Available Questions 450 found Page 9 of 23
Standalone Questions
#1258
Mathematics
Linear Programming
SA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Solve the following linear programming problem graphically: Maximise $z=4x+3y.$ subject to the constraints $x+y\le800$, $2x+y\le1000$, $x\le400$, $x,y\ge0$.
Key:
Sol:
Sol:
#1257
Mathematics
Definite Integrals
SA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Evaluate: $\int_{0}^{\pi/4}\frac{1}{sin~x+cos~x}dx$
Key:
Sol:
Sol:
#1256
Mathematics
Integrals
SA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Find: $\int\frac{2+sin~2x}{1+cos~2x}e^{x}dx$
Key:
Sol:
Sol:
#1255
Mathematics
Integrals
SA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Find: $\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx$
Key:
Sol:
Sol:
#1254
Mathematics
Applications of Derivatives
SA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Find the absolute maximum and absolute minimum values of the function f given by $f(x)=\frac{x}{2}+\frac{2}{x}$ , on the interval [1, 2].
Key:
Sol:
Sol:
#1253
Mathematics
Applications of Derivatives
SA
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Find the intervals in which the function $f(x)=\frac{log~x}{x}$ is strictly increasing or strictly decreasing.
Key:
Sol:
Sol:
#1252
Mathematics
Continuity and Differentiability
SA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
Find the value of a and b so that function f defined as : $$ f(x) = \begin{cases} \frac{x-2}{|x-2|} + a, & \text{if } x < 2 \\ a+b, & \text{if } x = 2 \\ \frac{x-2}{|x-2|} + b, & \text{if } x > 2 \end{cases} $$ is a continuous function.
Key:
Sol:
Sol:
#1251
Mathematics
Derivatives
SA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
3 Marks
If $x~cos(p+y)+cos~p~sin(p+y)=0$ prove that $cos~p\frac{dy}{dx}=-cos^{2}(p+y),$ where p is a constant.
Key:
Sol:
Sol:
#1250
Mathematics
Vector Algebra
VSA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Prove that $|\vec{a}\times\vec{b}|\le|\vec{a}||\vec{b}|$ . State the condition under which equality holds, i.e., $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|$
Key:
Sol:
Sol:
#1249
Mathematics
Vector Algebra
VSA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Find the position vector of point C which divides the line segment joining points A and B having position vectors $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $4:1$ externally. Further, find $|\vec{AB}|:|\vec{BC}|$ .
Key:
Sol:
Sol:
#1248
Mathematics
Integrals
VSA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Given $\frac{d}{dx}F(x)=\frac{1}{\sqrt{2x-x^{2}}}$ and $F(1)=0$, find $F(x)$.
Key:
Sol:
Sol:
#1247
Mathematics
Definite Integrals
VSA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Evaluate: $\int_{0}^{\pi/2}sin~2x~cos~3x~dx$
Key:
Sol:
Sol:
#1246
Mathematics
Continuity and Differentiability
VSA
REMEMBER
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Check the differentiability of $f(x)=\begin{cases}x^{2}+1,&0\le x<1\\ 3-x,&1\le x\le2\end{cases}$ at $x=1.$
Key:
Sol:
Sol:
#1245
Mathematics
Derivatives
VSA
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
If $x=e^{x/y}$, prove that $\frac{dy}{dx}=\frac{log~x-1}{(log~x)^{2}}$
Key:
Sol:
Sol:
#1244
Mathematics
Inverse Trigonometric Functions
VSA
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
2 Marks
Evaluate : $sec^{2}(tan^{-1}\frac{1}{2})+cosec^{2}(cot^{-1}\frac{1}{3})$
Key:
Sol:
Sol:
#985
Physics
Nuclei
VSA
APPLY
2025
KNOWLEDGE
2 Marks
State two important properties of the nuclear force.
Key:
Sol:
Sol:
hkdcdj ds
#982
Physics
Alternating Current
graph plot and analyze
#974
Mathematics
Continuity and Differentiability
ASSERTION_REASON
REMEMBER
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Assertion (A): $f(x) = \begin{cases} x\sin\frac{1}{x}, & x\neq 0 \\ 0, & x=0 \end{cases}$ is continuous at $x=0$.
Reason (R): When $x \to 0$, $\sin\frac{1}{x}$ is a finite value between $-1$ and $1$.
Key:
Sol:
Sol:
#969
Mathematics
Inverse Trigonometric Functions
MCQ_SINGLE
REMEMBER
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The following graph is a combination of : (A) $y = \sin^{-1} x$ and $y = \cos^{-1} x$
(B) $y = \cos^{-1} x$ and $y = \cos x$
(C) $y = \sin^{-1} x$ and $y = \sin x$
(D) $y = \cos^{-1} x$ and $y = \sin x$
Key: C
Sol:
Sol:
#945
Mathematics
Inverse Trigonometric Functions
VSA
APPLY
2024
KNOWLEDGE
2 Marks
Express \(\tan^{-1}(\frac{\cos~x}{1-\sin~x})\) where \(\frac{-\pi}{2}\lt x\lt \frac{\pi}{2}\) in the simplest form.
Key:
Sol:
Sol:
Let $y = \tan^{-1}\left(\frac{\cos x}{1-\sin x}\right)$.First, use half-angle identities to rewrite the numerator and denominator:$$\cos x = \cos^2\frac{x}{2} - \sin^2\frac{x}{2} = \left(\cos\frac{x}{2} - \sin\frac{x}{2}\right)\left(\cos\frac{x}{2} + \sin\frac{x}{2}\right)$$$$1 - \sin x = \cos^2\frac{x}{2} + \sin^2\frac{x}{2} - 2\sin\frac{x}{2}\cos\frac{x}{2} = \left(\cos\frac{x}{2} - \sin\frac{x}{2}\right)^2$$Substitute these back into the expression:$$\frac{\cos x}{1-\sin x} = \frac{\left(\cos\frac{x}{2} - \sin\frac{x}{2}\right)\left(\cos\frac{x}{2} + \sin\frac{x}{2}\right)}{\left(\cos\frac{x}{2} - \sin\frac{x}{2}\right)^2}$$Cancel the common term $\left(\cos\frac{x}{2} - \sin\frac{x}{2}\right)$:$$= \frac{\cos\frac{x}{2} + \sin\frac{x}{2}}{\cos\frac{x}{2} - \sin\frac{x}{2}}$$Divide the numerator and denominator by $\cos\frac{x}{2}$:$$= \frac{1 + \tan\frac{x}{2}}{1 - \tan\frac{x}{2}}$$Since $\tan\frac{\pi}{4} = 1$, we can rewrite this as:$$= \frac{\tan\frac{\pi}{4} + \tan\frac{x}{2}}{1 - \tan\frac{\pi}{4}\tan\frac{x}{2}}$$Using the identity $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$:$$= \tan\left(\frac{\pi}{4} + \frac{x}{2}\right)$$Therefore:$$y = \tan^{-1}\left(\tan\left(\frac{\pi}{4} + \frac{x}{2}\right)\right)$$Since $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$, the angle lies within the principal range, so:$$y = \frac{\pi}{4} + \frac{x}{2}$$