Board Paper of Class 10 2019 Maths Abroad(Set 3) - Solutions

Evaluate:                                    OR

Express (sin 67° + cos 75°) in terms of trigonometric ratios of the angle between 0° and 45°. VIEW SOLUTION

Find the solution of the pair of equation :
$\frac{3}{x}+\frac{8}{y}=-1;$  $\frac{1}{x}-\frac{2}{y}=2, x, y \ne 0$
                      OR
Find the value(s) of k for which the pair of equations $\left\{\begin{array}{l}kx+2y=3\\ 3x+6y=10\end{array}\right.$ has a unique solution. VIEW SOLUTION

Use Euclid's division algorithm to find the HCF of 255 and 867. VIEW SOLUTION

The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that $AR=\frac{3}{4}AB.$ Find the coordinates of R. VIEW SOLUTION

How many multiples of 4 lie between 10 and 205 ?
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Determine the A.P. whose third term is 16 and 7^th term exceeds the 5^th by 12. VIEW SOLUTION

Three different coins are tossed simultaneously. Find the probability of getting exactly one head. VIEW SOLUTION

A die is thrown once. Find the probability of getting.
(a) a prime number.
(b) an odd number VIEW SOLUTION

In Figure 1, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL² + CM²) = 5 BC².  
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals. VIEW SOLUTION

In Figure 2, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded region.
VIEW SOLUTION

A cone of height 24 cm and radius of base 6 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere and hence find the surface area of this sphere.
                                                                            OR
A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in his field which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/hr, how much time will the tank be filled ? VIEW SOLUTION

Calculate the mode of the following distribution :
 

Class :	10 − 15	15 − 20	20 − 25	25 − 30	30 − 35
Frequency :	4	7	20	8	1

VIEW SOLUTION

Show that $\frac{2+3\sqrt{2}}{7}$ is not a rational number, given that $\sqrt{2}$ is an irrational number. VIEW SOLUTION

Obtain all the zeroes of the polynomial 2x⁴ − 5x³ − 11x² + 20x + 12 when 2 and − 2 are two zeroes of the above polynomial VIEW SOLUTION

A motorboat whose speed is 18 km/hr in still water takes on hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. VIEW SOLUTION

Prove that:

(sin θ + 1 + cos θ) (sin θ − 1 + cos θ) . sec θ cosec θ = 2

                                 OR

Prove that :

$\sqrt{\frac{\sec \mathrm{\theta }-1}{\sec \mathrm{\theta }+1}+}\sqrt{\frac{\sec \mathrm{\theta }+1}{\sec \mathrm{\theta }-1}}=2\mathrm{cosec} \mathrm{\theta }$ VIEW SOLUTION

In what ratio does the point P(−4, y) divide the line segment joining the points A(−6, 10) and B(3, −8) ? Hence find the value of y.

                                                        OR

Find the value of p for which the points (−5, 1), (1, p) and (4, −2) are collinear. VIEW SOLUTION

ABC is a right triangle in which ∠B = 90°.  If AB = 8 cm and BC = 6 cm, find the diameter of the circle inscribed in the triangle. VIEW SOLUTION

In an A.P., the first term is −4, the last term is 29 and the sum of all its terms is 150. Find its common difference VIEW SOLUTION

Draw a circle of radius 4 cm. From a point 6 cm away from its centre, construct a pair of tangents to the circle and measure their lengths VIEW SOLUTION

Prove that :

2(sin⁶ θ + cos⁶ θ) − 3 (sin⁴ θ + cos⁴ θ) + 1 = 0 VIEW SOLUTION

Solve for x :$\frac{1}{2a+b+2x}=\frac{1}{2a}+\frac{1}{b}+\frac{1}{2x}; x\ne 0, x\ne \frac{-2a-b}{2}, a, b\ne 0$
 
                                                                   OR

The sum of the areas of two squares is 640 m². If the difference of their perimeters is 64 m, find the sides of the square. VIEW SOLUTION

In ∆ ABC (Figure 3), AD ⊥ BC. Prove that
AC² = AB² +BC² − 2BC × BD
VIEW SOLUTION

A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min.

                                                       OR

There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole. VIEW SOLUTION

Calculate the mean of the following frequency distribution :
 

Class :	10−30	30−50	50−70	70−90	90−110	110−130
Frequency :	5	8	12	20	3	2

                                                OR
The following table gives production yield in kg per hectare of wheat of 100 farms of a village :

Production yield (kg/hectare) :	40−45	45−50	50−55	55−60	60−65	65−70
Number of farms	4	6	16	20	30	24

Change the distribution to a 'more than type' distribution, and draw its ogive. VIEW SOLUTION

A container opened at the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container, at the rate of ₹ 50 per litre. Also find the cost of metal sheet used to make the container, if it costs ₹ 10 per 100 cm². (Take π = 3⋅14) VIEW SOLUTION