Available Questions 145 found Page 1 of 8
Standalone Questions
#1486
Mathematics
Three Dimensional Geometry
LA
REMEMBER
2025
AISSCE(Board Exam)
Competency
5 Marks
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).
Key:
Sol:
Sol:
#1483
Mathematics
Applications of Integrals
LA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
5 Marks
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.
Key:
Sol:
Sol:
#1475
Mathematics
Matrices and Determinants
SA
REMEMBER
2025
AISSCE(Board Exam)
Competency
3 Marks
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?
Key:
Sol:
Sol:
#1471
Mathematics
Probability
VSA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
2 Marks
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?
Key:
Sol:
Sol:
#1464
Mathematics
Three Dimensional Geometry
LA
REMEMBER
2025
AISSCE(Board Exam)
Competency
5 Marks
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.
Key:
Sol:
Sol:
#1461
Mathematics
Applications of Integrals
LA
REMEMBER
2025
AISSCE(Board Exam)
Competency
5 Marks
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 $y=0$.
Key:
Sol:
Sol:
#1458
Mathematics
Probability
SA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
3 Marks
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.
Key:
Sol:
Sol:
#1455
Mathematics
Linear Programming
SA
REMEMBER
2025
AISSCE(Board Exam)
Competency
3 Marks
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.
Key:
Sol:
Sol:
#1453
Mathematics
Relations and Functions
SA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
3 Marks
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.
Key:
Sol:
Sol:
#1444
Mathematics
Derivatives
VSA
APPLY
2025
AISSCE(Board Exam)
Competency
2 Marks
Differentiate $\sqrt{e^{\sqrt{2x}}}$ with respect to $e^{\sqrt{2x}}$ for $x>0$.
Key:
Sol:
Sol:
#1439
Mathematics
Derivatives
LA
REMEMBER
2025
AISSCE(Board Exam)
Competency
5 Marks
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.
Key:
Sol:
Sol:
#1438
Mathematics
Matrices and Determinants
LA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
5 Marks
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.
Key:
Sol:
Sol:
#1437
Mathematics
Applications of Integrals
LA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
5 Marks
Sketch a graph of $y=x^{2}$. Using integration, find the area of the region bounded by $y=9$, $x=0$ and $y=x^{2}$.
Key:
Sol:
Sol:
#1436
Mathematics
Probability
SA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
3 Marks
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).
Key:
Sol:
Sol:
#1430
Mathematics
Definite Integrals
SA
APPLY
2025
AISSCE(Board Exam)
Competency
3 Marks
Evaluate: $\int_{1}^{4}(|x-2|+|x-4|)dx$.
Key:
Sol:
Sol:
#1428
Mathematics
Applications of Derivatives
SA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
3 Marks
Find the value of 'a' for which $f(x)=\sqrt{3}\sin x-\cos x-2ax+6$ is decreasing in R.
Key:
Sol:
Sol:
#1416
Mathematics
Applications of Derivatives
LA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
5 Marks
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?
Key:
Sol:
Sol:
#1415
Mathematics
Matrices and Determinants
LA
ANALYZE
2025
AISSCE(Board Exam)
Competency
5 Marks
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 $A=\begin{bmatrix}2&-1&1\\ -1&2&-1\\ 1&-1&2\end{bmatrix}$.
Key:
Sol:
Sol:
#1414
Mathematics
Linear Programming
SA
REMEMBER
2025
AISSCE(Board Exam)
Competency
3 Marks
Solve the following Linear Programming Problem using graphical method: Maximise $Z=100x+50y$ subject to the constraints $3x+y\le600$, $x+y\le300$, $y\le x+200$, $x\ge0$, $y\ge0$.
Key:
Sol:
Sol:
#1413
Mathematics
Three Dimensional Geometry
SA
UNDERSTAND
2025
AISSCE(Board Exam)
Competency
3 Marks
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 $\frac{x-4}{5}=\frac{y-1}{2}=z$.
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: #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: