'A' \) (Reversal property)

4. \( (kA)' = kA' \)


Step 3: Detailed Explanation:

Now examine each statement:

(i) \( (A + B)' = A' + B' \): This follows the transpose rule for addition of matrices, so it is correct.

(ii) \( (A - B)' = B' - A' \): The correct identity is \( (A - B)' = A' - B' \). Since the order is reversed here, the statement is incorrect.

(iii) \( (AB)' = A'B' \): The correct rule states \( (AB)' = B'A' \). Because the order of matrices must reverse, this statement is incorrect.

(iv) \( (kAB)' = kB'A' \): Applying scalar and product transpose rules gives:
\[ (kAB)' = k(AB)' = k(B'A') \]

As \( k \) is a scalar, it remains unchanged. Hence, this statement is correct.

Therefore, statements (i) and (iv) are valid.


Step 4: Final Answer:

Statements (i) and (iv) are true.
Quick Tip: A key rule to remember is the \textbf{reversal property}:
For transpose, \( (AB)' = B'A' \), and for inverses, \( (AB)^{-1} = B^{-1}A^{-1} \).
In both cases, the order of matrices reverses when dealing with products.

Step 1: Understanding the Concept:

The determinant of a matrix can be evaluated either by expansion or by applying determinant properties.

For a diagonal matrix, the determinant equals the product of its diagonal entries.

Also, if two rows or columns of a determinant are interchanged, the value of the determinant changes sign.


Step 2: Key Formula or Approach:

For a \( 3 \times 3 \) diagonal matrix \( D = diag(a, b, c) \), the determinant is: \[ |D| = abc \]
If two rows are swapped, then: \[ R_i \leftrightarrow R_j \Rightarrow New determinant = -(Original determinant) \]


Step 3: Detailed Explanation:

First evaluate \( \Delta_1 \):

Since \( \Delta_1 \) is a diagonal matrix, its determinant is the product of the diagonal elements: \[ \Delta_1 = 1 \times 2 \times 3 = 6 \]

Next evaluate \( \Delta_2 \):

It is convenient to expand along the third row because it contains two zeros: \[ \Delta_2 = 6 \begin{vmatrix} 0 & 2
1 & 0 \end{vmatrix} \]

Now compute the \( 2 \times 2 \) determinant: \[ \begin{vmatrix} 0 & 2
1 & 0 \end{vmatrix} = (0 \times 0 - 2 \times 1) = -2 \]

Thus, \[ \Delta_2 = 6(-2) = -12 \]

Now compare the two determinants: \[ \frac{\Delta_2}{\Delta_1} = \frac{-12}{6} = -2 \]

Therefore, \[ \Delta_2 = -2\Delta_1 \]

This corresponds to option (B).


Step 4: Final Answer:
\[ \Delta_2 = -2\Delta_1 \] Quick Tip: Another approach is to transform \( \Delta_1 \) into \( \Delta_2 \).
First interchange \( R_1 \) and \( R_2 \), which changes the determinant sign to \( -6 \).
Then multiply \( R_3 \) by \( 2 \), doubling the determinant.
Thus the overall change is \( -1 \times 2 = -2 \), giving \( \Delta_2 = -2\Delta_1 \).

Step 1: Understanding the Concept:

The determinant of a \( 2 \times 2 \) matrix \( \begin{vmatrix} a & b
c & d \end{vmatrix} \)
is calculated as \( ad - bc \).

After expanding the determinant, the resulting expression forms a trigonometric equation.


Step 2: Key Formula or Approach:

1. Determinant expansion: \[ \begin{vmatrix} \cos x & \sin x
-\cos x & \sin x \end{vmatrix} = (\cos x)(\sin x) - (\sin x)(-\cos x) \]
2. Trigonometric identity: \[ 2\sin x \cos x = \sin(2x) \]


Step 3: Detailed Explanation:

Expand the determinant: \[ (\cos x)(\sin x) - (\sin x)(-\cos x) = 1 \]

Simplify the expression: \[ \sin x \cos x + \sin x \cos x = 1 \]
\[ 2\sin x \cos x = 1 \]

Using the double-angle identity for sine: \[ \sin(2x) = 1 \]

The sine function equals 1 for angles such as \( \frac{\pi}{2}, \frac{5\pi}{2}, \ldots \).

Taking the principal solution: \[ 2x = \frac{\pi}{2} \]

Dividing both sides by 2: \[ x = \frac{\pi}{4} \]

This corresponds to option (B).


Step 4: Final Answer:

A value of \( x \) that satisfies the equation is \[ x = \frac{\pi}{4} \] Quick Tip: When solving trigonometric equations, try to convert expressions into standard identities such as \( \sin 2x \) or \( \cos 2x \).
Also remember the general solution: \( \sin \theta = 1 \Rightarrow \theta = (4n+1)\frac{\pi}{2} \), where \( n \in \mathbb{Z} \).

Step 1: Understanding the Concept:

A matrix \( M \) is called skew-symmetric if its transpose satisfies the condition \( M' = -M \).

Since \( A \) and \( B \) are skew-symmetric matrices, we have: \[ A' = -A \quad and \quad B' = -B \]


Step 2: Key Formula or Approach:

Important transpose properties used here are: \[ (X \pm Y)' = X' \pm Y', \qquad (XY)' = Y'X' \]


Step 3: Detailed Explanation:

Now examine each option by taking its transpose.

(A) \( (AB)' = B'A' = (-B)(-A) = BA \).

This is generally not equal to \( -AB \). Hence, \( AB \) is not necessarily skew-symmetric.

(B) \( (AB + BA)' = (AB)' + (BA)' = B'A' + A'B' \).

Substituting \( A' = -A \) and \( B' = -B \): \[ (-B)(-A) + (-A)(-B) = BA + AB = AB + BA \]
Thus, the matrix equals its transpose, meaning it is symmetric, not skew-symmetric.

(C) \( ((A+B)^2)' = ((A+B)(A+B))' \).

Using the transpose rule for products: \[ (A+B)'(A+B)' \]
Since \( (A+B)' = A' + B' = -A - B = -(A+B) \), we get: \[ (-(A+B))(-(A+B)) = (A+B)^2 \]
Hence this matrix is symmetric.

(D) \( (A - B)' = A' - B' \).

Substituting \( A' = -A \) and \( B' = -B \): \[ (-A) - (-B) = -A + B = -(A - B) \]
Since the transpose equals the negative of the matrix, \( A - B \) is skew-symmetric.


Step 4: Final Answer:

The matrix \( A - B \) is skew-symmetric.
Quick Tip: Any linear combination of skew-symmetric matrices such as \( \alpha A + \beta B \) is also skew-symmetric.
Subtraction is a special case where \( \alpha = 1 \) and \( \beta = -1 \).

Step 1: Understanding the Concept:

To determine the absolute minimum (least value) of a continuous function over a closed interval \( [a,b] \), we evaluate the function at all critical points inside the interval and also at the endpoints \( a \) and \( b \).


Step 2: Key Formula or Approach:

1. Compute the derivative \( f'(x) \) and solve \( f'(x)=0 \) to obtain critical points.

2. Compare the values \( f(a), f(b) \), and \( f(c) \) for any critical point \( c \in (a,b) \).


Step 3: Detailed Explanation:

The function given is \( f(x) = x^3 - 12x \) for \( x \in [0,3] \).

First compute the derivative: \[ f'(x) = 3x^2 - 12 \]

Set the derivative equal to zero: \[ 3x^2 - 12 = 0 \] \[ 3x^2 = 12 \] \[ x^2 = 4 \] \[ x = \pm 2 \]

Among these, only \( x = 2 \) lies in the interval \( [0,3] \), so \( x=-2 \) is discarded.


Now evaluate the function at the relevant points:

At \( x=0 \): \[ f(0) = 0^3 - 12(0) = 0 \]

At \( x=2 \): \[ f(2) = 2^3 - 12(2) = 8 - 24 = -16 \]

At \( x=3 \): \[ f(3) = 3^3 - 12(3) = 27 - 36 = -9 \]

Comparing the values \( 0, -16, -9 \), the smallest value is \( -16 \).


Step 4: Final Answer:

The least value of the function on the interval \( [0,3] \) is \[ -16 \] Quick Tip: When finding the least value on a closed interval, check both critical points and the endpoints.
A local minimum inside the interval may not always be the absolute minimum.

Step 1: Understanding the Concept:

The given integral can be evaluated using the substitution method.

Observe that the derivative of the denominator is proportional to the numerator, which suggests using the logarithmic integration rule.


Step 2: Key Formula or Approach:

We use the standard result: \[ \int \frac{f'(x)}{f(x)}\,dx = \log|f(x)| + C \]


Step 3: Detailed Explanation:

Let \[ I = \int \frac{3ax}{b^2 + c^2x^2}\,dx \]

Set \[ u = b^2 + c^2x^2 \]

Differentiate with respect to \(x\): \[ du = 2c^2x\,dx \]

Thus, \[ x\,dx = \frac{du}{2c^2} \]

Substitute into the integral: \[ I = \int \frac{3a}{u} \cdot \frac{du}{2c^2} \]

Factor out the constants: \[ I = \frac{3a}{2c^2} \int \frac{1}{u}\,du \]

Integrate: \[ I = \frac{3a}{2c^2}\log|u| + K \]

Substitute \(u = b^2 + c^2x^2\): \[ I = \frac{3a}{2c^2}\log|b^2 + c^2x^2| + K \]

Comparing with the form \(A\log|b^2 + c^2x^2| + K\), we obtain: \[ A = \frac{3a}{2c^2} \]


Step 4: Final Answer:
\[ A = \frac{3a}{2c^2} \] Quick Tip: Look for integrals of the type \( \frac{f'(x)}{f(x)} \).
Here, the derivative of the denominator is \(2c^2x\) while the numerator is \(3ax\).
Their ratio \( \frac{3ax}{2c^2x} = \frac{3a}{2c^2} \) directly gives the coefficient of the logarithmic term.

Step 1: Understanding the Concept:

When the limits of a definite integral are symmetric about zero (i.e., from \( -a \) to \( a \)), it is useful to check whether the integrand is an even or an odd function.


Step 2: Key Formula or Approach:

Property: \[ \int_{-a}^{a} f(x)\,dx = 0 \quad if f(x) is an odd function (f(-x)=-f(x)). \]


Step 3: Detailed Explanation:

Let \[ f(x)=\frac{x^3}{x^2+2|x|+1}. \]

Now evaluate \( f(-x) \): \[ f(-x)=\frac{(-x)^3}{(-x)^2+2|-x|+1}. \]

Since \[ (-x)^3=-x^3,\quad (-x)^2=x^2,\quad |-x|=|x|, \]
we obtain \[ f(-x)=\frac{-x^3}{x^2+2|x|+1}. \]

Thus, \[ f(-x)=-\left(\frac{x^3}{x^2+2|x|+1}\right)=-f(x). \]

Hence, \( f(x) \) is an odd function.

Since the integral is taken over the symmetric interval \( [-1,1] \), the integral of an odd function over this interval is zero.


Step 4: Final Answer:
\[ \int_{-1}^{1}\frac{x^3}{x^2+2|x|+1}\,dx=0. \] Quick Tip: For symmetric limits \( [-a,a] \), always check parity.
Here \(x^3\) is an odd function and \(x^2+2|x|+1\) is even.
Odd \( \div \) Even \(=\) Odd, so the definite integral over \( [-1,1] \) is \(0\).

Step 1: Understanding the Concept:

The area bounded between a curve and the \(x\)-axis from \(x=a\) to \(x=b\) is given by \[ \int_a^b |y|\,dx. \]
Since area represents a physical quantity, it must always be non-negative.


Step 2: Key Formula or Approach:

Here the curve is \(y = x|x|\). Thus the required area is \[ Area=\int_{-1}^{1} |x|x||\,dx. \]

Using the definition of absolute value: \[ |x| = \begin{cases} -x, & x<0
x, & x\ge 0 \end{cases} \]

Hence, \[ y=x|x|= \begin{cases} x(-x)=-x^2, & x<0
x(x)=x^2, & x\ge 0 \end{cases} \]


Step 3: Detailed Explanation:

The area can therefore be written as \[ Area=\int_{-1}^{0}|-x^2|\,dx+\int_{0}^{1}|x^2|\,dx. \]

Since \(x^2\ge0\) for all \(x\), \[ Area=\int_{-1}^{0}x^2\,dx+\int_{0}^{1}x^2\,dx. \]

Evaluate each integral: \[ Area=\left[\frac{x^3}{3}\right]_{-1}^{0}+\left[\frac{x^3}{3}\right]_{0}^{1}. \]
\[ =\left(0-\left(-\frac{1}{3}\right)\right)+\left(\frac{1}{3}-0\right). \]
\[ =\frac{1}{3}+\frac{1}{3}=\frac{2}{3}. \]


Step 4: Final Answer:

The total area bounded by the curve and the \(x\)-axis is \[ \frac{2}{3} square units. \] Quick Tip: The function \(y=x|x|\) is symmetric about the origin.
Thus the area can also be computed as \[ Area=2\int_{0}^{1}x^2\,dx =2\left[\frac{x^3}{3}\right]_0^1 =\frac{2}{3}. \]

Step 1: Understanding the Concept:

A first-order linear differential equation in \(x\) (with \(x\) as the dependent variable) is written in the standard form \[ \frac{dx}{dy}+h(y)x=g(y). \]


Step 2: Key Formula or Approach:

For a differential equation in this form, the integrating factor (IF) is \[ IF=e^{\int h(y)\,dy}. \]


Step 3: Detailed Explanation:

The given differential equation is \[ R\frac{dx}{dy}+Px=Q. \]

To convert it into the standard form, divide the entire equation by \(R\): \[ \frac{dx}{dy}+\left(\frac{P}{R}\right)x=\frac{Q}{R}. \]

Comparing this with \[ \frac{dx}{dy}+h(y)x=g(y), \]
we identify \[ h(y)=\frac{P}{R}. \]

Hence the integrating factor becomes \[ IF=e^{\int h(y)\,dy} = e^{\int \frac{P}{R}\,dy}. \]

Thus the integrating factor is \(e^{\int \frac{P}{R}dy}\).


Step 4: Final Answer:
\[ Integrating Factor = e^{\int \frac{P}{R}\,dy}. \] Quick Tip: Before finding the integrating factor, always rewrite the equation so that the coefficient of the derivative term (such as \(dx/dy\) or \(dy/dx\)) is equal to 1.
The coefficient of the variable term then directly gives \(h(y)\) for the integrating factor formula.

Step 1: Understanding the Concept:

The order of a differential equation is the highest order derivative present in the equation.

The degree is the power of the highest order derivative after the equation is expressed as a polynomial in derivatives.


Step 2: Key Formula or Approach:

First expand the derivative \( \frac{d}{dx} \) using the chain rule to obtain the differential equation explicitly.


Step 3: Detailed Explanation:

Given: \[ \frac{d}{dx}(e^y)=0 \]

Differentiate using the chain rule: \[ \frac{d}{dx}(e^y)=e^y\frac{dy}{dx}. \]

Thus the equation becomes \[ e^y\frac{dy}{dx}=0. \]

Since \(e^y \neq 0\) for any real value of \(y\), divide both sides by \(e^y\): \[ \frac{dy}{dx}=0. \]

From this simplified equation:

The highest derivative present is \( \frac{dy}{dx} \), so the order is \(1\).
The power of this derivative is \(1\), so the degree is \(1\).


Hence the pair \((order,degree)=(1,1)\).


Step 4: Final Answer:

Order \(=1\), Degree \(=1\). Quick Tip: Always expand any indicated derivatives such as \( \frac{d}{dx}(\cdot) \) before determining order and degree.
Degree is defined only when the equation can be written as a polynomial in derivatives \(y',y'',\dots\).

Step 1: Understanding the Concept:

Two non-zero vectors are perpendicular (orthogonal) if and only if their scalar (dot) product is zero.


Step 2: Key Formula or Approach:

If \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \), then:
\( \vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = 0 \).


Step 3: Detailed Explanation:

Let \( \vec{a} = 1\hat{i} + 2\hat{j} + 3\hat{k} \) and \( \vec{b} = 2\hat{i} - p\hat{j} + 1\hat{k} \).

Set the dot product to zero:
\[ (1)(2) + (2)(-p) + (3)(1) = 0 \]
\[ 2 - 2p + 3 = 0 \]
\[ 5 - 2p = 0 \]
\[ 2p = 5 \]
\[ p = \frac{5}{2} \]

This matches option (C).


Step 4: Final Answer:

The value of \( p \) is \( \frac{5}{2} \).
Quick Tip: Dot product calculation: Multiply corresponding coefficients and add them up.
Condition for perpendicular vectors is always \( \vec{a} \cdot \vec{b} = 0 \).

Step 1: Understanding the Concept:

Three points \( A, B, C \) are collinear if the vectors formed by them, say \( \vec{AB} \) and \( \vec{BC} \), are parallel.

Parallel vectors have proportional direction ratios.


Step 2: Key Formula or Approach:

1. Find \( \vec{AB} = PV of B - PV of A \) and \( \vec{BC} = PV of C - PV of B \).

2. If \( \vec{AB} \parallel \vec{BC} \), then \( \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \).


Step 3: Detailed Explanation:

Let \( A(-1, -1, 2) \), \( B(2, m, 5) \), and \( C(3, 11, 6) \).

Vector \( \vec{AB} = (2 - (-1))\hat{i} + (m - (-1))\hat{j} + (5 - 2)\hat{k} = 3\hat{i} + (m+1)\hat{j} + 3\hat{k} \)

Vector \( \vec{BC} = (3 - 2)\hat{i} + (11 - m)\hat{j} + (6 - 5)\hat{k} = 1\hat{i} + (11-m)\hat{j} + 1\hat{k} \)

For collinearity, the ratio of components must be equal:
\[ \frac{3}{1} = \frac{m+1}{11-m} = \frac{3}{1} \]

Taking the equality:
\[ 3 = \frac{m+1}{11-m} \]
\[ 3(11 - m) = m + 1 \]
\[ 33 - 3m = m + 1 \]
\[ 32 = 4m \implies m = 8 \]

This corresponds to option (A).


Step 4: Final Answer:

The points are collinear for \( m = 8 \).
Quick Tip: Check the ratios of \( \hat{i} \) and \( \hat{k} \) components first.
Here, \( 3/1 = 3/1 \). This tells us the factor is 3.
Simply set: \( y-comp of \vec{AB} = 3 \times y-comp of \vec{BC} \).
\( (m+1) = 3(11-m) \implies m = 8 \).

Step 1: Understanding the Concept:

The magnitudes of the dot product and cross product are related through the angle between the vectors, or more directly by Lagrange's identity.


Step 2: Key Formula or Approach:

Lagrange's Identity: \( |\vec{a} \cdot \vec{b}|^2 + |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \).


Step 3: Detailed Explanation:

Let \( D = |\vec{a} \cdot \vec{b}| \) and \( C = |\vec{a} \times \vec{b}| \).

We are given: \( |\vec{a}| = 8, |\vec{b}| = 3 \), and \( C = 12 \).

Substitute these into the identity:
\[ D^2 + (12)^2 = (8)^2 \times (3)^2 \]
\[ D^2 + 144 = 64 \times 9 \]
\[ D^2 + 144 = 576 \]
\[ D^2 = 576 - 144 \]
\[ D^2 = 432 \]

Now, take the square root to find \( D \):
\[ D = \sqrt{432} = \sqrt{144 \times 3} = 12\sqrt{3} \]

This matches option (C).


Step 4: Final Answer:

The value of the dot product magnitude is \( 12\sqrt{3} \).
Quick Tip: Alternatively, find \( \sin \theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| |\vec{b}|} = \frac{12}{24} = \frac{1}{2} \).
Then \( \cos \theta = \frac{\sqrt{3}}{2} \).
Then \( |\vec{a} \cdot \vec{b}| = |\vec{a}| |\vec{b}| \cos \theta = 24 \times \frac{\sqrt{3}}{2} = 12\sqrt{3} \).

Step 1: Understanding the Concept:

The given line has direction ratios (1, 0, 0), which means it is parallel to the x-axis. Since it passes through (0,0,0), it is exactly the x-axis.


Step 2: Key Formula or Approach:

The perpendicular distance of a point \( (x_1, y_1, z_1) \) from the x-axis is given by \( \sqrt{y_1^2 + z_1^2} \).


Step 3: Detailed Explanation:

The point is \( P(2, 5, 7) \).

The line is the x-axis.

The foot of the perpendicular from \( P \) onto the x-axis is obtained by setting the \( y \) and \( z \) coordinates to 0. So, the foot is \( M(2, 0, 0) \).

The length of the perpendicular is the distance between \( P \) and \( M \):
\[ d = \sqrt{(2 - 2)^2 + (5 - 0)^2 + (7 - 0)^2} \]
\[ d = \sqrt{0^2 + 5^2 + 7^2} \]
\[ d = \sqrt{25 + 49} = \sqrt{74} \]

This matches option (C).


Step 4: Final Answer:

The length of the perpendicular is \( \sqrt{74} \).
Quick Tip: General shortcut: Distance of \( (x, y, z) \) from:
- x-axis: \( \sqrt{y^2 + z^2} \)
- y-axis: \( \sqrt{x^2 + z^2} \)
- z-axis: \( \sqrt{x^2 + y^2} \)

Step 1: Understanding the Concept:

According to the Corner Point Theorem, the optimal values of a linear objective function occur at the vertices of the bounded feasible region.


Step 2: Key Formula or Approach:

Calculate \( Z = 5x + 7y \) at each corner point.

Corners identified from graph: \( (0, 0), (7, 0), (3, 4), (0, 2) \).


Step 3: Detailed Explanation:

1. At \( (0, 0) \): \( Z = 5(0) + 7(0) = 0 \)

2. At \( (7, 0) \): \( Z = 5(7) + 7(0) = 35 \)

3. At \( (3, 4) \): \( Z = 5(3) + 7(4) = 15 + 28 = 43 \)

4. At \( (0, 2) \): \( Z = 5(0) + 7(2) = 14 \)

Maximum value \( = 43 \) (at \( (3, 4) \)).

Minimum value \( = 0 \) (at \( (0, 0) \)).

Difference \( = Max Z - Min Z = 43 - 0 = 43 \).

This matches option (D).


Step 4: Final Answer:

The required difference is \( 43 \).
Quick Tip: Always check the origin if it's part of the feasible region. For positive coefficients in \( Z \), the origin usually gives the minimum value.

Step 1: Understanding the Concept:

Linear programming problems (LPP) are named "linear" because all relationships—both constraints and the objective function—are linear equations or inequalities.


Step 2: Key Formula or Approach:

The objective function is typically written as \( Z = ax + by \).


Step 3: Detailed Explanation:

In the expression \( Z = ax + by \), the variables \( x \) and \( y \) are to the first power.

The highest power of variables in a linear expression is always 1.

Therefore, the degree of any linear objective function is always 1.


Step 4: Final Answer:

The degree is 1.
Quick Tip: The word "Linear" means degree 1. This applies to the objective function and all constraint inequalities in LPP.

Step 1: Understanding the Concept:

This problem deals with conditional probability, which measures the probability of an event given that another event has occurred.


Step 2: Key Formula or Approach:

Formula for conditional probability: \( P(A|B) = \frac{n(A \cap B)}{n(B)} \) for equally likely outcomes.


Step 3: Detailed Explanation:

Let \( S = \{1, 2, 3, 4, 5, 6\} \).

Event \( B \) (number is odd) \( = \{1, 3, 5\} \), so \( n(B) = 3 \).

Event \( A \) (number is prime) \( = \{2, 3, 5\} \).

The intersection \( A \cap B \) (odd prime numbers) \( = \{3, 5\} \), so \( n(A \cap B) = 2 \).

By the definition of conditional probability:
\[ P(A|B) = \frac{n(A \cap B)}{n(B)} = \frac{2}{3} \]

So, Assertion (A) is true.

Reason (R) states the standard conditional probability formula, which is the logic used to verify (A).

Thus, (R) is the correct explanation for (A).


Step 4: Final Answer:

The answer is (A).
Quick Tip: In conditional probability \( P(A|B) \), the set \( B \) becomes your "new sample space". Just count how many elements from \( A \) are inside this new set \( B \).

Step 1: Understanding the Concept:

Two lines are perpendicular if their direction vectors are perpendicular (dot product is zero).


Step 2: Key Formula or Approach:

Convert asymmetric line equations into symmetric form to find direction ratios.


Step 3: Detailed Explanation:

Line 1: \( x-q = py \) and \( z-s = ry \).

Rewriting: \( \frac{x-q}{p} = y = \frac{z-s}{r} \). Direction vector \( \vec{b_1} = (p, 1, r) \).

