Available Questions 832 found Page 26 of 42
Standalone Questions
#932
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2023
KNOWLEDGE
2 Marks
If \(f(x)=a(\tan x-\cot x)\), where \(a>0\), then find whether \(f(x)\) is increasing or decreasing function in its domain.
Key:
Sol:
Sol:
#931
Mathematics
Applications of Derivatives
VSA
UNDERSTAND
2023
KNOWLEDGE
2 Marks
Find the maximum and minimum values of the function given by \(f(x)=5+\sin 2x\).
Key:
Sol:
Sol:
#930
Mathematics
Applications of Integrals
LA
APPLY
2023
Competency
5 Marks
The area of the region bounded by the line \(y=mx (m>0)\), the curve \(x^{2}+y^{2}=4\) and the \(x\)-axis in the first quadrant is \(\frac{\pi}{2}\) units. Using integration, find the value of m.
Key:
Sol:
Sol:
#929
Mathematics
Applications of Integrals
LA
APPLY
2023
Competency
5 Marks
33. Using integration, find the area of the region bounded by the parabola $y^{2}=4ax$ and its latus rectum.
Key:
Sol:
Sol:
#928
Mathematics
Linear Programming
SA
APPLY
2023
Competency
3 Marks
Determine graphically the minimum value of the following objective function : $z=500x+400y$ subject to constraints $x+y\le200, x\ge20, y\ge4x, y\ge0$
Key:
Sol:
Sol:
#927
Mathematics
Linear Programming
SA
APPLY
2023
KNOWLEDGE
3 Marks
30. Solve the following linear programming problem graphically: Minimise: $z=-3x+4y$ subject to the constraints $x+2y\le8, 3x+2y\le12, x,y\ge0$
Key:
Sol:
Sol:
#926
Mathematics
Vector Algebra
VSA
APPLY
2023
KNOWLEDGE
2 Marks
If $\vec{r}=3\hat{i}-2\hat{j}+6\hat{k}$, find the value of $(\vec{r}\times\hat{j})\cdot(\vec{r}\times\hat{k})-12$
Key:
Sol:
Sol:
#925
Mathematics
Vector Algebra
VSA
APPLY
2023
KNOWLEDGE
2 Marks
24. If the projection of the vector $\hat{i}+\hat{j}+\hat{k}$ on the vector $p\hat{i}+\hat{j}-2\hat{k}$ is $\frac{1}{3}$, then find the value(s) of $p$.
Key:
Sol:
Sol:
#924
Mathematics
Matrices and Determinants
LA
APPLY
2023
KNOWLEDGE
5 Marks
If $A=\begin{bmatrix}1&2&-2\\ -1&3&0\\ 0&-2&1\end{bmatrix}$ and $B^{-1}=\begin{bmatrix}3&-1&1\\ -15&6&-5\\ 5&-2&2\end{bmatrix},$ find $(AB)^{-1}$.
OR Solve the following system of equations by matrix method :$ x+2y+3z=6, 2x-y+z=2, 3x+2y-2z=3.$
OR Solve the following system of equations by matrix method :$ x+2y+3z=6, 2x-y+z=2, 3x+2y-2z=3.$
Key:
Sol:
Sol:
#923
Mathematics
Relations and Functions
LA
APPLY
2023
Competency
5 Marks
Show that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined as $f(x)=\frac{5x-3}{4}$ is both one-one and onto.
Key:
Sol:
Sol:
#922
Mathematics
Relations and Functions
LA
APPLY
2023
Competency
5 Marks
Let $f : \mathbb{R} - \left\{ \frac{4}{3} \right\} \to \mathbb{R}$ be a function defined as:$$f(x) = \frac{4x}{3x+4}$$Show that $f$ is a one-one function. Also, check whether $f$ is an onto function or not.
Key:
Sol:
Sol:
#921
Mathematics
Relations and Functions
LA
APPLY
2023
Competency
5 Marks
34. (a) If N denotes the set of all natural numbers and R is the relation on $N \times N$ defined by $(a, b) R (c, d)$, if $ad(b+c)=bc(a+d)$. Show that R is an equivalence relation.
Key:
Sol:
Sol:
#920
Mathematics
Relations and Functions
LA
APPLY
2023
Competency
5 Marks
A relation $R$ is defined on a set of real numbers $\mathbb{R}$ as:$$R = \{(x, y) : x \cdot y \text{ is an irrational number}\}$$Check whether $R$ is reflexive, symmetric, and transitive or not.
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Sol:
Sol:
#919
Mathematics
Differential Equations
SA
APPLY
2023
KNOWLEDGE
3 Marks
Solve the following differential equation : $xe^{\frac{y}{x}}-y+x\frac{dy}{dx}=0$
Key:
Sol:
Sol:
#918
Mathematics
Differential Equations
SA
APPLY
2023
KNOWLEDGE
3 Marks
Find the general solution of the differential equation : $\frac{d}{dx}(xy^{2})=2y(1+x^{2})$
Key:
Sol:
Sol:
#917
Mathematics
Differential Equations
SA
APPLY
2023
KNOWLEDGE
3 Marks
Find the general solution of the differential equation \(e^{x}\tan y~dx+(1-e^{x})\sec^{2}y~dy=0\).
Key:
Sol:
Sol:
#916
Mathematics
Differential Equations
SA
APPLY
2023
KNOWLEDGE
3 Marks
29. (a) Find the particular solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}, y(1)=0$.
Key:
Sol:
Sol:
#915
Mathematics
Differential Equations
SA
APPLY
2023
KNOWLEDGE
3 Marks
Find the particular solution of the differential equation:$$\frac{dy}{dx} + \sec^{2}x \cdot y = \tan x \cdot \sec^{2}x$$given that $y(0) = 0$.
Key:
Sol:
Sol:
#914
Mathematics
Three Dimensional Geometry
VSA
APPLY
2023
KNOWLEDGE
2 Marks
Position vectors of the points A, B and C as shown in the figure below are a, $\vec{b}$ and $\vec{c}$ respectively. If $\vec{AC}=\frac{5}{4}\vec{AB}$ , express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$ .
OR Check whether the lines given by equations $x=2\lambda+2$, $y=7\lambda+1$, $z=-3\lambda-3$ and $x=-\mu-2,$ $y=2\mu+8,$ $z=4\mu+5$ are perpendicular to each other or not.
OR Check whether the lines given by equations $x=2\lambda+2$, $y=7\lambda+1$, $z=-3\lambda-3$ and $x=-\mu-2,$ $y=2\mu+8,$ $z=4\mu+5$ are perpendicular to each other or not.
Key:
Sol:
Sol:
#913
Mathematics
Relations and Functions
VSA
APPLY
2023
KNOWLEDGE
2 Marks
A function $f:A\rightarrow B$ defined as $f(x)=2x$ is both one-one and onto. If $A=\{1,2,3,4\}$, then find the set $B$.
OR
Evaluate : $\sin^{-1}(\sin\frac{3\pi}{4})+\cos^{-1}(\cos\frac{3\pi}{4})+\tan^{-1}(1)$
OR
Evaluate : $\sin^{-1}(\sin\frac{3\pi}{4})+\cos^{-1}(\cos\frac{3\pi}{4})+\tan^{-1}(1)$
Key:
Sol:
Sol:
To find the set $B$, we need to determine the range of the function $f$.Reasoning:Since the function $f: A \rightarrow B$ is given to be onto (surjective), every element in the codomain ($B$) must have a corresponding element in the domain ($A$). This implies that the set $B$ is equal to the range of $f$.
Calculation:
Given the domain $A = \{1, 2, 3, 4\}$ and the rule $f(x) = 2x$, we calculate the image for each element in $A$: For $x = 1$: $f(1) = 2(1) = 2$ For $x = 2$: $f(2) = 2(2) = 4$ For $x = 3$: $f(3) = 2(3) = 6$ For $x = 4$: $f(4) = 2(4) = 8$ Hence Collecting these values gives us the set $B$.$$B = \{2, 4, 6, 8\}$$