2025

AISSCE(Board Exam)

Full Paper Analysis & Solutions

Mathematics

Case Study / Passage Based

Read the following passage:
A technical company is designing a rectangular solar panel installation on a roof using 300 metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections.

Let the length of the side perpendicular to the partition be $x$ metres and with parallel to the partition be $y$ metres.,
1 COMPETENCY APPLY SUBJECTIVE 1 Mark(s) 2025
#701
Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of
x and y..
MEDIUM
2 COMPETENCY APPLY SUBJECTIVE 1 Mark(s) 2025
#702
Write the area of the solar panel as a function of $x$
MEDIUM
3 COMPETENCY APPLY SUBJECTIVE 1 Mark(s) 2025
#703
Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.
HARD
4 COMPETENCY APPLY SUBJECTIVE 2 Mark(s) 2025
#704
Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.
HARD
Read the following passage:
Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out on straight paths and decide that A and B after reaching their destinations will meet up with C at his predecided destination, following straight paths from A to C and B to C in such a way that $\overrightarrow{OA} = \vec{a}$, $\overrightarrow{OB} = \vec{b}$ and $\overrightarrow{OC} = 5\vec{a}-2\vec{b}$ respectively.
1 SUBJECTIVE 4 Mark(s) 2025
#965
(i) Complete the given figure to explain their entire movement plan along the respective vectors.
(ii) Find vectors $\vec{AC}$ and $\vec{BC}$.
(iii) (a) If $\vec{a} \cdot \vec{b} = 1$, distance of O to A is 1 km and that from O to B is 2 km, then find the angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$. Also, find $|
\vec{a} \times \vec{b}|$.
OR
(iii) (b) If $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and $\vec{b} = \hat{j} - \hat{k}$, then find a unit vector perpendicular to $(\vec{a}+\vec{b})$ and $(\vec{a}-\vec{b})$.
Read the following passage:
Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature.

A cylindrical camphor tablet whose height is equal to its radius (r) evaporates when exposed to air such that that the rate of reduction of its volume is proportional to its total surface area. Thus, $\frac{dV}{dt} = kS$ is the differential equation, where V is the volume, S is the surface area and t is the time in hours.

1 COMPETENCY REMEMBER SUBJECTIVE 1 Mark(s) 2025
#966
(i) Write the order and degree of the given differential equation.
(ii) Substituting $V = \pi r^3$ and $S = 2\pi r^2$, we get the differential equation $\frac{dr}{dt} = \frac{2}{3}k$. Solve it, given that $r(0) = 5$ mm.
(iii) (a) If it is given that $r = 3$ mm when $t = 1$ hour, find the value of k. Hence, find t for $r = 0$ mm.
OR
(iii) (b) If it is given that $r = 1$ mm when $t = 1$ hour, find the value of k. Hence, find t for $r = 0$ mm.
MEDIUM
Read the following passage:
Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
Let $A_1$: People with good health,
$A_2$: People with average health,
and $A_3$: People with poor health.
During a pandemic, the data expressed that the chances of people contracting the disease from category $A_1$, $A_2$ and $A_3$ are 25%, 35% and 50%, respectively.
1 COMPETENCY APPLY SUBJECTIVE 4 Mark(s) 2025
#967
(i) A person was tested randomly. What is the probability that he/she has contracted the disease ?
(ii) Given that the person has not contracted the disease, what is the probability that the person is from category $A_2$ ?
MEDIUM
Read the following passage:
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers.
1 COMPETENCY APPLY SUBJECTIVE 4 Mark(s) 2025
#972
(i) Form the equations required to solve the problem of finding the price of each item, and express it in the matrix form $AX = B$.
(ii) Find $|A|$ and confirm if it is possible to find $A^{-1}$.
(iii) (a) Find $A^{-1}$, if possible, and write the formula to find $X$.
OR
(iii) (b) Find $A^2 - 8I$, where $I$ is an identity matrix.
EASY
Read the following passage:
A shop selling electronic items sells smartphones of only three reputed companies A, B and C because chances of their manufacturing a defective smartphone are only 5%, 4% and 2% respectively. In his inventory he has 25% smartphones from company A, 35% smartphones from company B and 40% smartphones from company C.
1 COMPETENCY APPLY SUBJECTIVE 4 Mark(s) 2025
#973
A person buys a smartphone from this shop.
(i) Find the probability that it was defective.
(ii) What is the probability that this defective smartphone was manufactured by company B ?
MEDIUM
Read the following passage:
A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively.
1 COMPETENCY APPLY VSA 2 Mark(s) 2025
#699
What is the probability that a customer after availing the loan will default on the loan repayment?
MEDIUM
2 COMPETENCY APPLY VSA 2 Mark(s) 2025
#700
A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?
HARD

Multiple Choice Questions

1 APPLY MCQ_SINGLE 1 Mark(s) 2025
#561
If \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\) for any two vectors, then vectors \(\vec{a}\) and \(\vec{b}\) are:
(A)
orthogonal vectors
(B)
parallel to each other
(C)
unit vectors
(D)
collinear vectors
MEDIUM
2 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#562
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
MEDIUM
3 APPLY MCQ_SINGLE 1 Mark(s) 2025
#563
Let \(|\vec{a}|=5\) and \(-2\le\lambda\le1\). Then, the range of \(|\lambda\vec{a}|\) is:
(A)
[5, 10]
(B)
[-2, 5]
(C)
[2, 1]
(D)
[-10, 5]
MEDIUM
4 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#564
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
MEDIUM
5 APPLY MCQ_SINGLE 1 Mark(s) 2025
#565
Let \(\vec{a}\) be a position vector whose tip is the point \((2,-3)\). If \(\vec{AB}=\vec{a}\), where coordinates of A are \((-4, 5)\), then the coordinates of B are:
(A)
\((-2,-2)\)
(B)
\((2,-2)\)
(C)
\((-2,2)\)
(D)
\((2, 2)\)
MEDIUM
6 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#566
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
MEDIUM
7 APPLY MCQ_SINGLE 1 Mark(s) 2025
#567
Let \(\vec{p}\) and \(\vec{q}\) be two unit vectors and \(\alpha\) be the angle between them. Then \((\vec{p}+\vec{q})\) will be a unit vector for what value of \(\alpha\)?
(A)
\(\frac{\pi}{4}\)
(B)
\(\frac{\pi}{3}\)
(C)
\(\frac{\pi}{2}\)
(D)
\(\frac{2\pi}{3}\)
MEDIUM
8 APPLY MCQ_SINGLE 1 Mark(s) 2025
#568
The projection vector of vector \(\vec{a}\) on vector \(\vec{b}\) is
(A)
\((\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|^{2}})\vec{b}\)
(B)
\(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\)
(C)
\(\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}\)
(D)
\((\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|^{2}})\vec{b}\)
MEDIUM
9 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#569
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)
\(\dfrac{\pi}{2}\)
MEDIUM
10 APPLY MCQ_SINGLE 1 Mark(s) 2025
#570
If vector \(\vec{a} = 3\hat{i} + 2\hat{j} - \hat{k}\) and vector \(\vec{b} = \hat{i} - \hat{j} + \hat{k}\), then which of the following is correct ?
(A)
\(\vec{a} \parallel \vec{b}\)
(B)
\(\vec{a} \perp \vec{b}\)
(C)
\(|\vec{b}| > |\vec{a}|\)
(D)
\(|\vec{a}| = |\vec{b}|\)
MEDIUM
11 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#575
If \(f:N\rightarrow W\) is defined as \(f(n)=\begin{cases}\frac{n}{2},&if~n~is~even\\ 0,&if~n~is~odd\end{cases}\), then f is:
(A)
injective only
(B)
surjective only
(C)
a bijection
(D)
neither surjective nor injective
MEDIUM
12 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#576
For real x, let \(f(x)=x^{3}+5x+1\). Then:
(A)
f is one-one but not onto on R
(B)
f is onto on R but not one-one
(C)
f is one-one and onto on R
(D)
f is neither one-one nor onto on R
MEDIUM
13 APPLY MCQ_SINGLE 1 Mark(s) 2025
#583
Domain of \(f(x)=\cos^{-1}x+\sin x\) is :
(A)
R
(B)
\((-1, 1)\)
(C)
\([-1, 1]\)
(D)
\([-\pi/2, \pi/2]\)
EASY
14 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#584
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)
\(\frac{a}{y}\)
MEDIUM
15 APPLY MCQ_SINGLE 1 Mark(s) 2025
#585
\([\sec^{-1}(-\sqrt{2})-\tan^{-1}(\frac{1}{\sqrt{3}})]\) is equal to:
(A)
\(\frac{11\pi}{12}\)
(B)
\(\frac{5\pi}{12}\)
(C)
\(-\frac{5\pi}{12}\)
(D)
\(\frac{7\pi}{12}\)
MEDIUM
16 REMEMBER MCQ_SINGLE 1 Mark(s) 2025
#586
The principal value of \(\cot^{-1}(-\frac{1}{\sqrt{3}})\) is:
(A)
\(-\frac{\pi}{3}\)
(B)
\(-\frac{2\pi}{3}\)
(C)
\(\frac{\pi}{3}\)
(D)
\(\frac{2\pi}{3}\)
EASY
17 REMEMBER MCQ_SINGLE 1 Mark(s) 2025
#587
The principal value of \(\sin^{-1}(\sin(-\frac{10\pi}{3}))\) is:
(A)
\(-\frac{2\pi}{3}\)
(B)
\(-\frac{\pi}{3}\)
(C)
\(\frac{\pi}{3}\)
(D)
\(\frac{2\pi}{3}\)
MEDIUM
18 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#588
If \(y=\sin^{-1}x\), \(-1 \le x \le 0\), then the range of y is
(A)
\((\frac{-\pi}{2}, 0)\)
(B)
\([\frac{-\pi}{2}, 0]\)
(C)
\([\frac{-\pi}{2}, 0)\)
(D)
\((\frac{-\pi}{2}, 0]\)
EASY
19 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#589
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
HARD
20 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#592
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)
\(a=-3\), \(b=8\)
MEDIUM
21 COMPETENCY UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#593
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
MEDIUM
22 APPLY MCQ_SINGLE 1 Mark(s) 2025
#594
If \(f(x)=\{[x],x\in R\}\) is the greatest integer function, then the correct statement is:
(A)
f is continuous but not differentiable at \(x=2\).
(B)
f is neither continuous nor differentiable at \(x=2\).
(C)
f is continuous as well as differentiable at \(x=2\).
(D)
f is not continuous but differentiable at \(x=2\).
MEDIUM
23 APPLY MCQ_SINGLE 1 Mark(s) 2025
#595
If \(f(x)=\begin{cases}\frac{\sin^{2}ax}{x^{2}},&x\ne0\\ 1,&x=0\end{cases}\) is continuous at \(x=0\), then the value of a is:
(A)
1
(B)
-1
(C)
\(\pm1\)
(D)
0
MEDIUM
24 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#596
The function f defined by \(f(x)=\begin{cases}x,&if~x\le1\\ 5,&if~x>1\end{cases}\) is not continuous at:
(A)
\(x=0\)
(B)
\(x=1\)
(C)
\(x=2\)
(D)
\(x=5\)
MEDIUM
25 APPLY MCQ_SINGLE 1 Mark(s) 2025
#597