Line 2: \( \frac{x-q'}{p'} = y = \frac{z-s'}{r'} \). Direction vector \( \vec{b_2} = (p', 1, r') \).

For perpendicularity: \( \vec{b_1} \cdot \vec{b_2} = 0 \)
\[ p \cdot p' + 1 \cdot 1 + r \cdot r' = 0 \]
\[ pp' + rr' + 1 = 0 \implies pp' + rr' = -1 \]

Assertion (A) says the condition is \( pp' + rr' = 1 \). This is mathematically false.

Reason (R) correctly describes the condition for perpendicular lines in vector form.

Thus, (A) is false and (R) is true.


Step 4: Final Answer:

Option (D) is correct.
Quick Tip: Standard condition for these specific line forms is \( aa' + bb' + cc' = 0 \). Here the 'b' coefficients are both 1, so the condition must include a '+1' term.

Step 1: Understanding the Concept:

A function is continuous at \( x=a \) if \( \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) \).


Step 2: Key Formula or Approach:

For \( x < 3 \), \( |x-3| = -(x-3) \).


Step 3: Detailed Explanation:

1. Left Hand Limit (LHL):
\[ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} \frac{-(x-3)}{2(x-3)} = \lim_{x \to 3^-} \left(-\frac{1}{2}\right) = -0.5 \]

2. Right Hand Limit (RHL):
\[ \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} \frac{x-6}{6} = \frac{3-6}{6} = \frac{-3}{6} = -0.5 \]

3. Value of function \( f(3) \):

From the definition \( x \geq 3 \): \( f(3) = \frac{3-6}{6} = -0.5 \)

Since \( LHL = RHL = f(3) = -0.5 \), the function is continuous.


Step 4: Final Answer:

The function is continuous at \( x = 3 \).
Quick Tip: Always simplify modulus expressions based on the limit direction before evaluating the limit.

Step 1: Understanding the Concept:

Use implicit differentiation to find the derivative when \( y \) is not isolated.


Step 2: Key Formula or Approach:

Differentiate both sides with respect to \( x \). Use product rule for \( xy \).


Step 3: Detailed Explanation:
\[ \sqrt{3}(2x + 2y y') = 4(x y' + y) \]

Divide by 2: \( \sqrt{3}x + \sqrt{3}y y' = 2x y' + 2y \)

Substitute \( x = 1/2, y = \sqrt{3}/2 \):
\[ \sqrt{3}(1/2) + \sqrt{3}(\sqrt{3}/2) y' = 2(1/2) y' + 2(\sqrt{3}/2) \]
\[ \frac{\sqrt{3}}{2} + \frac{3}{2} y' = y' + \sqrt{3} \]
\[ \frac{3}{2} y' - y' = \sqrt{3} - \frac{\sqrt{3}}{2} \]
\[ \frac{1}{2} y' = \frac{\sqrt{3}}{2} \implies y' = \sqrt{3} \]


Step 4: Final Answer:
\( \frac{dy}{dx} = \sqrt{3} \).
Quick Tip: Plug in values immediately after differentiation to avoid complicated algebraic rearrangements.

Step 1: Understanding the Concept:

Related rates problem. Volume change is related to height change.


Step 2: Key Formula or Approach:
\( V = \frac{1}{3} \pi r^2 h \). Also \( r = h \tan \alpha \).


Step 3: Detailed Explanation:
\( \alpha = \pi/6 \implies r = h \tan(\pi/6) = h/\sqrt{3} \).
\( V = \frac{1}{3} \pi (\frac{h}{\sqrt{3}})^2 h = \frac{\pi h^3}{9} \).

Differentiate wrt \( t \): \( \frac{dV}{dt} = \frac{\pi h^2}{3} \frac{dh}{dt} \).

Given \( \frac{dV}{dt} = -1 \) and \( h = 10 \):
\[ -1 = \frac{\pi (100)}{3} \frac{dh}{dt} \implies \frac{dh}{dt} = -\frac{3}{100\pi} \]

Rate of drop is \( \frac{3}{100\pi} \).


Step 4: Final Answer:
\( \frac{3}{100\pi} \, mm/min \).
Quick Tip: In cone problems, always express \( r \) in terms of \( h \) first to simplify the volume formula.