Evaluate
Question: Evaluate $\lim _{x \rightarrow 1}\left(6 x^{2}-4 x+3\right)$ Solution: To evaluate: $\lim _{x \rightarrow 1}\left(x^{2}-4 x+3\right)$ Formula used: We have, $\lim _{x \rightarrow a} f(x)=f(a)$ As $x \rightarrow 1$, we have $\lim _{x \rightarrow 1}\left(x^{2}-4 x+3\right)=1^{2}-4(1)+3$ Thus, the value of $\lim _{x \rightarrow 1}\left(x^{2}-4 x+3\right)$ is 0 ....
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Question: Evaluate $\lim _{x \rightarrow 2}(5-x)$ Solution: To evaluate: $\lim _{x \rightarrow 2}(5-x)$ Formula used: We have, $\lim _{x \rightarrow a} f(x)=f(a)$ As $\mathrm{x} \rightarrow 2$, we have $\lim _{x \rightarrow 2}(5-x)=5-2$ $\lim _{x \rightarrow 2}(5-x)=3$ Thus, the value of $\lim _{x \rightarrow 2}(5-x)$ is3...
Read More →Find the standard deviation of the first n
Question: Find the standard deviation of the first n natural numbers. Solution: Given set of first $\mathrm{n}$ natural numbers Now we have to find the standard deviation Given first n natural numbers, we can write in table as shown below So, the sums of these are $\sum x_{i}=1+2+3+\cdots+n=\frac{n(n+1)}{2}$ And $\sum x_{i}^{2}=1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$ Therefore, the standard deviation can be written as, $\sigma=\sqrt{\frac{\sum x_{i}^{2}}{n}-\left(\frac{\sum x_{i}...
Read More →The midpoints of the sides of a triangle are
Question: The midpoints of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Find its vertices. Solution: The midpoints of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4). Let its vertices be $A\left(x_{1}, y_{1}, z_{1}\right), B\left(x_{2}, y_{2}, z_{2}\right), C\left(x_{3}, y_{3}, z_{3}\right)$. The mid point of AB is (1,5,-1), therefore $\frac{x_{2}+x_{1}}{2}=1$ $x_{1}+x_{2}=2 \ldots \ldots \ldots . . e q .1$ $\frac{y_{2}+y_{1}}{2}=5$ $y_{1}+y_{2}=10 \ldots \l...
Read More →Calculate the mean deviation about the mean
Question: Calculate the mean deviation about the mean of the set of first n natural numbers when n is an even number. Solution: Given set of first $n$ natural numbers when $n$ is an even number. Now we have to find the mean deviation about the mean We know first $\mathrm{n}$ natural numbers are $1,2,3 \ldots ., \mathrm{n}$. And given $\mathrm{n}$ is even number. So mean is, $\overline{\mathrm{x}}=\frac{1+2+3+\cdots+\mathrm{n}}{\mathrm{n}}=\frac{\frac{\mathrm{n}(\mathrm{n}+1)}{2}}{\mathrm{n}}=\fr...
Read More →Calculate the mean deviation about the mean
Question: Calculate the mean deviation about the mean of the set of first n natural numbers when n is an odd number. Solution: Given set of first $\mathrm{n}$ natural numbers when $\mathrm{n}$ is an odd number Now we have to find the mean deviation about the mean We know first n natural numbers are $1,2,3 \ldots . . \mathrm{n}$. And given $n$ is odd number. So mean is, $\overline{\mathrm{x}}=\frac{1+2+3+\cdots+\mathrm{n}}{\mathrm{n}}=\frac{\frac{\mathrm{n}(\mathrm{n}+1)}{2}}{\mathrm{n}}=\frac{(\...
Read More →If the origin is the centroid of triangle
Question: If the origin is the centroid of triangle ABC with vertices A(a, 1, 3), B(-2, b, - 5) and C(4, 7, c), find the values of a, b, c. Solution: Since, centroid of a triangle is found by $\left(\frac{x_{2}+x_{1}+x_{1}}{3}, \frac{y_{2}+y_{1}+y_{2}}{3}, \frac{z_{2}+z_{1}+z_{3}}{3}\right)$ The points are $A(a, 1,3)$ and $B(-2, b,-5)$, and its centroid is $(0,0,0)$ and its third vertex $\mathrm{C}(4,7, \mathrm{c})$. Using the formula, we get $=\left(\frac{-2+4+a}{3}, \frac{1+7+b}{3}, \frac{3-5+...
Read More →Find the mean deviation about the median
Question: Find the mean deviation about the median of the following distribution: Solution: Given data distribution Now we have to find the mean deviation about the median Let us make a table of the given data and append other columns after calculations Now, here N=20, which is even. Here median, $M=\frac{1}{2}\left[\left(\frac{N}{2}\right)^{\text {th }}\right.$ observation $+\left(\frac{N}{2}+1\right)^{\text {th }}$ observation $]$ $M=\frac{1}{2}\left[\left(\frac{20}{2}\right)^{\text {th }}\rig...
Read More →Find the mean deviation about
Question: Find the mean deviation about the mean of the distribution: Solution: Given data distribution Now we have to find the mean deviation about the mean of the distribution Construct a table of the given data We know that mean, $\overline{\mathrm{X}}=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}=\frac{433}{20}=21.65$ To find mean deviation we have to construct another table Hence Mean Deviation becomes, M. $\mathrm{D}=\frac{\sum \mathrm{f}_{\mathr...
Read More →Two vertices of a triangle ABC are A
Question: Two vertices of a triangle ABC are A(2, -4, 3) and B(3, -1, -2), and its centroid is (1, 0, 3). Find its third vertex C. Solution: Since the centroid of a triangle $=\left(\frac{x_{2}+x_{1}+x_{1}}{3}, \frac{y_{2}+y_{1}+y_{2}}{3}, \frac{z_{2}+z_{1}+z_{3}}{3}\right)$ The points are $A(2,-4,3)$ and $B(3,-1,-2)$, and its centroid is $(1,0,3)$. And let its third vertex $\mathrm{C}(\mathrm{a}, \mathrm{b}, \mathrm{c})$ Using the formula, we get $=\left(\frac{2+3+a}{3}, \frac{-4-1+b}{3}, \frac...
Read More →If the three consecutive vertices of a parallelogram be
Question: If the three consecutive vertices of a parallelogram be A(3, 4, -3), B(7, 10, -3) and C(5, -2, 7), find the fourth vertex D. Solution: the vertices of the parallelogram be A(3, 4, -3), B(7, 10, -3) and C(5, -2, 7), and the fourth coordinate be D(a,b,c). the property of parallelogram is the diagonal bisect each other. Therefore, diagonal $\mathrm{AC}$ and $\mathrm{BD}$ will bisect each other, and the bisecting point will be equal to the two diagnals. By using section formula, we get $\l...
Read More →The vertices of a triangle ABC
Question: The vertices of a triangle ABC are A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3). The bisector AD of A meets BC at D, find the fourth vertex D. Solution: The given co-ordinates: A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) Now, $A B=\sqrt{(5-3)^{2}+(3-2)^{2}+(2-0)^{2}}=\sqrt{4+1+4}=3$ Also, $A C=\sqrt{(-9-3)^{2}+(6-2)^{2}+(-3-0)^{2}}=\sqrt{144+16+9}=13$ Now, we have, $\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{3}{13}$ By the property of internal angle bisector, $\frac{\mathrm{AB}}{\mathrm{AC}}=\frac{\...
Read More →Which of the following is not a statement
Question: Which of the following is not a statement (A) Smoking is injurious to health. (B) 2 + 2 = 4 (C) 2 is the only even prime number. (D) Come here. Solution: (D) Come here. Explanation: To given order like Come here, Go there are not statements....
Read More →Which of the following is a statement.
Question: Which of the following is a statement. (A) x is a real number. (B) Switch off the fan. (C) 6 is a natural number. (D) Let me go. Solution: (C) 6 is a natural number. Explanation: A statement is an assertive (declarative) sentence if it is either true or false but not both. Here, 6 is a natural number is true...
Read More →Using contrapositive method prove that
Question: Using contrapositive method prove that if n2 is an even integer, then n is also an even integers. Solution: Let us assume p: n2is an even integer. ~p: n is not an even integer q: n is also an even integer ~q=n is not an even integer. Since, in the contrapositive, a conditional statement is logically equivalent to its contrapositive. Therefore, ~q~p = If n is not an even integer then n2is not an even integer. Hence, ~q is true~p is true. Objective type questions:...
Read More →Prove by direct method that for any
Question: Prove by direct method that for any real numbers x, y if x = y, then x2= y2. Solution: Given for any real number x, y if x=y Now we have to find x2= y2 Let us assume p: x=y where x and y are real number On squaring both sides we get x2= y2: q (Assumption) Therefore, p = q Hence, Proved...
Read More →Find the ratio in which the plane
Question: Find the ratio in which the plane x 2y + 3z = 5 divides the join of A(3, -5, 4) and B(2, 3, -7). Find the coordinates of the point of intersection of the line and the plane. Solution: Let the plane x 2y + 3z = 5 divides the join of A(3, -5, 4) and B(2, 3, -7) in ratio k:1. The point which will come by section formula will be in the plane. Putting that in the plane equation will give the point coordinates. The points are $A(3,-5,4)$ and $B(2,3,-7)$. Using section formula, $\left(\frac{\...
Read More →Prove the following statement by contradiction method.
Question: Prove the following statement by contradiction method. p: The sum of an irrational number and a rational number is irrational Solution: Let p is false, as the sum of an irrational number and a rational number is irrational. Letis irrational and n is rational number + n = r = r n But, we know thatis irrational whereas (r-n) is rational which is contradiction. Here, Our Assumption is False Hence, P is true....
Read More →Check the validity of the following statement.
Question: Check the validity of the following statement. (i) p: 125 is divisible by 5 and 7. (ii) q: 131 is a multiple of 3 or 11. Solution: (i) p: 125 is divisible by 5 and 7. p: 125 is divisible by 5 and 7 Let, q: 125 is divisible by 5. r: 125 is divisible 7. Here, q is true and r is false. Therefore, q ᴧ r is False Hence, p is not valid. (ii) q: 131 is a multiple of 3 or 11. Given q : 131 is a multiple of 3 or 11 Let, P: 131 is a multiple of 3. Q: 131 is a multiple of 11. Here, P is false and...
Read More →Find the coordinates of the point where the line joining
Question: Find the coordinates of the point where the line joining A(3, 4, 1) and B(5, 1, 6) crosses the xy-plane. Solution: Let the plane $X Y$ divides the points $A(3,4,1)$ and $B(5,1,6)$ in ratio $k: 1$. Hence, using section formula $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$, we get $=\left(\frac{\mathrm{k} \times 5+1 \times 3}{\mathrm{k}+1}, \frac{\mathrm{k} \times 1+1 \times 4}{\mathrm{k}+1}, \frac{\mathrm{k} \times 6+1 \times 1}{\ma...
Read More →Prove by direct method that for
Question: Prove by direct method that for any integer n, n3 n is always even. Solution: Given n3-n Let us assume, n is even Let n = 2k, where k is natural number n3 n = (2k)3 (2k) n3 n = 2k (4k2-1) Let k (4k2 1) = m n3 n = 2m Therefore, (n3-n) is even. Now, let us assume n is odd Let n = (2k + 1), where k is natural number n3 n = (2k + 1)3 (2k + 1) n3 n = (2k + 1) [(2k + 1)2 1] n3 n = (2k + 1) [(4k2+ 4k + 1 1)] n3 n = (2k + 1) [(4k2+ 4k)] n3 n = 4k (2k + 1) (k + 1) n2 n = 2.2k (2k + 1) (k + 1) L...
Read More →Identify the Quantifiers in the following statements.
Question: Identify the Quantifiers in the following statements. (i) There exists a triangle which is not equilateral. (ii) For all real numbers x and y, xy = y x. (iii) There exists a real number which is not a rational number. (iv) For every natural number x, x + 1 is also a natural number. (v) For all real numbers x with x 3, x 2 is greater than 9. (vi) There exists a triangle which is not an isosceles triangle. (vii) For all negative integers x, x 3 is also a negative integers. (viii) There e...
Read More →Find the ratio in which the line segment having the end points
Question: Find the ratio in which the line segment having the end points A(-1, -3, 4) and B(4, 2, -1) is divided by the xz-plane. Also, find the coordinates of the point of division Solution: Let the plane $X Z$ divides the points $A(-1,-3,4)$ and $B(4,2,-1)$ in ratio $k: 1$. Hence, using section formula $\left(\frac{\mathrm{mx}_{2}+\mathrm{nx}_{1}}{\mathrm{~m}+\mathrm{n}}, \frac{\mathrm{my}_{2}+\mathrm{ny}_{1}}{\mathrm{~m}+\mathrm{n}}, \frac{\mathrm{mz}_{2}+\mathrm{nz}_{1}}{\mathrm{~m}+\mathrm{...
Read More →Find the ratio in which the point
Question: Find the ratio in which the point C(5, 9, -14) divides the join of A(2, -3, 4) and B(3, 1, -2). Solution: Let the ratio be k:1 in which point R divides point P and point Q. Using $\left(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}, \frac{m z_{2}+n z_{1}}{m+n}\right)$, we get, Here $m$ and $n$ are $k$ and 1 . The point which this formula gives is already given, i.e. $R(5,9,-14)$ and the joining points are $P(2,-3,4)$ and $Q(3,1,-2)$. $(5,9,-14)=\left(\frac{\mathrm{k} \times ...
Read More →Write down the converse of following statements :
Question: Write down the converse of following statements : (i) If a rectangle R is a square, then R is a rhombus. (ii) If today is Monday, then tomorrow is Tuesday. (iii) If you go to Agra, then you must visit Taj Mahal. (iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled. (v) If all three angles of a triangle are equal, then the triangle is equilateral. (vi) If x: y = 3 : 2, then 2x = 3y. (vii) If S is a cy...
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