Available Questions 450 found Page 11 of 23
Standalone Questions
#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:
#892
Mathematics
Integrals
SA
APPLY
2023
KNOWLEDGE
3 Marks
(a) Evaluate: $\int_0^{2\pi} \frac{1}{1 + e^{\sin x}} dx $
OR
(b) Find: $\int \frac{x⁴} { ((x-1)(x²+1))}dx.$
OR
(b) Find: $\int \frac{x⁴} { ((x-1)(x²+1))}dx.$
Key:
Sol:
Sol:
#890
Mathematics
Matrices and Determinants
LA
APPLY
2023
KNOWLEDGE
5 Marks
If $A=\begin{bmatrix}1 & 0 & 2\\ 0 & 2 & 1\\ 2 & 0 & 3\end{bmatrix}$, then show that $A^{3}-6A^{2}+7A+2I=O$
Key:
Sol:
Sol:
#888
Mathematics
Matrices and Determinants
SA
APPLY
2023
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
If $A=\begin{bmatrix}1 & 2 & 3\\ 3 & -2 & 1\\ 4 & 2 & 1\end{bmatrix}$, then show that A³ - 23A - 40I = O.
Key:
Sol:
Sol:
#885
Mathematics
Three Dimensional Geometry
VSA
APPLY
2023
KNOWLEDGE
2 Marks
If the angle between the lines $\frac{x-5}{\alpha}=\frac{y+2}{-5}=\frac{z+\frac{24}{5}}{\beta}$ and $\frac{x}{1}=\frac{y}{0}=\frac{z}{1}$ is $\frac{\pi}{4}$, find the relation between $\alpha$ and $\beta$.
Key:
Sol:
Sol: