The number of points in the Cartesian plane with integral co-ordinates satisfying the inequalities |x| $less or equal than$ k, |y| $less or equal than$ k, |x – y| $less or equal than$ k ; is-

The straight lines I₁, I₂, I₃ are parallel and lie in the same plane. A total number of m points are taken on I₁ ; n points on I₂, k points on I₃. The maximum number of triangles formed with vertices at these points are -

If the line $l x plus m y equals 1$ is a normal to the hyperbola $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ then $fraction numerator a to the power of 2 end exponent over denominator l to the power of 2 end exponent end fraction minus fraction numerator b to the power of 2 end exponent over denominator m to the power of 2 end exponent end fraction equals$

If the tangents drawn from a point on the hyperbola $x to the power of 2 end exponent minus y to the power of 2 end exponent equals a to the power of 2 end exponent minus b to the power of 2 end exponent$ to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ make angles α and β with the transverse axis of the hyperbola, then

So here we understood the concept of hyperbola and the normal lines.In analytic geometry, a hyperbola is a conic section created when a plane meets a double right circular cone at an angle that overlaps both cone halves. So the correct relation is $tan alpha space tan beta equals 1$

If the tangents drawn from a point on the hyperbola $x to the power of 2 end exponent minus y to the power of 2 end exponent equals a to the power of 2 end exponent minus b to the power of 2 end exponent$ to the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction minus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ make angles α and β with the transverse axis of the hyperbola, then

The eccentricity of the hyperbola whose latus rectum subtends a right angle at centre is

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r²t⁴s², then the number of ordered pair (p, q) is –

Finding the smallest common multiple between any two or more numbers is done using the least common multiple (LCM) approach. A number that is a multiple of two or more other numbers is said to be a common multiple. Here we understood the concept of LCM and the pairs, so the total pairs can be 225.

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r²t⁴s², then the number of ordered pair (p, q) is –

A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length

Here we used the concept of number system and the rectangle, we can also solve it by permutation and combination. herefore, we get the number of rectangles possible with odd side length = m²n².

A rectangle has sides of (2m – 1) & (2n – 1) units as shown in the figure composed of squares having edge length one unit then no. of rectangles which have odd unit length

Here we used the concept of number system and the rectangle, we can also solve it by permutation and combination. herefore, we get the number of rectangles possible with odd side length = m²n².

ⁿC_r + ²ⁿC_r+1 + ⁿC^r+2 is equal to (2 $less or equal than$ r $less or equal than$ n)

The coefficient of $x to the power of n$ in $fraction numerator x plus 1 over denominator left parenthesis x minus 1 right parenthesis squared left parenthesis x minus 2 right parenthesis end fraction$ is

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which not two S are adjacent ?

The value of ⁵⁰C₄ + $not stretchy sum subscript r equals 1 end subscript superscript 6 end superscript 56 minus r C subscript 3 end subscript$ is -

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is-

physics-

A mass of $100 blank g$ strikes the wall with speed $5 blank m divided by s$ at an angle as shown in figure and it rebounds with the same speed. If the contact time is $2 cross times 10 to the power of negative 3 end exponent s e c$ , what is the force applied on the mass by the wall

physics-General

If the locus of the mid points of the chords of the ellipse $fraction numerator x to the power of 2 end exponent over denominator a to the power of 2 end exponent end fraction plus fraction numerator y to the power of 2 end exponent over denominator b to the power of 2 end exponent end fraction equals 1$ , drawn parallel to $y equals m subscript 1 end subscript x$ is $y equals m subscript 2 end subscript x$ then $m subscript 1 end subscript m subscript 2 end subscript equals$

If ⁿC_r denotes the number of combinations of n things taken r at a time, then the expression ⁿC_r+1 + ⁿC_{r –1} + 2 × ⁿC_r equals-