If $h(x) = 4^x$ and $h^{-1}(x) = 2$, then value of $x$ is:
- (a)$-4$
- (b)$4$
- (c)$-16$
- (d)$16$
No answer yet.
ISC Mathematics 2025 Competency Focused Questions (CFQ) with Answers
All 134 questions of the ISC Mathematics 2025 Competency Focused Questions (CFQ), in printed order, in full. Tap "Show answer" under a question to see its answer.
If $a + \frac{\pi}{2} < 2\tan^{-1} x + 3\cot^{-1} x < b$, then $a$ and $b$ are respectively:
- (a)$\frac{\pi}{2}$ and $2\pi$
- (b)$\frac{\pi}{2}$ and $-\frac{\pi}{2}$
- (c)$0$ and $\pi$
- (d)$0$ and $2\pi$
No answer yet.
Which one of the following is true?
- (a)$\sin(\cos^{-1} x) = \cos(\sin^{-1} x)$
- (b)$\sec(\tan^{-1} x) = \tan(\sec^{-1} x)$
- (c)$\cos(\tan^{-1} x) = \tan(\cot^{-1} x)$
- (d)$\tan(\sin^{-1} x) = \sin(\tan^{-1} x)$
No answer yet.
If a matrix $A = [a_{ij}]_{2 \times 2}$, where $a_{ij} = \begin{cases} 1, & i \neq j \\ 0, & i = j \end{cases}$, then $A^{-1}$ is:
- (a)$I$
- (b)$A$
- (c)$-A$
- (d)$-I$
No answer yet.
If the value of 3rd order determinant is $5$, then the value of determinant formed by replacing its element by its co-factor is:
- (a)$5$
- (b)$\frac{1}{5}$
- (c)$125$
- (d)$25$
No answer yet.
If $A = \begin{bmatrix} 0 & 5 & -y \\ -5 & 0 & x \\ y & -x & 0 \end{bmatrix}$, then the value of $A^{-1} \cdot (\operatorname{adj} A) A$ is:
- (a)$A^2$
- (b)$I$
- (c)$0$
- (d)$A$
No answer yet.
If $D = \begin{vmatrix} p & p & p \\ p & p+x & p \\ p & p & p+y \end{vmatrix}$ for $p \neq 0, x \neq 0, y \neq 0$ then $D$ is divisible by:
- (a)only $p$
- (b)$p$ and $x$ but not $y$
- (c)$p$ and $y$ but not $x$
- (d)$p, x$ and $y$
No answer yet.
If $\operatorname{adj}(A) = \begin{bmatrix} 2 & 3 & 5 \\ x & 5 & 1 \\ 3 & 3 & 4 \end{bmatrix}$ and $|A| = 4$, then the value of $x$ is:
- (a)$16$
- (b)$12$
- (c)$32$
- (d)$10$
No answer yet.
Which of the following could be a sketch of the function $y = \frac{d}{dx}(x \log x)$?
- (a)Figure (a)
- (b)Figure (b)
- (c)Figure (c)
- (d)Figure (d)
No answer yet.
If $\int \left(\frac{2-x}{(x-1)^2}\right) e^x \, dx = e^x f(x) + c$, then $f(x)$ will be:
- (a)$\frac{1}{x-1}$
- (b)$\frac{1}{2-x}$
- (c)$\frac{1}{3-x}$
- (d)$\frac{1}{1-x}$
No answer yet.
An integrating factor of the differential equation: $x \frac{dy}{dx} + y \cdot P = x \cdot e^x \cdot x^{-1/2} \log x$, where $P$ is a function of $x$, $x > 0$, is $(\sqrt{e})^{(\log x)^2}$, then $P$ is:
- (a)$(\log x)^2$
- (b)$\frac{1}{\log x}$
- (c)$\log x$
- (d)$e^{\log x}$
No answer yet.
Identify the correct answer of $\int_0^\pi \sin^{2024} x \cos^{2023} x \, dx$:
- (a)$0$
- (b)$1$
- (c)$2$
- (d)$3$
No answer yet.
Three fair dice are thrown. What is the probability of getting a total of 15 given that they exhibit three different numbers that are in arithmetic progression (A.P.)?
- (a)$\frac{1}{4}$
- (b)$\frac{1}{6}$
- (c)$\frac{1}{8}$
- (d)$\frac{1}{12}$
No answer yet.
Rohit and Vishal, two below-average students in a class, are attempting a Mathematics problem during revision classes. Their respective probabilities of solving the sum correctly are $\frac{1}{6}$ and $\frac{1}{8}$ respectively. Their previous experience shows that while solving the same question, the probability of a common mistake is $\frac{1}{10}$. What is the probability that they obtain the same answer?
- (a)$\frac{3}{4}$
- (b)$\frac{7}{48}$
- (c)$\frac{11}{96}$
- (d)$\frac{9}{96}$
No answer yet.
The value of $\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j})$ is:
- (a)$-3$
- (b)$-2$
- (c)$-1$
- (d)$0$
No answer yet.
Four students are playing a game. In a box, there are four strips of paper with four expressions written on each one. The student who picks up the meaningless expression will be out of the game.
• Swati picks up the expression $\vec{a} \cdot (\vec{b} \times \vec{c})$.
• Imran picks up the expression $\vec{a} \times (\vec{b} \times \vec{c})$.
• Aryan picks up the expression $(\vec{a} \cdot \vec{b}) \times (\vec{c} \cdot \vec{d})$.
• Maria picks up the expression $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})$.
Who is out of the game?
- (a)Swati
- (b)Imran
- (c)Aryan
- (d)Maria
No answer yet.
If $L$ is the foot of the perpendicular drawn from a point $P(a, b, c)$ on the $XY$ plane, what is the co-ordinate of $L$?
- (a)$(a, b, 0)$
- (b)$(0, 0, c)$
- (c)$(0, b, 0)$
- (d)$(a, 0, 0)$
No answer yet.
Find the equation of the plane passing through the point $(2, -1, 3)$ and containing the line: $\frac{x-1}{3} = \frac{2-y}{4} = \frac{z+1}{2}$
- (a)$2x - 2y + z = 9$
- (b)$2x + 2y + z = 5$
- (c)$11x + 2y + z + 21 = 0$
- (d)$11x - 2y + z + 17 = 0$
No answer yet.
Rohit joins a career counselling institute as a counsellor. The manager says, “Within a year I want a breakeven point. For that, I will give you ₹ 24,000 fixed salary per month and the variable salary will be 25% of the revenue recovered on hiring students at the rate of ₹ 800/- charged from every student.” Find how many students should be admitted by Rohit in a year in the institute to fulfil his manager's condition.
- (a)$30$
- (b)$40$
- (c)$80$
- (d)$100$
No answer yet.
The fixed cost of a new product is ₹ 200 and the variable cost per unit is ₹ $x$. If the demand function $p(x) = 30$, then what are the value(s) of $x$ that result in a loss given that $x$ units of the product are sold?
- (a)$10 < x < 20$
- (b)$x > 20$
- (c)$x < 10$
- (d)$x < 10\text{ or } x > 20$
No answer yet.
In statistical modelling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion.
Source: https://en.wikipedia.org/wiki/Regression_analysis
If plotted on a graph, the independent variable is represented along _________
- (a)Depends on the dataset.
- (b)Y axis.
- (c)X axis.
- (d)None of the above.
No answer yet.
The function represented by the given graph is not differentiable at which of the following points:
- (a)$-1, 0, 1$
- (b)$-1, 1$
- (c)$0$
- (d)$1$
No answer yet.
In an examination, a candidate takes three tests namely $\alpha, \beta, \gamma$ in succession and the probability of failing the first test $\alpha$ is $\frac{1}{2}$. The probability of passing each succeeding test is $\frac{1}{2}$ or $\frac{1}{4}$ according to whether he passes or fails in the preceding one. The candidate is selected, if he passes at least two tests. What is the probability that candidate is selected?
- (a)$\frac{3}{8}$
- (b)$\frac{1}{8}$
- (c)$\frac{5}{8}$
- (d)$\frac{3}{4}$
No answer yet.
In the picture given above, take a look at the double-arrowed lines drawn on the overpass. This is an example of skew lines in the real world.
Based on this, which of these statements is INCORRECT?
- (a)These lines are not parallel.
- (b)These lines are intersecting.
- (c)These lines are not coplanar.
- (d)These lines can only exist in 3 or higher dimensional space.
No answer yet.
The graph below shows the two lines of regressions for a certain set of $x$ and $y$ values. What is the mean of $x$ and mean of $y$?
- (a)$5$ and $8$
- (b)$8$ and $5$
- (c)$(1, 0)$
- (d)The mean of $x$ and mean of $y$ cannot be read from the graph.
No answer yet.
A point $x = c$ is called the critical point of a function if:
- (a)$f'(c) = 0$
- (b)$f$ is not differentiable at $x = c$.
- (c)both (a) & (b).
- (d)none of the above.
No answer yet.
Let $f(x)$ be a function such that $f'(x) = g(x)$ and $f''(x) = -f(x)$.
Let $h(x) = \{f(x)\}^2 + \{g(x)\}^2$.
Then, consider the following statements:
Statement I: $h'(2024) = 0$.
Statement II: $h(2) = h\left(\frac{1}{2}\right)$
Which of the statements given above is/are correct?
- (a)I only.
- (b)II only
- (c)Both I and II
- (d)Neither I nor II.
No answer yet.
Statement I: For any two real numbers $a$ and $b$, we define $a\text{ R }b$ if $\sec^2 a - \tan^2 b = 1$. Then, R is transitive.
Statement II: The relation R on the set $\{2, 3, 4\}$ defined by $\text{R} = \{(2, 2)\}$ is not symmetric.
Which of the following options is correct?
- (a)Both the statements are true.
- (b)Both the statements are false.
- (c)Statement I is true, and Statement II is false.
- (d)Statement I is false, and Statement II is true.
No answer yet.
Statement I: $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \frac{1}{x} - 2$, is neither injective nor surjective.
Statement II: $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = \sqrt[3]{x^9}$, is neither injective nor surjective.
- (a)Both the statements are true.
- (b)Both the statements are false.
- (c)Statement I is true, and Statement II is false.
- (d)Statement I is false, and Statement II is true.
No answer yet.
In the third-order matrix, $a_{ij}$ denotes the element of the $i^{\text{th}}$ row and $j^{\text{th}}$ column:
$a_{ij} = \begin{cases} 0, & i \neq j \\ 1, & i = j \end{cases}$
Statement I: A Matrix is an upper triangular matrix.
Statement II: The determinant of the matrix is equal to 1.
Which of the above statement/s is/are correct?
- (a)Only I.
- (b)Only II.
- (c)Both I and II.
- (d)Neither I nor II.
No answer yet.
$y = x^5 + x^3, x \in \mathbb{R}$ is a function. Rina, Abha and Saurabh have given their opinions about the function in the following statements:
• Statement 1: Rina says that 'y' is an increasing function for all values of $x$.
• Statement 2: Abha says that 'y' is an odd function.
• Statement 3: Saurabh says that 'y' is symmetrical about the origin.
Related to the above statements, which of the following option is true?
- (a)Rina, Abha and Saurabh are correct.
- (b)Rina and Abha are correct, but Saurabh is wrong.
- (c)Rina and Saurabh are correct, but Abha is wrong.
- (d)Saurabh and Abha are correct, but Rina is wrong.
No answer yet.
Statement I: $f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$ but $f'(x)$ is not continuous at $x = 0$.
Statement II: The derivative of a continuous function need not be a continuous function.
- (a)Both (I) and (II) are correct and (II) is the correct explanation of (I).
- (b)Both (I) and (II) are correct and (II) is not the correct explanation of (I).
- (c)(I) is correct but (II) is incorrect.
- (d)(II) is correct but (I) is incorrect.
No answer yet.
Statement I: $\int_{-2}^2 \frac{x^2}{1 + 2^x} \, dx = \frac{8}{3}$.
Statement II: $\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx$, if $f(x)$ is an even function.
- (a)Both the statements are true.
- (b)Both the statements are false.
- (c)Statement I is false, and statement II is true.
- (d)Statement I is true, and statement II is false.
No answer yet.
For any given events A:
Statement I: Event A and null event $\emptyset$ are always independent.
Statement II: Event A and sure event S are always independent.
- (a)Both the statements are true.
- (b)Both the statements are false.
- (c)Statement I is false, and statement II is true.
- (d)Statement I is true, and statement II is false.
No answer yet.
Which of the following statement(s) DOES NOT/DO NOT hold true related to regression analysis:
Statement I: $|r|$ is the geometric mean of $b_{yx}$ and $b_{xy}$.
Statement II: $b_{xy}, b_{yx}$ and $r$ all are of the same sign.
Statement III: The two regression lines do not intersect at $(\bar{x}, \bar{y})$.
Statement IV: $-1 \le b_{yx} \times b_{xy} \le 1$.
- (a)III and IV only.
- (b)IV only.
- (c)III only.
- (d)II and III only.
No answer yet.
Assertion: The relation $f: \{m, n, p, q\} \to \{11, 12, 13, 14\}$ defined by $f = \{(m, 11), (n, 12), (p, 13)\}$ is a bijective function.
Reason: The function $f: \{m, n, p\} \to \{11, 12, 13, 14\}$ such that $f = \{(m, 11), (n, 12), (p, 13)\}$ is one-one.
Assertion (A): The relation $f: \{m, n, p, q\} \to \{11, 12, 13, 14\}$ defined by $f = \{(m, 11), (n, 12), (p, 13)\}$ is a bijective function.
Reason (R): The function $f: \{m, n, p\} \to \{11, 12, 13, 14\}$ such that $f = \{(m, 11), (n, 12), (p, 13)\}$ is one-one.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true; Reason (R) is false.
- (d)Assertion (A) is false; Reason (R) is true.
No answer yet.
Assertion: Let $A = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$, then $A^{-1} = \begin{bmatrix} d_1^{-1} & 0 & 0 \\ 0 & d_2^{-1} & 0 \\ 0 & 0 & d_3^{-1} \end{bmatrix}$
Reason: If $A$ is a diagonal matrix, then $A^{-1}$ exists, it is also a diagonal matrix.
Assertion (A): Let $A = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$, then $A^{-1} = \begin{bmatrix} d_1^{-1} & 0 & 0 \\ 0 & d_2^{-1} & 0 \\ 0 & 0 & d_3^{-1} \end{bmatrix}$
Reason (R): If $A$ is a diagonal matrix, then $A^{-1}$ exists, it is also a diagonal matrix.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation for Assertion (A).
- (c)Assertion (A) is true, and Reason (R) is false.
- (d)Assertion (A) is false, and Reason (R) is true.
No answer yet.
Assertion: If $E_1$ and $E_2$ are two mutually exclusive events associated with a random experiment and $E$ is an event such that $P(E) \neq 0$, then $P\left(\frac{E_1 \cup E_2}{E}\right) = P\left(\frac{E_1}{E}\right) + P\left(\frac{E_2}{E}\right)$.
Reason: For two mutually exclusive events $E_1$ and $E_2$, $P(E_1 \cap E_2) = 0$.
If $E_1$ and $E_2$ are two mutually exclusive events associated with a random experiment and $E$ is an event such that: $P(E) \neq 0$.
Assertion (A): $P\left(\frac{E_1 \cup E_2}{E}\right) = P\left(\frac{E_1}{E}\right) + P\left(\frac{E_2}{E}\right)$.
Reason (R): For two mutually exclusive events $E_1$ and $E_2$, $P(E_1 \cap E_2) = 0$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (A) is not the correct explanation for Assertion (A).
- (c)Assertion (A) is true, and Reason (R) is false.
- (d)Assertion (A) is false, and Reason (R) is true.
No answer yet.
Assertion: $\vec{PQ} \times (\vec{RS} + \vec{ST}) \neq \vec{0}$.
Reason: $\vec{PQ} \times \vec{RS} = \vec{0}$ and $\vec{PQ} \times \vec{ST} \neq \vec{0}$.
The vectors $\vec{PQ}, \vec{QR}, \vec{RS}, \vec{ST}, \vec{TU}$ and $\vec{UP}$ represent the sides of a regular hexagon.
Assertion (A): $\vec{PQ} \times (\vec{RS} + \vec{ST}) \neq \vec{0}$.
Reason (R): $\vec{PQ} \times \vec{RS} = \vec{0}$ and $\vec{PQ} \times \vec{ST} \neq \vec{0}$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of A.
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
Assertion: A company uses a demand function $p = \frac{a}{x+b} - c$, where $a, b, c \in \mathbb{R}$ and $x = \text{number of units}$. The Marginal Revenue decreases with the increase of $x$.
Reason: $\frac{d}{dx}(MR) < 0$, where $0 < a < b$.
Assertion (A): A company uses a demand function $p = \frac{a}{x+b} - c$, where $a, b, c \in \mathbb{R}$ and $x = \text{number of units}$. The Marginal Revenue decreases with the increase of $x$.
Reason (R): $\frac{d}{dx}(MR) < 0$, where $0 < a < b$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
Assertion: The curve in the graph below is not a one-one function:
Reason: If any straight line parallel to y-axis does not cut the curve at more than one point, then that curve represents a function.
Assertion (A): The curve in the graph below is not a one-one function:
Reason (R): If any straight line parallel to y-axis does not cut the curve at more than one point, then that curve represents a function.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
Assertion: Let $f(x)$ be a polynomial function of degree 7 such that $\frac{d}{dx}(f(x)) = (x-2)^3(x+1)^2(7x-2)$ has a local minimum at $x = -1$.
Reason: Let $f$ have first derivative at $c$ such that $f'(c) = 0$ and $f'(x) > 0, \forall x \in (c-\delta, c)$, $f'(x) < 0, \forall x \in (c, c+\delta)$, then $c$ is a point of local minimum.
Assertion (A): Let $f(x)$ be a polynomial function of degree 7 such that $\frac{d}{dx}(f(x)) = (x-2)^3(x+1)^2(7x-2)$ has a local minimum at $x = -1$.
Reason (R): Let $f$ have first derivative at $c$ such that $f'(c) = 0$ and $f'(x) > 0, \forall x \in (c-\delta, c)$, $f'(x) < 0, \forall x \in (c, c+\delta)$, then $c$ is a point of local minimum.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
Assertion: If $y = \sin^{-1}(x\sqrt{x})$, then $\frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}$.
Reason: $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, |x| \le 1$.
Assertion (A): If $y = \sin^{-1}(x\sqrt{x})$, then $\frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}$.
Reason (R): $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, |x| \le 1$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of A.
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
Assertion: Degree of the differential equation $a\left(\frac{dy}{dx}\right)^2 + b\frac{dy}{dx} = c$ cannot be determined.
Reason: If each term involving derivatives of a differential equation is a polynomial (or can be expressed as a polynomial) then the highest exponent of the highest order derivative is called the degree of the differential equation.
Assertion (A): Degree of the differential equation $a\left(\frac{dy}{dx}\right)^2 + b\frac{dy}{dx} = c$ cannot be determined.
Reason (R): If each term involving derivatives of a differential equation is a polynomial (or can be expressed as a polynomial) then the highest exponent of the highest order derivative is called the degree of the differential equation.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
Assertion: Maximum value of the function is 0.
Reason: Minimum value of the function approaches $\infty$.
Shown below the graph of $f(x) = -2|x-3|$.
Assertion (A): Maximum value of the function is 0.
Reason (R): Minimum value of the function approaches $\infty$.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
No answer yet.
If the domain of the function $f(x) = \sqrt{\cos^{-1}(3x) + \frac{\pi}{4}}$ is $[a, b]$, then find the value of $a+b$.
No answer yet.
Each triangular face of the pyramid of Peace of Kazakhstan is made up of 25 smaller triangles as shown in figure below:
Using the above information and concept of determinant answer the following question:
If $(1,2)$ and $(3,6)$ are the coordinates of the two vertices of one of the smaller triangles and its area is 5 square cm, then find the equation of line on which the third vertex of the triangle lies.
No answer yet.
The figure given below shows the graph of a function $y = f(x)$. What is the derivative of the function?
No answer yet.
Find the angle between the normal to the planes:
$\vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 3$ and $\vec{r} \cdot (3\hat{i} + 2\hat{j} - \hat{k}) + 5 = 0$.
No answer yet.
The Average Revenue (AR) and Marginal Revenue (MR) of a company on selling $x$ units (on x-axis) is shown in the following graph:
At how many units, should the manufacturer limit production so that each additional unit DOES NOT bring losses?
No answer yet.
$f(x) = x^2 + 2x - 3$ for $x \ge -1$.
Given that the minimum value of $x^2 + 2x - 3$ occurs when $x = -1$, explain why $f(x)$ has an inverse.
No answer yet.
Explain why $x + 4y = 12$, $x, y \in \mathbb{N}$ is NOT a symmetric relation.
No answer yet.
If $\cos^{-1}\frac{x}{a} - \cos^{-1}\frac{y}{b} = \frac{\pi}{3}$ and $\sin^{-1}\frac{x}{a} + \sin^{-1}\frac{y}{b} = \frac{2\pi}{3}$, then find the value of $4\frac{x^2}{a^2} + \frac{y^2}{b^2}$.
No answer yet.
The matrix $A = \begin{bmatrix} a & 6 & b \\ 2c & x+1 & 8 \\ -1 & -8 & y-3 \end{bmatrix}$ is a skew symmetric matrix, then find the value of $ab + bc - xy$.
No answer yet.
The displacement, $x\text{ m}$ of a particle from a fixed point at time '$t$' seconds is given by $x = 6\cos\left(3t + \frac{\pi}{3}\right)$.
Find the acceleration of the particle, when $t = \frac{2\pi}{3}$.
No answer yet.
Evaluate: $\int_{-\pi/2}^{\pi/2} \sin|x| \, dx$.
No answer yet.
Rahul, a class XII student, has a probability of getting a grade A in the examination of three subjects namely Mathematics, Physics and Chemistry are $0.2$, $0.3$ and $0.5$ respectively. Find the probability that he gets a grade A in none of the subjects.
No answer yet.
If $A$ and $B$ are two events such that $P(\bar{A}) = 0.3$, $P(B) = 0.4$, $P(A \cap \bar{B}) = 0.5$, then find the value of $P(B / (A \cup \bar{B}))$.
No answer yet.
There are 10 cookies in a box. Six have chocolate centres and four have jam-filled centres. Shweta randomly chooses a cookie from the box and eats it. Then, Ali randomly chooses and eats one of the remaining cookies.
What is the probability that Shweta and Ali choose cookies with different centres?
No answer yet.
A biased dice is tossed and the respective probabilities for various faces to turn up are:
If an odd face has turned up, then what is the probability that it is face 1 or face 3?
| Face | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Probability | 0.10 | 0.24 | 0.19 | 0.18 | 0.15 | 0.14 |
No answer yet.
If $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|\vec{c}| = 3$ and $\vec{a} + \vec{b} + \vec{c} = 0$, then verify $\vec{a}, \vec{b}, \vec{c}$ are NOT mutually perpendicular.
No answer yet.
If $\vec{p} \times \vec{q} = \vec{0}$ and $\vec{p} \cdot \vec{q} = 0$, then what conclusion can we draw?
No answer yet.
A line makes angles $\alpha, \beta$ and $\gamma$ with the x, y and z axes respectively. Given that $\alpha + \beta = \frac{\pi}{2}$, then what is the value of $\gamma$?
No answer yet.
A variable plane at unit distance from the origin cuts the co-ordinate axes at A, B and C. The centroid of the triangle formed by joining the points A, B and C satisfies the relation $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = k$, then what is the value of $k$?
No answer yet.
The angle between the line $\frac{x+1}{1} = \frac{y-1}{2} = \frac{z-2}{2}$ and the plane $2x - y + \sqrt{k}z + 4 = 0$ is $\alpha$. Given $\sin\alpha = \frac{1}{3}$, what will be the value of $k$?
No answer yet.
The cost function of a product is given by $C(x) = \frac{x^3}{3} - 45x^2 - 900x$ where $x$ is the number of units produced. What will be the slope of marginal cost?
No answer yet.
A company paid ₹ 8475 towards the rent of the building and ₹ 7625 interest on the loan. The cost of producing one unit of a product is ₹ 20. If each unit is sold for ₹ 27, find the break-even point.
No answer yet.
The total cost of producing and marketing $x$ units of bulbs by a wholesaler is given by $C(x) = \frac{1}{3}x^2 + e^{2x} + 3e$.
What will be the average cost of producing 3 units of bulbs?
No answer yet.
Nikhil is investigating the growth patterns of a certain species of slug and measures their thickness, $t$, and length, $l$. The results are shown below:
Using scatter diagram, explain why a line of best fit should not be used for this data?
| Thickness $t$ (mm) | Length $l$ (cm) |
|---|---|
| 9 | 5.9 |
| 8 | 4.1 |
| 3 | 2.1 |
| 6 | 1.8 |
| 4 | 3.0 |
| 10 | 7.8 |
No answer yet.
If the covariance of $x$ and $y$ is $58.08$, the variance of $x$ is $121$ and the standard deviation of $y$ is $8$ then, find the regression coefficient of $x$ and $y$.
No answer yet.
The values of two functions $f$ and $g$ for certain values of $x$ are given in the following table:
| $x$ | -2 | 0 | 3 |
|---|---|---|---|
| $f(x)$ | -12 | -4 | 8 |
| $g(x)$ | 0 | -12 | 30 |
(a)
Find the value of $f^{-1}(8)$?
(b)
Given that $f(x)$ is a linear function, find $f(x)$?
No answer yet.
“A function $f$ is called a self-inverse function if $f^{-1}(x) = f(x)$ for all values of $x$ in the domain.”
Let $f(x) = \frac{\pi^2}{x}$, where $x \neq 0, x \in \mathbb{R}$.
Show that $f(x)$ is a self-inverse function.
No answer yet.
Rahul sits on a speed boat in an island to do water sports, which is moving along a curve $y = \frac{1}{\cos x \sin x}$ in water. Find the number of stationary points.
No answer yet.
The function defined by $f(x) = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is differentiable at $x = 0$.
Is this statement true or false? Give reason for your answer.
No answer yet.
Seema enjoys a roller coaster ride in Ferrari world by first going downwards and then upwards to the maximum height. The relation between the distance travelled (cm) with respect to the time taken to complete the side by Seema is given by the following equation:
$y = 4x - \frac{1}{2}x^2$
where $x = \text{time in seconds}$.
(a)
What is the rate of change of displacement with respect to the time?
(b)
How many seconds it will take her to go to its maximum height?
No answer yet.
A child is solving an equation $x^2 + y^2 = 1$ and finds that this is the solution of the following differential equation: $1 + y y'' + (y')^2 = 0$. Justify his answer.
No answer yet.
Given $2f(x) = \log|x|^a - bx^2 + x$, if $f(x)$ has extreme values at $x = -1$ and $x = 2$, find the value of $a$ and $b$.
No answer yet.
The solution of $x\,dy - y\,dx = 0$ represents a family of straight lines passing through the origin. Justify.
No answer yet.
Find the function ‘f’ which satisfies the equation $\frac{df}{dx} = 2f$, given that $f(0) = e^3$.
No answer yet.
Evaluate: $\int 2^{2^{2^x}} \cdot 2^{2^x} \cdot 2^x \, dx$.
No answer yet.
Evaluate: $\int \{f(ax+b)\}^n \cdot f'(ax+b) \, dx, n \neq -1$.
No answer yet.
Evaluate: $\int \frac{dx}{x^{1/2} - x^{1/3}}$.
No answer yet.
Derive a condition for which given, $\vec{b} = \vec{c}$, we can say $\vec{a} \times \vec{b} = \vec{c} \times \vec{a}$?
No answer yet.
If the angle between $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ is acute and the angle between $\vec{b}$ and the x-axis lies between $\frac{\pi}{2}$ and $\pi$. Find the range of values for $x$.
No answer yet.
To launch a new product in his company, Mr. Rajesh spends ₹ 1 lakh on the infrastructure and the variable cost of the product is estimated as ₹ 150/- per unit. The sale price per unit is fixed as ₹ 200/-. Additionally, Mr Rajesh also spends ₹ 0.5 per unit squared for marketing.
Find the profit function. Hence, draw an inference regarding the breakeven point.
No answer yet.
Anuj and Ashish launched a new fountain pen in their pen-factory which is consisting of ₹ 6400/- as overheads, ₹ 35/- per pen as the cost of material and labour cost, ₹ $\frac{x^2}{100}$ for $x$ items produced. Find the values of $x$ for which average cost is increasing.
No answer yet.
If $\tan^{-1}\left(\frac{1}{1+1\cdot 2}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \dots + \tan^{-1}\left(\frac{1}{111}\right) = S$, then find $\tan S$.
No answer yet.
The adjacent figure is the graph of function $y = f(x)$. Give answers to the following questions.
(a)
Name the type of discontinuity of the function.
(b)
Give reason for your answer.
No answer yet.
If $|\vec{a} - \vec{b}| = |\vec{a} + \vec{b}|$ in the given figure, what inference can you draw?
No answer yet.
In Z Square Mall in Kanpur, there is a space to keep 300 cars and the entry fees per car is ₹ 20. It is estimated that if the entry fee is decreased by ₹ 5, then 50 additional cars can be adjusted in the same parking.
Justify that the Marginal Revenue (MR) decreases at a higher rate than the Average Revenue (AR).
No answer yet.
Consider the functions $f(x) = -(x-h)^2 + 2k$ and $g(x) = e^{x-2} + k$, where $h, k \in \mathbb{R}$.
(a)
Find $f'(x)$.
(b)
The graphs of $f$ and $g$ have a common tangent at $x = 3$. Show that: $2h = e + 6$.
No answer yet.
Evaluate: $\int 2^x [f'(x) + f(x)\log 2] \, dx$.
No answer yet.
Let $f(x) = \frac{\log 5x}{kx}$, where $x > 0, k \in \mathbb{R}^+$.
(a)
Show that $f'(x) = \frac{1-\log 5x}{kx^2}$.
(b)
The graph of $f$ has exactly one maximum point at P. Find the $x$-coordinate of P.
(c)
Find $f''(x)$.
(d)
Find the value of $x$ for which $f''(x)$ vanishes.
No answer yet.
A linear programming problem (LPP) is given as:
Maximize $Z = x + 2y$ subject to the constraints
$x - y \ge 0, 2 \ge 2y - x, x \ge 0, y \ge 0$
Based on the above information, answer the following questions.
(a)
Find the corner points of the feasible region.
(b)
Find the corner point where maximum occurs.
(c)
Optimum solution does not exist. Justify your answer.
No answer yet.
There are two curves given in the first quadrant as:
$x^2 + y^2 = \pi^2, y = \sin x$.
(a)
What are the points of intersection of both the given curves?
(b)
What is the value of K, if $\int_0^\pi \sqrt{\pi^2 - x^2} \, dx = \frac{\pi^3}{K}$?
(c)
Sketch the region enclosed by the given curves in the first quadrant and the y–axis.
(d)
Find the area of the region enclosed by the given curves in the first quadrant and the y–axis.
No answer yet.
Let $\vec{\alpha} = 3\hat{i} + \hat{j}$ and $\vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k}$. If $\vec{\beta} = \vec{\beta}_1 - \vec{\beta}_2$ where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular to $\vec{\alpha}$.
(a)
Find $\vec{\beta}_1$.
(b)
Find $\vec{\beta}_2$.
(c)
Hence, find $\vec{\beta}_1 \times \vec{\beta}_2$.
No answer yet.
Given $f(x) = 2\log(x-2) - x^2 + 4x + 1$ and $f'(x) = \frac{-k(x-p)(x-q)}{(x-k)}$.
(a)
Find $k + p + q$.
(b)
The Statement “$f(x)$ is strictly increasing in $(-\infty, 1] \cup (2, 3]$” is false. Justify.
(c)
Hence, find the interval(s) in which the function is strictly increasing.
No answer yet.
Case Study:
The length of the perimeter of a slice of a pizza in the form of a sector of circle is 20 cm.
$r$ be the radius of the circle, sectorial angle be $\theta$ radian and $l$ be the length of the arc.
Based on the above information, answer the following questions.
(a)
Express the radius of the sector in terms of sectorial angle $\theta$ radian.
(b)
Let A be the area of the slice. Then, express A in terms of r.
(c)
For the maximum value of A, find the value of the sectorial angle.
(d)
Maximum area of the slice of the pizza is ____.
No answer yet.
In a classroom, a teacher explains the properties of a particular curve by saying that this particular curve has beautiful ups and downs. It starts and heads down until $\pi$ radian, and then heads up again and is closely related to sine function. Both follow each other at exactly $\frac{\pi}{2}$ radians apart as shown in the figure given below:
Based on the above information, answer the questions that follow.
(a)
Name the curve that the teacher explained in the classroom.
(b)
Find the area of the curve explained in the passage from 0 to $\frac{\pi}{2}$.
(c)
Find the area of the shaded region.
No answer yet.
Answer the following questions:
(a)
If a real-valued function is given by: $f(x) = \sqrt{25 - x^2}$ is an onto function, then find the co-domain for $f(x)$.
(b)
If the domain is given to be $[-5, 5]$, is $f(x)$ a one-one function?
(c)
Find all possible values of ‘a’ for which $f(a) = 4$.
No answer yet.
Find the value of:
$\cot \left[\sum_{n=1}^{25} \cot^{-1} \left(1 + \sum_{k=1}^n 2k\right)\right]$.
No answer yet.
If $f: [1, \infty) \to [2, \infty)$ is given by $f(x) = x + \frac{1}{x}$ then, find $\frac{d}{dx}f^{-1}(x)$.
No answer yet.
Find the acute angle between the curves $y = |x^2 - 1|$ and $y = |x^2 - 3|$ at their point of intersection when $x > 0$.
No answer yet.
$y = \ln(x+1) - \ln x$ is a curve. The tangent to the curve at the point $P(1, \ln 2)$ meets x-axis at A and y-axis at B. The normal to the curve at P meets the x-axis at C and y-axis at D.
(a)
Find the slope of tangent at P and find the slope of normal at P.
(b)
Find the equation of tangent at P.
(c)
Find the equation of normal at P.
(d)
Find the co-ordinates of A and C in terms of $\ln 2$.
No answer yet.
Solve the following differential equation:
$\cos^2 x \frac{dy}{dx} + y = \tan x$, given that $y(0) = 1$. Hence, find $y\left(\frac{\pi}{4}\right)$.
No answer yet.
Evaluate: $\int \frac{dx}{\sqrt[4]{(x-1)^3 (x+2)^5}}$.
No answer yet.
For any function $f(x)$, we have:
$\int_a^b f(x) \, dx = \int_a^{c_1} f(x) \, dx + \int_{c_1}^{c_2} f(x) \, dx + \dots + \int_{c_n}^b f(x) \, dx$
where $a < c_1 < c_2 < \dots < c_n < b$.
Based on above information, evaluate: $\int_0^2 |x^2 + 2x - 3| \, dx$.
No answer yet.
A pot contains 5 red and 2 green balls. At random a ball is drawn from this pot. If a drawn ball is green, then put a red ball in the pot. If a drawn ball is red, then put a green ball in the pot. While drawn ball is not replaced in the pot. Now, we draw another ball randomly. What is the probability that the second ball drawn is a red ball?
No answer yet.
Rahul and Divya were playing the snakes and ladders board game. Each one had their own dice to play the game. Rahul was using a red dice, whereas Divya was using a black dice. In the beginning of the game, they were using their own dice to play. After some time, in order to play the game faster they both started using both the dice together for playing.
When Divya rolled both red and black dice together then:
(a)
find the conditional probability of obtaining sum greater than 9, given that black dice resulted in a 5.
(b)
find the conditional probability that sum of the number on the dice is not 4, given that the numbers on the both the dice are different.
No answer yet.
Imagine you are at a point A, a café you visit often. Your friend is at the point B, a bookstore a few blocks away on a straight road. You want to meet your friend at a point on the line joining the café and the bookstore. Another friend, who is at home on the other side of the same road represented by point P, also wants to join. You decide to determine the exact meeting point by finding the foot of the perpendicular from P on the line joining the café and the bookstore.
Given that co-ordinates of the café (A) are $(1, 2, 4)$, of the bookstore (B) are $(3, 4, 5)$ and the home are $(2, 1, 3)$, find the location of the meeting point.
No answer yet.
Two friends are planning a road trip. One friend stays in City A represented by the position vector $(-2\hat{i} + 3\hat{j} + 5\hat{k})$. The trip will start from City A and proceed towards the City B represented by the position vector $(\hat{i} + 2\hat{j} + 3\hat{k})$. The friend living in City C represented by the position vector $7\hat{i} - \hat{k}$ will join when the first friend passes through her city.
(a)
Find the vector equation for the straight path between the cities A and B.
(b)
Hence, find out whether the three cities lie on the same straight path.
(c)
If the two friends now plan to travel $\sqrt{126}$ units along the vector $\vec{AB}$ from the City C, find the position vector of the destination point.
No answer yet.
In the beautiful town of Darjeeling in the Himalayan foothills, the city planning committee wants to construct two major roads to connect the various neighbourhoods. The two roads are represented by the equations $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ and $\frac{x-2}{1} = \frac{y-4}{k} = \frac{z-6}{7}$. As an in charge of the planning committee, ensure that these roads lie on the same plane to facilitate efficient urban planning and infrastructure development.
(a)
For what value of k, will the construction meet the requirement?
(b)
Hence, find the equation of the plane containing these lines.
No answer yet.
The plane $px + y + pz = r$ intersects the co-ordinate axes at A, B and C.
(a)
Find the co-ordinates of the points A, B and C.
(b)
Find the co-ordinates of orthocentre of the triangle ABC.
No answer yet.
From any point $P(2, 1, 2)$ perpendiculars PM and PN are drawn to ZX and XY planes.
(a)
If O is the origin, find the equation of the plane OMN.
(b)
Find $\theta$, if $\theta$ is the angle made by OP with the plane OMN.
(c)
If $\alpha, \beta$ and $\gamma$ are the angles made by OP with the co-ordinate planes, prove that $\csc^2 \theta = \csc^2 \alpha + \csc^2 \beta + \csc^2 \gamma$.
No answer yet.
Draw a rough sketch and find the area enclosed by the curve $y = -x^2$ and the line $x + y + 2 = 0$.
No answer yet.
Find the area of the region bounded by the x-axis, part of the curve $y = 1 + \frac{8}{x^2}$, and the ordinates $x = 2$ and $x = 4$. If the ordinate at $x = a$ divides the area into two equal parts, find the value of $a$.
No answer yet.
Find the area enclosed between the co-ordinate axis and the curve $y^2 = 4a(x + \lambda)$ in the second quadrant.
No answer yet.
A study was conducted to investigate the relationship between the number of hours a student studies per week (X) and their scores on a standardized test (Y). The following statistical data was collected from a sample of 50 students:
The correlation coefficient between X and Y is 0.65.
| X | Y | |
|---|---|---|
| Mean | 15 | 75 |
| Standard Deviation (SD) | 4 | 10 |
(a)
Estimate the test score for a student who studies 20 hours per week.
(b)
If the pass mark is 40, then how many hours does a student need to study to pass?
No answer yet.
The corner points of the feasible region determined by the system of linear constraints are as shown below:
Answer the following questions.
(a)
Let $Z = 3x - 4y$ be the objective function. Find the maximum and minimum value of $Z$ and also the corresponding points at which the maximum and minimum value occurs.
(b)
Let $Z = px + qy$ where $p, q > 0$ be the objective function. Find the condition on $p$ and $q$ so that the maximum value of $Z$ occurs at $B(4,10)$ and $C(6,8)$.
(c)
State the number of optimal solutions in this case.
No answer yet.
A movie cinema is considering significantly reducing the price of their popcorn as they believe their customers spend more on drinks when they buy popcorn. They recorded the following data of the daily revenue from popcorn, ‘x’, and the daily revenue from drinks, ‘y’ over 8 randomly selected days:
| Popcorn revenue (‘x’) | Drinks revenue (‘y’) |
|---|---|
| 14 | 22 |
| 12 | 23 |
| 12 | 17 |
| 14 | 24 |
| 16 | 18 |
| 10 | 25 |
| 13 | 23 |
| 12 | 24 |
(a)
Find $\bar{x}, \bar{y}$.
(b)
Using $\bar{x}, \bar{y}$, find regression coefficient of y on x.
(c)
The equation of the regression line y on x is in the form $y = a + bx$. Calculate the values of $a$ and $b$.
No answer yet.
A part of the graph of the function $f(x) = 2x^3 - 3x^2 - 12x + 8, x \in \mathbb{R}$ is shown below:
Answer the following questions.
(a)
Explain why ‘f’ does not have an inverse.
(b)
The domain of ‘f’ is now restricted to $a \le x \le b$ where $a < 0$ and $b > 0$. $a$ and $b$ are chosen so that f has an inverse and the interval $[a, b]$ is as large as possible. Find the domain and range of $f^{-1}$.
No answer yet.
In a Kabaddi league, two matches are being played between Jaipur and Delhi. It is assumed that the outcomes of two games are independent. The probability of Jaipur winning, drawing and losing the game against Delhi are $\frac{1}{2}, \frac{3}{10}$, and $\frac{1}{5}$ respectively.
Each team gets 5 points for win, 3 points for draw and 0 point for loss in a game. After two games, find the probability that:
(a)
Jaipur has more points than Delhi.
(b)
Jaipur and Delhi have equal points.
No answer yet.
Two drones are being used for soil analysis over an area of farmland. Drone A has been programmed to fly on the path given by $\vec{r} = 6\hat{i} + 2\hat{j} + 2\hat{k} + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})$ and drone B has been programmed to fly on the path $\vec{r} = -4\hat{i} - \hat{k} + \mu(3\hat{i} - 2\hat{j} - 2\hat{k})$.
At what points on their respective paths should they reach, so that they will be closest to each other?
No answer yet.
A manufacturing company produces two types of cell phones, Android and iOS.
The company has resources to make at the most 300 sets a week. It takes ₹ 1800 to make an Android set and ₹ 2700 to make an iOS set. The company cannot spend more than ₹ 648000 a week to make cell phones. The company makes a profit of ₹ 510 per Android and ₹ 675 per iOS set. If $x$ and $y$ denote, respectively, the number of Android sets and iOS sets made each week, then formulate this problem as a Linear Programming Problem (LPP) given that the objective is to maximize the profit. Based on it, answer the questions that follow.
(a)
What will be the maximum profit function on $x$ and $y$ sets? (write your objective based on the above data).
(b)
What will be the values of your objective function in the feasible region? (at corner points)
(c)
At what point the maximum profit will occur?
(d)
What’s the weekly cost (in ₹) of manufacturing the sets?
No answer yet.
Rajesh wants to purchase some fruits from fruit market. 4 kilograms (kgs) apples, 3 kgs grapes and 2 kgs oranges cost him ₹ 600, 2 kgs apples, 4 kgs grapes and 6 kgs oranges cost him ₹ 900, and 6 kgs apples, 2 kgs grapes and 3 kgs oranges cost him ₹ 700.
Using the given information, answer the following questions.
(a)
Express the given data in the form of a set of simultaneous equation.
(b)
Solve the set of simultaneous equations formed in sub part (a) by matrix method.
(c)
Hence, find how much Rajesh has to pay per kilogram (Kg) for each fruit.
No answer yet.
In a raffle draw, 1000 raffle tickets are sold for ₹ 1 each. Each has an equal chance of winning. First prize is ₹ 300, second prize is ₹ 200, and third prize is ₹ 100. Let X denote the net gain from the purchase of one ticket.
(a)
Construct the probability distribution of X.
(b)
Find the probability of winning any money in the purchase of one ticket.
(c)
Find the expected value of X and interpret its meaning.
No answer yet.
Ram and Shyam play a game with a coin. Ram stakes ₹ 1.00 and throws the coin 4 times. If he throws 4 heads, he gets his stake and ₹ 3.00 from Shyam. If he throws only three heads and they are consecutive, he gets his stake and ₹ 2.00 from Shyam. If he throws only 2 heads and they are consecutive, he gets his stake and ₹ 1.00 from Shyam. In all other cases, Shyam takes the stake money. Is this game fair? Provide reasons for your answer.
No answer yet.
The fuel cost per hour for running a ship is proportional to the square of the speed generated in knots. The fuel cost is ₹ 75/h at 10 knots and the fixed charges amount to ₹ 1000/h.
(a)
Given that the fuel cost per hour is k times the square of the speed, the ship generates in km per hour, then what is the value of ‘k’?
(b)
What will be the cost per unit distance?
(c)
Determine the most economical speed to run the ship.
No answer yet.
Evaluate: $\int \frac{\tan\frac{x}{4}}{1 - \sin\frac{x}{4}} \, dx$.
No answer yet.
Evaluate the following integral: $\int \left(\frac{\cot x + \cot^3 x}{1 + \cot^3 x}\right) \, dx$.
No answer yet.
Let $f(x) = \begin{cases} a + \sin^{-1}(x+b), & x \ge 1 \\ x, & x < 1 \end{cases}$, and $f'(1)$ exists.
(a)
The statement “$f(x)$ is continuous at $x = 1$” is true. Justify.
(b)
Hence, find a relation between $a$ and $b$.
(c)
Find $f'(x)$.
(d)
Hence, find the values of $a$ and $b$.
No answer yet.
Let $f(x) = \cos x + \sqrt{3}\sin x, 0 \le x \le 2\pi$.
The following diagram shows the graph of $f$.
The $y$-intercept is at $(0, 1)$ and intersects $x$-axis at C and D. There is a minimum point at $A(p, q)$ and a maximum point at B.
Based on the above information, answer the questions that follow.
(a)
Write $f'(x)$ in the form of $\lambda\cos(x+\mu)$.
(b)
Find the value of $q$.
(c)
Find the coordinate of the point B.
(d)
Find the interval $f(x)$ is decreasing.
(e)
Find the slope of the tangent to the curve at D.
(f)
Find the slope of the normal to the curve at C.
No answer yet.
Consider the following differential equation and answer the questions:
$\left[x\cos\left(\frac{y}{x}\right) + y\sin\left(\frac{y}{x}\right)\right] y\,dx - \left[y\sin\left(\frac{y}{x}\right) - x\cos\left(\frac{y}{x}\right)\right] x\,dy = 0$
(a)
Transform the above equation in the form $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$.
(b)
Use appropriate substitution to transform it into variable separable form.
(c)
Write the differential equation in variable separable form.
(d)
Prove that the solution of the differential equation is $\sec\left(\frac{y}{x}\right) = cxy$.
(e)
Find the solution if $x = 1, y = 1$.
No answer yet.
Sonia watches a painting which has its bottom edge 2 meters (m) above eye level and its top edge is 3 m above eye level as shown in the diagram.
Based on the above information answer the questions that follow.
(a)
Given $\alpha$ and $\theta$ as shown in the diagram, find $\tan\alpha$ and $\tan(\alpha+\theta)$.
(b)
Find $\theta$ in terms of $x$ only.
(c)
Find $\frac{d\theta}{dx}$.
(d)
Find $x$ so that $\frac{d\theta}{dx} = 0$.
(e)
Use 1st derivative test, find the distance Sonia should stand from the wall to maximize her viewing angle of the painting.
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