Date: March 2019

Duration: 3h

For what values of k does the quadratic equation 4x2 − 12x − k = 0 has no real roots?

Chapter: [2.03] Quadratic Equations

Find the distance between the points (a, b) and (−a, −b).

Chapter: [4.01] Heights and Distances

Find a rational number between `sqrt(2) "and" sqrt(7)`.

Chapter: [1.01] Real Numbers

Write the number of zeroes in the end of a number whose prime factorization is 2^{2 }× 5^{3} × 3^{2} × 17.

Chapter: [2.03] Quadratic Equations

Let ∆ ABC ∽ ∆ DEF and their areas be respectively, 64 cm^{2} and 121 cm^{2}. If EF = 15⋅4 cm, find BC.

Chapter: [2.03] Quadratic Equations

Evaluate:

`(tan 65^circ)/(cot 25^circ)`

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

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

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

Find the number of terms in the A.P.: 18, `15 1/2, 13, ...., -47.`

Chapter: [2.02] Arithmetic Progressions

A bag contains 15 balls, out of which some are white and the others are black. If the probability of drawing a black ball at random from the bag is `2/3`, then find how many white balls are there in the bag.

Chapter: [5.01] Probability [5.01] Probability

A card is drawn at random from a pack of 52 playing cards. Find the probability of drawing a card that is neither a spade nor a king.

Chapter: [5.01] Probability [5.01] Probability

Find the solution of the pair of the equation :`3/x + 8/y = - 1; 1/x - 2/y = 2`, x, y ≠ 0

Chapter: [2.01] Pair of Linear Equations in Two Variables

Find the value (s) of k for which the pair of equations

Chapter: [2.01] Pair of Linear Equations in Two Variables

How many multiples of 4 lie between 10 and 205?

Chapter: [2.02] Arithmetic Progressions

Determine the A.P. whose third term is 16 and 7^{th} term exceeds the 5^{th} by 12.

Chapter: [2.02] Arithmetic Progressions

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

Chapter: [1.01] Real Numbers

The point R divides the line segment AB, where A(−4, 0) and B(0, 6) such that AR=34AB.">AR = `3/4`AB. Find the coordinates of R.

Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)

Prove that:

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

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

Prove that:

`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

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

Chapter: [6.01] Lines (In Two-dimensions)

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

Chapter: [6.01] Lines (In Two-dimensions)

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.

Chapter: [3.01] Circles [3.01] Circles

In the given figure, BL and CM are medians of a ∆ABC right-angled at A. Prove that 4 (BL^{2} + CM^{2}) = 5 BC^{2}.

Chapter: [3.02] Triangles

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals

Chapter: [3.02] Triangles

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.

Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles

Calculate the mode of the following distribution:

Class |
10 − 15 | 15 − 20 | 20 − 25 | 25 − 30 | 30 − 35 |

Frequency |
4 | 7 | 20 | 8 | 1 |

Chapter: [5.02] Statistics

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.

Chapter: [7.02] Surface Areas and Volumes

A farmer connects a pipe of internal diameter 20 cm form 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/h, in how much time will the tank be filled?

Chapter: [7.02] Surface Areas and Volumes

Prove that `2 + 3sqrt(3)` is an irrational number when it is given that `sqrt(3)` is an irrational number.

Chapter: [1.01] Real Numbers

Sum of the areas of two squares is 157 m^{2}. If the sum of their perimeters is 68 m, find the sides of the two squares.

Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles

Find the quadratic polynomial, sum, and product of whose zeroes are −1 and −20 respectively. Also, find the zeroes of the polynomial so obtained.

Chapter: [2.04] Polynomials [2.04] Polynomials

A plane left 30 minutes later than the scheduled time and in order to reach its destination 1500 km away on time, it has to increase its speed by 250 km/hr from its usual speed. Find the usual speed of the plane.

Chapter: [7.01] Areas Related to Circles

Find the dimensions of a rectangular park whose perimeter is 60 m and area 200 m^{2}.

Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles

Find the value of x, when in the A.P. given below 2 + 6 + 10 + ... + x = 1800.

Chapter: [2.02] Arithmetic Progressions

If sec θ + tan θ = m, show that `(m^2 - 1)/(m^2 + 1) = sin theta`

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

In ∆ ABC, AD ⊥ BC.

Prove that AC^{2} = AB^{2} +BC^{2} − 2BC x BD

Chapter: [3.02] Triangles

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.

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

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.

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

Construct a triangle with sides 5 cm, 6 cm, and 7 cm and then another triangle whose sides are `3/5` of the corresponding sides of the first triangle.

Chapter: [3.02] Triangles

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 |

Chapter: [5.02] Statistics

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.

Chapter: [6.01] Lines (In Two-dimensions)

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 cm^{2}. (Take π = 3⋅14)

Chapter: [7.02] Surface Areas and Volumes

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