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Standalone Questions
#2101
Biology
Biotechnology and its Applications
asfdsadf
#2100
Mathematics
Applications of Integrals
SUBJECTIVE
APPLY
2026
AISSCE(Board Exam)
Competency
4 Marks
Roundabouts are often made on busy roads to ease the traffic and avoid red lights.One such round-about is made such that equation representing its boundary is given by $C_1 : x^2 + y^2 = 64$. There is a circular pond with a fountain in the middle of the roundabout whose equation is given by $C_2 : x^2 + y^2 = 4$.Based on the given information, answer the following questions :
(i) Represent the given equations $C_1$ and $C_2$ with the help of a diagram. [1 mark]
(ii) Express y as a function of $x$, ($y = f(x)$), for both $C_1$ an $C_2$. [1 mark]
(iii) (a) Using integration find the area of region covered by the roundabout. [2 marks]
OR
(iii) (b) Using integration, find the area of region covered by circular pond. [2 marks]
Key:
Sol:
Sol:
#2099
Mathematics
Applications of Derivatives
SUBJECTIVE
APPLY
2026-27
CBSE Sample QP
Competency
4 Marks
Water is essential for life, and rivers are vital water sources across the world. The Yamuna Action Plan (YAP), one of India's largest river restoration projects, was launched to clean the Yamuna, but its effectiveness (WQI) the Water Quality Index must be measured through cost-benefit, economic and environmental analysis. Therefore, comprehensive pollutant treatment, prevention of further contamination, and community awareness are crucial for sustainable river management.Implementing the same concept to the river Yamuna, if an initial base line WQI is 35, a quadratic expression is formed to measure the impact of investment of ₹$x$ crore on the WQI i.e., $W(x)$ as $W(x) = 35 + 1.6x - 0.02x^2$, where $x$ is in Rupees in crores Use the given $W(x)$ to answer the following questions:
(i) Find the value of $x$ at the stationary point for the function $W(x)$. [1 mark]
(ii) Find the maximum value of the WQI using the second derivative test? [1 mark]
(iii) (a) If the investment is ₹ 35 crores, find by how much the WQI is short of its maximum achievable value.[2 marks]
Or
(b) Evaluate the WQI when the investment is ₹60 crore. Determine whether this indicates an enhancement or a diminishing return in our objective when compared with the optimum WQI. [2 marks]
Key:
Sol:
Sol:
#2098
Mathematics
Applications of Integrals
LA
APPLY
2026-27
CBSE Sample QP
KNOWLEDGE
5 Marks
Using integration, find the area of the region bounded by the curve $y^2 = 2x$ and the line $x - 2y = 0$.
Key:
Sol:
Sol:
#2097
Mathematics
Applications of Integrals
LA
APPLY
2026-27
CBSE Sample QP
Competency
5 Marks
Using Integration, find the area of the region $\{(x, y) : 0 \leq y \leq \sqrt{4 - x^2}, x \geq 2 - y\}$.
Key:
Sol:
Sol:
#2096
Mathematics
Continuity and Differentiability
ASSERTION_REASON
EVALUATE
Competency
1 Marks
If a function is continuous on a closed interval, then it must be differentiable on that interval.
Continuity ensures smoothness, which guarantees differentiability.
Key:
Sol:
Sol:
#2095
Mathematics
Differential Equations
VSA
REMEMBER
KNOWLEDGE
2 Marks
The general solution of the differential equation $xdy - (1+x^2)dx = dx$
Key:
Sol:
Sol:
#2094
Mathematics
Differential Equations
SA
APPLY
2026-27
CBSE Sample QP
Competency
3 Marks
Find the particular solution of the differential equation $(x^2 + y^2)dy = xydx$, given that $y(0) = 1$
Key:
Sol:
Sol:
#2093
Mathematics
Differential Equations
SA
APPLY
2026-27
CBSE Sample QP
Competency
3 Marks
Find the particular solution of the differential equation $x dy + (y - x^2 e^x) dx = 0$, given that $y(1)=0$
Key:
Sol:
Sol:
#2092
Mathematics
Differential Equations
ASSERTION_REASON
APPLY
2026-27
CBSE Sample QP
Competency
1 Marks
The general solution of a differential equation represents a family of curves.
The derivative of an arbitrary constant is zero.
Key:
Sol:
Sol:
#2091
Mathematics
Vector Algebra
MCQ_SINGLE
REMEMBER
2026-27
CBSE Sample QP
KNOWLEDGE
1 Marks
Angle made by the vector $\hat{i} + \hat{j} + \sqrt{2}\hat{k}$ with positive direction of z-axis is
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
Key: B
Sol:
Sol:
#2090
Mathematics
Integrals
MCQ_SINGLE
APPLY
2026-27
CBSE Sample QP
Competency
1 Marks
If $\int \frac{1}{1+e^{2x}} dx = x - P \log(1+e^{2x}) + C$, then 'P' is equal to
(A) $-\frac{1}{2}$
(B) $-\frac{1}{3}$
(C) $\frac{1}{3}$
(D) $\frac{1}{2}$
Key: D
Sol:
Sol:
#2089
Mathematics
Applications of Derivatives
SA
APPLY
2026
CBSE Sample QP
Competency
3 Marks
The circular area of vision on the ground by a surveillance drone fitted with camera increases at the rate of $44 𝑚^2$ per metre as it gains vertical height. The radius of the circle on the ground increases at the rate of 6 m/s. Find the rate at which the height of the drone increases with time, at the instant when the radius of the circular area on the ground is 14 m. (𝑈𝑠𝑒 𝜋 =22/7)
Key:
Sol:
Sol:
#2088
Mathematics
Continuity and Differentiability
VSA
APPLY
2026
CBSE Sample QP
Competency
2 Marks
If $y = \sin(x+y)$ and $\frac{dy}{dx} = 0$, then prove that $y = \pm 1$.
Key:
Sol:
Sol:
#2087
Physics
Electromagnetic Waves
MCQ_SINGLE
REMEMBER
2026
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
A plane electromagnetic wave travels through a medium and the magnetic field associated with it is given by
$$B = 5 \times 10^{-8} \sin (3 \times 10^{10} t - 150 x) T$$
where x is in metres and t is in seconds.
The velocity of the wave is :
$$B = 5 \times 10^{-8} \sin (3 \times 10^{10} t - 150 x) T$$
where x is in metres and t is in seconds.
The velocity of the wave is :
(A) $$2.0 \times 10^{8} ms^{-1}$$
(B) $$4.5 \times 10^{7} ms^{-1}$$
(C) $$3.5 \times 10^{7} ms^{-1}$$
(D) $$2.5 \times 10^{8} ms^{-1}$$
Key: A
Sol:
Sol:
Compare with the Standard Wave Equation:The general equation for a plane electromagnetic wave travelling along the x-axis is given by:$$B = B_0 \sin(\omega t - kx)$$Comparing this with the given equation:$$B = 5 \times 10^{-8} \sin(3 \times 10^{10} t - 150 x)$$Extract the Wave Parameters:Angular frequency ($\omega$) = $3 \times 10^{10} \text{ rad s}^{-1}$Propagation constant / Wave number ($k$) = $150 \text{ rad m}^{-1}$Calculate the Velocity of the Wave:The phase velocity ($v$) of an electromagnetic wave in a medium is related to $\omega$ and $k$ by the formula:$$v = \frac{\omega}{k}$$Substitute the values:$$v = \frac{3 \times 10^{10}}{150} = \frac{3 \times 10^{10}}{1.5 \times 10^2} = 2.0 \times 10^8 \text{ ms}^{-1}$$
#2086
Physics
Electromagnetic Induction
MCQ_SINGLE
APPLY
2026
AISSCE(Board Exam)
Competency
1 Marks
A square loop of side 50 cm is placed in a uniform magnetic field of $3\cdot 0$ T acting perpendicular to the plane of the loop. If the loop is rotated through an angle of $90^{\circ}$ in $0\cdot 3$ s, the value of emf induced in the loop would be :
(A) $2\cdot 5$ V
(B) $0\cdot 50$ V
(C) $0\cdot 75$ V
(D) $1\cdot 0$ V
Key: A
Sol:
Sol:
#2084
Physics
Electric Charges and Fields
MCQ_SINGLE
APPLY
2026
AISSCE(Board Exam)
Competency
1 Marks
A particle of mass m and charge q starts from rest and moves in an electric field $\vec{E} = E_0 \hat{i}$. After travelling a distance x in the field along x-axis, the kinetic energy of the particle will be :
(A) $q E_0 x^2$
(B) $q E_0 x$
(C) $q^2 E_0 x$
(D) $q^2 E_0^2 x$
Key: B
Sol:
Sol:
Force on the charged particle:When a particle of charge $q$ is placed in a uniform electric field $\vec{E} = E_0 \hat{i}$, the electrostatic force acting on it is given by:$$F = q E_0$$Acceleration of the particle:Using Newton's second law ($F = ma$), the acceleration $a$ of the particle of mass $m$ is:$$a = \frac{F}{m} = \frac{q E_0}{m}$$Kinetic Energy calculation:Since the particle starts from rest ($u = 0$) and travels a distance $x$, we can use the third equation of motion ($v^2 = u^2 + 2ax$):$$v^2 = 0 + 2 \left(\frac{q E_0}{m}\right) x = \frac{2 q E_0 x}{m}$$Now, substitute $v^2$ into the formula for Kinetic Energy ($K = \frac{1}{2} m v^2$):$$K = \frac{1}{2} m \left(\frac{2 q E_0 x}{m}\right) = q E_0 x$$
#2082
Mathematics
Derivatives
SUBJECTIVE
APPLY
AI Import
Competency
4 Marks
Amusement parks heavily rely on mathematical curves to design roller coaster tracks, balancing intense thrill with absolute safety. For a safe ride, the track must have no breaks (it must be continuous) and no sharp, sudden corners (it must be differentiable/smooth).
An engineering team is designing a transition segment of a new coaster. The height $h(x)$ (in meters) of the track at a horizontal distance $x$ (in meters) from the starting platform is modeled by the following piecewise function:
$$h(x) = \begin{cases} \frac{1}{2}x^2 + 2x, & 0 \leq x < 2 \\ ax + b, & 2 \leq x \leq 5 \end{cases}$$
Based on the given information, answer the following questions:
(i) For the track to have no broken rails at the transition point $x = 2$, it must be continuous. Formulate the mathematical equation relating $a$ and $b$ to ensure this continuity. (1 Mark)
(ii) Find the left-hand derivative (the slope of the track just before the transition point) at $x = 2$. (1 Mark)
(iii) To ensure the transition at $x = 2$ is perfectly smooth (differentiable), calculate the exact values of $a$ and $b$ that the engineers must use. (2 Marks)
— OR —
(iii) If a junior engineer mistakenly designed the second track segment using $a = 3$ and $b = 0$, verify whether the track is mathematically safe (smooth and differentiable) at $x = 2$. Justify your answer. (2 Marks)
An engineering team is designing a transition segment of a new coaster. The height $h(x)$ (in meters) of the track at a horizontal distance $x$ (in meters) from the starting platform is modeled by the following piecewise function:
$$h(x) = \begin{cases} \frac{1}{2}x^2 + 2x, & 0 \leq x < 2 \\ ax + b, & 2 \leq x \leq 5 \end{cases}$$
Based on the given information, answer the following questions:
(i) For the track to have no broken rails at the transition point $x = 2$, it must be continuous. Formulate the mathematical equation relating $a$ and $b$ to ensure this continuity. (1 Mark)
(ii) Find the left-hand derivative (the slope of the track just before the transition point) at $x = 2$. (1 Mark)
(iii) To ensure the transition at $x = 2$ is perfectly smooth (differentiable), calculate the exact values of $a$ and $b$ that the engineers must use. (2 Marks)
— OR —
(iii) If a junior engineer mistakenly designed the second track segment using $a = 3$ and $b = 0$, verify whether the track is mathematically safe (smooth and differentiable) at $x = 2$. Justify your answer. (2 Marks)
Key:
Sol:
Sol:
#1894
Physics
Alternating Current
MCQ_SINGLE
APPLY
Competency
1 Marks
Mr. Iyer takes a grinder rated 220 V, 50 Hz to the U.S where supply is 110 V, 60 Hz. He needs to use a
(A) step - downtransformer with turn ratio 2: 1
(B) step - up transformer with turn ratio 1 : 3
(C) step - up transformer with turn ratio 1 : 2
(D) step - downtransformer with turn ratio 5: 2
Key: C
Sol:
Sol:
#1889
Chemistry
Chemical Kinetics
VSA
APPLY
2005
SNP
Competency
1 Marks
Find out the rate constant for the reaction $\text{C(s)} + \text{O}_2\text{(g)} \rightarrow \text{CO}_2\text{(g)}$
Key:
Sol:
Sol:
Case-Based Questions
CASE ID: #118
Cl: CBSE Class 12
Mathematics
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.
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
4 Marks
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 ?
(i) Find the probability that it was defective.
(ii) What is the probability that this defective smartphone was manufactured by company B ?
Key:
Sol:
Sol:
CASE ID: #117
Cl: CBSE Class 12
Mathematics
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.
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
4 Marks
(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.
(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.
Key:
Sol:
Sol:
CASE ID: #116
Cl: CBSE Class 12
Mathematics
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.
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
4 Marks
(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 $A_2$ ?
(ii) Given that the person has not contracted the disease, what is the probability that the person is from category $A_2$ ?
Key:
Sol:
Sol:
Based on the information provided, here are the step-by-step calculations:Let the events be defined as:$A_1$: Person has good health$A_2$: Person has average health$A_3$: Person has poor health$E$: Person contracts the diseaseFrom the data given:$P(A_1) = \frac{700}{1000} = 0.7$$P(A_2) = \frac{200}{1000} = 0.2$$P(A_3) = \frac{100}{1000} = 0.1$The conditional probabilities of contracting the disease are:$P(E|A_1) = 25\% = 0.25$$P(E|A_2) = 35\% = 0.35$$P(E|A_3) = 50\% = 0.50$(i) Probability that a randomly selected person has contracted the diseaseWe use the Law of Total Probability:$$P(E) = P(E|A_1)P(A_1) + P(E|A_2)P(A_2) + P(E|A_3)P(A_3)$$Substitute the values:$$P(E) = (0.25 \times 0.7) + (0.35 \times 0.2) + (0.50 \times 0.1)$$$$P(E) = 0.175 + 0.07 + 0.05$$$$P(E) = 0.295$$Answer: The probability that the person has contracted the disease is 0.295 (or 29.5%).(ii) Probability that the person is from category $A_2$, given that they have not contracted the diseaseLet $E'$ be the event that the person has not contracted the disease.We need to find $P(A_2|E')$.First, calculate the probability of not contracting the disease, $P(E')$:$$P(E') = 1 - P(E) = 1 - 0.295 = 0.705$$Next, calculate the probability of not contracting the disease given the person is in category $A_2$:$$P(E'|A_2) = 1 - P(E|A_2) = 1 - 0.35 = 0.65$$Now, apply Bayes' Theorem:$$P(A_2|E') = \frac{P(E'|A_2) \cdot P(A_2)}{P(E')}$$Substitute the values:$$P(A_2|E') = \frac{0.65 \times 0.2}{0.705}$$$$P(A_2|E') = \frac{0.13}{0.705}$$$$P(A_2|E') = \frac{130}{705} = \frac{26}{141}$$$$P(A_2|E') \approx 0.1844$$Answer: The probability that the person is from category $A_2$ given they have not contracted the disease is $\frac{26}{141}$ or approximately 0.184.
CASE ID: #115
Cl: CBSE Class 12
Mathematics
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.
SUBJECTIVE
REMEMBER
2025
AISSCE(Board Exam)
Competency
1 Marks
(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.
(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.
Key:
Sol:
Sol:
CASE ID: #114
Cl: CBSE Class 12
Mathematics
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.
SUBJECTIVE
2025
AISSCE(Board Exam)
4 Marks
(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 $(\vec{a}-\vec{b})$.
(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 $(\vec{a}-\vec{b})$.
Key:
Sol:
Sol:
CASE ID: #109
Cl: CBSE Class 12
Mathematics
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.,
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of
x and y..
x and y..
Key:
Sol:
Sol:
k
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
Write the area of the solar panel as a function of $x$
Key:
Sol:
Sol:
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
1 Marks
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.
Key:
Sol:
Sol:
SUBJECTIVE
APPLY
2025
AISSCE(Board Exam)
Competency
2 Marks
Using first derivative test, calculate the maximum area the company can enclose with the 300 metres of boundary material, considering the parallel partition.
Key:
Sol:
Sol:
CASE ID: #108
Cl: CBSE Class 12
Mathematics
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.
VSA
APPLY
2025
AISSCE(Board Exam)
Competency
2 Marks
What is the probability that a customer after availing the loan will default on the loan repayment?
Key:
Sol:
Sol:
VSA
APPLY
2025
AISSCE(Board Exam)
Competency
2 Marks
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?
Key:
Sol:
Sol: