Available Questions Page 2 of 14
Standalone Questions
#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 $B$.$$B = \{2, 4, 6, 8\}$$
#910
Mathematics
Probability
SA
APPLY
2023
KNOWLEDGE
3 Marks
31. From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.
Key:
Sol:
Sol:
#907
Mathematics
Applications of Derivatives
VSA
APPLY
2023
KNOWLEDGE
2 Marks
Show that the function $f(x)=\frac{16\sin x}{4+\cos x}-x$, is strictly decreasing in $(\frac{\pi}{2},\pi)$
Key:
Sol:
Sol:
#905
Mathematics
Integrals
#904
Mathematics
Integrals
#903
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
Evaluate $\int_{1}^{e}\frac{1}{\sqrt{4x^{2}-(x\log x)^{2}}}dx$
Key:
Sol:
Sol:
#902
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
Evaluate: $\int_{1}^{3}\frac{\sqrt{4-x}}{\sqrt{x}+\sqrt{4-x}}dx$
Key:
Sol:
Sol:
#901
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
28. (b) OR: Evaluate: $\int_{0}^{\pi/2}\sqrt{\sin x}\cos^{5}x~dx$
Key:
Sol:
Sol:
#900
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
28. (a) Find: $\int\frac{e^{x}}{\sqrt{5-4e^{x}-e^{2x}}}dx$
Key:
Sol:
Sol:
#899
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
27. (b) OR: Evaluate: $\int_{-2}^{2}\frac{x^{2}}{1+5^{x}}dx$
Key:
Sol:
Sol:
#898
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
27. (a) Evaluate: $\int_{\pi/4}^{\pi/2}e^{2x}(\frac{1-\sin 2x}{1-\cos 2x})dx$
Key:
Sol:
Sol:
#897
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
26. Find: $\int\frac{x^{2}+x+1}{(x+1)^{2}(x+2)}dx$
Key:
Sol:
Sol:
#896
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
Evaluate : $\int_{0}^{\frac{\pi}{2}}e^{x }\sin x~dx$.
OR
Find: $\int\frac{1}{\cos(x-a)\cos(x-b)}dx$
OR
Find: $\int\frac{1}{\cos(x-a)\cos(x-b)}dx$
Key:
Sol:
Sol:
#895
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
Find:$\int \frac{1}{\sqrt{x}(\sqrt{x}+1)(\sqrt{x}+2)} \, dx$
Key:
Sol:
Sol:
#894
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
Evaluate $\int_{0}^{\frac{\pi}{2}}[\log(\sin~x)-\log(2\cos~x)]dx.$
Key:
Sol:
Sol:
#893
Mathematics
Integrals
LA
APPLY
2023
KNOWLEDGE
5 Marks
Evaluate: $\int_ 0 ^{π/2} [\sin 2x \tan⁻¹(\sin x)] dx$
Key:
Sol:
Sol: