Available Questions Page 4 of 5
Standalone Questions
#610
Mathematics
Applications of Derivatives
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
If the sides of a square are decreasing at the rate of \(1.5~cm/s\) the rate of decrease of its perimeter is:
(A) \(1.5~cm/s\)
(B) \(6~cm/s\)
(C) \(3~cm/s\)
(D) \(2.25~cm/s\)
Key:
Sol:
Sol:
Let the side of the square be $s$.
The perimeter of the square is $P = 4s$.
We are given that the rate of decrease of the side is $\frac{ds}{dt} = -1.5~cm/s$.
We need to find the rate of decrease of its perimeter, which is $\frac{dP}{dt}$.
Differentiate the perimeter equation with respect to time $t$:
$\frac{dP}{dt} = \frac{d}{dt}(4s)$
$\frac{dP}{dt} = 4 \frac{ds}{dt}$
Substitute the given value of $\frac{ds}{dt}$:
$\frac{dP}{dt} = 4 (-1.5~cm/s)$
$\frac{dP}{dt} = -6~cm/s$
The rate of decrease of its perimeter is $6~cm/s$.
The final answer is $\boxed{6~cm/s}$.
#607
Mathematics
Applications of Derivatives
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
Given a curve \(y=7x-x^{3}\) and x increases at the rate of 2 units per second. The rate at which the slope of the curve is changing, when \(x=5\) is:
(A) \(-60~units/sec\)
(B) \(60~units/sec\)
(C) \(-70~units/sec\)
(D) \(-140~units/sec\)
Key:
Sol:
Sol:
#603
Mathematics
Derivatives
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
The derivative of \(2^{x}\) w.r.t. \(3^{x}\) is:
(A) \((\frac{3}{2})^{x}\frac{log~2}{log~3}\)
(B) \((\frac{2}{3})^{x}\frac{log~3}{log~2}\)
(C) \((\frac{2}{3})^{x}\frac{log~2}{log~3}\)
(D) \((\frac{3}{2})^{x}\frac{log~3}{log~2}\)
Key:
Sol:
Sol:
#602
Mathematics
Derivatives
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
Derivative of \(e^{2x}\) with respect to \(e^{x}\), is:
(A) \(e^{x}\)
(B) \(2e^{x}\)
(C) \(2e^{2x}\)
(D) \(2e^{3x}\)
Key:
Sol:
Sol:
#600
Mathematics
Derivatives
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
Derivative of \(e^{\sin^{2}x}\) with respect to cos x is:
(A) \(sin~x~e^{sin^{2}x}\)
(B) \(cos~x~e^{sin^{2}x}\)
(C) \(-2~cos~x~e^{sin^{2}x}\)
(D) \(-2~sin^{2}x~cos~x~e^{sin^{2}x}\)
Key: C
Sol:
Sol:
#599
Mathematics
Continuity and Differentiability
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If f(x)=∣x∣+∣x−1∣, then which of the following is correct?
(A) f(x) is both continuous and differentiable, at x=0 and x=1.
(B) f(x) is differentiable but not continuous, at x=0 and x=1.
(C) f(x) is continuous but not differentiable, at x=0 and x=1.
(D) f(x) is neither continuous nor differentiable, at x=0 and x=1.
Key: C
Sol:
Sol:
#593
Mathematics
Continuity and Differentiability
MCQ_SINGLE
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
1 Marks
If \(f(x)=\begin{cases}\frac{\log(1+ax)+\log(1-bx)}{x},&for~x\ne0\\ k&,for~x=0\end{cases}\) is continuous at \(x=0\), then the value of k is:
(A) a
(B) \(a+b\)
(C) \(a-b\)
(D) b
Key: C
Sol:
Sol:
#592
Mathematics
Continuity and Differentiability
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If \( f(x) = \begin{cases} 1, & \text{if } x \leq 3 \\ ax + b, & \text{if } 3 < x < 5 \\ 7, & \text{if } x \geq 5 \end{cases} \) is continuous for all real numbers, then find the values of \(a\) and \(b\):
(A) \(a=3\), \(b=-8\)
(B) \(a=3\), \(b=8\)
(C) \(a=-3\), \(b=-8\)
(D) \(a=-3\), \(b=8\)
Key:
Sol:
Sol:
#589
Mathematics
Inverse Trigonometric Functions
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
The Graph of a trigonomertic finction is as shown. Which of the following will represent graphs of its inverse?(A) A
(B) B
(C) C
(D) D
Key: C
Sol:
Sol:
#584
Mathematics
Inverse Trigonometric Functions
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If \(\tan^{-1}(x^{2}-y^{2})=a\), where 'a' is a constant, then \(\frac{dy}{dx}\) is:
(A) \(\frac{x}{y}\)
(B) \(-\frac{x}{y}\)
(C) \(\frac{a}{x}\)
(D) \(\frac{a}{y}\)
Key:
Sol:
Sol:
#569
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), \(|\vec{a}| = \sqrt{37}\), \(|\vec{b}| = 3\) and \(|\vec{c}| = 4\), then the angle between \(\vec{b}\) and \(\vec{c}\) is
(A) \(\dfrac{\pi}{6}\)
(B) \(\dfrac{\pi}{4}\)
(C) \(\dfrac{\pi}{3}\)
(D) \(\dfrac{\pi}{2}\)
Key: C
Sol:
Sol:
#566
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
If the sides AB and AC of \(\triangle ABC\) are represented by vectors \(\hat{j}+\hat{k}\) and \(3\hat{i}-\hat{j}+4\hat{k}\) respectively, then the length of the median through A on BC is:
(A) \(2\sqrt{2}\) units
(B) \(\sqrt{18}\) units
(C) \(\frac{\sqrt{34}}{2}\) units
(D) \(\frac{\sqrt{48}}{2}\) units
Key: C
Sol:
Sol:
The position vector of the midpoint \(D\) is the average of the position vectors of \(B\) and \(C\) relative to \(A\):\[\vec{AD} = \frac{\vec{AB} + \vec{AC}}{2}\]
\[\vec{AD} = \frac{3}{2}\hat{i} + \frac{5}{2}\hat{k}\]
The length of the median is the magnitude of the vector \(\vec{AD}\), denoted as \(|\vec{AD}|\):\[|\vec{AD}| = \sqrt{\left(\frac{3}{2}\right)^2 + \left(0\right)^2 + \left(\frac{5}{2}\right)^2}\]
\[|\vec{AD}| = \frac{\sqrt{34}}{2}\]
#564
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
The respective values of \(|\vec{a}|\) and \(|\vec{b}|\), if given \((\vec{a}-\vec{b})\cdot(\vec{a}+\vec{b})=512\) and \(|\vec{a}|=3|\vec{b}|\), are:
(A) 48 and 16
(B) 3 and 1
(C) 24 and 8
(D) 6 and 2
Key:
Sol:
Sol:
#562
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector \(3\hat{i}+15\hat{j}+6\hat{k}\) and the other is along the vector \(2\hat{i}+10\hat{j}+\lambda\hat{k}\), then the value of \(\lambda\) is :
(A) 6
(B) 1
(C) \(\frac{1}{4}\)
(D) 4
Key:
Sol:
Sol:
#553
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
The vectors \(\vec{a}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\) and \(\vec{c}=-3\hat{i}+4\hat{j}+4\hat{k}\) represents the sides of
(A) an equilateral triangle
(B) an obtuse-angled triangle
(C) an isosceles triangle
(D) a right-angled triangle
Key:
Sol:
Sol:
#552
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
Let \(\vec{a}\) be any vector such that \(|\vec{a}|=a\) The value of \(|\vec{a}\times\hat{i}|^{2}+|\vec{a}\times\hat{j}|^{2}+|\vec{a}\times\hat{k}|^{2}\) is:
(A) \(a^{2}\)
(B) \(2a^{2}\)
(C) \(3a^{2}\)
(D) 0
Key:
Sol:
Sol:
#551
Mathematics
Vector Algebra
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
Competency
1 Marks
The position vectors of points P and Q are \(\vec{p}\) and \(\vec{q}\) respectively. The point R divides line segment PQ in the ratio 3:1 and S is the mid-point of line segment PR. The position vector of S is:
(A) \(\frac{\vec{p}+3\vec{q}}{4}\)
(B) \(\frac{\vec{p}+3\vec{q}}{8}\)
(C) \(\frac{5\vec{p}+3\vec{q}}{4}\)
(D) \(\frac{5\vec{p}+3\vec{q}}{8}\)
Key:
Sol:
Sol: