If two adjacent angles of a parallelogram
Question: If two adjacent angles of a parallelogram are in the ratio 2 : 3, then the measure of angles are (a) 72, 108 (b) 36, 54 (c) 80, 120 (d) 96, 144 Solution: (a) Let the angles be 2x and 3x. Then, 2x + 3x = 180 [ adjacent angles of a parallelogram are supplementary] = 5x = 180 = x = 36 Hence, the measures of angles are 2x = 2 x 36= 72 and 3x = 336= 108...
Read More →Two cars start together from the same place in the same direction.
Question: Two cars start together from the same place in the same direction. The first go with a uniform speed of 60 km/hr. The second goes at a speed of 48 km/hr in the first hour and increases the speed by 1 km each succeeding hour. After how many hours will the second car overtake the first car if both cars go non - stop Solution: Given : Two cars start together from the same place and move in the same direction. The first car moves with a uniform speed of 60km/hr. The second car moves with 4...
Read More →If the solve the problem
Question: If $y=\left(\tan ^{-1} x\right)^{2}$, then prove that $(1-x 2)^{2} y_{2}+2 x\left(1+x^{2}\right) y_{1}=2$ 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) $\frac{\mathrm{d}\left(\tan ^{-1} \mathrm{x}\right)}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{2}}$ (iii) $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{x}^{\mathrm{n}}=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ Given: - $Y=\left(\tan ^{-1} x\right)^{2...
Read More →In a parallelogram PQRS,
Question: In a parallelogram PQRS, if P = 60o, then other three angles are (a) 45o, 135o, 120o (b) 60o, 120o, 120o (c) 60o, 135o, 135o (d) 45o, 135o, 135o Solution: (b) 60o, 120o, 120o In parallelogram P and Q are supplementary. We know that, sum of supplementary angle is equal to 180o. Then, P + Q = 180o 60o+ Q = 180o P = 180o 60o P = 120o And also, opposite angles P and R are equal in parallelogram. So, P = R = 60o Q = S = 120o Therefore, the other three angles of parallelograms are 60o, 120oa...
Read More →If the solve the problem
Question: If $y=\left\{\log \left(x+\sqrt{x}^{2}+1\right)^{2}\right.$, show that $\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}=2$ Solution: Formula: - (i) $\frac{d y}{d x}=y_{1}$ and $\frac{d^{2} y}{d x^{2}}=y_{2}$ (ii) $\frac{\mathrm{d}(\log \mathrm{x})}{\mathrm{dx}}=\frac{1}{\mathrm{x}}$ (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}(\text { wou })}{\mathrm{d...
Read More →There are 30 trees at equal distances of 5 metres in a line with a well,
Question: There are 30 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A Gardner waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the Gardner will cover in order to water all the trees. Solution: Hint: Distances between trees and well are in A.P. Given: The distance of well from its nearest tree is 10 metres Dista...
Read More →The number of sides of a regular polygon
Question: The number of sides of a regular polygon where each exterior angle has a measure of 45ois (a) 8 (b) 10 (c) 4 (d) 6 Solution: (a) 8 Now let us assume number of sides of a regular polygon be n. WKT, sum of all exterior angles of all polygons is equal to 360o. Form the question it is given that each exterior angle has a measure of 45o. Then, n 45o= 360o n = 360o/45o n = 8...
Read More →A quadrilateral has three acute angles.
Question: A quadrilateral has three acute angles. If each measures 80, then the measure of the fourth angle is (a) 150o (b) 120o (c) 105o (d) 140o Solution: (b) 120o We know that, sum of all interior angle of quadrilaterals is equal to 360o. Let us assume the fourth angle be x Then, 80o+ 80o+ 80o+ x = 360o 240o+ x = 360o x = 360o 240o x = 120o...
Read More →The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4.
Question: The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. The smallest angle is (a) 72o (b) 144o (c) 36o (d) 18o Solution: (c) 36o We know that, sum of all interior angle of quadrilaterals is equal to 360o. Let us assume the angles be x, 2x, 3x, and 4x Then, x + 2x + 3x + 4x = 360o 10x = 360o x = 360/10 x = 36 Therefore the angles are x = 36o 2x = 2 36 = 72o 3x = 3 36 = 108o 4x = 4 36 = 144o...
Read More →Which one has all the properties
Question: Which one has all the properties of a kite and a parallelogram? (a) Trapezium (b) Rhombus (c) Rectangle (d) Parallelogram Solution: (b) In a kite Two pairs of equal sides. Diagonals bisect at 90. One pair of opposite angles are equal. In a parallelogram Opposite sides are equal. Opposite angles are equal. Diagonals bisect each other. So, from the given options, all these properties are satisfied by rhombus....
Read More →A circle is completely divided into n sectors in such a way that the angles
Question: A circle is completely divided into n sectors in such a way that the angles of the sectors are in AP. If the smallest of these angles is 80 and the largest is 720 , calculate n and the angle in the fifth sector. Solution: A circle is divided into n sectors. Given, Angles are in A.P Smallest angle $=a=8^{\circ}$ Largest angle $=1=72^{\circ}$ Final term of last term of an A.P series is l = a + (n - 1)d So, $72^{\circ}=8^{\circ}+(n-1) \times d$ $(n-1) \times d=64^{\circ} \longrightarrow(1...
Read More →Which of the following is an equiangular
Question: Which of the following is an equiangular and equilateral polygon? (a) Square (b) Rectangle (c) Rhombus (d) Right triangle Solution: (a) In a square, all the sides and all the angles are equal. Hence, square is an equiangular and equilateral polygon....
Read More →If the solve the problem
Question: If $y=\tan ^{-1} x$, show that $\left(1+x^{2}\right) \frac{d^{2} y}{d x^{2}}+2 x \frac{d y}{d x}=0$ 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) $\frac{\mathrm{d}\left(\tan ^{-1} \mathrm{x}\right)}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{2}}$ (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...
Read More →The sum of all exterior angles
Question: The sum of all exterior angles of a triangle is (a) 180 (b) 360 (c) 540 (d) 720 Solution: (b) We know that the sum of exterior angles, taken in order of any polygon is 360 and triangle is also a polygon. Hence, the sum of all exterior angles of a triangle is 360....
Read More →If the diagonals of a quadrilateral are equal
Question: If the diagonals of a quadrilateral are equal and bisect each other, then the quadrilateral is a (a) rhombus (b) rectangle (c) square (d) parallelogram Solution: (b) Since, diagonals are equal and bisect each other, therefore it will be a rectangle....
Read More →If the adjacent sides of a parallelogram
Question: If the adjacent sides of a parallelogram are equal, then parallelogram is a (a) rectangle (b) trapezium (c) rhombus (d) square Solution: (c)We know that, in a parallelogram, opposite sides are equal. But according to the question, adjacent sides are also equal. Thus, the parallelogram in which all the sides are equal is known as rhombus....
Read More →How many diagonals does a hexagon
Question: How many diagonals does a hexagon have? (a) 9 (b) 8 (c) 2 (d) 6 Solution: (a) 9 We know that, The number of diagonals in a polygon of n sides is n(n 3)/2 Where n = 6 Then, = 6 (6 3)/2 = 6 3/2 = 18/2 = 9...
Read More →A quadrilateral whose all sides,
Question: A quadrilateral whose all sides, diagonals and angles are equal is a (a) square (b) trapezium (c) rectangle (d) rhombus Solution: (a) These are the properties of a square, i.e. in a square, all sides, diagonals and angles are equal....
Read More →The interior angles of a polygon are in AP.
Question: The interior angles of a polygon are in AP. The smallest angle is 520, and the common difference is 80 . Find the number of sides of the polygon. Solution: Given: Interior angles of a polygon are in A.P Smallest angle $=a=52^{\circ}$ Common difference $=\mathrm{d}=8^{\circ}$ Let the number of sides of a polygon $=n$ Angles are in the following order $52^{\circ}, 52^{\circ}+d, 52^{\circ}+2 d, \ldots \ldots . ., 52^{\circ}+(n-1) \times d$ Sum of $n$ terms in $A . P=s=\frac{n}{2}\{2 a+(n-...
Read More →A quadrilateral whose opposite sides
Question: A quadrilateral whose opposite sides and all the angles are equal is a (a) rectangle (b) parallelogram (c) square (d) rhombus Solution: (a) We know that, in a rectangle, opposite sides and all the angles are equal....
Read More →A quadrilateral whose all sides are equal,
Question: A quadrilateral whose all sides are equal, opposite angles are equal and the diagonals bisect each other at-right angles is a . (a) rhombus (b) parallelogram (c) square (d) rectangle Solution: (a) We know that, in rhombus, all sides are equal, opposite angles are equal and diagonals bisect each other at right angles....
Read More →If two adjacent angles of a parallelogram
Question: If two adjacent angles of a parallelogram are (5x 5)oand (10x + 35)o, then the ratio of these angles is (a) 1 : 3 (b) 2 : 3 (c) 1 : 4 (d) 1 : 2 Solution: (a) 1: 3...
Read More →What is the sum of all angles
Question: What is the sum of all angles of a hexagon? (a) 180 (b) 360 (c) 540 (d) 720 Solution: (d) Sum of all angles of a n-gon is (n 2) x 180. In hexagon, n = 6, therefore the required sum = (6 2) x 180 = 4 x 180 = 720...
Read More →A carpenter was hired to build 192 window frames.
Question: A carpenter was hired to build 192 window frames. The first day he made 5 frames and each day, thereafter he made 2 more frames than he made the day before. How many days did he take to finish the job? Solution: Let the carpenter take n days to finish the job. To Find: n He builds 5 frames on day 1, 7 on day 2, 9 on day 3 and so on. So it forms an AP 5, 7, 9, 11, and so on. We need to find the number of terms in this AP such that the sum of the AP will be equal to 192 Given: Sum of AP ...
Read More →What is the sum of all the angles
Question: What is the sum of all the angles of a pentagon? (a) 180 (b) 360 (c) 540 (d) 720 Solution: (c) We know that, the sum of angles of a polygon is (n 2) x 180, where n is the number of sides of the polygon. In pentagon, n = 5 Sum of the angles = (n 2) x 180 = (5 2) x 180 = 3 x 180= 540...
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