Available Questions 559 found Page 24 of 28
Standalone Questions
#684
Mathematics
Linear Programming
MCQ_SINGLE
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If the feasible region of a linear programming problem with objective function \(Z = ax + by\), is bounded, then which of the following is correct?
(A) It will only have a maximum value.
(B) It will only have a minimum value.
(C) It will have both maximum and minimum values.
(D) It will have neither maximum nor minimum value.
Key: C
Sol:
Sol:
#683
Mathematics
Linear Programming
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function \(Z=5x+7y,\) where x and y are the number of units of X and Y respectively sold. Which of the following statement is correct?
(A) The objective function maximizes the difference of the profit earned from products X and Y.
(B) The objective function measures the total production of products X and Y.
(C) The objective function maximizes the combined profit earned from selling X and Y.
(D) The objective function ensures the company produces more of product X than product Y.
Key: C
Sol:
Sol:
The objective function maximizes the combined profit earned from selling products X and Y.
#678
Mathematics
Linear Programming
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
A linear programming problem deals with the optimization of a/an:
(A) logarithmic function
(B) linear function
(C) quadratic function
(D) exponential function
Key: B
Sol:
Sol:
#677
Mathematics
Linear Programming
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The number of corner points of the feasible region determined by constraints \(x\ge0, y\ge0, x+y\ge4\) is:
(A) 0
(B) 1
(C) 2
(D) 3
Key: C
Sol:
Sol:
#676
Mathematics
Linear Programming
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The common region determined by all the constraints of a linear programming problem is called :
(A) an unbounded region
(B) an optimal region
(C) a bounded region
(D) a feasible region
Key: D
Sol:
Sol:
#675
Mathematics
Linear Programming
MCQ_SINGLE
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called :
(A) feasible solutions
(B) constraints
(C) optimal solutions
(D) infeasible solutions
Key: B
Sol:
Sol:
#672
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The line \(x=1+5\mu\), \(y=-5+\mu\), \(z=-6-3\mu\) passes through which of the following point ?
(A) \((1, -5, 6)\)
(B) \((1, 5, 6)\)
(C) \((1, -5, -6)\)
(D) \((-1, -5, 6)\)
Key: C
Sol:
Sol:
\((1, -5, -6)\)
#671
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If P is a point on the line segment joining (3, 6, -1) and (6, 2, -2) and y-coordinate of P is 4, then its z-coordinate is:
(A) \(-\frac{3}{2}\)
(B) 0
(C) 1
(D) \(\frac{3}{2}\)
Key: A
Sol:
Sol:
We are given two points, \(A = (3, 6, -1)\) and \(B = (6, 2, -2)\). The point \(P\) lies on the line segment \(AB\), and its \(y\)-coordinate is \(4\). We need to find its \(z\)-coordinate.
Let \(P\) divide the line segment \(AB\) in the ratio \(\lambda:1\).
Step 1: Find the Ratio (\(\lambda\))
We use the section formula for the \(y\)-coordinate, where \(y=4\), \(y_1=6\), and \(y_2=2\):
\[y = \frac{\lambda y_2 + y_1}{\lambda + 1}\]
\[4 = \frac{\lambda(2) + 6}{\lambda + 1}\]
\[4(\lambda + 1) = 2\lambda + 6\]
\[4\lambda + 4 = 2\lambda + 6\]
\[2\lambda = 2\]
\[\mathbf{\lambda = 1}\]
The point \(P\) is the **midpoint** of the segment \(AB\) since \(\lambda = 1\).
Step 2: Find the \(z\)-coordinate (\(z\))
Now, we use the section formula for the \(z\)-coordinate with \(\lambda=1\), \(z_1=-1\), and \(z_2=-2\):
\[z = \frac{\lambda z_2 + z_1}{\lambda + 1}\]
\[z = \frac{(1)(-2) + (-1)}{1 + 1}\]
\[z = \frac{-2 - 1}{2}\]
\[\mathbf{z = -\frac{3}{2}}\]
#670
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If the direction cosines of a line are \(\sqrt{3}k, \sqrt{3}k\), \(\sqrt{3}k,\) then the value of k is:
(A) \(\pm1\)
(B) \(\pm\sqrt{3}\)
(C) \(\pm3\)
(D) \(\pm\frac{1}{3}\)
Key: D
Sol:
Sol:
#669
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The distance of point \(P(a,b,c)\) from y-axis is :
(A) b
(B) \(b^{2}\)
(C) \(\sqrt{a^{2}+c^{2}}\)
(D) \(a^{2}+c^{2}\)
Key: C
Sol:
Sol:
#668
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The coordinates of the foot of the perpendicular drawn from the point \((0, 1, 2)\) on the x-axis are given by:
(A) \((1,0,0)\)
(B) \((2,0,0)\)
(C) \((\sqrt{5},0,0)\)
(D) \((0,0,0)\)
Key: D
Sol:
Sol:
#667
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Direction ratios of a vector parallel to line \(\frac{x-1}{2}=-y=\frac{2z+1}{6}\) are:
(A) \(2,-1,6\)
(B) \(2, 1, 6\)
(C) \(2, 1, 3\)
(D) \(2,-1, 3\)
Key: D
Sol:
Sol:
#666
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If a line makes an angle of \(30^{\circ}\) with the positive direction of x-axis, \(120^{\circ}\) with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:
(A) \(90^{\circ}\)
(B) \(120^{\circ}\)
(C) \(60^{\circ}\)
(D) \(0^{\circ}\)
Key: A
Sol:
Sol:
#665
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If \(\alpha\), \(\beta\) and \(\gamma\) are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the following is not true?
(A) \(cos^{2}\alpha+cos^{2}\beta+cos^{2}\gamma=1\)
(B) \(sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma=2\)
(C) \(cos~2\alpha+cos~2\beta+cos~2\gamma=-1\)
(D) \(cos~\alpha+cos~\beta+cos~\gamma=1\)
Key: D
Sol:
Sol:
#664
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:
(A) 0
(B) \(\frac{\pi}{4}\)
(C) \(\frac{\pi}{2}\)
(D) \(\pi\)
Key: C
Sol:
Sol:
#663
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The vector equation of a line passing through the point (1, -1, 0) and parallel to Y-axis is :
(A) \(\vec{r}=\hat{i}-\hat{j}+\lambda(\hat{i}-\hat{j})\)
(B) \(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{j}\)
(C) \(\vec{r}=\hat{i}-\hat{j}+\lambda\hat{k}\)
(D) \(\vec{r}=\lambda\hat{j}\)
Key: B
Sol:
Sol:
#662
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The lines \(\frac{1-x}{2}=\frac{y-1}{3}=\frac{z}{1}\) and \(\frac{2x-3}{2p}=\frac{y}{-1}=\frac{z-4}{7}\) are perpendicular to each other for p equal to:
(A) \(-\frac{1}{2}\)
(B) \(\frac{1}{2}\)
(C) 2
(D) 3
Key: C
Sol:
Sol:
#661
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The angle which the line \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}\) makes with the positive direction of Y-axis is:
(A) \(\frac{5\pi}{6}\)
(B) \(\frac{3\pi}{4}\)
(C) \(\frac{5\pi}{4}\)
(D) \(\frac{7\pi}{4}\)
Key: B
Sol:
Sol:
#660
Mathematics
Three Dimensional Geometry
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is:
(A) \(\frac{x-1}{2}=\frac{y+3}{0}=\frac{z-2}{-1}\)
(B) \(\frac{x+1}{1}=\frac{y-3}{1}=\frac{z+2}{2}\)
(C) \(\frac{x+1}{2}=\frac{y-3}{0}=\frac{z+2}{-1}\)
(D) \(\frac{x-1}{1}=\frac{y+3}{1}=\frac{z-2}{2}\)
Key: D
Sol:
Sol:
#659
Mathematics
Differential Equations
MCQ_SINGLE
UNDERSTAND
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Which of the following is not a homogeneous function of \(x\) and \(y\) ?
(A) \(y^2 - xy\)
(B) \(x - 3y\)
(C) \(\sin^2 \frac{y}{x} + \frac{y}{x}\)
(D) \(\tan x - \sec y\)
Key: D
Sol:
Sol:
A function $f(x, y)$ is said to be homogeneous of degree $k$ if $f(tx, ty) = t^k f(x, y)$ for some real number $k$ and all $t > 0$.
To identify a function that is not homogeneous, we look for functions where this scaling property does not hold. Common characteristics of non-homogeneous functions include:
1. **Presence of a constant term:** A term that does not involve $x$ or $y$.
2. **Terms with different total degrees:** For polynomial functions, if the sum of the powers of $x$ and $y$ in each term is not the same.
3. **Transcendental functions (e.g., exponential, logarithmic) that do not scale appropriately:** For instance, $e^x$, $\log(x+y)$, or $\sin(x)$ are generally not homogeneous unless their arguments are homogeneous of degree 0.
Let's consider a common example of a function that is not homogeneous:
The function $f(x, y) = x^2 + y + 1$ is not a homogeneous function.
**Step-by-step verification:**
1. **Recall the definition of a homogeneous function:** $f(tx, ty) = t^k f(x, y)$.
2. **Substitute $tx$ for $x$ and $ty$ for $y$ into the function $f(x, y) = x^2 + y + 1$:**
$$f(tx, ty) = (tx)^2 + (ty) + 1$$
3. **Simplify the expression:**
$$f(tx, ty) = t^2x^2 + ty + 1$$
4. **Compare $f(tx, ty)$ with $t^k f(x, y)$:**
If $f(x, y)$ were homogeneous, we should be able to factor out $t^k$ such that the remaining expression is $x^2 + y + 1$.
However, we cannot write $t^2x^2 + ty + 1$ as $t^k(x^2 + y + 1)$ for any constant $k$. For instance, the term $1$ does not scale with $t$, and the term $ty$ scales differently from $t^2x^2$.
Since $f(tx, ty) \ne t^k f(x, y)$ for any real number $k$, the function $f(x, y) = x^2 + y + 1$ is not a homogeneous function.