The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1.
Question: The 4thterm of an A.P. is three times the first and the 7thterm exceeds twice the third term by 1. Find the first term and the common difference. Solution: In the given problem, let us take the first term asaand the common difference asd Here, we are given that, $a_{4}=3 a$.......(1) $a_{7}=2 a_{3}+1$........(2) We need to findaandd So, as we know, $a_{s}=a+(n-1) d$ For the $4^{\text {th }}$ term $(n=4)$, $a_{4}=a+(4-1) d$ $3 a=a+3 d$ $3 a-a=3 d$ $2 a=3 d$ $a=\frac{3}{2} d$ Similarly, ...
Read More →The number of diagonals that can be drawn by joining the vertices of an octagon is
Question: The number of diagonals that can be drawn by joining the vertices of an octagon is (a) 20 (b) 28 (c) 8 (d) 16 Solution: (a) 20 An octagon has 8 vertices. The number of diagonals of a polygon is given by $\frac{n(n-3)}{2}$ $\therefore$ Number of diagonals of an octagon $=\frac{8(8-3)}{2}=20$...
Read More →Solve the following
Question: If43Cr 6=43C3r+ 1, then the value ofris (a) 12 (b) 8 (c) 6 (d) 10 (e) 14 Solution: (a) 12 $r-6+3 r+1=43 \quad\left[\because{ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow n=x+y\right.$ or $\left.x=y\right]$ $\Rightarrow 4 r-5=43$ $\Rightarrow 4 r=48$ $\Rightarrow r=12$...
Read More →Mark the correct alternative in the following question:
Question: Mark the correct alternative in the following question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be given by $f(x)=\tan x .$ Then, $f^{-1}(1)$ is (a) $\frac{\pi}{4}$ (b) $\left\{n \pi+\frac{\pi}{4}: n \in \mathbf{Z}\right\}$ (c) does not exist (d) none of these Solution: We have, $f: \mathbf{R} \rightarrow \mathbf{R}$ is given by $f(x)=\tan x$ $\Rightarrow f^{-1}(x)=\tan ^{-1} x$ $\therefore f^{-1}(1)=\tan ^{-1} 1=\left\{n \pi+\frac{\pi}{4}: n \in \mathbf{Z}\right\}$ Hence, the corre...
Read More →A is of the same age as B and C is of the same age as B. Euclid's which axiom illustrates the relative ages of A and C?
Question: A is of the same age as B and C is of the same age as B. Euclid's which axiom illustrates the relative ages of A and C?(a) First axiom(b) second axiom(c) Third axiom(d) Fourth axiom Solution: Euclid's first axiom states that the things which are equal to the same thing are equal to one another.It is given that, the age of A is equal to the age of B and the age of C is equal to the age of B.UsingEuclid's first axiom, we conclude that the age of A is equal to the age of C.Thus,Euclid's f...
Read More →How many different committees of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve?
Question: How many different committees of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve? (a) 6 (b) 20 (c) 60 (d) 120 Solution: (d) 120 Number of committes that can be formed $={ }^{6} C_{3} \times{ }^{4} C_{2}$ $=\frac{6 !}{3 ! 3 !} \times \frac{4 !}{2 ! 2 !}$ $=\frac{6 \times 5 \times 4}{3 \times 2} \times \frac{4 \times 3}{2}$ = 120...
Read More →Euclid's which axiom illustrates the statement that
Question: Euclid's which axiom illustrates the statement that whenx + y= 15, thenx + y + z= 15 +z?(a) first(b) second(c) third(d) fourth Solution: Euclid's second axiom states that if equals be added to equals, the wholes are equal.x + y= 15Addingzto both sides, we getx + y + z= 15 +zThus, Euclid's second axiom illustrates the statement that whenx + y= 15, thenx + y + z= 15 +z.Hence, the correct answer is option (b)....
Read More →Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle.
Question: Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given points is (a) 216 (b) 156 (c) 172 (d) none of these Solution: (b) 156 (b) 156 We need at least three points to draw a circle that passes through them. Now, number of circles formed out of 11 points by taking three points at a time =11C3= 165 Number of circles formed out of 5 points by taking three points at...
