Every kite is a parallelogram.
Question: Every kite is a parallelogram. Solution: False Kite is not a parallelogram as its opposite sides are not equal and parallel....
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Question: If $x=\cos t+\log \tan \frac{t}{2}, y=\sin t$, then find the value of $\frac{d^{2} y}{d t^{2}}$ and $\frac{d^{2} y}{d x^{2}}$ at $t=\frac{\pi}{4}$. 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{\mathrm{d}}{\mathrm{dx}} \log \mathrm{x}=\f...
Read More →Diagonals of rectangle bisect
Question: Diagonals of rectangle bisect each other at right angles. Solution: False Diagonals of a rectangle does not bisect each other....
Read More →Diagonals of a rectangle are equal.
Question: Diagonals of a rectangle are equal. Solution: True The diagonals of a rectangle are equal....
Read More →Diagonals of a rhombus are equal
Question: Diagonals of a rhombus are equal and perpendicular to each other. Solution: False As diagonals of a rhombus are perpendicular to each other but not equal....
Read More →Insert six arithmetic means between 11 and -10
Question: Insert six arithmetic means between 11 and -10 Solution: To find: Six arithmetic means between 11 and -10 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 11 and -10 Using Formula, $d=\frac{b-a}{n+1}$ $d=\frac{-10-(11)}{6+1}$ $d=\frac{-21}{7}$ $d=-3$ Using Formula, $A_{n}=a+n d$ First arithmetic mean, $A_{1}=a+d$ $=11+(-3)$ $=8$ Second arithmetic mean, $\mathrm{A}_{2}=\mathrm{a}+2 \mathrm{~d}$ $=11...
Read More →If opposite angles of a quadrilateral
Question: If opposite angles of a quadrilateral are equal, it must be a parallelogram. Solution: True If opposite angles are equal, it has to be a parallelogram....
Read More →If diagonals of a quadrilateral are equal,
Question: If diagonals of a quadrilateral are equal, it must be a rectangle. Solution: True If diagonals are equal, then it is definitely a rectangle....
Read More →Rectangle is a regular quadrilateral.
Question: Rectangle is a regular quadrilateral. Solution: False As its all sides are not equal....
Read More →A polygon is regular,
Question: A polygon is regular, if all of its sides are equal. Solution: False By definition of a regular polygon, we know that, a polygon is regular, if all sides and all angles are equal....
Read More →If the sum of interior angles is double
Question: If the sum of interior angles is double the sum of exterior angles taken in an order of a polygon, then it is a hexagon. Solution: True Since the sum of exterior angles of a hexagon is 360 and the sum of interior angles of a hexagon is 720, i.e. double the sum of exterior angles....
Read More →The sum of interior angles and the sum
Question: The sum of interior angles and the sum of exterior angles taken in an order are equal in case of quadrilaterals only. Solution: True Since the sum of interior angles as well as of exterior angles of a quadrilateral are 360....
Read More →A kite is not a convex quadrilateral.
Question: A kite is not a convex quadrilateral. Solution: False A kite is a convex quadrilateral as the line segment joining any two opposite vertices inside it, lies completely inside it....
Read More →Triangle is a polygon whose sum of exterior
Question: Triangle is a polygon whose sum of exterior angles is double the sum of interior angles. Solution: True As the sum of interior angles of a triangle is 180 and the sum of exterior angles is 360, i.e. double the sum of interior angles....
Read More →If the solve the problem
Question: If $y=\operatorname{cosec}^{-1} x, x1$, then show that $x\left(x^{2}-1\right) \frac{d^{2} y}{d x^{2}}+\left(2 x^{2}-1\right) \frac{d y}{d x}=0$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{d\left(\operatorname{cosec}^{-1} x\right)}{d x}=\frac{-1}{|x| \sqrt{x^{2}-1}}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{\mathrm{n}}=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ (iv) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d}(\tex...
Read More →A quadrilateral has two diagonals.
Question: A quadrilateral has two diagonals. Solution: True A quadrilateral has two diagonals....
Read More →Sum of all the angles of a quadrilateral
Question: Sum of all the angles of a quadrilateral is 180. Solution: False Since sum of all the angles of a quadrilateral is 360....
Read More →Insert three arithmetic means between 23 and 7.
Question: Insert three arithmetic means between 23 and 7. Solution: To find: Three arithmetic means between 23 and 7 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 23 and 7 Using Formula, $d=\frac{b-a}{n+1}$ $d=\frac{7-23}{3+1}$ $d=\frac{-16}{4}$ $d=-4$ Using Formula, $A_{n}=a+n d$ First arithmetic mean, $A_{1}=a+d$ $=23+(-4)$ $=19$ Second arithmetic mean, $\mathrm{A}_{2}=\mathrm{a}+2 \mathrm{~d}$ $=23+2(-...
Read More →All rhombuses are square.
Question: All rhombuses are square. Solution: False As in a rhombus, each angle is not a right angle, so rhombuses are not squares....
Read More →All rectangles are parallelograms.
Question: All rectangles are parallelograms. Solution: True Since rectangles satisfy all theproperties of parallelograms. Therefore, we can say that, all rectangles are parallelograms but vice-versa is not true....
Read More →All kites are squares.
Question: All kites are squares. Solution: False As kites do not satisfy all the properties of a square. e.g. In square, all the angles are of 90 but in kite, it is not the case....
Read More →All squares are rectangles.
Question: All squares are rectangles. Solution: True Since squares possess all the properties of rectangles. Therefore, we can say that, all squares are rectangles but vice-versa is not true....
Read More →All angles of a trapezium
Question: All angles of a trapezium are equal. Solution: False As all angles of a trapezium are not equal....
Read More →If the solve the problem
Question: If $\mathrm{y}=\left(\cot ^{-1} \mathrm{x}\right)^{2}$, prove that $\mathrm{y}_{2}\left(\mathrm{x}^{2}+1\right)^{2}+2 \mathrm{x}\left(\mathrm{x}^{2}+1\right) \mathrm{y}_{1}=2$ Solution: (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}}{\mathrm{dx}} \cot ^{-1} \mathrm{x}=\frac{-1}{1+\mathrm{x}^{2}}$ (iii) $\frac{d}{d x} x^{n}=n x^{n-1}$ (iv) chain rule $\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{d} \text { (wou) }}{\mathrm{dt}} \cdot \frac{\mathr...
Read More →Insert four arithmetic means between 4 and 29
Question: Insert four arithmetic means between 4 and 29 Solution: To find: Four arithmetic means between 4 and 29 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 4 and 29 Using Formula, $d=\frac{b-a}{n+1}$ $d=\frac{29-4}{4+1}$ $d=\frac{25}{5}$ $d=5$ Using Formula, $A_{n}=a+n d$ First arithmetic mean, $A_{1}=a+d$ = 4 + 5 = 9 Second arithmetic mean, $A_{2}=a+2 d$ $=4+2(5)$ $=4+10$ $=14$ Third arithmetic mean,...
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