If \(f(x)=\begin{cases}3x-2,&0 \lt x\le 1\\ 2x^{2}+ax,&1\lt x\lt 2\end{cases}\) is continuous for \(x\in(0,2)\), then a is equal to:
(A)
-4
(B)
\(-\frac{7}{2}\)
(C)
-2
(D)
-1
MEDIUM
26 APPLY MCQ_SINGLE 1 Mark(s) 2025
#598
If A denotes the set of continuous functions and B denotes set of differentiable functions, then which of the following depicts the correct relation between set A and B ?

(A)
A
(B)
B
(C)
C
(D)
D
MEDIUM
27 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#599
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.
MEDIUM
28 APPLY MCQ_SINGLE 1 Mark(s) 2025
#606
If \(f(x)=-2x^{8}\) then the correct statement is :
(A)
\(f^{\prime}(\frac{1}{2})=f^{\prime}(-\frac{1}{2})\)
(B)
\(f^{\prime}(\frac{1}{2})=-f^{\prime}(-\frac{1}{2})\)
(C)
\(-f^{\prime}(\frac{1}{2})=f(-\frac{1}{2})\)
(D)
\(f(\frac{1}{2})=-f(-\frac{1}{2})\)
MEDIUM
29 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#613
A spherical ball has a variable diameter \(\frac{5}{2}(3x+1).\) The rate of change of its volume w.r.t. x, when \(x=1\), is :
(A)
\(225\pi\)
(B)
\(300\pi\)
(C)
\(375\pi\)
(D)
\(125\pi\)
MEDIUM
30 APPLY MCQ_SINGLE 1 Mark(s) 2025
#614
Let \(f(x)=|x|\), \(x\in R\). Then, which of the following statements is **incorrect**?
(A)
f has a minimum value at \(x=0\).
(B)
f has no maximum value in R.
(C)
f is continuous at \(x=0\).
(D)
f is differentiable at \(x=0\).
MEDIUM
31 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#615
The slope of the curve \(y=-x^{3}+3x^{2}+8x-20\) is maximum at:
(A)
(1,-10)
(B)
(1,10)
(C)
(10, 1)
(D)
(-10, 1)
MEDIUM
32 APPLY MCQ_SINGLE 1 Mark(s) 2025
#616
If \(f(x)=2x+\cos x\), then f(x):
(A)
has a maxima at \(x=\pi\)
(B)
has a minima at \(x=\pi\)
(C)
is an increasing function
(D)
is a decreasing function
MEDIUM
33 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#617
The values of \(\lambda\) so that \(f(x)=\sin x-\cos x-\lambda x+C\) decreases for all real values of x are:
(A)
\(1\lt\lambda\lt\sqrt{2}\)
(B)
\(\lambda\ge1\)
(C)
\(\lambda\ge\sqrt{2}\)
(D)
\(\lambda\lt1\)
HARD
34 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#618
A cylindrical tank of radius \(10\) cm is being filled with sugar at the rate of \(100~\pi~cm^{3}/s\). The rate, at which the height of the sugar inside the tank is increasing, is:
(A)
\(0.1~cm/s\)
(B)
\(0.5~cm/s\)
(C)
\(1~cm/s\)
(D)
\(1.1~cm/s\)
MEDIUM
35 APPLY MCQ_SINGLE 1 Mark(s) 2025
#619
The function \(f(x)=x^{2}-4x+6\) is increasing in the interval
(A)
\((0, 2)\)
(B)
\((-\infty, 2]\)
(C)
\([1, 2]\)
(D)
\([2, \infty)\)
MEDIUM
36 APPLY MCQ_SINGLE 1 Mark(s) 2025
#620
The absolute maximum value of function \( f(x) = x^3 - 3x + 2 \) in [0, 2] is:
(A)
0
(B)
2
(C)
4
(D)
5
MEDIUM
37 APPLY MCQ_SINGLE 1 Mark(s) 2025
#622
\(\int\frac{e^{x}}{\sqrt{4-e^{2x}}}dx\) is equal to:
(A)
\(\frac{1}{2}\cos^{-1}(e^{x})+C\)
(B)
\(\frac{1}{2}\sin^{-1}(e^{x})+C\)
(C)
\(\frac{e^{x}}{2}+C\)
(D)
\(\sin^{-1}(\frac{e^{x}}{2})+C\)
MEDIUM
38 APPLY MCQ_SINGLE 1 Mark(s) 2025
#623
For a function \(f(x)\) which of the following holds true?
(A)
\(\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\)
(B)
\(\int_{-a}^{a}f(x)dx=0,\) if f is an even function
(C)
\(\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx,\) if f is an odd function
(D)
\(\int_{0}^{2a}f(x)dx=\int_{0}^{a}f(x)dx-\int_{0}^{a}f(2a+x)dx\)
MEDIUM
39 APPLY MCQ_SINGLE 1 Mark(s) 2025
#624
\(\int\frac{x+5}{(x+6)^{2}}e^{x}dx\) is equal to:
(A)
\(\log(x+6)+C\)
(B)
\(e^{x}+C\)
(C)
\(\frac{e^{x}}{x+6}+C\)
(D)
\(\frac{-1}{(x+6)^{2}}+C\)
MEDIUM
40 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#625
\(\int\sqrt{1+\sin x}dx\) is equal to :
(A)
\(2(-\sin\frac{x}{2}+\cos\frac{x}{2})+C\)
(B)
\(2(\sin\frac{x}{2}-\cos\frac{x}{2})+C\)
(C)
\(-2(\sin\frac{x}{2}+\cos\frac{x}{2})+C\)
(D)
\(2(\sin\frac{x}{2}+\cos\frac{x}{2})+C\)
MEDIUM
41 APPLY MCQ_SINGLE 1 Mark(s) 2025
#626
\(\int\frac{\cos 2x-\cos 2\alpha}{\cos x-\cos \alpha}dx\) is equal to:
(A)
\(2(\sin x+x\cos\alpha)+C\)
(B)
\(2(\sin x-x\cos\alpha)+C\)
(C)
\(2(\sin x+2x\cos\alpha)+C\)
(D)
\(2(\sin x+\sin\alpha)+C\)
MEDIUM
42 APPLY MCQ_SINGLE 1 Mark(s) 2025
#627
If
\(
\int \frac{2^\frac{1}{x}}{x^2} dx = k \cdot 2^{\frac{1}{x}} + C,
\)
then \(k\) is equal to
(A)
\(\dfrac{-1}{\log 2}\)
(B)
\(-\log 2\)
(C)
-1
(D)
\(\dfrac{1}{2}\)
MEDIUM
43 APPLY MCQ_SINGLE 1 Mark(s) 2025
#628
\(\int_{-1}^{1} \frac{|x|}{x} \, dx, x \ne 0 \text{ is equal to}\)
(A)
\(-1\)
(B)
0
(C)
1
(D)
2
MEDIUM
44 APPLY MCQ_SINGLE 1 Mark(s) 2025
#634
\(\int_{0}^{\pi/2}\cos x\cdot e^{\sin x}dx\) is equal to:
(A)
0
(B)
\(1-e\)
(C)
\(e-1\)
(D)
e
MEDIUM
45 APPLY MCQ_SINGLE 1 Mark(s) 2025
#635
The value of \(\int_{0}^{1}\frac{dx}{e^{x}+e^{-x}}\) is:
(A)
\(-\frac{\pi}{4}\)
(B)
\(\frac{\pi}{4}\)
(C)
\(\tan^{-1}e-\frac{\pi}{4}\)
(D)
\(\tan^{-1}e\)
MEDIUM
46 APPLY MCQ_SINGLE 1 Mark(s) 2025
#636
If \(f(2a-x)=f(x)\), then \(\int_{0}^{2a}f(x)dx\) is
(A)
\(\int_{0}^{2a}f(\frac{x}{2})dx\)
(B)
\(\int_{0}^{a}f(x)dx\)
(C)
\(2\int_{a}^{0}f(x)dx\)
(D)
\(2\int_{0}^{a}f(x)dx\)
MEDIUM
47 APPLY MCQ_SINGLE 1 Mark(s) 2025
#637
The area of the region bounded by the curve \(y^{2}=x\) between \(x=0\) and \(x=1\) is:
(A)
\(\frac{3}{2}\) sq units
(B)
\(\frac{2}{3}\) sq units
(C)
3 sq units
(D)
\(\frac{4}{3}\) sq units
MEDIUM
48 APPLY MCQ_SINGLE 1 Mark(s) 2025
#638
The area of the shaded region bounded by the curve \(y^2=x,x=4\) and the \(x-\)axis is given by

