mathematics Question Bank - Sarthaks eConnect

Let p and q be positive integers and p > q.

Statement 1 : The equation `{::}_()^pC_qx^2−{::}_()^(p+1)C_qx+{::}_()^pC_(q−1)=0` has rational roots.
Statement 2 : `{::}_()^pC_q` and `{::}_()^pC_(q-1)` are rational numbers.

For any two sets A and B, we write `AtriangleB` for `(AuuB)` - `(AnnB)` , i.e., `AtriangleB` is the symmetric difference of the two sets A and B.

Statement 1 : If P(A) denotes the probability of a random event A, then `P(AtriangleB)` `<=` |P(a) – P(B)|
Statement 2 : `P(AtriangleB)` = P(A – B) + P(B – A).

Let ƒ : R `->` R be defined by ƒ(x) = |x – 2n| if 2n – 1 £ x £ 2n + 1 and n Î Z.

Statement 1 : `int_(0)^(2n+1) f(x)dx=n+1` for all positive integer n.

Statement 2 : ƒ is periodic with period 2.

The root of the equation `cos^7 theta – sin^4 theta` = 1 that lie in the interval `[0, 2pi]` is :–

The number of solutions in isosceles triangle ABC, tan A + tan B + tan C = `lamda, lamda` < 0, is :–

If `(alpha,beta)` is a point of intersection of the lines x cos `theta` + y sin `theta` = 3 and x sin `theta` – y cos `theta` = 4 where `theta` is parameter, then the maximum value of `2^((alpha+beta)/sqrt2)` is :–

If a, b and c are in AP and one root of the equation `ax^2 + bx + c = 0` is 2, then the other root is :–

Let ABC be an equilateral triangle, let KLMN be a rectangle with K, L on BC, M on AC and N on AB. Suppose `(AN)/(NB) = 2` and the area of triangle BKN is 6. The area of the triangle ABC is :–

If S is the sume of infinity of a decreasing G.P. with the common ratio x such that |x| < 1; x `!=` 0. The ratio fo the fourth term to the secnod term is `1/16` and the ratio fo the thrid term to the square of the second term is `1/9`, Then the value of S is :–

The probability of a six-digit number N whose six digits are 1, 2, 3, 4, 5 and 5 written in random order and is divisible by 6 is :–

The number of permutations of 1, 2, 3, 4, 5, 6, 7, 8 and 9 taken all at a time such that digit 1 appearing somewhere to the left of 2, 3 appearing to the left of 4, and 5 somehere to the left of 6 is(eg 815723945 would be one such permutation) :–

lx + my = 1 is the equation of the chord PQ of `y^2` = 4x whose focus is S. If PS and QS meet the prabola again at R and T, respectively, then the slope of RT is :–

The value of a for which all extremum of functino `f(x) = x^3 – 3ax^2 + 3(a^2 –1)x + 1` lie in the interval (2, 4) is :–

If A=`[[1,1],[0,1]]` and B=`[[sqrt3/2,1/2],[-1/2,sqrt3/2]]` then `(BB^TA)^5` is equal to :–

If `lim_(x->a)[f(x) g(x)] =2` and `lim_(x->a)[f(x) g(x)]` , then `lim_(x->a)f(x)g(x)`

If `theta_1=sin^(-1) (4/5)+sin^(-1) (1/3)` and `theta_2 =cos^(-1) (4/5)+cos^(-1) (1/3)` , then :-

If f(x)= `inte^x(tan^(-1)x+(2x)/(1+x^2)^2)dx` , ƒ(0) = 0, then the value of ƒ(1) is :–

If `e_1` and `e_2` are the roots of the equation `x^2 – ax + 2 = 0`, where `e_1` and `e_2` are the eccentricities of an ellipse and hyperbola, respectively, then the value of a belongs to :–

Let ƒ(x) = `sin^2x + cos^4x + 2` and g(x) = cos (cos x) +cos (sin x). Also let period of ƒ(x) and g(x) be `T_1` and `T_2` respectively then :–

If the normal at any point P of the ellipse `X^2/16+Y^2/9=1` meets the corrdinates axes at M and N, respectively, then |PM|:|PN| equals :–

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