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Mathematics Exam Preparation
English 37 leveled MCQs Free
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5 marks Long answer
The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side.
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x = 41
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5 marks Long answer
Prove that: $\tan 4x = \frac{4 \tan x (1-\tan^2 x)}{1-6 \tan^2 x + \tan^4 x}$.
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Proved using trigonometric identities.
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5 marks Long answer
If $A = \{3, 5, 7, 9, 11\}$, $B = \{7, 9, 11, 13\}$, $C = \{11, 13, 15\}$ and $D = \{15, 17\}$; find (i) $A \cap (B \cup C)$ (ii) $A \cap D$ (iii) $A \cap (B \cup D)$ (iv) $(A \cap B) \cap (B \cup C)$ (v) $(A \cup D) \cap (B \cup C)$ and draw the venn diagram for each operation.
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Results obtained and Venn diagrams drawn.
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5 marks Long answer
Find the modulus of $(1 + \frac{2+i}{3+i})$.
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1
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5 marks Long answer
Raj works at a book store. While arranging some books on the book shelf, he observed that there are 5 History books, 3 Mathematics books and 4 Science books which are to be arranged on the shelf. (i) In how many ways can he select either a History book or a Maths book? (ii) If he selects 2 History books, 1 Maths book and 1 Science book to arrange them, then find the number of ways in which selection can be made. (iii) Find the number of ways, if the books of same subject are put together. (iv) Find the number of arrangements, if he selects 3 History books, 2 Maths Books, 2 Science books.
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1) 8; 2) 120; 3) 31104; 4) 907200
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5 marks Long answer
A man questioned his maths teacher that "if he has more than one linear inequality, then what will be the solution?" The teacher replies, "the solution set of a system of linear inequations in one variable is the intersection of the solution sets of the linear inequalities in given system." After that, the teacher gave that man to solve. Based on the above information, answer the following: 1. What is the solution of the linear inequality $2(2x+3)-10<6(x-2)$. 2. What is the solution of the linear inequality $\frac{2x-3}{4}+6 \ge 2+\frac{4x}{3}$. 3. The solution of the linear inequalities given by (i) and (ii) above.
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1) $(4, \infty)$; 2) $(-\infty, 39/10]$; 3) $(4, 39/10]$
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5 marks Long answer
To make himself self-dependent and to earn his living, a person decided to setup a small scale business of manufacturing hand sanitizers. He estimated a fixed cost of Rs. 15000 per month and a cost of Rs. 30 per unit to manufacture. (a) If x units of hand sanitizers are manufactured per month. What is the cost function? (b) If each unit is sold for Rs. 45. What is the selling (revenue) function? (c) What is the profit function? (d) For Break even (No Profit, no loss situation) in a month, how many units should be manufactured and sold?
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(a) $30x + 15000$; (b) $45x$; (c) $15x - 15000$; (d) $x=1000$
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3 marks Medium answer
Let $A = \{1, 2, 3, 4, 6\}$. Let R be the relation on A defined by $R = \{(a, b): a, b \in A, b \text{ is exactly divisible by } a\}. (i) Write R in roster form (ii) Find the domain of R (iii) Find the range of R.
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R = \{(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(6,6)\}; Domain = A; Range = \{1,2,3,4,6\}
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3 marks Medium answer
How many committee of five persons with a chairperson can be selected from 12 persons?
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3960
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3 marks Medium answer
Write the given complex number $(1-i) - (-1+i6)$ in the form $a+ib$.
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2+(-i7)
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3 marks Medium answer
The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.
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(7.77, 8.67)
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3 marks Medium answer
Find the radian measure corresponding to $-47^\circ 30'$
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-19\pi / 72
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3 marks Medium answer
Assume that $A = \{1, 2, 3, ..., 14\}$. Define a relation R in roster form from A to A by $R = \{(x, y): 3x - y = 0, \text{ such that } x, y \in A\}$. Determine and write down its range, domain, and codomain.
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Domain = \{1, 2, 3, 4\}; Range = \{3, 6, 9, 12\}; Codomain = A
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3 marks Medium answer
A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. Find the number of different ways of doing questions.
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780
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3 marks Medium answer
Let A, B and C be sets. Then show that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.
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Proved using set identities.
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3 marks Medium answer
Find $(a+b)^4 - (a-b)^4$. Hence evaluate $(\sqrt{3}+\sqrt{2})^4 - (\sqrt{3}-\sqrt{2})^4$.
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40\sqrt{6}
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3 marks Medium answer
If $(\frac{1+i}{1-i})^3 - (\frac{1-i}{1+i})^3 = x+iy$. Then find $x, y$.
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0, -2
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1 marks Short answer
How many elements are there in the complement of set A?
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None of these
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1 marks Short answer
Order of the power set P(A) of a set A of order n is equal to:
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$2^n$
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1 marks Short answer
If $f(x) = x^3 - \left(\frac{1}{x^3}\right)$, then $f(x) + f\left(\frac{1}{x}\right)$ is equal to
1 marks Short answer
If $f(x) = x^2 + 2$, $x \in R$, then the range of $f(x)$ is
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$[2, \infty)$
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1 marks Short answer
If $f(x) = ax + b$, where a and b are integers, $f(-1) = -5$ and $f(3) = 3$, then a and b are equal to
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$a = 2, b = -3$
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1 marks Short answer
If $\tan A = 1/2$ and $\tan B = 1/3$, then the value of $A + B$ is
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$\pi/6$
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1 marks Short answer
The value of $\cos 1^\circ \cos 2^\circ \cos 3^\circ \dots \cos 179^\circ$ is
1 marks Short answer
The value of $\sin (45^\circ + \theta) - \cos (45^\circ - \theta)$ is
1 marks Short answer
The value of $1 + i^2 + i^4 + i^6 + \dots + i^{2n}$ is
1 marks Short answer
The simplified value of $\frac{(1-i)^3}{(1-i^3)}$ is
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-2
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1 marks Short answer
The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then
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breadth > 20 cm
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1 marks Short answer
If $ -3x + 17 < -13$, then
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$x \in (10, \infty)$
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1 marks Short answer
Given that x, y and b are real numbers and x < y, b < 0, then
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$x/b < y/b$
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1 marks Short answer
The value of $P(n, n-1)$ is
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n
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1 marks Short answer
The number of ways 4 boys and 3 girls can be seated in a row so that they are alternate is
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144
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1 marks Short answer
A coin is tossed n times, the number of all the possible outcomes is
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$2^n$
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1 marks Short answer
The coefficient of the middle term in the expansion of $(2+3x)^4$ is:
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216
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1 marks Short answer
The fourth term in the expansion of $(x-2y)^{12}$ is:
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$-1760 x^9 y^3$
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1 marks Short answer
Assertion: The equation $x^2 + 1 = 0$ has no real solutions. Reason: The square of any real number is always non-negative.
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assertion and reason are true, reason is a correct explanation for assertion.
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1 marks Short answer
Assertion: The inequality $ax + by < 0$ is strict inequality. Reason: The inequality $ax + by \ge 0$ is slack inequality.
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assertion is true, reason is false.
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