(A)
\( \int_{0}^{4} x \, dx\)
(B)
\( \int_{0}^{2} y^2 \, dy\)
(C)
\( 2 \int_{0}^{4} \sqrt{x} \, dx\)
(D)
\( \int_{0}^{4} \sqrt{x} \, dx\)
MEDIUM
49 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#640
The area of the region enclosed between the curve \(y=x|x|\), x-axis, \(x=-2\) and \(x=2\) is:
(A)
\(\frac{8}{3}\)
(B)
\(\frac{16}{3}\)
(C)
0
(D)
8
MEDIUM
50 APPLY MCQ_SINGLE 1 Mark(s) 2025
#641
The area of the region enclosed by the curve \(y=\sqrt{x}\) and the lines \(x=0\) and \(x=4\) and x-axis is:
(A)
\(\frac{16}{9}\) sq. units
(B)
\(\frac{32}{9}\) sq. units
(C)
\(\frac{16}{3}\) sq. units
(D)
\(\frac{32}{3}\) sq. units
MEDIUM
51 APPLY MCQ_SINGLE 1 Mark(s) 2025
#642
The area of the shaded region (figure) represented by the curves \(y=x^{2}\), \(0\le x\le2\) and y-axis is given by
(A)
\(\int_{0}^{2} x^2 dx\)
(B)
\(\int_{0}^{2} \sqrt{y} dy\)
(C)
\(\int_{0}^{4} x^2 dx\)
(D)
\(\int_{0}^{4} \sqrt{y} dy\)
MEDIUM
52 APPLY MCQ_SINGLE 1 Mark(s) 2025
#651
The solution for the differential equation \(\log(\frac{dy}{dx})=3x+4y\) is:
(A)
\(3e^{4y}+4e^{-3x}+C=0\)
(B)
\(e^{3x+4y}+C=0\)
(C)
\(3e^{-3y}+4e^{4x}+12C=0\)
(D)
\(3e^{-4y}+4e^{3x}+12C=0\)
MEDIUM
53 APPLY MCQ_SINGLE 1 Mark(s) 2025
#652
The order and degree of the following differential equation are, respectively: \(-\frac{d^{4}y}{dx^{4}}+2e^{dy/dx}+y^{2}=0\)
(A)
4, 1
(B)
4, not defined
(C)
1, 1
(D)
4, 1
MEDIUM
54 APPLY MCQ_SINGLE 1 Mark(s) 2025
#653
Let \(f^{\prime}(x)=3(x^{2}+2x)-\frac{4}{x^{3}}+5,\) and \(f(1)=0\). Then, \(f(x)\) is:
(A)
\(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x+11\)
(B)
\(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x-11\)
(C)
\(x^{3}+3x^{2}-\frac{2}{x^{2}}+5x-11\)
(D)
\(x^{3}-3x^{2}-\frac{2}{x^{2}}+5x-11\)
MEDIUM
55 APPLY MCQ_SINGLE 1 Mark(s) 2025
#654
The sum of the order and degree of the differential equation \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) is:
(A)
2
(B)
\(\frac{5}{2}\)
(C)
3
(D)
4
MEDIUM
56 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#655
The integrating factor of the differential equation \((\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}})\frac{dx}{dy}=1\) is:
(A)
\(e^{-1/\sqrt{x}}\)
(B)
\(e^{2/\sqrt{x}}\)
(C)
\(e^{2\sqrt{x}}\)
(D)
\(e^{-2\sqrt{x}}\)
HARD
57 APPLY MCQ_SINGLE 1 Mark(s) 2025
#656
The order and degree of the differential equation \((\frac{d^{2}y}{dx^{2}})^{2}+(\frac{dy}{dx})^{2}=x\sin(\frac{dy}{dx})\) are:
(A)
order 2, degree 2
(B)
order 2, degree 1
(C)
order 2, degree not defined
(D)
order 1, degree not defined
MEDIUM
58 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#657
If \(p\) and \(q\) are respectively the order and degree of the differential equation \(\frac{d}{dx}(\frac{dy}{dx})^{3}=0,\) then \((p-q)\) is
(A)
0
(B)
1
(C)
2
(D)
3
EASY
59 APPLY MCQ_SINGLE 1 Mark(s) 2025
#658
The integrating factor of the differential equation \((x + 2y^3)\dfrac{dy}{dx} = 2y\) is
(A)
\(e^{\frac{y^2}{2}}\)
(B)
\(\dfrac{1}{\sqrt{y}}\)
(C)
\(\dfrac{1}{y^2}\)
(D)
\(e^{-\frac{1}{y^2}}\)
MEDIUM
60 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#659
Which of the following is not a homogeneous function of \(x\) and \(y\) ?

(A)
\(y^2 - xy\)
(B)
\(x - 3y\)
(C)
\(\sin^2 \frac{y}{x} + \frac{y}{x}\)
(D)
\(\tan x - \sec y\)
MEDIUM
61 APPLY MCQ_SINGLE 1 Mark(s) 2025
#671
If P is a point on the line segment joining (3, 6, -1) and (6, 2, -2) and y-coordinate of P is 4, then its z-coordinate is:
(A)
\(-\frac{3}{2}\)
(B)
0
(C)
1
(D)
\(\frac{3}{2}\)
MEDIUM
62 APPLY MCQ_SINGLE 1 Mark(s) 2025
#672
The line \(x=1+5\mu\), \(y=-5+\mu\), \(z=-6-3\mu\) passes through which of the following point ?
(A)
\((1, -5, 6)\)
(B)
\((1, 5, 6)\)
(C)
\((1, -5, -6)\)
(D)
\((-1, -5, 6)\)
MEDIUM
63 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#673
The equation of a line parallel to the vector \(3\hat{i}+\hat{j}+2\hat{k}\) and passing through the point \((4, -3, 7)\) is:
(A)
\(x=4t+3, y=-3t+1, z=7t+2\)
(B)
\(x=3t+4, y=t+3, z=2t+7\)
(C)
\(x=3t+4, y=t-3, z=2t+7\)
(D)
\(x=3t+4, y=-t+3, z=2t+7\)
MEDIUM
64 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#679
In a Linear Programming Problem (LPP), the objective function \(Z=2x+5y\) is to be maximised under the following constraints: \(x+y\le4\), \(3x+3y\ge18\), \(x, y\ge0\). Study the graph and select the correct option. The solution of the given LPP:

[Image Missing]
(A)
lies in the shaded unbounded region.
(B)
lies in \(\Delta AOB\).
(C)
does not exist.
(D)
lies in the combined region of \(\Delta AOB\) and unbounded shaded region.
MEDIUM
65 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#680
For a Linear Programming Problem (LPP), the given objective function is \(Z=x+2y\). The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph. \(P\equiv(\frac{3}{13},\frac{24}{13})\) \(Q\equiv(\frac{3}{2},\frac{15}{4})\) \(R\equiv(\frac{7}{2},\frac{3}{4})\) \(S\equiv(\frac{18}{7},\frac{2}{7})\). Which of the following statements is correct?
(A)
Z is minimum at \(S(\frac{18}{7},\frac{2}{7})\)
(B)
Z is maximum at \(R(\frac{7}{2},\frac{3}{4})\)
(C)
(Value of Z at P) > (Value of Z at Q)
(D)
(Value of Z at Q) < (Value of Z at R)
MEDIUM
66 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#681
For a Linear Programming Problem (LPP), the given objective function \(Z=3x+2y\) is subject to constraints: \(x+2y\le10\), \(3x+y\le15\), \(x, y\ge0\). The correct feasible region is:
(A)
ABC
(B)
AOEC
(C)
CED
(D)
Open unbounded region BCD
MEDIUM
67 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#682
The corner points of the feasible region of a Linear Programming Problem are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). If \(Z=ax+by;\) (a, \(b>0)\) be the objective function, and maximum value of Z is obtained at (0, 2) and (3, 0), then the relation between a and b is:
(A)
\(a=b\)
(B)
\(a=3b\)
(C)
\(b=6a\)
(D)
\(3a=2b\)
MEDIUM
68 APPLY MCQ_SINGLE 1 Mark(s) 2025
#683
A factory produces two products X and Y. The profit earned by selling X and Y is represented by the objective function \(Z=5x+7y,\) where x and y are the number of units of X and Y respectively sold. Which of the following statement is correct?

(A)
The objective function maximizes the difference of the profit earned from products X and Y.
(B)
The objective function measures the total production of products X and Y.
(C)
The objective function maximizes the combined profit earned from selling X and Y.
(D)
The objective function ensures the company produces more of product X than product Y.
EASY
69 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#684
If the feasible region of a linear programming problem with objective function \(Z = ax + by\), is bounded, then which of the following is correct?
(A)
It will only have a maximum value.
(B)
It will only have a minimum value.
(C)
It will have both maximum and minimum values.
(D)
It will have neither maximum nor minimum value.
MEDIUM
70 APPLY MCQ_SINGLE 1 Mark(s) 2025
#685
The corner points of the feasible region in graphical representation of a L.P.P. are \((2, 72)\), \((15, 20)\) and \((40, 15)\). If \(Z = 18x + 9y\) be the objective function, then
(A)
\(Z\) is maximum at \((2, 72)\), minimum at \((15, 20)\)
(B)
\(Z\) is maximum at \((15, 20)\), minimum at \((40, 15)\)
(C)
\(Z\) is maximum at \((40, 15)\), minimum at \((15, 20)\)
(D)
\(Z\) is maximum at \((40, 15)\), minimum at \((2, 72)\)
MEDIUM
71 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#690
A coin is tossed and a card is selected at random from a well shuffled pack of 52 playing cards. The probability of getting head on the coin and a face card from the pack is :
(A)
\(\frac{2}{13}\)
(B)
\(\frac{3}{26}\)
(C)
\(\frac{19}{26}\)
(D)
\(\frac{3}{13}\)
MEDIUM
72 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#691
If \(P(A)=\frac{1}{7}\), \(P(B)=\frac{5}{7}\) and \(P(A\cap B)=\frac{4}{7},\) then \(P(\overline{A}|B)\) is:
(A)
\(\frac{6}{7}\)
(B)
\(\frac{3}{4}\)
(C)
\(\frac{4}{5}\)
(D)
\(\frac{1}{5}\)
MEDIUM
73 REMEMBER MCQ_SINGLE 1 Mark(s) 2025
#692
A box has 4 green, 8 blue and 3 red pens. A student picks up a pen at random, checks its colour and replaces it in the box. He repeats this process 3 times. The probability that at least one pen picked was red is:
(A)
\(\frac{124}{125}\)
(B)
\(\frac{1}{125}\)
(C)
\(\frac{61}{125}\)
(D)
\(\frac{64}{125}\)
MEDIUM
74 APPLY MCQ_SINGLE 1 Mark(s) 2025
#693
If \(P(A\cup B)=0.9\) and \(P(A\cap B)=0\cdot4,\) then \(P(\overline{A})+P(\overline{B})\) is:
(A)
0.3
(B)
1
(C)
1.3
(D)
0.7
MEDIUM
75 UNDERSTAND MCQ_SINGLE 1 Mark(s) 2025
#694
If E and F are two events such that \(P(E)>0\) and \(P(F)\ne1,\) then \(P(\overline{E}/\overline{F})\) is
(A)
\(\frac{P(\overline{E})}{P(\overline{F})}\)
(B)
\(1-P(\overline{E}/F)\)
(C)
\(1-P(E/F)\)
(D)
\(\frac{1-P(E\cup F)}{P(\overline{F})}\)
MEDIUM
76 APPLY MCQ_SINGLE 1 Mark(s) 2025
#695
If E and F are two independent events such that \( P(E) = \frac{2}{3} \), \( P(F) = \frac{3}{7} \), then \(\mathbf{P(E \mid \overline{F})}\) is equal to:
(A)
\( \frac{1}{6} \)
(B)
\( \frac{1}{2} \)
(C)
\( \frac{2}{3} \)
(D)
\( \frac{7}{9} \)
MEDIUM
77 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#761
Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify \(4~AB+3(AB+BA)-4~BA,\) where A and B are both matrices of order \(2\times2\). It is known that \(A\ne B\ne I\) and \(A^{-1}\ne B\). Their answers are given as:
Abhay: \(6 AB\),
Bina : \(7 AB-BA\),
Chhaya: \(8 AB\),
Devesh: \(7 BA - AB\).
Who answered it correctly?
(A)
Abhay
(B)
Bina
(C)
Chhaya
(D)
Devesh
MEDIUM
78 APPLY MCQ_SINGLE 1 Mark(s) 2025
#764
If \(A\) and \(B\) are invertible matrices, then which of the following is not correct?
(A)
\((A + B)^{-1} = B^{-1} + A^{-1}\)
(B)
\((AB)^{-1} = B^{-1} A^{-1}\)
(C)
\(\text{adj}(A) = |A| A^{-1}\)
(D)
\(|A^{-1}| = |A|^{-1}\)
MEDIUM
79 APPLY MCQ_SINGLE 1 Mark(s) 2025
#765
If
\(
A = \begin{pmatrix}
7 & 0 & x \\
0 & 7 & 0 \\
0 & 0 & y
\end{pmatrix}
\)
is a scalar matrix, then \(y^x\) is equal to
(A)
\(0\)
(B)
\(1\)
(C)
\(7\)
(D)
\(\pm 7\)
MEDIUM
80 COMPETENCY ANALYZE MCQ_SINGLE 1 Mark(s) 2025
#968
The given graph illustrates :
(A)
$y = \tan^{-1}x$
(B)
$y = \csc^{-1}x$
(C)
$y = \cot^{-1}x$
(D)
$y = \sec^{-1}x$
MEDIUM
81 REMEMBER MCQ_SINGLE 1 Mark(s) 2025
#969
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 $y = \sin x$
MEDIUM
82 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#970
For a Linear Programming Problem (LPP), the given objective function is $Z = x + 2y$. The feasible region PQRS determined by the set of constraints is shown as a shaded region in the graph.
Which of the following statements is correct ?
(A)
Z is minimum at $(\frac{18}{7}, \frac{2}{7})$
(B)
Z is maximum at R$(\frac{7}{2}, \frac{3}{4})$
(C)
(Value of Z at P) > (Value of Z at Q)
(D)
(Value of Z at Q) < (Value of Z at R)
MEDIUM
83 COMPETENCY APPLY MCQ_SINGLE 1 Mark(s) 2025
#971
In a Linear Programming Problem (LPP), the objective function $Z = 2x + 5y$ is to be maximised under the following constraints :
$x + y \leq 4$, $3x + 3y \geq 18$, $x, y \geq 0$
Study the graph and select the correct option.
(A)
lies in the shaded unbounded region.
(B)
lies in $\triangle AOB$.
(C)
does not exist.
(D)
lies in the combined region of $\triangle AOB$ and unbounded shaded region.
HARD

Assertion & Reason

1 REMEMBER ASSERTION_REASON 1 Mark(s) 2025
#974
Assertion: 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: Reason (R): When $x \to 0$, $\sin\frac{1}{x}$ is a finite value between $-1$ and $1$.
(A)
Both A and R are true and R is the correct explanation of A.
(B)
Both A and R are true but R is NOT the correct explanation of A.
(C)
A is true but R is false.
(D)
A is false but R is true.
EASY
2 COMPETENCY REMEMBER ASSERTION_REASON 1 Mark(s) 2025
#975
Assertion: Assertion (A) : Set of values of $\sec^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is a null set.
Reason: Reason (R) : $\sec^{-1}$ x is defined for $x \in \mathbb{R}-(-1, 1)$.
(A)
Both A and R are true and R is the correct explanation of A.
(B)
Both A and R are true but R is NOT the correct explanation of A.
(C)
A is true but R is false.
(D)
A is false but R is true.
EASY
3 COMPETENCY APPLY ASSERTION_REASON 1 Mark(s) 2025
#976
Assertion: Assertion (A) : If $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=256$ and $|\vec{b}|=8$, then $|\vec{a}|=2$.
Reason: Reason (R) : $\sin ^{2} \theta+\cos ^{2} \theta=1$ and $|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta$ and $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta$.
(A)
Both A and R are true and R is the correct explanation of A.
(B)
Both A and R are true but R is NOT the correct explanation of A.
(C)
A is true but R is false.
(D)
A is false but R is true.
MEDIUM
4 REMEMBER ASSERTION_REASON 1 Mark(s) 2025
#977
Assertion: Assertion (A): Let $f(x) = e^{x}$ and $g(x) = \log x$. Then $(f + g)x = e^{x} + \log x$ where domain of $(f + g)$ is $\mathbb{R}$.
Reason: Reason (R): $\text{Dom}(f + g) = \text{Dom}(f) \cap \text{Dom}(g)$.
(A)
Both A and R are true and R is the correct explanation of A.
(B)
Both A and R are true but R is NOT the correct explanation of A.
(C)
A is true but R is false.
(D)
A is false but R is true.
MEDIUM

VSA

1 UNDERSTAND VSA 2 Mark(s) 2025
#1355
Differentiate $2^{\cos^{2}x}$ w.r.t $\cos^{2}x$.
EASY
2 APPLY VSA 2 Mark(s) 2025
#1356
If $\tan^{-1}(x^{2}+y^{2})=a^{2}$, then find $\frac{dy}{dx}$.
MEDIUM
3 UNDERSTAND VSA 2 Mark(s) 2025
#1357
Evaluate: $\tan^{-1}\left[2\sin\left(2\cos^{-1}\frac{\sqrt{3}}{2}\right)\right]$.
MEDIUM
4 REMEMBER VSA 2 Mark(s) 2025
#1358
The diagonals of a parallelogram are given by $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$. Find the area of the parallelogram.
MEDIUM
5 UNDERSTAND VSA 2 Mark(s) 2025
#1359
Find the intervals in which function $f(x)=5x^{\frac{3}{2}}-3x^{\frac{5}{2}}$ is (i) increasing (ii) decreasing.
MEDIUM
6 COMPETENCY REMEMBER VSA 2 Mark(s) 2025
#1360
Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$. Determine the angle formed between the kite strings. Assume there is no slack in the strings.
EASY
7 REMEMBER VSA 2 Mark(s) 2025
#1361
Find a vector of magnitude 21 units in the direction opposite to that of $\vec{AB}$ where A and B are the points $A(2,1,3)$ and $B(8,-1,0)$ respectively.
MEDIUM
8 REMEMBER VSA 2 Mark(s) 2025
#1377
A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.
MEDIUM
9 REMEMBER VSA 2 Mark(s) 2025
#1378
If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $QR=\frac{3}{2}QP$.
MEDIUM
10 UNDERSTAND VSA 2 Mark(s) 2025
#1379
Evaluate: $\int_{0}^{\frac{\pi}{4}}\sqrt{1+\sin 2x}dx$.
MEDIUM
11 UNDERSTAND VSA 2 Mark(s) 2025
#1380
Find the values of 'a' for which $f(x)=\sin x-ax+b$ is increasing on R.
MEDIUM
12 REMEMBER VSA 2 Mark(s) 2025
#1381
If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then find x, such that $\vec{\alpha}=(x-2)\vec{a}+\vec{b}$ and $\vec{\beta}=(3+2x)\vec{a}-2\vec{b}$ are collinear.
MEDIUM
13 UNDERSTAND VSA 2 Mark(s) 2025
#1382
If $x=e^{\frac{x}{y}}$, then prove that $\frac{dy}{dx}=\frac{x-y}{x\log x}$.
MEDIUM
14 VSA 2 Mark(s) 2025
#1383
Check the differentiability of f(x) at $x=-2$ if $f(x)=\begin{cases}2x-3,-3\le x\le-2\\ x+1,-2
15 REMEMBER VSA 2 Mark(s) 2025
#1399
Let A and B be two square matrices of order 3 such that $\det(A) = 3$ and $\det(B) = -4$. Find the value of $\det(-6AB)$.
EASY
16 UNDERSTAND VSA 2 Mark(s) 2025
#1400
Find the least value of 'a' so that $f(x)=2x^{2}-ax+3$ is an increasing function on $[2, 4]$.
MEDIUM
17 UNDERSTAND VSA 2 Mark(s) 2025
#1401
If $f(x)=x+\frac{1}{x}$, $x\ge1$, show that f is an increasing function.
EASY
18 UNDERSTAND VSA 2 Mark(s) 2025
#1404
Calculate the area of the region bounded by the curve $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ and the x-axis using integration.
MEDIUM
19 COMPETENCY UNDERSTAND VSA 2 Mark(s) 2025
#1405
For the curve $y=5x-2x^{3}$ if x increases at the rate of $2\text{ units/s}$, then how fast is the slope of the curve changing when $x=2$?
MEDIUM
20 REMEMBER VSA 2 Mark(s) 2025
#1422
Surface area of a balloon (spherical), when air is blown into it, increases at a rate of $5\text{ mm}^{2}/\text{s}$. When the radius of the balloon is 8 mm, find the rate at which the volume of the balloon is increasing.
MEDIUM
21 UNDERSTAND VSA 2 Mark(s) 2025
#1423
Differentiate $\frac{\sin x}{\sqrt{\cos x}}$ with respect to x.
MEDIUM
22 UNDERSTAND VSA 2 Mark(s) 2025
#1424
If $y=5\cos x-3\sin x$, prove that $\frac{d^{2}y}{dx^{2}}+y=0$.
EASY
23 REMEMBER VSA 2 Mark(s) 2025
#1425
Find a vector of magnitude 5 which is perpendicular to both the vectors $3\hat{i}-2\hat{j}+\hat{k}$ and $4\hat{i}+3\hat{j}-2\hat{k}$.
MEDIUM
24 UNDERSTAND VSA 2 Mark(s) 2025
#1426
Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}$ and $\vec{a}\times\vec{b}=\vec{a}\times\vec{c}$, $\vec{a}\ne\vec{0}$. Show that $\vec{b}=\vec{c}$.
MEDIUM
25 REMEMBER VSA 2 Mark(s) 2025
#1427
A man needs to hang two lanterns on a straight wire whose end points have coordinates $A(4,1,-2)$ and $B(6,2,-3)$. Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.
MEDIUM
26 COMPETENCY APPLY VSA 2 Mark(s) 2025
#1444
Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.
MEDIUM
27 UNDERSTAND VSA 2 Mark(s) 2025
#1445
If $(x)^{y}=(y)^{x}$, then find $\frac{dy}{dx}$.
MEDIUM
28 UNDERSTAND VSA 2 Mark(s) 2025
#1446
Determine the values of x for which $f(x)=\frac{x-4}{x+1}$, $x\ne-1$ is an increasing or a decreasing function.
MEDIUM
29 REMEMBER VSA 2 Mark(s) 2025
#1447
If $\vec{a}$ and $\vec{b}$ are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that $BC=3BA$.
MEDIUM
30 REMEMBER VSA 2 Mark(s) 2025
#1448
Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.
MEDIUM
31 REMEMBER VSA 2 Mark(s) 2025
#1449
Determine if the lines $\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3\hat{i}-\hat{j})$ and $\vec{r}=(4\hat{i}-\hat{k})+\mu(2\hat{i}+3\hat{k})$ intersect with each other.
MEDIUM
32 UNDERSTAND VSA 2 Mark(s) 2025
#1465
Let $f:A\rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$ where $A=R-\{3\}$ and $B=R-\{1\}$. Discuss the bijectivity of the function.
MEDIUM
33 REMEMBER VSA 2 Mark(s) 2025
#1466
If $A=\begin{bmatrix}2&3\\ -1&2\end{bmatrix}$, then show that $A^{2}-4A+7I=0$.
EASY
34 APPLY VSA 2 Mark(s) 2025
#1467
Differentiate $\left(\frac{5^{x}}{x^{5}}\right)$ with respect to x.
MEDIUM
35 UNDERSTAND VSA 2 Mark(s) 2025
#1468
If $-2x^{2}-5xy+y^{3}=76$, then find $\frac{dy}{dx}$.
MEDIUM
36 VSA 2 Mark(s) 2025
#1469
In a Linear Programming Problem, the objective function $Z=5x+4y$ needs to be maximised under constraints $3x+y\le6$, $x\le1$, $x, y\ge0$. Express the LPP on the graph and shade the feasible region and mark the corner points.
37 REMEMBER VSA 2 Mark(s) 2025
#1470
10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.
MEDIUM
38 COMPETENCY UNDERSTAND VSA 2 Mark(s) 2025
#1471
In a village of 8000 people, 3000 go out of the village to work and 4000 are women. It is noted that 30% of women go out of the village to work. What is the probability that a randomly chosen individual is either a woman or a person working outside the village?
MEDIUM

SA

1 REMEMBER SA 3 Mark(s) 2025
#1362
The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm?
MEDIUM
2 REMEMBER SA 3 Mark(s) 2025
#1363
Solve the following linear programming problem graphically: Maximise $Z=x+2y$ Subject to the constraints: $x-y\ge0$, $x-2y\ge-2$, $x\ge0$, $y\ge0$.
MEDIUM
3 APPLY SA 3 Mark(s) 2025
#1364
Find: $\int\frac{x+\sin x}{1+\cos x}dx$.
MEDIUM
4 SA 3 Mark(s) 2025
#1365
Evaluate: $\int_{0}^{\frac{\pi}{4}}\frac{dx}{\cos^{3}x\sqrt{2\sin 2x}}$.
5 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1366
Verify that lines given by $\vec{r}=(1-\lambda)\hat{i}+(\lambda-2)\hat{j}+(3-2\lambda)\hat{k}$ and $\vec{r}=(\mu+1)\hat{i}+(2\mu-1)\hat{j}-(2\mu+1)\hat{k}$ are skew lines. Hence, find shortest distance between the lines.
HARD
6 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1367
During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.
MEDIUM
7 REMEMBER SA 3 Mark(s) 2025
#1368
The probability distribution for the number of students being absent in a class on a Saturday is as follows: X: 0, 2, 4, 5; $P(X)$: p, 2p, 3p, p. Where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.
EASY
8 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1369
For the vacancy advertised in the newspaper, 3000 candidates submitted their applications. From the data it was revealed that two third of the total applicants were females and other were males. The selection for the job was done through a written test. The performance of the applicants indicates that the probability of a male getting a distinction in written test is 0.4 and that a female getting a distinction is 0.35. Find the probability that the candidate chosen at random will have a distinction in the written test.
MEDIUM
9 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1370
Sketch the graph of $y=|x+3|$ and find the area of the region enclosed by the curve, x-axis, between $x=-6$ and $x=0$, using integration.
MEDIUM
10 APPLY SA 3 Mark(s) 2025
#1384
Solve the differential equation $2(y+3)-xy\frac{dy}{dx}=0;$ given $y(1)=-2$.
MEDIUM
11 UNDERSTAND SA 3 Mark(s) 2025
#1385
Solve the following differential equation: $(1+x^{2})\frac{dy}{dx}+2xy=4x^{2}$.
MEDIUM
12 REMEMBER SA 3 Mark(s) 2025
#1386
Let R be a relation defined over N, where N is set of natural numbers, defined as "mRn if and only if m is a multiple of n, m, $n\in N$." Find whether R is reflexive, symmetric and transitive or not.
MEDIUM
13 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1387
Solve the following linear programming problem graphically: Minimise $Z=x-5y$ subject to the constraints: $x-y\ge0$, $-x+2y\ge2$, $x\ge3$, $y\le4$, $y\ge0$.
MEDIUM
14 REMEMBER SA 3 Mark(s) 2025
#1388
If $y=\log(\sqrt{x}+\frac{1}{\sqrt{x}})^{2}$, then show that $x(x+1)^{2}y_{2}+(x+1)^{2}y_{1}=2$.
HARD
15 UNDERSTAND SA 3 Mark(s) 2025
#1389
If $x\sqrt{1+y}+y\sqrt{1+x}=0$, $-1
MEDIUM
16 REMEMBER SA 3 Mark(s) 2025
#1390
A die with number 1 to 6 is biased such that probability of $P(2)=\frac{3}{10}$ and probability of other numbers is equal. Find the mean of the number of times number 2 appears on the dice, if the dice is thrown twice.
MEDIUM
17 REMEMBER SA 3 Mark(s) 2025
#1391
Two dice are thrown. Defined are the following two events A and B: $A=\{(x,y):x+y=9\}$, $B=\{(x,y):x\ne3\}$ where (x, y) denote a point in the sample space. Check if events A and B are independent or mutually exclusive.
MEDIUM
18 UNDERSTAND SA 3 Mark(s) 2025
#1392
Find: $\int\frac{1}{x}\sqrt{\frac{x+a}{x-a}}dx$.
HARD
19 REMEMBER SA 3 Mark(s) 2025
#1406
If $f:R^{+}\rightarrow R$ is defined as $f(x) = \log_{a} x$ ($a > 0$ and $a\ne1$), prove that f is a bijection. ($R^{+}$ is a set of all positive real numbers.)
MEDIUM
20 REMEMBER SA 3 Mark(s) 2025
#1407
Let $A=\{1,2,3\}$ and $B=\{4,5,6\}$. A relation R from A to B is defined as $R=\{(x,y):x+y=6, x\in A, y\in B\}$. (i) Write all elements of R. (ii) Is R a function? Justify. (iii) Determine domain and range of R.
EASY
21 REMEMBER SA 3 Mark(s) 2025
#1408
Find k so that $f(x)=\begin{cases}\frac{x^{2}-2x-3}{x+1},&x\ne-1\\ k,&x=-1\end{cases}$ is continuous at $x=-1$.
EASY
22 UNDERSTAND SA 3 Mark(s) 2025
#1409
Check the differentiability of function $f(x)=x|x|$ at $x=0$.
MEDIUM
23 UNDERSTAND SA 3 Mark(s) 2025
#1410
Evaluate: $\int_{\pi/2}^{\pi}e^{x}\left(\frac{1-\sin x}{1-\cos x}\right)dx$.
HARD
24 APPLY SA 3 Mark(s) 2025
#1411
Find the probability distribution of the number of boys in families having three children, assuming equal probability for a boy and a girl.
MEDIUM
25 REMEMBER SA 3 Mark(s) 2025
#1412
A coin is tossed twice. Let X be a random variable defined as number of heads minus number of tails. Obtain the probability distribution of X and also find its mean.
MEDIUM
26 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1413
Find the distance of the point $(-1, -5, -10)$ from the point of intersection of the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$.
MEDIUM
27 COMPETENCY REMEMBER SA 3 Mark(s) 2025
#1414
Solve the following Linear Programming Problem using graphical method: Maximise $Z=100x+50y$ subject to the constraints $3x+y\le600$, $x+y\le300$, $y\le x+200$, $x\ge0$, $y\ge0$.
MEDIUM
28 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1428
Find the value of 'a' for which $f(x)=\sqrt{3}\sin x-\cos x-2ax+6$ is decreasing in R.
MEDIUM
29 UNDERSTAND SA 3 Mark(s) 2025
#1429
Find: $\int\frac{2x}{(x^{2}+3)(x^{2}-5)}dx$.
MEDIUM
30 COMPETENCY APPLY SA 3 Mark(s) 2025
#1430
Evaluate: $\int_{1}^{4}(|x-2|+|x-4|)dx$.
MEDIUM
31 REMEMBER SA 3 Mark(s) 2025
#1431
Find the particular solution of the differential equation $\left[x\sin^{2}\left(\frac{y}{x}\right)-y\right]dx+x~dy=0$ given that $y=\frac{\pi}{4}$ when $x=1$.
HARD
32 REMEMBER SA 3 Mark(s) 2025
#1432
In the Linear Programming Problem (LPP), find the point/points giving maximum value for $Z=5x+10y$ subject to constraints $x+2y\le120$, $x+y\ge60$, $x-2y\ge0$, $x, y\ge0$.
MEDIUM
33 REMEMBER SA 3 Mark(s) 2025
#1433
If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ such that $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$, then find the angle between $\vec{a}$ and $\vec{b}$.
MEDIUM
34 REMEMBER SA 3 Mark(s) 2025
#1434
If $\vec{a}$ and $\vec{b}$ are unit vectors inclined with each other at an angle $\theta$, then prove that $\frac{1}{2}|\vec{a}-\vec{b}|=\sin\frac{\theta}{2}$.
MEDIUM
35 REMEMBER SA 3 Mark(s) 2025
#1435
The probability that a student buys a colouring book is 0.7 and that she buys a box of colours is 0.2. The probability that she buys a colouring book, given that she buys a box of colours, is 0.3. Find the probability that the student: (i) Buys both the colouring book and the box of colours. (ii) Buys a box of colours given that she buys the colouring book.
MEDIUM
36 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1436
A person has a fruit box that contains 6 apples and 4 oranges. He picks out a fruit three times, one after the other, after replacing the previous one in the box. Find: (i) The probability distribution of the number of oranges he draws. (ii) The expectation of the random variable (number of oranges).
MEDIUM
37 REMEMBER SA 3 Mark(s) 2025
#1450
Let $A=\begin{bmatrix}1\\ 4\\ -2\end{bmatrix}$ and $C=\begin{bmatrix}3&4&2\\ 12&16&8\\ -6&-8&-4\end{bmatrix}$ be two matrices. Then, find the matrix B if $AB=C$.
MEDIUM
38 UNDERSTAND SA 3 Mark(s) 2025
#1451
Differentiate $y=\sin^{-1}(3x-4x^{3})$ w.r.t. x, if $x\in[-\frac{1}{2},\frac{1}{2}]$.
MEDIUM
39 REMEMBER SA 3 Mark(s) 2025
#1452
Differentiate $y=\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ with respect to x, when $x\in(0,1)$.
MEDIUM
40 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1453
A student wants to pair up natural numbers in such a way that they satisfy the equation $2x+y=41$, $x, y\in N$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.
MEDIUM
41 REMEMBER SA 3 Mark(s) 2025
#1454
Show that the function $f:N\rightarrow N$, where N is a set of natural numbers, given by $f(n) = n-1$, if n is even, $n+1$, if n is odd, is a bijection.
MEDIUM
42 COMPETENCY REMEMBER SA 3 Mark(s) 2025
#1455
Consider the Linear Programming Problem, where the objective function $Z=(x+4y)$ needs to be minimized subject to constraints $2x+y\ge1000$, $x+2y\ge800$, $x,y\ge0$. Draw a neat graph of the feasible region and find the minimum value of Z.
MEDIUM
43 REMEMBER SA 3 Mark(s) 2025
#1456
Find the distance of the point $P(2,4,-1)$ from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.
MEDIUM
44 REMEMBER SA 3 Mark(s) 2025
#1457
Let the position vectors of the points A, B and C be $3\hat{i}-\hat{j}-2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+5\hat{j}+3\hat{k}$ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.
MEDIUM
45 COMPETENCY UNDERSTAND SA 3 Mark(s) 2025
#1458
A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection: (i) in both committees (ii) in only one committee.
MEDIUM
46 UNDERSTAND SA 3 Mark(s) 2025
#1472
Show that the function $f:R\rightarrow R$ defined by $f(x)=4x^{3}-5$, $\forall x\in R$ is one-one and onto.
MEDIUM
47 REMEMBER SA 3 Mark(s) 2025
#1473
Let R be a relation defined on a set N of natural numbers such that $R=\{(x,y): xy \text{ is a square of a natural number, } x, y\in N\}$. Determine if the relation R is an equivalence relation.
MEDIUM
48 REMEMBER SA 3 Mark(s) 2025
#1474
Let $2x+5y-1=0$ and $3x+2y-7=0$ represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.
MEDIUM
49 COMPETENCY REMEMBER SA 3 Mark(s) 2025
#1475
A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹150 (Chemistry), ₹175 (Physics) and ₹180 (Maths), then find his total sale in two days, using matrix method. If cost price of all the books together is ₹35,000, what profit did he earn after the sale of two days?
MEDIUM
50 REMEMBER SA 3 Mark(s) 2025
#1476
Differentiate $y=\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}$ with respect to x.
HARD
51 APPLY SA 3 Mark(s) 2025
#1477
Amongst all pairs of positive integers with product as 289, find which of the two numbers add up to the least.
EASY
52 REMEMBER SA 3 Mark(s) 2025
#1478
In the Linear Programming Problem for objective function $Z=18x+10y$ subject to constraints $4x+y\ge20$, $2x+3y\ge30$, $x,y\ge0$ find the minimum value of Z.
MEDIUM
53 REMEMBER SA 3 Mark(s) 2025
#1479
The scalar product of the vector $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ with a unit vector along sum of vectors $\vec{b}=2\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}=\lambda\hat{i}-2\hat{j}-3\hat{k}$ is equal to 1. Find the value of $\lambda$.
MEDIUM
54 REMEMBER SA 3 Mark(s) 2025
#1480
Find the shortest distance between the lines: $\vec{r}=(2\hat{i}-\hat{j}+3\hat{k})+\lambda(\hat{i}-2\hat{j}+3\hat{k})$ and $\vec{r}=(\hat{i}+4\hat{k})+\mu(3\hat{i}-6\hat{j}+9\hat{k})$.
MEDIUM

LA

1 REMEMBER LA 5 Mark(s) 2025
#1371
If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, then prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$.
HARD
2 REMEMBER LA 5 Mark(s) 2025
#1372
If $x=a\left(\cos\theta+\log\tan\frac{\theta}{2}\right)$ and $y=\sin\theta$, then find $\frac{d^{2}y}{dx^{2}}$ at $\theta=\frac{\pi}{4}$.
HARD
3 UNDERSTAND LA 5 Mark(s) 2025
#1373
Find the absolute maximum and absolute minimum of function $f(x)=2x^{3}-15x^{2}+36x+1$ on $[1, 5]$.
MEDIUM
4 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1374
Find the image A' of the point $A(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Also, find the equation of the line joining A and A'.
HARD
5 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1375
Find a point P on the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$ such that its distance from point $Q(2,4,-1)$ is 7 units. Also, find the equation of line joining P and Q.
HARD
6 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1376
A school wants to allocate students into three clubs Sports, Music and Drama, under following conditions: The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.
HARD
7 REMEMBER LA 5 Mark(s) 2025
#1393
Using integration, find the area of the region bounded by the line $y=5x+2$, the x-axis and the ordinates $x=-2$ and $x=2$.
MEDIUM
8 LA 5 Mark(s) 2025
#1394
Find: $\int\frac{x^{2}+x+1}{(x+2)(x^{2}+1)}dx$.
9 REMEMBER LA 5 Mark(s) 2025
#1395
Find the shortest distance between the lines: $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ and $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$.
MEDIUM
10 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1396
Find the image A' of the point A(2, 1, 2) in the line $l:\vec{r}=4\hat{i}+2\hat{j}+2\hat{k}+\lambda(\hat{i}-\hat{j}-\hat{k})$. Also, find the equation of line joining AA'. Find the foot of perpendicular from point A on the line l.
HARD
11 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1397
Given $A=\begin{bmatrix}-4&4&4\\ -7&1&3\\ 5&-3&-1\end{bmatrix}$ and $B=\begin{bmatrix}1&-1&1\\ 1&-2&-2\\ 2&1&3\end{bmatrix}$, find AB. Hence, solve the system of linear equations: $x-y+z=4$, $x-2y-2z=9$, $2x+y+3z=1$.
HARD
12 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1398
If $A=\begin{bmatrix}1&2&0\\ -2&-1&-2\\ 0&-1&1\end{bmatrix}$, then find $A^{-1}$. Hence, solve the system of linear equations: $x-2y=10$, $2x-y-z=8$, $-2y+z=7$.
HARD
13 COMPETENCY ANALYZE LA 5 Mark(s) 2025
#1415
If A is a $3\times3$ invertible matrix, show that for any scalar $k\ne0$, $(kA)^{-1}=\frac{1}{k}A^{-1}$. Hence calculate $(3A)^{-1}$, where $A=\begin{bmatrix}2&-1&1\\ -1&2&-1\\ 1&-1&2\end{bmatrix}$.
MEDIUM
14 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1416
The relation between the height of the plant (y cm) with respect to exposure to sunlight is governed by the equation $y=4x-\frac{1}{2}x^{2}$, where x is the number of days exposed to sunlight. (i) Find the rate of growth of the plant with respect to sunlight. (ii) In how many days will the plant attain its maximum height? What is the maximum height?
MEDIUM
15 UNDERSTAND LA 5 Mark(s) 2025
#1417
Find: $\int\frac{\cos x}{(4+\sin^{2}x)(5-4\cos^{2}x)}dx$.
HARD
16 UNDERSTAND LA 5 Mark(s) 2025
#1418
Evaluate: $\int_{0}^{\pi}\frac{dx}{a^{2}\cos^{2}x+b^{2}\sin^{2}x}$.
MEDIUM
17 REMEMBER LA 5 Mark(s) 2025
#1419
Show that the area of a parallelogram whose diagonals are represented by $\vec{a}$ and $\vec{b}$ is given by $\frac{1}{2}|\vec{a}\times\vec{b}|$. Also find the area of a parallelogram whose diagonals are $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}-\hat{k}$.
MEDIUM
18 REMEMBER LA 5 Mark(s) 2025
#1420
Find the equation of a line in vector and cartesian form which passes through the point $(1,2,-4)$ and is perpendicular to the lines $\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}$ and $\vec{r}=15\hat{i}+29\hat{j}+5\hat{k}+\mu(3\hat{i}+8\hat{j}-5\hat{k})$.
MEDIUM
19 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1437
Sketch a graph of $y=x^{2}$. Using integration, find the area of the region bounded by $y=9$, $x=0$ and $y=x^{2}$.
MEDIUM
20 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1438
A furniture workshop produces three types of furniture chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is 45. It was also found that production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using matrix method.
MEDIUM
21 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1439
For a positive constant 'a', differentiate $a^{t+\frac{1}{t}}$ with respect to $(t+\frac{1}{t})^{a}$ where t is a non-zero real number.
HARD
22 UNDERSTAND LA 5 Mark(s) 2025
#1440
Find $\frac{dy}{dx}$ if $y^{x}+x^{y}+x^{x}=a^{b}$, where a and b are constants.
HARD
23 REMEMBER LA 5 Mark(s) 2025
#1441
Find the foot of the perpendicular drawn from the point (1, 1, 4) on the line $\frac{x+2}{5}=\frac{y+1}{2}=\frac{-z+4}{-3}$.
MEDIUM
24 REMEMBER LA 5 Mark(s) 2025
#1442
Find the point on the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-4}{3}$ at a distance of $2\sqrt{2}$ units from the point (-1, -1, 2).
MEDIUM
25 UNDERSTAND LA 5 Mark(s) 2025
#1459
Find: $\int\frac{x^{2}+1}{(x-1)^{2}(x+3)}dx$.
HARD
26 UNDERSTAND LA 5 Mark(s) 2025
#1460
Evaluate: $\int_{0}^{\pi/2}\frac{x}{\sin x+\cos x}dx$.
HARD
27 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1461
Draw a rough sketch for the curve $y=2+|x+1|$. Using integration, find the area of the region bounded by the curve $y=2+|x+1|$, $x=-4$, $x=3$ and $y=0$.
MEDIUM
28 UNDERSTAND LA 5 Mark(s) 2025
#1462
Solve the differential equation: $x^{2}y~dx-(x^{3}+y^{3})dy=0$.
MEDIUM
29 REMEMBER LA 5 Mark(s) 2025
#1463
Solve the differential equation $(1+x^{2})\frac{dy}{dx}+2xy-4x^{2}=0$ subject to initial condition $y(0)=0$.
MEDIUM
30 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1464
Let the polished side of the mirror be along the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{2z-4}{6}$. A point $P(1,6,3)$, some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image.
HARD
31 UNDERSTAND LA 5 Mark(s) 2025
#1481
Find: $\int\frac{x^{2}+1}{(x^{2}+2)(2x^{2}+1)}dx$
HARD
32 UNDERSTAND LA 5 Mark(s) 2025
#1482
Evaluate: $\int_{0}^{\pi}\frac{x\tan x}{\sec x+\tan x}dx$
HARD
33 COMPETENCY UNDERSTAND LA 5 Mark(s) 2025
#1483
A woman discovered a scratch along a straight line on a circular table top of radius 8 cm. She divided the table top into 4 equal quadrants and discovered the scratch passing through the origin inclined at an angle $\frac{\pi}{4}$ anticlockwise along the positive direction of x-axis. Find the area of the region enclosed by the x-axis, the scratch and the circular table top in the first quadrant, using integration.
MEDIUM
34 UNDERSTAND LA 5 Mark(s) 2025
#1484
Solve the differential equation $\frac{dy}{dx}=\cos x-2y$.
MEDIUM
35 REMEMBER LA 5 Mark(s) 2025
#1485
Find the point Q on the line $\frac{2x+4}{6}=\frac{y+1}{2}=\frac{-2z+6}{-4}$ at a distance of $3\sqrt{2}$ from the point $P(1,2,3)$.
HARD
36 COMPETENCY REMEMBER LA 5 Mark(s) 2025
#1486
Find the image of the point (-1,5,2) in the line $\frac{2x-4}{2}=\frac{y}{2}=\frac{2-z}{3}$. Find the length of the line segment joining the points (given point and the image point).
HARD