Read More →A point C is said to lie between the points A and B if
Question: A pointCis said to lie between the pointsAandBif(a)AC=CB(b)AC+CB=AB(c) pointsA,CandBare collinear(d) None of these Solution: (c) pointsA,CandBare collinear...
Read More →Let f : R→R be given by f(x)
Question: Let $f: R \rightarrow R$ be given by $f(x)=x^{2}-3$. Then, $f^{-1}$ is given by (a) $\sqrt{x+3}$ (b) $\sqrt{x}+3$ (c) $x+\sqrt{3}$ (d) None of these Solution: (d) Let $f^{-1}(x)=y$ $f(y)=x$ $y^{2}-3=x$ $y^{2}=x+3$ $y=\pm \sqrt{x+3}$ So, the answer is (d)....
Read More →A point C is called the mid-point of a line segment
Question: A point $C$ is called the mid-point of a line segment $\overline{A B}$ if (a)Cis an interior point ofAB(b)AC=CB (c) $C$ is an interior point of $A B$, such that $\overline{A C}=\overline{C B}$ (d)AC+CB=AB Solution: (c)Cis an interior point ofAB,such thatAC=CB...
Read More →Find the 12th term from the end of the following arithmetic progressions:
Question: Find the 12thterm from the end of the following arithmetic progressions: (i) 3, 5, 7, 9, ... 201(ii) 3, 8, 13, ..., 253(iii) 1, 4, 7, 10, ..., 88 Solution: In the given problem, we need to find the 12thterm from the end for the given A.P. (i) 3, 5, 7, 9 201 Here, to find the 12thterm from the end let us first find the total number of terms. Let us take the total number of terms asn. So, First term (a) = 3 Last term (an) = 201 Common difference $(d)=5-3$ $=2$ Now, as we know, $a_{s}=a+(...
Read More →The number of ways in which a host lady can invite for a party of 8 out of 12 people of whom two do not want to attend the party together is
Question: The number of ways in which a host lady can invite for a party of 8 out of 12 people of whom two do not want to attend the party together is (a) 2 11C7+10C8 (b)10C8+11C7 (c)12C810C6 (d) none of these Solution: (c) ${ }^{12} C_{8}-{ }^{10} C_{6}$ A host lady can invite 8 out of 12 people in12C812C8ways. Two out of these 12 people do not want to attend the party together. Number of ways =12C810C6...
Read More →Let f(x)
Question: Let $f(x)=x^{3}$ be a function with domain $\{0,1,2,3\}$. Then domain of $f^{-1}$ is (a) $\{3,2,1,0\}$ (b) $\{0,-1,-2,-3\}$ (c) $\{0,1,8,27\}$ (d) $\{0,-1,-8,-27\}$ Solution: (c) $\{0,1,8,27\}$ $f(x)$ $=x^{3}$ Domain $=$ $\{0,1,2,3\}$ Range $=$ $\left\{0^{3}, 1^{3},\right.$, $2^{3}, 3^{3}$ \}$=$ $\{0,1,8,$, $27\}$ So, $f$ $=$ $\{(0,0),$, $(1,1)$, $(2,8)$, $(3,27)\}$ $\mathrm{f}^{-1}=$ $\{(0,0),$, $(1,1)$, $(8,2)$, $(27,3)\}$ Domain of $f^{-1}=\{0,1,8,27\}$...
Read More →Solve the following
Question: IfC0+C1+C2+ ... +Cn= 256, then2nC2is equal to (a) 56 (b) 120 (c) 28 (d) 91 Solution: (b) 120 If set $S$ has $n$ elements, then $C(n, k)$ is the number of ways of choosing $k$ elements from $S$. Thus, the number of subsets of $S$ of all possible values is given by $C(n, 0)+C(n, 1)+C(n, 3)+\ldots+C(n, n)=2^{n}$ Comparing the given equation with the above equation: $2^{n}=256$ $\Rightarrow 2^{n}=2^{8}$ $\Rightarrow n=8$ $\therefore{ }^{2 n} C_{2}={ }^{16} C_{2}$ $\Rightarrow{ }^{16} C_{2}...
Read More →Which of the following is a false statement?
Question: Which of the following is a false statement?(a) An infinite number of lines can be drawn through a given point.(b) A unique line can be drawn through two given points. (c) Ray $\underset{A B}{\longrightarrow}=$ ray $\overrightarrow{B A}$. (d) A ray has one end-point. Solution: (c) Ray $\overrightarrow{A B}=$ Ray $\overrightarrow{B A}$...
Read More →Which of the following is a true statement?
Question: Which of the following is a true statement?(a) Only a unique line can be drawn through a given point.(b) Infinitely many lines can be drawn through two given points.(c) If two circles are equal, then their radii are equal.(d) A line has a definite length. Solution: (c)If two circles are equal, then their radii are equal....
Read More →There are 13 players of cricket, out of which 4 are bowlers.
Question: There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of eleven be selected from them so as to include at least two bowlers? (a) 72 (b) 78 (c) 42 (d) none of these Solution: (b) 78 4 out of 13 players are bowlers. In other words, 9 players are not bowlers. A team of 11 is to be selected so as to include at least 2 bowlers. $\therefore$ Number of ways $={ }^{4} C_{2} \times{ }^{9} C_{9}+{ }^{4} C_{3} \times{ }^{9} C_{8}+{ }^{4} C_{4} \times{ }^{9} C_{7}$ $...
Read More →If f : R→R is given by
Question: If $f: R \rightarrow R$ is given by $f(x)=x^{3}+3$, then $f^{-1}(x)$ is equal to (a) $x^{1 / 3}-3$ (b) $x^{1 / 3}+3$ (c) $(x-3)^{1 / 3}$ (d) $x+3^{1 / 3}$ Solution: (c) Let $f^{-1}(x)=y$ $f(y)=x$ $\Rightarrow y^{3}+3=x$ $\Rightarrow y^{3}=x-3$ $\Rightarrow y=\sqrt[3]{x-3}$ $\Rightarrow y=(x-3)^{\frac{1}{3}}$ So, the answer is (c)....
Read More →Which of the following is a true statement?
Question: Which of the following is a true statement?(a) The floor and a wall of a room are parallel planes.(b) The ceiling and a wall of a room are parallel planes.(c) The floor and the ceiling of a room are parallel planes.(d) Two adjacent walls of a room are parallel planes. Solution: (c) The floor and the ceiling of a room are parallel planes....
Read More →There are 10 points in a plane and 4 of them are collinear.
Question: There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is (a) 45 (b) 40 (c) 39 (d) 38 Solution: (b) 40 Number of straight lines formed by joining the 10 points if we take 2 points at a time $={ }^{10} C_{2}=\frac{10}{2} \times \frac{9}{1}=45$ Number of straight lines formed by joining the 4 points if we take 2 points at a time $={ }^{4} C_{2}=\frac{4}{2} \times \frac{3}{1}=6$ But, 4 collinear points, when joined in pairs, give o...
Read More →Axioms are assumed
Question: Axioms are assumed(a) definitions(b) theorems(c) universal truths specific to geometry(d) universal truths in all branches of mathematics Solution: (d) universal truths in all branches of mathematics...
Read More →If f(x)
Question: If $f(x)=\sin ^{2} x$ and the composite function $g(f(x))=|\sin x|$, then $g(x)$ is equal to (a) $\sqrt{x-1}$ (b) $\sqrt{x}$ (c) $\sqrt{x+1}$ (d) $-\sqrt{x}$ Solution: (b) If we take $g(x)=\sqrt{x}$, then $g(f(x))=g\left(\sin ^{2} x\right)=\sqrt{\sin ^{2} x}=\pm \sin x=|\sin x|$ So, the answer is (b)....
Read More →The number of planes passing through 3 non-collinear points is
Question: The number of planes passing through 3 non-collinear points is(a) 4(b) 3(c) 2(d) 1 Solution: (d) 1...
Read More →In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one women?
Question: In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one women? (a) 246 (b) 222 (c) 186 (d) none of these Solution: (a) 246 Required number of ways $={ }^{4} C_{1} \times{ }^{6} C_{4}+{ }^{4} C_{2} \times{ }^{6} C_{3}+{ }^{4} C_{3} \times{ }^{6} C_{2}+{ }^{4} C_{4} \times{ }^{6} C_{1}$ $=60+120+60+6$ = 246...
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