Two adjacent angles of a parallelogram
Question: Two adjacent angles of a parallelogram are in the ratio 1 : 3. Find its angles. Solution: Let the adjacent angles of a parallelogram be x and 8c. Then, we have x + (3 x) = 180 [adjacent angles of parallelogram are supplementary] = 4 x = 180 = x = 45 Thus, the angles are 45, 135. Hence, the angles are 45, 135, 45, 135. [ opposite angles in a parallelogram are equal]...
Read More →Insert five numbers between 11 and 29 such that the resulting sequence is an AP.
Question: Insert five numbers between 11 and 29 such that the resulting sequence is an AP. Solution: To find: Five numbers between 11 and 29, which are in A.P. Given: (i) The numbers are 11 and 29 Formula used: (i) $A_{n}=a+(n-1) d$ Let the five numbers be $A_{1}, A_{2}, A_{3}, A_{4}$ and $A_{5}$ According to question $11, A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ and 29 are in A.P. We can see that the number of terms in this series is 7 For the above series:- $a=11, n=7$ $\mathrm{A}_{7}=29$ Using form...
Read More →A rhombus can be constructed
Question: A rhombus can be constructed uniquely, if both diagonals are given. Solution: True A rhombus can be constructed uniquely, if both diagonals are given....
Read More →A parallelogram can be constructed uniquely,
Question: A parallelogram can be constructed uniquely, if both diagonals and the angle between them is given. Solution: True We can draw a unique parallelogram, if both diagonals and the angle between them is given....
Read More →Insert arithmetic means between 16 and 65 such that the 5th AM is 51.
Question: Insert arithmetic means between 16 and 65 such that the 5th AM is 51. Find the number of arithmetic means. Solution: To find: The number of arithmetic means Given: (i) The numbers are 16 and 65 (ii) $5^{\text {th }}$ arithmetic mean is 51 Formula used: (i) $d=\frac{b-a}{n+1}$, where, $d$ is the common difference n is the number of arithmetic means (ii) $A_{n}=a+n d$ We have 16 and 65, Using Formula, $d=\frac{b-a}{n+1}$ $d=\frac{65-16}{n+1}$ $d=\frac{49}{n+1}$ Using Formula, $A_{n}=a+n ...
Read More →If the solve the problem
Question: If $\mathrm{x}=\mathrm{a}\left(\cos \mathrm{t}+\log \tan \frac{\mathrm{t}}{2}\right), \mathrm{y}=\mathrm{a} \sin \mathrm{t}$, evaluate $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ at $\mathrm{t}=\frac{\pi}{3}$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}=\sin \mathrm{x}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \sin \mathrm{x}=-\cos \mathrm{x}$ (iv) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x...
Read More →A quadrilateral can be constructed uniquely,
Question: A quadrilateral can be constructed uniquely, if three angles and any two included sides are given. Solution: True We can construct a unique quadrilateral with given three angles given and two included sides....
Read More →If diagonals of a quadrilateral
Question: If diagonals of a quadrilateral bisect each other, it must be a parallelogram. Solution: True It is the property of a parallelogram....
Read More →A quadrilateral can be drawn,
Question: A quadrilateral can be drawn, if three sides and two diagonals are given. Solution: True A quadrilateral can be drawn, if three sides and two diagonals are given....
Read More →A quadrilateral can be drawn,
Question: A quadrilateral can be drawn, if all four sides and one angle is known. Solution: True A quadrilateral can be drawn, if all four sides and one angle is known....
Read More →A quadrilateral can be drawn,
Question: A quadrilateral can be drawn, when all the four angles and one side is given. Solution: False We cannot draw a unique-quadrilateral, if four angles and one side is known....
Read More →A quadrilateral can be drawn,
Question: A quadrilateral can be drawn, if all four sides and one diagonal is known. Solution: True A quadrilateral can be constructed uniquely, if four sides and one diagonal is known....
Read More →A quadrilateral can have
Question: A quadrilateral can have all four angles as obtuse. Solution: False If all angles will be obtuse, then their sum will exceed 360. This is not possible in case of a quadrilateral....
Read More →If the solve the problem
Question: If $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$, then find the value of $\frac{d^{2} y}{d x^{2}}$ at $t=\frac{\pi}{4}$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}=\sin \mathrm{x}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \sin \mathrm{x}=-\cos \mathrm{x}$ (iv) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{\mathrm{n}}=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ (v) chain rule $\frac{\mathrm{df}}{\math...
Read More →A quadrilateral can be drawn
Question: A quadrilateral can be drawn if only measures of four sides are given. Solution: False As we require at least five measurements to determine a quadrilateral uniquely....
Read More →Every rhombus is a trapezium.
Question: Every rhombus is a trapezium. Solution: True Since a rhombus satisfies all the properties of a trapezium. So, we can say that, every rhombus is a trapezium but vice-versa is not true....
Read More →Every square is a trapezium.
Question: Every square is a trapezium. Solution: True As a square has all the properties of a trapezium. So, we can say that, every square is a trapezium but vice-versa is not true....
Read More →Every square is a parallelogram.
Question: Every square is a parallelogram. Solution: True Every square is also a parallelogram as it has all the properties of a parallelogram but vice-versa is not true....
Read More →If the solve the problem
Question: If $x=a \sin t$ and $y=a\left(\cos t+\log \tan \frac{t}{2}\right)$, find $\frac{d^{2} y}{d x^{2}}$ Solution: Formula: - (i) $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}_{1}$ and $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{y}_{2}$ (ii) $\frac{\mathrm{d}}{\mathrm{dx}} \cos \mathrm{x}=\sin \mathrm{x}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \sin \mathrm{x}=-\cos \mathrm{x}$ (iv) $\frac{d}{d x} x^{n}=n x^{n-1}$ (v) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\t...
Read More →Every square is a rhombus.
Question: Every square is a rhombus. Solution: True As a square possesses all the properties of a rhombus. So, we can say that, every square is a rhombus but vice-versa is not true....
Read More →Every rectangle is a trapezium.
Question: Every rectangle is a trapezium. Solution: True As a rectangle satisfies all the properties of a trapezium. So, we can say that, every rectangle is a trapezium but vice-versa is not true....
Read More →There is n arithmetic means between 9 and 27.
Question: There is n arithmetic means between 9 and 27. If the ratio of the last mean to the first mean is 2 : 1, find the value of n. Solution: To find: The value of n Given: (i) The numbers are 9 and 27 Formula used: (i) $d=\frac{b-a}{n+1}$, where, $d$ is the common difference n is the number of arithmetic means (ii) $A_{n}=a+n d$ We have 9 and 27, Using Formula, $d=\frac{b-a}{n+1}$ $d=\frac{27-9}{n+1}$ $d=\frac{18}{n+1}$ Using Formula, $A_{n}=a+n d$ First mean i.e., $A_{1}=9+$ (1) $\left(\fra...
Read More →Every trapezium is a rectangle.
Question: Every trapezium is a rectangle. Solution: False Since in a rectangle, opposite sides are equal and parallel but in a trapezium, it is not so....
Read More →Every parallelogram is a rectangle.
Question: Every parallelogram is a rectangle. Solution: False As in a parallelogram, all angles are not right angles, while in a rectangle, all angles are equal and are right angles....
Read More →Every trapezium is a parallelogram.
Question: Every trapezium is a parallelogram. Solution: False Since in a trapezium, only one pair of sides is parallel....
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